# Brownian motion

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The Concepts and Practice of Mathematical Finance by Mark S. Joshi

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The general case follows from approximating by piecewise constant functions. 8.6 The joint distribution of minimum and terminal value for a Brownian motion with drift In Section 8.4, we derived the joint law of the minimum and terminal value for a Brownian motion without drift. In this section, we combine that result with our results on Girsanov's theorem to derive the joint law for a Brownian motion with drift. Let Wt be a Brownian motion. Let Yt = aWt, and my be the minimum of Yt up to time t. We then have for y < 0 and x > y that ?(Yt > x, m ' < y) = I1(Yt < 2y - x). (8.37) This follows from the result for Brownian motion, (8.20), as the volatility term makes no real difference. We wish to prove an analogous result for a Brownian motion with drift. Let Zt = vdt + 6dWt (8.38) and mZ denotes its minimum up to time t. Our main result is Theorem 8.2 If y < 0 and x > y, then IED(Zt > x, mZ < y) = e2vy6-2IED(Zt < 2y - x + 2vt) = e2vyv-2N (2y - x + Vt 1 0- "It .

A quanto option is an option that whose pay-off is transformed into another currency at pre-determined rate. A multi-dimensional Brownian motion is a vector of processes which have jointly normal increments and is a Brownian motion in each dimension. Correlated Brownian motions can be constructed by adding together multiples of one-dimensional Brownian motions. The Ito calculus goes over to higher dimensions with the additional rule dWjdWk = pjkdt where Pjk is the correlation between Wj and Wk. When adding correlation Brownian motions we can find the volatility of the new process by treating the original processes as vectors. We can change the drift of a multi-dimensional Brownian motion by using Girsanov's theorem. 11.11 Exercises 281 No arbitrage will occur if and only if the discounted price processes can be made driftless by a change of measure.

If we interpret the correlation coefficient as being the cosine of the angle between the two Brownian motions, then this means that the new volatility is just the length of the vector obtained by summing the vectors for each Brownian motion. 266 Multiple sources of risk More generally, we could construct a Brownian motion from any vector a=(a1,...,ak) Ek=1 ajX( ). with a? = 1, by taking When we have a process driven by k > 2 Brownian motions, we obtain a similar expression to (11.20). The volatility becomes n Y" Ors 6j Pig N i,J=1 and we can write dYt = µdt + adW, (11.22) for a Brownian motion W constructed from the old one. We can similarly regard our processes as vectors in 1R which add according to their directions. Example 11.1 Suppose the stocks Xt and Yt follow correlated geometric Brownian motions. Show that XtYt also follows a geometric Brownian motion and compute its drift and volatility. Solution We write dXt = aXtdt + o XtdWW 1), dYt = ,BYtdt + vYdWt(2), and take the correlation coefficient to be p. We compute d(XtYt) = XtdYt + YtdXt + dXt.dYt, = XtYt (f3dt + vdW 2) + adt + vdWr 1) + avpdt) , = XtYt ((a +,B + QVp)dt + o'dW(1) + vdWr2)) .

Monte Carlo Simulation and Finance by Don L. McLeish

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To motivate this family of distributions, let us GENERATING RANDOM NUMBERS FROM NON-UNIFORM CONTINUOUS DISTRIBUTIONS159 250 200 150 100 50 0 -50 -100 -150 -200 -250 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 Figure 3.18: A Symmetric Stable Random Walk with index α = 1.7 suppose that stock returns follow a Brownian motion process but with respect to a random time scale possibly dependent on volume traded and other external factors independent of the Brownian motion itself. After one day, say, the return on the stock is the value of the Brownian motion process at a random time, τ, independent of the Brownian motion. Assume that this random time has the Inverse Gaussian distribution having probability density function (θ − t)2 θ exp{− } g(t) = √ 2c2 t c 2πt3 (3.21) for parameters θ > 0, c > 0. This is the distribution of a first passage time for Brownian motion. In particular consider a Brownian motion process B(t) having drift 1 and diﬀusion coeﬃcient c. Such a process is the solution to the stochastic diﬀerential equation dB(t) = dt + cdW (t), B(0) = 0.

The high dm is generated by moving horizontally to the right an even number of steps until just before exiting the histogram. This is above the value d2m−u and dm is between du and d2m−u . A similar result is available for Brownian motion and Geometric Brownian motion. A justification of these results can be made by taking a limit in the discrete case as the time steps and the distances dj − dj−1 all approach zero. If we do this, the parameter θ is analogous to the drift of the Brownian motion. The result for Brownian motion is as follows: Theorem 44 Suppose St is a Brownian motion process dSt = µdt + σdWt , S0 = 0, ST = C, H = max{St ; 0 · t · T } and L = min{St ; 0 · t · T }. If f0 denotes the Normal(0,σ 2 T ) probability density function, the distribution of SIMULATING BARRIER AND LOOKBACK OPTIONS 277 C under drift µ = 0, then f0 (2H − C) is distributed as U [0, 1] independently of C, f0 (C) f0 (2L − C) UL = is distributed as U [0, 1] independently of C. f0 (C) 1 ZH = H(H − C) is distributed as Exponential ( σ 2 T ) independently of C, 2 1 2 ZL = L(L − C) is distributed as Exponential ( σ T )independently of C. 2 UH = We will not prove this result since it is a special case of Theorem 46 below.

The vertical √ scale is to be decreased by a factor 1/ n and the horizontal scale by a factor n−1 . The theorem concludes that the sequence of processes 1 Yn (t) = √ Snt n converges weakly to a standard Brownian motion process as n → ∞. In practice this means that a process with independent stationary increments tends to look like a Brownian motion process. As we shall see, there is also a wide variety of non-stationary processes that can be constructed from the Brownian motion process by integration. Let us use the above limiting result to render some of the properties of the Brownian motion more plausible, since a serious proof is beyond our scope. Consider the question of continuity, for example. Since Pn(t+h) |Yn (t + h) − Yn (t)| ≈ | √1n i=nt Xi | and this is the absolute value of an asymptotically normally(0, h) random variable by the central limit theorem, it is plausible that the limit as h → 0 is zero so the function is continuous at t.

pages: 443 words: 51,804

Handbook of Modeling High-Frequency Data in Finance by Frederi G. Viens, Maria C. Mariani, Ionut Florescu

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In this chapter, we consider the continuous-time long-memory stochastic volatility (LMSV) model by Comte and Renault (1998): If Xt are the log-returns of the price process St and Yt is the volatility process, then ⎧ ⎨ dXt ⎩ dYt σ 2 (Yt ) = μ− dt + σ (Yt ) dWt , 2 = α Yt dt + β dBtH , (8.1) where Wt is a standard Brownian motion and BtH is a fractional Brownian motion with Hurst index H ∈ (0, 1]. The fractional Brownian motion (fBm) with Hurst parameter H ∈ (0, 1] is a Gaussian process with almost surely continuous paths and covariance structure given by Cov(BtH , BsH ) = 1 2H |t| + |s|2H − |t − s|2H . 2 For H = 1/2 the process is the well-known standard Brownian motion. Formally, we say that a process exhibits long-range dependence when the series of the autocorrelation function is nonsummable, that is +∞ n=1 ρ(n) = +∞. 8.1 Introduction 221 From the covariance function of the fractional Brownian motion, we can easily deduce that the autocorrelation function of the increments of fBm is of order n2H −2 .

Dayanik and Karatzas (2003), Proposition 5.7 STANDING ASSUMPTION 11.15 l0 = x ↓ 0 h+ (x) h+ (x) < ∞, and l = x ↑ ∞ . ∞ x ρ− x ρ+ Let us denote by τx inf {t ≥ 0 / Y (t) = x}, x > 0, the ﬁrst hitting time of level x. The reduction of the optimal stopping problem to the Brownian motion case has been studied by Dayanik and Karatzas in the case when the diffusion was driven by one-dimensional Brownian motion. Their results hold also for the case in which we have a m-dimensional Brownian motion with the only modiﬁcation in the equation solved by the Green functions ψ and ϕ. Still, the problem is solved explicitly for the Brownian motion case by using a graphical method. By taking advantage of the theory developed in Dayanik and Karatzas (2003), in terms of the existence of an optimal stopping time and provided that we can either solve or determine the shape of intervals that form C0 {y > 0/Lg(y) > 0}, (11.59) then the value function is the supremum over stopping times which are exit times from open intervals that contain C0 .

In fact, the parameter α of the Levy distribution is inversely proportional to the Hurst parameter. The Hurst parameter is an indicator of the memory effects coming from the fractional Brownian motion, which has correlated increments. Furthermore, the TLF maintains statistical properties that are indistinguishable from the Levy ﬂights [15]. 6.2.2 RESCALED RANGE ANALYSIS Hurst [27] initially developed the Rescaled range analysis (R/S analysis). He observed many natural phenomena that followed a biased random walk, that is, every phenomenon showed a pattern. He measured the trend using an exponent now called the Hurst exponent. Mandelbrot [28,29] later introduced a generalized form of the Brownian motion model, the fractional Brownian motion to model the Hurst effect. The numerical procedure to estimate the Hurst exponent H by using the R/S analysis is presented next (for more details, please see [27] and references therein). 1.

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The Misbehavior of Markets by Benoit Mandelbrot

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Instead, it is best to go beyond cartoons. My current best model of how a market works is fractional Brownian motion of multifractal time. It has been called the Multifractal Model of Asset Returns. The basic ideas are similar to the cartoon versions above—though far more intricate, mathematically. The cartoon of Brownian motion gets replaced by an equation that a computer can calculate. The trading-time process is expressed by another mathematical function, called f(α), that can be tuned to fit a wide range of market behavior. My model redistributes time. It compresses it in some places, stretches it out in others. The result appears very wild, very random. The two functions, of time and Brownian motion, work together in what mathematicians call a compound manner: Price is a function of trading time, which in turn is a function of clock time.

Here, major news—approval of a medicine by the Food and Drug Administration, an unexpected takeover offer, or a windfall legal victory—caused market indigestion; sell and buy orders did not match, and market-makers had to raise or lower their price quotes until they did. To cope, some exchanges license “specialist” broker-dealers to step into the breach and trade when others will not. These specialists, while risking much, also profit greatly. Discontinuity, far from being an anomaly best ignored, is an essential ingredient of markets that helps set finance apart from the natural sciences. 4. Assumption: Price changes follow a Brownian motion. Theory: Brownian motion, again, is a term borrowed from physics for the motion of a molecule in a uniformly warm medium. Bachelier had suggested that this process can also describe price variation. Several critical assumptions come together in this idea. First, independence: Each change in price—whether a five-cent uptick or a \$26 collapse—appears independently from the last, and price changes last week or last year do not influence those today.

By repeating this process over and over, a jagged, complex chart gradually appeared. By careful design, the specific kind of chart shown before was of a Brownian motion—the standard model underlying conventional financial theory. What made it so was the specific shape of the generator: Starting at the point (0, 0), it rose to the point (4/9, 2/3), fell to the point (5/9, 1/3), and ended up at (1, 1). A key observation regards the size of the three segments of the generator. Their widths were 4/9, 1/9, 4/9. The heights: 2/3, -1/3 (minus, because the line falls), and 2/3. Look closely at those six numbers. Each width is the square of each height. It is a nice, tidy relationship—just the kind of thing you would expect from a well-mannered Brownian motion. A cartoon of discontinuity. There are many ways to illustrate the crucial concepts of fat tails and discontinuity—and this one employs the kind of fractal process used earlier in this book.

pages: 313 words: 34,042

Tools for Computational Finance by Rüdiger Seydel

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. — The Example 1.15 with a system of three SDEs is taken from [HPS92]. [KP92] in Section 4.4 gives a list of SDEs that are analytically solvable or reducible. The model of a geometric Brownian motion of equation (1.33) is the classical model describing the dynamics of stock prices. It goes back to Samuelson (1965; Nobel Prize in economics in 1970). Already in 1900 Bachelier had suggested to model stock prices with Brownian motion. Bachelier used the arithmetic version, which can be characterized by replacing the lefthand side of (1.33) by the absolute change dS. This amounts to the process St = S0 + µt + σWt . Here the stock price can become negative. Main advantages of the geometric Brownian motion are the success of the approaches of Black, Merton and Scholes, which is based on that motion, and the existence of moments (as the expectation).

. = equal except for rounding errors ≡ identical =⇒ implication ⇐⇒ equivalence Landau-symbol: for h → 0 O(hk ) is bounded f (h) = O(hk ) ⇐⇒ fh(h) k normal distributed with expectation µ and variance σ 2 ∼ N (µ, σ 2 ) ∼ U[0, 1] uniformly distributed on [0, 1] XVIII Notations ∆t tr C 0 [a, b] ∈ C k [a, b] D ∂D L2 H [0, 1]2 Ω f + := max{f, 0} u̇ small increment in t transposed; Atr is the matrix where the rows and columns of A are exchanged. set of functions that are continuous on [a, b] k-times continuously diﬀerentiable set in IRn or in the complex plane, D̄ closure of D, D◦ interior of D boundary of D set of square-integrable functions Hilbert space, Sobolev space (Appendix C3) unit square sample space (in Appendix B1) time derivative du dt of a function u(t) integers: i, j, k, l, m, n, M, N, ν various variables: Xt , X, X(t) Wt y(x, τ ) w h ϕ ψ 1D random variable Wiener process, Brownian motion (Deﬁnition 1.7) solution of a partial diﬀerential equation for (x, τ ) approximation of y discretization grid size basis function (Chapter 5) test function (Chapter 5) indicator function: = 1 on D, = 0 elsewhere. abbreviations: BDF CFL Dow FTBS FTCS GBM MC ODE OTC PDE PIDE PSOR QMC SDE SOR Backward Diﬀerence Formula, see Section 4.2.1 Courant-Friedrichs-Lewy, see Section 6.5.1 Dow Jones Industrial Average Forward Time Backward Space, see Section 6.5.1 Forward Time Centered Space, see Section 6.4.2 Geometric Brownian Motion, see (1.33) Monte Carlo Ordinary Diﬀerential Equation Over The Counter Partial Diﬀerential Equation Partial Integro-Diﬀerential Equation Projected Successive Overrelaxation Quasi Monte Carlo Stochastic Diﬀerential Equation Successive Overrelaxation Notations TVD i.i.d. inf sup supp(f ) XIX Total Variation Diminishing independent and identical distributed inﬁmum, largest lower bound of a set of numbers supremum, least upper bound of a set of numbers support of a function f : {x ∈ D : f (x) = 0} hints on the organization: (2.6) (A4.10) −→ number of equation (2.6) (The ﬁrst digit in all numberings refers to the chapter.) equation in Appendix A; similarly B, C, D hint (for instance to an exercise) Chapter 1 Modeling Tools for Financial Options 1.1 Options What do we mean by option?

The ∆ of (1.16) is the hedge parameter delta, which eliminates the risk exposure of our portfolio caused by the written option. In multi-period models and continuous models ∆ must be adapted dynamically. The general deﬁnition is ∂V (S, t) ; ∆ = ∆(S, t) = ∂S the expression (1.16) is a discretized version. 1.6 Stochastic Processes Brownian motion originally meant the erratic motion of a particle (pollen) on the surface of a ﬂuid, caused by tiny impulses of molecules. Wiener suggested a mathematical model for this motion, the Wiener process. But earlier Bachelier had applied Brownian motion to model the motion of stock prices, which instantly respond to the numerous upcoming informations similar as pollen react to the impacts of molecules. The illustration of the Dow in Figure 1.14 may serve as motivation. A stochastic process is a family of random variables Xt , which are deﬁned for a set of parameters t (−→ Appendix B1).

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Market Risk Analysis, Quantitative Methods in Finance by Carol Alexander

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The process Wt is called a Wiener process, also called a Brownian motion. It is a continuous process that has independent increments dWt and each increment has a normal distribution with mean 0 and variance dt.19 On adding uncertainty to the exponential price path (I.1.31) the price process (I.1.29) becomes dSt = dt + dWt (I.1.32) St This is an example of a diffusion process. Since the left-hand side has the proportional change in the price at time t, rather than the absolute change, we call (I.1.32) geometric Brownian motion. If the left-hand side variable were dSt instead, the process would be called arithmetic Brownian motion. The diffusion coefficient is the coefficient of dWt, which is a constant in the case of geometric Brownian motion. This constant is called the volatility of the process.

Probability and Statistics 139 Application of Itô’s lemma with f = ln S shows that a continuous time representation of geometric Brownian motion that is equivalent to the geometric Brownian motion (I.3.143) but is translated into a process for log prices is the arithmetic Brownian motion, d ln St = − 21 2 dt + dWt (I.3.145) We already know what a discretization of (I.3.145) looks like. The change in the log price is the log return, so using the standard discrete time notation Pt for a price at time t we have d ln St → ln Pt Hence the discrete time equivalent of (I.3.145) is ln Pt = + \$t \$t ∼ NID 0 2 (I.3.146) where = − 21 2 . This is equivalent to a discrete time random walk model for the log prices, i.e. ln Pt = + ln Pt−1 + \$t \$t ∼ NID 0 2 (I.3.147) To summarize, the assumption of geometric Brownian motion for prices in continuous time is equivalent to the assumption of a random walk for log prices in discrete time.

A Wiener process, also called a Brownian motion, describes the stochastic part when the process does not jump. Thus a Wiener process is a continuous process with stationary, independent normally distributed increments. The increments are normally distributed so we often use any of the notations Wt Bt or Zt for such a process. The increments of the process are the total change in the process over an infinitesimally small time period; these are denoted dWt dBt or dZt, with EdW = 0 and VdW = dt Now we are ready to write the equation for the dynamics of a continuous time stochastic process X t as the following SDE: dXt = dt + dZt (I.3.141) where is called the drift of the process and is called the process volatility. The model (I.3.141) is called arithmetic Brownian motion. Arithmetic Brownian motion is the continuous time version of a random walk.

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Derivatives Markets by David Goldenberg

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398 11.7 Further Implications of European Put-Call Parity 11.7.1 Synthesizing Forward Contract from Puts and Calls 399 399 11.8 Financial Innovation using European Put-Call Parity 401 11.8.1 Generalized Forward Contracts 401 11.8.2 American Put-Call Parity (No Dividends) 403 11.9 Postscript on ROP CHAPTER 12 OPTION TRADING STRATEGIES, PART 2 405 415 12.1 Generating Synthetic Option Strategies from European Put-Call Parity 416 12.2 The Covered Call Hedging Strategy 419 12.2.1 Three Types Of Covered Call Writes 420 DETAILED CONTENTS xvii 12.2.2 Economic Interpretation of the Covered Call Strategy 12.3 The Protective Put Hedging Strategy 426 427 12.3.1 Puts as Insurance 427 12.3.2 Economic Interpretation of the Protective Put Strategy 429 CHAPTER 13 MODEL-BASED OPTION PRICING IN DISCRETE TIME, PART 1: THE BINOMIAL OPTION PRICING MODEL (BOPM, N=1) 435 13.1 The Objective of Model-Based Option Pricing (MBOP) 437 13.2 The Binomial Option Pricing Model, Basics 437 13.2.1 Modeling Time in a Discrete Time Framework 437 13.2.2 Modeling the Underlying Stock Price Uncertainty 438 13.3 The Binomial Option Pricing Model, Advanced 440 13.3.1 Path Structure of the Binomial Process, Total Number of Price Paths 440 13.3.2 Path Structure of the Binomial Process, Total Number of Price Paths Ending at a Specific Terminal Price 442 13.3.3 Summary of Stock Price Evolution for the N-Period Binomial Process 444 13.4 Option Valuation for the BOPM (N=1) 445 13.4.1 Step 1, Pricing the Option at Expiration 445 13.4.2 Step 2, Pricing the Option Currently (time t=0) 446 13.5 Modern Tools for Pricing Options 448 13.5.1 Tool 1, The Principle of No-Arbitrage 448 13.5.2 Tool 2, Complete Markets or Replicability, and a Rule of Thumb 449 13.5.3 Tool 3, Dynamic and Static Replication 450 xviii DETAILED CONTENTS 13.5.4 Relationships between the Three Tools 13.6 Synthesizing a European Call Option 450 453 13.6.1 Step 1, Parameterization 454 13.6.2 Step 2, Defining the Hedge Ratio and the Dollar Bond Position 455 13.6.3 Step 3, Constructing the Replicating Portfolio 456 13.6.4 Step 4, Implications of Replication 462 13.7 Alternative Option Pricing Techniques 464 13.8 Appendix: Derivation of the BOPM (N=1) as a Risk-Neutral Valuation Relationship 467 CHAPTER 14 OPTION PRICING IN DISCRETE TIME, PART 2: DYNAMIC HEDGING AND THE MULTI-PERIOD BINOMIAL OPTION PRICING MODEL, N>1 14.1 Modeling Time and Uncertainty in the BOPM, N>1 473 475 14.1.1 Stock Price Behavior, N=2 475 14.1.2 Option Price Behavior, N=2 476 14.2 Hedging a European Call Option, N=2 477 14.2.1 Step 1, Parameterization 477 14.2.2 Step 2, Defining the Hedge Ratio and the Dollar Bond Position 478 14.2.3 Step 3, Constructing the Replicating Portfolio 478 14.2.4 The Complete Hedging Program for the BOPM, N=2 484 14.3 Implementation of the BOPM for N=2 485 14.4 The BOPM, N>1 as a RNVR Formula 490 14.5 Multi-period BOPM, N>1: A Path Integral Approach 493 DETAILED CONTENTS xix 14.5.1 Thinking of the BOPM in Terms of Paths 493 14.5.2 Proof of the BOPM Model for general N 499 CHAPTER 15 EQUIVALENT MARTINGALE MEASURES: A MODERN APPROACH TO OPTION PRICING 15.1 Primitive Arrow–Debreu Securities and Option Pricing 507 508 15.1.1 Exercise 1, Pricing B(0,1) 510 15.1.2 Exercise 2, Pricing ADu() and ADd() 511 15.2 Contingent Claim Pricing 514 15.2.1 Pricing a European Call Option 514 15.2.2 Pricing any Contingent Claim 515 15.3 Equivalent Martingale Measures (EMMs) 517 15.3.1 Introduction and Examples 517 15.3.2 Definition of a Discrete-Time Martingale 521 15.4 Martingales and Stock Prices 15.4.1 The Equivalent Martingale Representation of Stock Prices 15.5 The Equivalent Martingale Representation of Option Prices 521 524 526 15.5.1 Discounted Option Prices 527 15.5.2 Summary of the EMM Approach 528 15.6 The Efficient Market Hypothesis (EMH), A Guide To Modeling Prices 529 15.7 Appendix: Essential Martingale Properties 533 CHAPTER 16 OPTION PRICING IN CONTINUOUS TIME 539 16.1 Arithmetic Brownian Motion (ABM) 540 16.2 Shifted Arithmetic Brownian Motion 541 16.3 Pricing European Options under Shifted Arithmetic Brownian Motion with No Drift (Bachelier) 542 xx DETAILED CONTENTS 16.3.1 Theory (FTAP1 and FTAP2) 542 16.3.2 Transition Density Functions 543 16.3.3 Deriving the Bachelier Option Pricing Formula 547 16.4 Defining and Pricing a Standard Numeraire 551 16.5 Geometric Brownian Motion (GBM) 553 16.5.1 GBM (Discrete Version) 553 16.5.2 Geometric Brownian Motion (GBM), Continuous Version 559 16.6 Itô’s Lemma 562 16.7 Black–Scholes Option Pricing 566 16.7.1 Reducing GBM to an ABM with Drift 567 16.7.2 Preliminaries on Generating Unknown Risk-Neutral Transition Density Functions from Known Ones 570 16.7.3 Black–Scholes Options Pricing from Bachelier 571 16.7.4 Volatility Estimation in the Black–Scholes Model 582 16.8 Non-Constant Volatility Models 585 16.8.1 Empirical Features of Volatility 585 16.8.2 Economic Reasons for why Volatility is not Constant, the Leverage Effect 586 16.8.3 Modeling Changing Volatility, the Deterministic Volatility Model 586 16.8.4 Modeling Changing Volatility, Stochastic Volatility Models 16.9 Why Black–Scholes is Still Important CHAPTER 17 RISK-NEUTRAL VALUATION, EMMS, THE BOPM, AND BLACK–SCHOLES 17.1 Introduction 17.1.1 Preliminaries on FTAP1 and FTAP2 and Navigating the Terminology 587 588 595 596 596 DETAILED CONTENTS xxi 17.1.2 Pricing by Arbitrage and the FTAP2 597 17.1.3 Risk-Neutral Valuation without Consensus and with Consensus 598 17.1.4 Risk-Neutral Valuation without Consensus, Pricing Contingent Claims with Unhedgeable Risks 599 17.1.5 Black–Scholes’ Contribution 601 17.2 Formal Risk-Neutral Valuation without Replication 601 17.2.1 Constructing EMMs 601 17.2.2 Interpreting Formal Risk-Neutral Probabilities 602 17.3 MPRs and EMMs, Another Version of FTAP2 605 17.4 Complete Risk-Expected Return Analysis of the Riskless Hedge in the (BOPM, N=1) 607 17.4.1 Volatility of the Hedge Portfolio 608 17.4.2 Direct Calculation of S 611 17.4.3 Direct Calculation of C 612 17.4.4 Expected Return of the Hedge Portfolio 616 17.5 Analysis of the Relative Risks of the Hedge Portfolio’s Return 618 17.5.1 An Initial Look at Risk Neutrality in the Hedge Portfolio 618 17.5.2 Role of the Risk Premia for a Risk-Averse Investor in the Hedge Portfolio 620 17.6 Option Valuation Index 624 17.6.1 Some Manipulations 624 17.6.2 Option Valuation Done Directly by a Risk-Averse Investor 626 17.6.3 Option Valuation for the Risk-Neutral Investor 631 637 This page intentionally left blank FIGURES 1.1 1.2 1.3 1.4 2.1 2.2 2.3 2.4 3.1 3.2 3.3 3.4 5.1 5.2 5.3 5.4 6.1 6.2 6.3 6.4 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 7.9 Canada/US Foreign Exchange Rate Intermediation by the Clearing House Offsetting Trades Gold Fixing Price in London Bullion Market (USD\$) Graphical Method to Get Hedged Position Profits Payoff Per Share to a Long Forward Contract Payoff Per Share to a Short Forward Contract Profits per bu. for the Unhedged Position Profits Per Share to a Naked Long Spot Position Payoffs Per Share to a Naked Long Spot Position Payoffs (=Profits) Per Share to a Naked Long Forward Position Payoffs Per Share to a Naked Long Spot Position and to a Naked Long Forward Position Order Flow Process (Pit Trading) The Futures Clearing House Offsetting Trades Overall Profits for Example 2 Long vs.

We used the fact that X1() is independent of W0 in the fourth equality. Our conclusion is that the property E(W1()|W0)=W0 holds. CHAPTER 16 OPTION PRICING IN CONTINUOUS TIME 16.1 Arithmetic Brownian Motion (ABM) 540 16.2 Shifted Arithmetic Brownian Motion 541 16.3 Pricing European Options under Shifted Arithmetic Brownian Motion with No Drift (Bachelier) 542 16.3.1 Theory (FTAP1 and FTAP2) 542 16.3.2 Transition Density Functions 543 16.3.3 Deriving the Bachelier Option Pricing Formula 547 16.4 Defining and Pricing a Standard Numeraire 551 16.5 Geometric Brownian Motion (GBM) 553 16.5.1 GBM (Discrete Version) 553 16.5.2 Geometric Brownian Motion (GBM), Continuous Version 559 16.6 Itô’s Lemma 562 16.7 Black–Scholes Option Pricing 566 16.7.1 Reducing GBM to an ABM with Drift 567 16.7.2 Preliminaries on Generating Unknown Risk-Neutral Transition Density Functions from Known Ones 570 16.7.3 Black–Scholes Options Pricing from Bachelier 571 16.7.4 Volatility Estimation in the Black–Scholes Model 582 16.8 Non-Constant Volatility Models 585 16.8.1 Empirical Features of Volatility 585 16.8.2 Economic Reasons for why Volatility is not Constant, the Leverage Effect 586 540 OPTIONS 16.8.3 Modeling Changing Volatility, the Deterministic Volatility Model 586 16.8.4 Modeling Changing Volatility, Stochastic Volatility Models 16.9 Why Black–Scholes Is Still Important 587 588 In this chapter we are going to give an introduction to continuous-time ﬁnance.

Black–Scholes (1973), which is a continuous time, continuous state space model even shows up in some jump option pricing models, such as Merton’s (1976) jump model. Thus, Black–Scholes may indeed survive as a component of some option pricing models. We should probably keep it around for these reasons. n n n 590 n OPTIONS KEY CONCEPTS 1. Arithmetic Brownian Motion (ABM). 2. Shifted Arithmetic Brownian Motion. 3. Pricing European Options under Shifted Arithmetic Brownian Motion (Bachelier). 4. Theory (FTAP1 and FTAP2). 5. Transition Density Functions. 6. Deriving the Bachelier Option Pricing Formula. 7. Deﬁning and Pricing a Standard Numeraire. 8. Geometric Brownian Motion (GBM). 9. GBM (Discrete Version). 10. Geometric Brownian Motion (GBM), Continuous Version. 11. Itô’s Lemma. 12. Black–Scholes Option Pricing. 13. Reducing GBM to an ABM with Drift. 14. Preliminaries on Risk-Neutral Transition Density Functions. 15. Black–Scholes Pricing from Bachelier. 16.

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Frequently Asked Questions in Quantitative Finance by Paul Wilmott

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Short Answer Girsanov’s theorem is the formal concept underlying the change of measure from the real world to the risk-neutral world. We can change from a Brownian motion with one drift to a Brownian motion with another. Example The classical example is to start withdS = µS dt + σ S dWt with W being Brownian motion under one measure (the real-world measure) and converting it to under a different, the risk-neutral, measure. Long Answer First a statement of the theorem. Let Wt be a Brownian motion with measure and sample space Ω. If γt is a previsible process satisfying the constraint then there exists an equivalent measure on Ω such that is a Brownian motion. It will be helpful if we explain some of the more technical terms in this theorem. Sample space: All possible future states or outcomes.

Figure 2-5: Certainty equivalent as a function of the risk-aversion parameter for example in the text. References and Further Reading Ingersoll, JE Jr 1987 Theory of Financial Decision Making. Rowman & Littlefield What is Brownian Motion and What are its Uses in Finance? Short Answer Brownian Motion is a stochastic process with stationary independent normally distributed increments and which also has continuous sample paths. It is the most common stochastic building block for random walks in finance. Example Pollen in water, smoke in a room, pollution in a river, are all examples of Brownian motion. And this is the common model for stock prices as well. Long Answer Brownian motion (BM) is named after the Scottish botanist who first described the random motions of pollen grains suspended in water. The mathematics of this process were formalized by Bachelier, in an option-pricing context, and by Einstein.

This idea of the random walk has permeated many scientific fields and is commonly used as the model mechanism behind a variety of unpredictable continuous-time processes. The lognormal random walk based on Brownian motion is the classical paradigm for the stock market. See Brown (1827). 1900 Bachelier Louis Bachelier was the first to quantify the concept of Brownian motion. He developed a mathematical theory for random walks, a theory rediscovered later by Einstein. He proposed a model for equity prices, a simple normal distribution, and built on it a model for pricing the almost unheard of options. His model contained many of the seeds for later work, but lay ‘dormant’ for many, many years. It is told that his thesis was not a great success and, naturally, Bachelier’s work was not appreciated in his lifetime. See Bachelier (1995). 1905 Einstein Albert Einstein proposed a scientific foundation for Brownian motion in 1905. He did some other clever stuff as well.

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The Physics of Wall Street: A Brief History of Predicting the Unpredictable by James Owen Weatherall

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Over the course of the next six years, French physicist Jean-Baptiste Perrin developed an experimental method to track particles suspended in a fluid with enough precision to show that they indeed followed paths of the sort Einstein predicted. These experiments were enough to persuade the remaining skeptics that atoms did indeed exist. Lucretius’s contribution, meanwhile, went largely unappreciated. The kind of paths that Einstein was interested in are examples of Brownian motion, named after Scottish botanist Robert Brown, who noted the random movement of pollen grains suspended in water in 1826. The mathematical treatment of Brownian motion is often called a random walk — or sometimes, more evocatively, a drunkard’s walk. Imagine a man coming out of a bar in Cancun, an open bottle of sunscreen dribbling from his back pocket. He walks forward for a few steps, and then there’s a good chance that he will stumble in one direction or another. He steadies himself, takes another step, and then stumbles once again.

They get interested in a stock, they make a lot of trades and send the volume of trades way up, and then they gradually stop paying attention and volume decreases. If you allow for variations in volume, you have to change the underlying assumptions of the random walk model, and you get a new, more accurate model of how stock prices evolve, which Osborne called the “extended Brownian motion” model. In the mid-sixties, Osborne and a collaborator showed that at any instant, the chances that a stock will go up are not necessarily the same as the chances that the stock will go down. This assumption, you’ll recall, was an essential part of the Brownian motion model, where a step in one direction is assumed to be just as likely as a step in the other. Osborne showed that if a stock went up a little bit, its next motion was much more likely to be a move back down than another move up. Likewise, if a stock went down, it was much more likely to go up in value in its next change.

Although discussing such debates is far from the scope of this book, I should note that the arguments offered here for how one should think of the status of mathematical models in finance are closely connected to more general discussions concerning the status of mathematical or physical theories quite generally. “. . . named after Scottish botanist Robert Brown . . .”: Brown’s observations were published as Brown (1828). “The mathematical treatment of Brownian motion . . .”: More generally, Brownian motion is an example of a random or “stochastic” process. For an overview of the mathematics of stochastic processes, see Karlin and Taylor (1975, 1981). “. . . it was his 1905 paper that caught Perrin’s eye”: Einstein published four papers in 1905. One of them was the one I refer to here (Einstein 1905b), but the other three were equally remarkable. In Einstein (1905a), he first suggests that light comes in discrete packets, now called quanta or photons; in Einstein (1905c), he introduces his special theory of relativity; and in Einstein (1905d), he proposes the famous equation e = mc2

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Mathematics of the Financial Markets: Financial Instruments and Derivatives Modelling, Valuation and Risk Issues by Alain Ruttiens

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Coming back to the starting point of our non-deterministic description of financial products, in Chapter 8, Section 8.2, we have defined the Brownian motion, or standard Wiener process (also called white noise) as per Eq. 8.2: where y(t) is distributed as (E = 0, V = 1), so that Z(t) is distributed as (E = 0, V = t), that is, with STD = √ t. Because of the nature of y(t), successive values of Z are independent; in discrete time, and abandoning the random subscript “∼”, we have where is a so-called “random number”, actually a number randomly selected from a normal density distribution. For two different times t and t′, the covariance between two Brownian motions is necessarily 0, that is (cf. Eq. 8.5), E[dZ(t).dZ(t′)] = 0. Now, we can generalize the Brownian motion to a fractional Brownian motion BH, H ∈ (0, 1) having as covariance function The parameter H is called the Hurst coefficient.5 As a particular case, if H = , covH(t, t′) = 0 and B1/2 is our standard Brownian motion: the corresponding time series is (pure) random.

Passing from discrete to continuous time, and thus from discrete time (or “finite”) intervals Δt to infinitely short, « infinitesimal » or « instantaneous » time intervals noted dt, Eq. 8.1 becomes (8.2) called a standard Wiener process, or a Brownian process or Brownian motion.1 This process is also called (although improperly2) white noise, by analogy with the very light but permanent scratching behind a sound produced electronically. From Eq. 8.2 we may deduct that the (t)s are independently distributed and stationary. They are normally distributed, with E = 0, V = dt (or STD = dt). Furthermore, (t) is distributed according to a Gaussian distribution of parameters E = 0, V = t (or STD = t). We see now the reason of the presence of a in Eq. 8.1 and 2: the process allows us to consider that it is the variance V of the process that is proportional to time. Formally speaking, a process (X(t), t ≥ 0) is a standard Wiener or Brownian motion if: P[X(0)] = 0: the Brownian motion starts from the origin, in t0 = 0; ∀ s ≤ t, X(t) − X(s) is a real variable, normally distributed, centered on its mean, and with a variance equal to (t − s): the successive increases of the process are stationary; ∀ n, ∀ ti, 0 ≤ t1… ≤ tn, the variables X(tn) − X(tn−1), …, X(t1) − X(t0), X(t0), are independent: the successive increases of the process are independent.

Analysis of Financial Time Series by Ruey S. Tsay

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Furthermore, for any given time indexes 0 ≤ t1 < t2 < · · · < tk , the random vector (wt1 , wt2 , . . . , wtk ) follows a multivariate normal distribution. Finally, a Brownian motion is standard if w0 = 0 almost surely, µ = 0, and σ 2 = 1. Remark: An important property of Brownian motions is that their paths are not differentiable almost surely. In other words, for a standard Brownian motion wt , it can be shown that dwt /dt does not exist for all elements of except for elements in a subset 1 ⊂ such that P(1 ) = 0. As a result, we cannot use the usual intergation in calculus to handle integrals involving a standard Brownian motion when we consider the value of an asset over time. Another approach must be sought. This is the purpose of discussing Ito’s calculus in the next section. 6.2.2 Generalized Wiener Processes The Wiener process is a special stochastic process with zero drift and variance proportional to the length of time interval.

If {et } is a white noise series with finite moments of order slightly greater than 2, then the DF-statistic converges to a function of the standard Brownian motion as T → ∞; see Chan and Wei (1988) and Phillips (1987) for more information. If φ0 is zero but Eq. (2.36) is employed anyway, then the resulting t ratio for testing φ1 = 1 will converge to another nonstandard asymptotic distribution. In either case, simulation is used to obtain critical values of the test statistics; see Fuller (1976, Chapter 8) for selected critical values. Yet if φ0 = 0 and Eq. (2.36) is used, then the t ratio for testing φ1 = 1 is asymptotically normal. However, large sample sizes are needed for the asymptotic normal distribution to hold. Standard Brownian motion is introduced in Chapter 6. 2.8 SEASONAL MODELS Some financial time series such as quarterly earning per share of a company exhibits certain cyclical or periodic behavior.

Using the notation of the general Ito’s process in Eq. (6.2), we have µ(xt , t) = µxt and σ (xt , t) = σ xt , where xt = Pt . Such a special process is referred to as a geometric Brownian motion. We now apply the Ito’s lemma to obtain a continuous-time model for the logarithm of the stock price Pt . Let G(Pt , t) = ln(Pt ) be the log price of the underlying stock. Then we have ∂G 1 = , ∂ Pt Pt ∂G = 0, ∂t 1 −1 1 ∂2G = . 2 2 ∂ Pt 2 Pt2 Consequently, via Ito’s lemma, we obtain d ln(Pt ) = 1 1 −1 2 2 σ2 1 µPt + σ Pt dt + σ Pt dwt = µ − dt + σ dwt . Pt 2 Pt2 Pt 2 ITO ’ S LEMMA 229 This result shows that the logarithm of a price follows a generalized Wiener Process with drift rate µ − σ 2 /2 and variance rate σ 2 if the price is a geometric Brownian motion. Consequently, the change in logarithm of price (i.e., log return) between current time t and some future time T is normally distributed with mean (µ−σ 2 /2)(T − t) and variance σ 2 (T −t).

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The Rise of the Quants: Marschak, Sharpe, Black, Scholes and Merton by Colin Read

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He also allowed a drift of zero mean of the security price and assumed that the variance of the price drift is proportional to the length of time of the random walk. In combination, he had described what we now call a Weiner process. The Times 105 While Bachelier was the first to apply Brownian motion to finance, the methodology is now commonplace. The term “Brownian motion” originated in 1828 from the observations of the botanist Robert Brown, who discovered that pollen suspended in water seemed to experience unusual and random jumps when observed under a microscope. The renowned MIT mathematician Norbert Weiner described the mathematics of Brownian motion in his 1918 PhD thesis. Bachelier had already discovered this, though. His statement that stock prices could be modeled as a random walk according to a Weiner process was amenable to empirical verification. Alfred Cowles, who would found the Cowles Commission, and Herbert Jones explored and subsequently vindicated this notion that there is no memory effect in the price of stocks in a 1937 paper together.8 While the notion of the random walk has since been replaced with the less restrictive concept of a martingale process, much of finance pricing theory still retains the random walk because of its simple first and second moment characterization of price movements.

Alternative approaches Once Black and Scholes’ paper was published in the prestigious Journal of Political Economy, and with the opening of the CBOE, options pricing theory had arrived as the most sophisticated and potentially most valuable tool for financial market analysts. As theorists struggled to understand and interpret their result, and as Merton soon published his complementary work, a renewed interest developed in Bachelier’s work from 70 years earlier, and even Einstein’s theory of Brownian motion. Within a few years, Brownian motion, Markov processes, martingales, and stochastic calculus had begun to be integrated into the finance discipline. The quantitative school of finance had taken root. At that time, those most schooled in stochastic calculus were trained in physics and applied mathematics. With the advent of the statistical approach to physics that arose from theories of thermodynamics and quantum mechanics in the early twentieth century, physicists had become accustomed to the path of processes buffeted by random shocks.

It too is of a value that follows a Weiner process because all assets the manager uses to fund his or her portfolio would follow a Weiner process. Then, the difference between the proportional drift of the sum of assets S used to fund this portfolio and the portfolio payout V will also follow a Weiner process: dS S − dV V = (μ − α )dt + θdB, The Theory 155 where µ is the mean return on the funding portfolio, ␣ is the mean return on the payout portfolio, and the Brownian motion term at the end is a combination of the standard deviation on the funding portfolio and the underlying stock multiplied by the Brownian motion of the underlying asset. Merton then established that the tracking error on such a funding portfolio made up of market securities is uncorrelated with the tracking error on the underlying security. This is a consequence of the choice of a funding portfolio that minimizes its correlation with the tracking errors so that one could not capitalize on predictable differences.

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The Quants by Scott Patterson

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The formula has many components, one of which is the assumption that the future movement of a stock—its volatility—is random and leaves out the likelihood of large swings (see fat tails). Brownian motion: First described by Scottish botanist Robert Brown in 1827 when observing pollen particles suspended in water, Brownian motion is the seemingly random vibration of molecules. Mathematically, the motion is a random walk in which the future direction of the movement—left, right, up, down—is unpredictable. The average of the motion, however, can be predicted using the law of large numbers, and is visually captured by the bell curve or normal distribution. Quants use Brownian motion mathematics to predict the volatility of everything from the stock market to the risk of a multinational bank’s balance sheet. Credit default swap: Created in the early 1990s, these contracts essentially provide insurance on a bond or a bundle of bonds.

After testing a range of other plant specimens, even the ground dust of rocks, and observing similar herky-jerky motion, he concluded that he was observing a phenomenon that was completely and mysteriously random. (The mystery remained unsolved for decades, until Albert Einstein, in 1905, discovered that the strange movement, by then known as Brownian motion, was the result of millions of microscopic particles buzzing around in a frantic dance of energy.) The connection between Brownian motion and market prices was made in 1900 by a student at the University of Paris named Louis Bachelier. That year, he’d written a dissertation called “The Theory of Speculation,” an attempt to create a formula that would capture the movement of bonds on the Paris stock exchange. The first English translation of the essay, which had lapsed into obscurity until it resurfaced again in the 1950s, had been included in the book about the market’s randomness that Thorp had read in New Mexico.

It’s much more likely that the confused drunkard will sway randomly in many directions as the night progresses (samples that would fall in the middle of the curve) than that he will move continuously in a straight line, or spin in a circle (samples that would fall in the ends of the curve, commonly known as the tails of the distribution). In a thousand coin flips, it’s more likely that the sample will contain roughly five hundred heads and five hundred tails (falling in the curve’s middle) than nine hundred heads and one hundred tails (outer edge of the curve). Thorp, already well aware of Einstein’s 1905 discovery, was familiar with Brownian motion and rapidly grasped the connection between bonds and warrants. Indeed, it was in a way the same statistical rule that had helped Thorp win at blackjack: the law of large numbers (the more observations, the more coin flips, the greater the certainty of prediction). While he could never know if he’d win every hand at blackjack, he knew that over time he’d come out on top if he followed his card-counting strategy.

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Financial Modelling in Python by Shayne Fletcher, Christopher Gardner

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Throughout this section we denote the function of the one-dimensional Brownian motion4 X T by yT := f (X T ) and the filtratio at time t by Ft . A full mathematical definitio of Brownian motion can be found in [16] but for the purposes of what follows all we need to know is that X T − X t is independent of X t for t < T and is normally distributed with zero mean and variance T − t. The aim of this subsection is to explain how to compute the following conditional expectations: yt := E[yT |Ft ] yt+ := E[max(yT , 0)|Ft ] (4.42) (4.43) Because the increments of a Brownian motion are independent of each other, the above conditional expectations can be written as yt (x) = E[ f (x + X T − X t )|xt = x] yt+ (x) = E[max( f (x + X T − X t ), 0)|xt = x] (4.44) (4.45) We donote the discrete grid of states for the Brownian motion increment X T − X t by xtT .

In the case of the rollback method the integrator is simply the integral member function of the distribution class represented by ftT, and in the case of the rollback max method the integrator is the integral max member function of ftT. class semi analytic domain integrator: def 4 create cached moments(self, x, f): In 1828 the Scottish botanist Robert Brown observed that pollen grains suspended in liquid performed an irregular motion – Brownian motion. 58 Financial Modelling in Python n = x.shape[0] self. ys = numpy.zeros([n, 4]) self. ys[2] = f.moments(4, x[2]) # cubic for j in range(2, n-2): self. ys[j+1] = f.moments(4, x[j+1]) # cubic def rollback (self, t, T, xt, xT, xtT, yT, regridder, integrator): if len(xt.shape) <> len(xT.shape) or \ len(xT.shape) <> len(yT.shape) or \ len(xt.shape) <> 1 or len(xtT.shape) <> 1: raise RuntimeError, ’expected one dimensional arrays’ nt = xt.shape[0] nT = xT.shape[0] ntT = xtT.shape[0] if nt <> nT or ntT <> nT: raise RuntimeError, ’expected array to be of same size’ if yT.shape[0] <> nT: raise RuntimeError, \ ‘array yT has different number of points to xT’ yt = numpy.zeros(nt) cT = piecewise cubic fit(xT, yT) for i in range(nt): # regrid regrid xT = numpy.zeros(nT) xti = xt[i] for j in range(nT): regrid xT[j] = xti+xtT[j] regrid yT = regridder(xT, cT, regrid xT) # polynomial fit cs = piecewise cubic fit(xtT, regrid yT) # perform expectation sum = 0 xl = xtT[2] for j in range(2, nT-2): # somehow this should be enscapsulated xh = xtT[j+1] sum = sum + integrator(cs[:, j-2], xl, xh, self. ys[j], self. ys[j+1]) xl = xh yt[i] = sum if t == 0.0: for j in range(1, nt): yt[j] = yt[0] break return yt Basic Mathematical Tools 59 def rollback(self, t, T, xt, xT, xtT, ftT, yT): # create cache of moments self. create cached moments(xtT, ftT) return self. rollback (t, T, xt, xT, xtT, yT, ftT.regrid, ftT.integral) def rollback max(self, t, T, xt, xT, xtT, ftT, yT): # create cache of moments self. create cached moments(xtT, ftT) return self. rollback (t, T, xt, xT, xtT, yT, ftT.regrid, ftT.integral max) Note that we precompute the moments in the function create cached moments() prior to performing the rollback.

Indeed Boost.Python offers many more features to help the C++ programmer to seamlessly expose C++ classes to Python and embed Python into C++. 218 Financial Modelling in Python Note that, as expected, the stochastic discount factor is a Q-martingale, in fact it is an exponential martingale, whereas the zero coupon bond price is not a Q-martingale because, as can be seen below, its SDE has a non-zero drift. d P(t, T ) = P(t, T ) (r (t)dt + (φ(t) − φ(T )) C(t)dW (t)) . (C.11) For non path-dependent pricing problems it is normally convenient to work in the so-called forward QT -measure. In this measure the numeraire at time t is simply P(t, T) and Girsanov’s theorem implies that W̄ (t), as define below, is a QT -Brownian motion d W̄ (t) = dW (t) + (φ(T ) − φ(t)) C(t)dt. Substitution of equation (C.12) into equation (C.10) yields t P(t, T ) P(0, T ) C(s)d W̄ (s) = exp − φ(T ) − φ(T ) P(t, T ) P(0, T ) 0 2 t 1 − φ(T ) − φ(T ) C(s)2 ds , ∀t ≤ T ≤ T. 2 0 (C.12) (C.13) In other words, the numeraire-rebased zero coupon bond in the forward QT -measure is a QT martingale. This to be expected in complete markets, where all numeraire-rebased tradables are martingales.

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Asset and Risk Management: Risk Oriented Finance by Louis Esch, Robert Kieffer, Thierry Lopez

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In ﬁnancial modelling, several speciﬁc cases of Itô process are used, and a geometric Brownian motion is therefore obtained when: at (Xt ) = a · Xt bt (Xt ) = b · Xt An Ornstein–Uhlenbeck process corresponds to: at (Xt ) = a · (c − Xt ) bt (Xt ) = b and the square root process is such that: √ bt (Xt ) = b Xt at (Xt ) = a · (c − Xt ) 2.3.3 Stochastic differential equations Expressions of the type dXt = at (Xt ) · dt + bt (Xt ) · dwt cannot simply be handled in the same way as the corresponding determinist expressions, because wt cannot be derived. It is, however, possible to extend the deﬁnition to a concept of stochastic differential, through the theory of stochastic integral calculus.8 As the stochastic process zt is deﬁned within the interval [a; b], the stochastic integral of zt is deﬁned within [a; b] with respect to the standard Brownian motion wt by: a 7 8 b zt dwt = lim n→∞ δ→0 n−1 ztk (wtk+1 − wtk ) k=0 The root function presents a vertical tangent at the origin.

We no longer have a stationary random model, such as Sharpe’s example, but a model that combines the random and temporal elements; this is known as a stochastic process. An example of this type of model is the Black–Scholes model for equity options (see Section 5.3.2), where the price p is a function of various variables (price of underlying asset, realisation price, maturity, volatility of underlying asset, risk-free interest rate). In this model, the price of the underlying asset is itself modelled by a stochastic process (standard Brownian motion). 3 Equities 3.1 THE BASICS An equity is a ﬁnancial asset that corresponds to part of the ownership of a company, its value being indicative of the health of the company in question. It may be the subject of a sale and purchase, either by private agreement or on an organised market. The law of supply and demand on this market determines the price of the equity. The equity can also give rise to the periodic payment of dividends. 3.1.1 Return and risk 3.1.1.1 Return on an equity Let us consider an equity over a period of time [t − 1; t] the duration of which may be one day, one week, one month or one year.

Note that this property is a generalisation for the random case of the determinist formula St = S0 · (1 + i)t . 3.4.2.2 Continuous model The method of equity value change shown in the binomial model is of the random walk type. At each transition, two movements are possible (rise or fall) with unchanged probability. When the period between each transaction tends towards 0, this type of random sequence converges towards a standard Brownian motion or SBM.52 Remember that we are looking at a stochastic process wt (a random variable that is a function of time), which obeys the following processes: • w0 = 0. • wt is a process with independent increments : if s < t < u, then wu − wt is independent of wt − ws . • wt is a process with stationary increments : the random variables wt+h − wt and wh are identically distributed. • Regardless of what t may be, the random variable √ wt is distributed according to a normal law of zero mean and standard deviation t: fwt (x) = √ 1 2πt e−x 2 /2t The ﬁrst use of this process for modelling the development in the value of a ﬁnancial asset was produced by L.

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Life at the Speed of Light: From the Double Helix to the Dawn of Digital Life by J. Craig Venter

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Einstein’s notion was eventually confirmed with careful experiments conducted in Paris by Jean Baptiste Perrin (1870–1942), who was rewarded for this and other work with the Nobel Prize in Physics in 1926. Brownian motion has profound consequences when it comes to understanding the workings of living cells. Many of the vital components of a cell, such as DNA, are larger than individual atoms but still small enough to be jostled by the constant pounding of the surrounding sea of atoms and molecules. So while DNA is indeed shaped like a double helix, it is a writhing, twisting, spinning helix as a result of the forces of random Brownian motion. The protein robots of living cells are only able to fold into their proper shapes because their components are mobile chains, sheets, and helices that are constantly buffeted within the cell’s protective membrane. Life is driven by Brownian motion, from the kinesin protein trucks that pull tiny sacks of chemicals along microtubules to the spinning ATP synthase.31 Critically, the amount of Brownian motion depends on temperature: too low and there is not enough motion; too high and all structures become randomized by the violent motion.

Life is driven by Brownian motion, from the kinesin protein trucks that pull tiny sacks of chemicals along microtubules to the spinning ATP synthase.31 Critically, the amount of Brownian motion depends on temperature: too low and there is not enough motion; too high and all structures become randomized by the violent motion. Thus life can only exist in a narrow temperature range. Within this range, the equivalent of a Richter 9 earthquake rages continuously inside cells. “You would not need to even pedal your bicycle: you would simply attach a ratchet to the wheel preventing it from going backwards and shake yourselves forward,” according to George Oster and Hongyun Wang, of the Department of Molecular and Cellular Biology at the University of California, Berkeley.32 Protein robots accomplish a comparable feat by using ratchets and power strokes to harness the power of Brownian motion. Due to the incessant random movement and vibrations of molecules, diffusion is very rapid over short distances, which enables biological reactions to occur with tiny quantities of reactants in the extremely confined volumes of most cells.

A protein with one hundred amino acids can fold in myriad ways, such that the number of alternate structures ranges from 2100 to 10100 possible conformations. For each protein to try every possible conformation would require on the order of ten billion years. But built into the linear protein code are the folding instructions, which are in turn determined by the linear genetic code. As a result, with the help of Brownian motion, the incessant molecular movement caused by heat energy, these processes happen very quickly—in a few thousandths of a second. They are driven by the fact that a correctly folded protein has the lowest possible free energy, so that, like water flowing to the lowest point, the protein naturally achieves its favored shape. The correctly folded conformation that ensures that the enzyme can work properly involves moving from a high degree of entropy and free energy to the thermodynamically stable state of decreased entropy and free energy.

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The Myth of the Rational Market: A History of Risk, Reward, and Delusion on Wall Street by Justin Fox

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Another famous example came in the mid-1920s when the young founder of Moscow’s Business Cycle Institute, Nikolai Kondratiev, proposed that economic activity moved in half-century-long “waves.”35 As the study of statistics progressed and the mathematics of random processes such as Brownian motion became more widely understood, those on the frontier of this work began to question these apparent cycles. In his November 1925 presidential address to Great Britain’s Royal Statistical Society, Cambridge professor George Udny Yule demonstrated that random Brownian motion could, with a little tweaking, produce dramatic patterns that didn’t look random at all.36 A few years later, a mathematician working for Kondratiev in Moscow penned what came to be seen as the definitive debunking of the pattern finders. “Almost all of the phenomena of economic life,” wrote Eugen Slutsky, “occur in sequences of rising and falling movements, like waves.”

The mathematical expectation of the speculator was the expected return of the stock or of the overall market, around which the actual return would fluctuate randomly. It is conceivable that Bachelier and Poincaré were aware of these flaws in 1900, and didn’t bother correcting them because Bachelier’s formula was meant to look only an “instant” into the future. It didn’t matter that Bachelier’s Brownian motion would eventually lead prices where they could not go, because he had made explicit that it was not to be used for purposes of long-term prediction anyway. Beset by no such qualms, Samuelson revised Bachelier’s formula. He introduced what he variously called “geometric,” “economic,” or “logarithmic” Brownian motion, which avoided negative prices by describing percentage moves of stock prices, not dollars and cents. And he depicted stock market investing as a bet in which the payoffs fluctuated randomly around the expected return. SAMUELSON BEGAN TALKING UP THESE ideas around MIT and in visits to other universities in the late 1950s.

After some study of the market, he concluded: It was a game of competitive gambling. In it some were smart and some were not so smart, and the players changed sides so often that it was a picture of financial chaos or bedlam. As I had some experience in molecular chaos as a physicist studying statistical mechanics, the analogies were very clear to me indeed.7 The analogy that was clearest to him was that of Brownian motion. As Samuelson had already noticed, straight arithmetic Brownian motion couldn’t possibly fit the data. Instead, Osborne used the same percentage-change version as Samuelson had, then published his findings in the March–April 1959 issue of the journal Operations Research. As soon as the article came out, letters pointing out similarities to the stock market work of Bachelier, Maurice Kendall, and others came pouring in. Osborne hadn’t known about any of that beforehand.

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Capital Ideas: The Improbable Origins of Modern Wall Street by Peter L. Bernstein

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If stock prices vary according to the square root of time, they bear a remarkable resemblance to molecules randomly colliding with one another as they move in space. An English physicist named Robert Brown discovered this phenomenon early in the nineteenth century, and it is generally known as Brownian motion. Brownian motion was a critical ingredient of Einstein’s theory of the atom. The mathematical formula that describes this phenomenon was one of Bachelier’s crowning achievements. Over time, in the literature on finance, Brownian motion came to be called the random walk, which someone once described as the path a drunk might follow at night in the light of a lamppost. No one knows who first used this expression, but it became increasingly familiar among academics during the 1960s, much to the annoyance of financial practitioners.

Second, again unaware that he was echoing Bachelier, Osborne argues that prices represent decisions at those moments—and only at those moments—when the buyer expects a stock to rise in price and the seller expects it to fall: Transactions take place only when there is a difference of opinion. This means that, for the market as a whole, the expected price change is zero. The market is as likely to go up x percent as down x percent. Third, Osborne’s mathematical manipulations show that the range within which prices tend to fluctuate will “increase as the square root of the time interval”26—Brownian motion, precisely as Bachelier had prophesied. Fourth, in a set of experiments with actual stock market data, Osborne confirms the hypothesis of Brownian motion, including percentage price changes over intervals of a day, a week, a month, two months, up to twelve years. He also finds that Cowles’s long history of stock prices follows “. . . the square root of time diffusion law very nicely indeed.”27 Finally, he finds that distributions of the monthly changes in the Dow Jones Industrial Average from 1925 to 1956 “are quite comparable.”28 The fifth finding made by Osborne’s imaginary statistician is another example of how Osborne, working all by himself, confirmed the research conclusions of others.

See also Wells Fargo Bank Barr Rosenberg Associates (BARRA) Battle for Investment Survival, The (Loeb) “Behavior of Stock Prices, The” (Fama) Bell Journal Bell Laboratories Beta: see Risk, systematic “Beta Revolution: Learning to Live with Risk” Black Monday (October, 1987, crash) Black/Scholes formula Block trading Boeing Bond(s) convertible discount rates and government high-grade interest rates international junk liquidity maturity risk treasury: see Bond(s), government zero-coupon Bond market Boston Company Brokerage commissions. See also Transaction costs Brownian motionBrownian Motion in the Stock Market” (Osborne) Butterfly swaps Buy and hold strategy California Public Employees Retirement System Calls: see Options Capital cost of optimal structure of preserving strategy “Capital Asset Prices: A Theory of Market Equilibrium Under Conditions of Risk” (Sharpe) Capital Asset Pricing Model (CAPM) non-stock applicability risk/return ratio in time analysis and Capital gains tax Capital Guardian Capital markets theory competition and corporate investment and debt/equity ratios and research CAPM: see Capital Asset Pricing Model CDs CEIR Center for Research in Security Prices (CRSP).

Mathematical Finance: Core Theory, Problems and Statistical Algorithms by Nikolai Dokuchaev

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The initial price S0>0 is a given non-random value, and the evolution of S(t) is described by the following Ito equation: dS(t)=S(t)(a(t)dt+σ(t)dw(t)). (5.1) Here w(t) is a (one-dimensional) Wiener process, and a and σ are market parameters. Sometimes in the literature S(t) is called a geometric Brownian motion (for the case of non-random and constant a, σ), sometimes ln S(t) is also said to be a Brownian motion. Mathematicians prefer to use the term ‘Brownian motion’ for w(t) only (i.e., Brownian motion is the same as a Wiener process). Definition 5.1 In (5.1), a(t) is said to be the appreciation rate, σ(t) is said to be the volatility. Note that, in terms of more general stochastic differential equations, the coefficient for dt (i.e., a(t)S(t)) is said to be the drift (or the drift coefficient), and the coefficient for dw(t) (i.e., σ(t)S(t)) is said to be the diffusion coefficient.

Prove that there exists an American option (Definition 3.43) such that its fair price is equal to the fair price of the option from Problem 3.44. © 2007 Nikolai Dokuchaev 4 Basics of Ito calculus and stochastic analysis This chapter introduces the stochastic integral, stochastic differential equations, and core results of Ito calculus. 4.1 Wiener process (Brownian motion) Let T>0 be given, Definition 4.1 We say that a continuous time random process w(t) is a (onedimensional) Wiener process (or Brownian motion) if (i) w(0)=0; (ii) w(t) is Gaussian with Ew(t)=0, Ew(t)2=t, i.e., w(t) is distributed as N(0, t); (iii) w(t+τ)−w(t) does not depend on {w(s), s≤t} for all t≥0, τ>0. Theorem 4.2 (N. Wiener). There exists a probability space such that there exists a pathwise continuous process with these properties. This is why we call it the Wiener process.

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Money Changes Everything: How Finance Made Civilization Possible by William N. Goetzmann

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It also required the time period for which the option is granted (the “maturity” of the option). Bachelier presented his book, Théorie de la Spéculation, as his doctoral thesis in mathematics at the Sorbonne in 1900. In working through the problem of option pricing, Bachelier had to devise a precise definition of how a stock price moved randomly through time. We now refer to this as Brownian motion. Interestingly, Albert Einstein developed a Brownian motion model in 1905, evidently later and independently from Bachelier. Bachelier’s answer to the option pricing problem turned out to be an equation beyond the knowledge of market participants at the time. This presented an interesting philosophical issue. If option prices conformed to a complex, nonlinear multivariate function that was undiscovered until 1900, how then did the invisible hand—the process of speculation—drive them toward efficiency?

Evidently none knew of Bachelier, and thus they had to retrace the mathematical logic of fair prices and random walks when they began work on the problem of option pricing in the late 1960s. Like Bachelier, they relied on a model of variation in prices—Brownian motion—although unlike Bachelier, they chose one that did not allow prices to become negative—a limitation of Bachelier’s work. The Black-Scholes formula, as it is now referred to, was mathematically sophisticated, but at its heart it contained a novel economic—as opposed to mathematical—insight. They discovered that the invisible hand setting option prices was risk-neutral. Option payoffs could be replicated risklessly, provided one could trade in an ideal, frictionless market in which stocks behaved according to Brownian motion. Later researchers4 developed a simple framework called a “binomial model” that was able to match the payoff of a put or a call by trading just the stock and a bond through time.

Paul Lévy formalized these prior random walk models into a very general family of stochastic processes referred to as Lévy processes. Brownian motion was just one process in the family of Lévy processes—and perhaps the best behaved of them. Other stochastic processes have such things as discontinuous jumps and unusually large shocks (which might, for example, explain the crash of 1987, when the US stock market lost 22.6% of its value in a single day). In the 1960s, Benoit Mandelbrot began to investigate whether Lévy processes described economic time series like cotton prices and stock prices. He found that the ones that generated jumps and extreme events better described financial markets. He developed a mathematics around these unusual Lévy processes that he called “fractal geometry.” He argued that unusual events—Taleb’s black swan—were in fact much more common phenomena than Brownian motion would suggest.

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The Drunkard's Walk: How Randomness Rules Our Lives by Leonard Mlodinow

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Then, in a deathblow to his wishful interpretation of the discovery, Brown also observed the motion when looking at inorganic particles—of asbestos, copper, bismuth, antimony, and manganese. He knew then that the movement he was observing was unrelated to the issue of life. The true cause of Brownian motion would prove to be the same force that compelled the regularities in human behavior that Quételet had noted—not a physical force but an apparent force arising from the patterns of randomness. Unfortunately, Brown did not live to see this explanation of the phenomenon he observed. The groundwork for the understanding of Brownian motion was laid in the decades that followed Brown’s work, by Boltzmann, Maxwell, and others. Inspired by Quételet, they created the new field of statistical physics, employing the mathematical edifice of probability and statistics to explain how the properties of fluids arise from the movement of the (then hypothetical) atoms that make them up.

But actually it did take an Einstein to finally convince the scientific world of the need for that new approach to physics. Albert Einstein did it in 1905, the same year in which he published his first work on relativity. And though hardly known in popular culture, Einstein’s 1905 paper on statistical physics proved equally revolutionary. In the scientific literature, in fact, it would become his most cited work.32 EINSTEIN’S 1905 WORK on statistical physics was aimed at explaining a phenomenon called Brownian motion. The process was named for Robert Brown, botanist, world expert in microscopy, and the person credited with writing the first clear description of the cell nucleus. Brown’s goal in life, pursued with relentless energy, was to discover through his observations the source of the life force, a mysterious influence believed in his day to endow something with the property of being alive. In that quest, Brown was doomed to failure, but one day in June 1827, he thought he had succeeded.

But most physicists are practical, and so the most important roadblock to acceptance was that although the theory reproduced some laws that were known, it made few new predictions. And so matters stood until 1905, when long after Maxwell was dead and shortly before a despondent Boltzmann would commit suicide, Einstein employed the nascent theory to explain in great numerical detail the precise mechanism of Brownian motion.34 The necessity of a statistical approach to physics would never again be in doubt, and the idea that matter is made of atoms and molecules would prove to be the basis of most modern technology and one of the most important ideas in the history of physics. The random motion of molecules in a fluid can be viewed, as we’ll note in chapter 10, as a metaphor for our own paths through life, and so it is worthwhile to take a little time to give Einstein’s work a closer look.

pages: 282 words: 89,436

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But that result was just the overture of his grand symphony of scientific revelations. Another major paper Einstein published in 1905 concerned a phenomenon called Brownian motion, named after the Scottish botanist Robert Brown, involving tiny random fluctuations of small particles. In 1827, Brown had observed the agitated motion of particles found in pollen grains immersed in water. He failed to find a credible explanation for their erratic behavior. Building upon his doctoral thesis, Einstein decided to model the movements of particles bashed around by water molecules and discovered precisely the kind of haphazard jig seen by Brown. By explaining Brownian motion as the zigzag result of 35 Einstein’s Dice and Schrödinger’s Cat myriad particle collisions, Einstein furnished important evidence for the existence of atoms.

Nothing in this practical work would hint at the explosion of ideas about to be ignited. In the spring of that year, Einstein took aim. Staring classical physics straight in the face, he lit the fuses and launched his grenades. He submitted four papers to a prestigious journal, Annalen der Physik. One was a version of his dissertation. The other three articles— addressing the photoelectric effect, Brownian motion, and the special theory of relativity—would shake the foundations of physical science. Einstein’s paper on the photoelectric effect cemented Planck’s quantum idea by making it tangible and eminently measurable. It considers what would happen if a researcher shines a light on a metal, supplying enough energy to release an electron. If light were purely a wave, theory suggested, its amount of energy would depend mainly on its 34 The Clockwork Universe brightness.

(Schrödinger), 208 Aristotle, 77, 215 Arkani-Hamed, Nima, 232 Ashtekar, Abhay, 232 Aspect, Alain, 210 Assembly of German Natural Scientists and Physicians (Vienna conference of 1913), 44–51 Atomic bomb Einstein and, 168–169, 183 Germany’s efforts to develop, 178–180 Atomic model, 35 Bohr, 46–48, 73, 81, 82, 84–85 Bohr-Sommerfeld, 83, 98 Rutherford, 45 Atomism, 28, 80 Atoms argument over reality of, 22–24 photoelectric effect and, 35 as probabilistic mechanism, 102 Autiero, Dario, 235 Axioms, 18 Aydelotte, Frank, 168 Bailey, Herbert, 204, 205 Ball-in-the-box thought experiment, 139, 141 Balmer series, 48 Bär, Richard, 154, 155 Bargmann, Sonja, 204 Bargmann, Valentine “Valya,” 149, 168, 169 Barnett, Lincoln, 204 Bauer, Alexander, 16 Bauer, Minnie, 16 Bauer-Bohm, Hansi, 143–144 Becquerel, Henri, 28 Bell, John, 210 Berg, Moe, 179 255 Index Heisenberg and, 86–87, 88 probabilistic interpretation and, 105 Schrödinger wave equation and, 5, 6 wavefunction as ghost field and, 99–100 Bose, Satyendra, 90, 92, 225 Bose-Einstein statistics, 90 Bosonic strings, 230–231 Bosons, 225–227, 230 Bottom quarks, 227 Braunizer, Andreas, 217 Braunizer, Arnulf, 217 Braunizer, Ruth, 219–220 Brecht, Bertolt, 109 Brout, Robert, 226 Brown, Robert, 35 Brownian motion, Einstein on, 34, 35–36 Buddhism/Buddha, 77, 80 Bergmann, Peter, 149, 168, 169, 213, 231 Berlin, 109, 131 Bertel, Annemarie “Anny,” 76, 77 Bertolucci, Sergio, 235 Besso, Michele, 29, 55–56 Beta decay, 136 “Big Bang,” 63 Birch, Francis, 192 BKS (Bohr-Kramers-Slater) theory, 103 Blackbody radiation, 31–32, 90 Black holes, 59 Black Mountain College, 169 Bohm, David, 209–210 Bohm-Aharonov version of EPR thought experiment, 209–210 Bohr, Margrethe, 102 Bohr, Niels, 1, 110, 215 atomic model, 35, 46–48, 73, 81, 82, 84–85 Eddington’s theory and, 148 Einstein and, 137, 168, 200 EPR paper and, 138 escape from Denmark, 179 uncertainty principle and, 106 “Bohr Festival,” 82, 84–85 Bohr’s Institute for Theoretical Physics, 88, 100–102 Bohr-Sommerfeld atomic model, 83, 98 Bohr-Sommerfeld quantization rules, 93 Boltzmann, Ludwig, 22–23, 24, 30, 81, 92 Bondi, Hermann, 157 Borges, Jorge Luis, 62 Born, Max, 1, 82, 87, 89, 144 dismissal and exile of, 128, 130 Einstein and, 103–104, 134 as go-between in de Valera offer to Schrödinger, 154, 155 Calabi, Eugenio, 231 Calabi-Yau manifolds, 231 Callaway, Joseph, 208 Caltech, Einstein’s visits to, 122–123 “Can Quantum Mechanical Description of Physical Reality Be Considered Complete?”

Mathematics for Finance: An Introduction to Financial Engineering by Marek Capinski, Tomasz Zastawniak

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N tN As N → ∞, we have tN → t and N tN → ∞, so that wN (tN ) → W (t) √ in distribution, where W (t) = tX. The last equality means that W (t) is normally distributed with mean 0 and variance t. This argument, based on the Central Limit Theorem, works for any single ﬁxed time t > 0. It is possible to extend the result to obtain a limit for all times t ≥ 0 simultaneously, but this is beyond the scope of this book. The limit W (t) is called the Wiener process (or Brownian motion). It inherits many of the properties of the random walk, for example: 1. W (0) = 0, which corresponds to wN (0) = 0. 2. E(W (t)) = 0, corresponding to E(wN (t)) = 0 (see the solution of Exercise 3.25). 3. Var(W (t)) = t, with the discrete counterpart Var(wN (t)) = t (see the solution of Exercise 3.25). 4. The increments W (t3 ) − W (t2 ) and W (t2 ) − W (t1 ) are independent for 0 ≤ t1 ≤ t2 ≤ t3 ; so are the increments wN (t3 ) − wN (t2 ) and wN (t2 ) − wN (t1 ). 2 See, for example, Capiński and Zastawniak (2001). 70 Mathematics for Finance 5.

Our treatment of continuous time is a compromise lacking full mathematical rigour, which would require a systematic study of Stochastic Calculus, a topic 186 Mathematics for Finance treated in detail in more advanced texts. In place of this, we shall exploit an analogy with the discrete time case. As a starting point we take the continuous time model of stock prices developed in Chapter 3 as a limit of suitably scaled binomial models with time steps going to zero. In the resulting continuous time model the stock price is given by (8.5) S(t) = S(0)emt+σW (t) , where W (t) is the standard Wiener process (Brownian motion), see Section 3.3.2. This means, in particular, that S(t) has the log normal distribution. Consider a European option on the stock expiring at time T with payoﬀ f (S(T )). As in the discrete-time case, see Theorem 8.4, the time 0 price D(0) of the option ought to be equal to the expectation of the discounted payoﬀ e−rT f (S(T )), (8.6) D(0) = E∗ e−rT f (S(T )) , under a risk-neutral probability P∗ that turns the discounted stock price process e−rt S(t) into a martingale.

Glossary of Symbols A B β c C C CA CE E C Cov delta div div0 D D DA E E∗ f F gamma Φ k K i m ﬁxed income (risk free) security price; money market account bond price beta factor covariance call price; coupon value covariance matrix American call price European call price discounted European call price covariance Greek parameter delta dividend present value of dividends derivative security price; duration discounted derivative security price price of an American type derivative security expectation risk-neutral expectation futures price; payoﬀ of an option; forward rate forward price; future value; face value Greek parameter gamma cumulative binomial distribution logarithmic return return coupon rate compounding frequency; expected logarithmic return 305 306 Mathematics for Finance M m µ N N k ω Ω p p∗ P PA PE P E PA r rdiv re rF rho ρ S S σ t T τ theta u V Var VaR vega w w W x X y z market portfolio expected returns as a row matrix expected return cumulative normal distribution the number of k-element combinations out of N elements scenario probability space branching probability in a binomial tree risk-neutral probability put price; principal American put price European put price discounted European put price present value factor of an annuity interest rate dividend yield eﬀective rate risk-free return Greek parameter rho correlation risky security (stock) price discounted risky security (stock) price standard deviation; risk; volatility current time maturity time; expiry time; exercise time; delivery time time step Greek parameter theta row matrix with all entries 1 portfolio value; forward contract value, futures contract value variance value at risk Greek parameter vega symmetric random walk; weights in a portfolio weights in a portfolio as a row matrix Wiener process, Brownian motion position in a risky security strike price position in a ﬁxed income (risk free) security; yield of a bond position in a derivative security Index admissible – portfolio 5 – strategy 79, 88 American – call option 147 – derivative security – put option 147 amortised loan 30 annuity 29 arbitrage 7 at the money 169 attainable – portfolio 107 – set 107 183 basis – of a forward contract 128 – of a futures contract 140 basis point 218 bear spread 208 beta factor 121 binomial – distribution 57, 180 – tree model 7, 55, 81, 174, 238 Black–Derman–Toy model 260 Black–Scholes – equation 198 – formula 188 bond – at par 42, 249 – callable 255 – face value 39 – ﬁxed-coupon 255 – ﬂoating-coupon 255 – maturity date 39 – stripped 230 – unit 39 – with coupons 41 – zero-coupon 39 Brownian motion 69 bull spread 208 butterﬂy 208 – reversed 209 call option 13, 181 – American 147 – European 147, 188 callable bond 255 cap 258 Capital Asset Pricing Model 118 capital market line 118 caplet 258 CAPM 118 Central Limit Theorem 70 characteristic line 120 compounding – continuous 32 – discrete 25 – equivalent 36 – periodic 25 – preferable 36 conditional expectation 62 contingent claim 18, 85, 148 – American 183 – European 173 continuous compounding 32 continuous time limit 66 correlation coeﬃcient 99 coupon bond 41 coupon rate 249 307 308 covariance matrix 107 Cox–Ingersoll–Ross model 260 Cox–Ross–Rubinstein formula 181 cum-dividend price 292 delta 174, 192, 193, 197 delta hedging 192 delta neutral portfolio 192 delta-gamma hedging 199 delta-gamma neutral portfolio 198 delta-vega hedging 200 delta-vega neutral portfolio 198 derivative security 18, 85, 253 – American 183 – European 173 discount factor 24, 27, 33 discounted stock price 63 discounted value 24, 27 discrete compounding 25 distribution – binomial 57, 180 – log normal 71, 186 – normal 70, 186 diversiﬁable risk 122 dividend yield 131 divisibility 4, 74, 76, 87 duration 222 dynamic hedging 226 eﬀective rate 36 eﬃcient – frontier 115 – portfolio 115 equivalent compounding 36 European – call option 147, 181, 188 – derivative security 173 – put option 147, 181, 189 ex-coupon price 248 ex-dividend price 292 exercise – price 13, 147 – time 13, 147 expected return 10, 53, 97, 108 expiry time 147 face value 39 ﬁxed interest 255 ﬁxed-coupon bond 255 ﬂat term structure 229 ﬂoating interest 255 ﬂoating-coupon bond 255 ﬂoor 259 ﬂoorlet 259 Mathematics for Finance forward – contract 11, 125 – price 11, 125 – rate 233 fundamental theorem of asset pricing 83, 88 future value 22, 25 futures – contract 134 – price 134 gamma 197 Girsanov theorem 187 Greek parameters 197 growth factor 22, 25, 32 Heath–Jarrow–Morton model hedging – delta 192 – delta-gamma 199 – delta-vega 200 – dynamic 226 in the money 169 initial – forward rate 232 – margin 135 – term structure 229 instantaneous forward rate interest – compounded 25, 32 – ﬁxed 255 – ﬂoating 255 – simple 22 – variable 255 interest rate 22 interest rate option 254 interest rate swap 255 261 233 LIBID 232 LIBOR 232 line of best ﬁt 120 liquidity 4, 74, 77, 87 log normal distribution 71, 186 logarithmic return 34, 52 long forward position 11, 125 maintenance margin 135 margin call 135 market portfolio 119 market price of risk 212 marking to market 134 Markowitz bullet 113 martingale 63, 83 Index 309 martingale probability 63, 250 maturity date 39 minimum variance – line 109 – portfolio 108 money market 43, 235 no-arbitrage principle 7, 79, 88 normal distribution 70, 186 option – American 183 – at the money 169 – call 13, 147, 181, 188 – European 173, 181 – in the money 169 – interest rate 254 – intrinsic value 169 – out of the money 169 – payoﬀ 173 – put 18, 147, 181, 189 – time value 170 out of the money 169 par, bond trading at 42, 249 payoﬀ 148, 173 periodic compounding 25 perpetuity 24, 30 portfolio 76, 87 – admissible 5 – attainable 107 – delta neutral 192 – delta-gamma neutral 198 – delta-vega neutral 198 – expected return 108 – market 119 – variance 108 – vega neutral 197 positive part 148 predictable strategy 77, 88 preferable compounding 36 present value 24, 27 principal 22 put option 18, 181 – American 147 – European 147, 189 put-call parity 150 – estimates 153 random interest rates random walk 67 rate – coupon 249 – eﬀective 36 237 – forward 233 – – initial 232 – – instantaneous 233 – of interest 22 – of return 1, 49 – spot 229 regression line 120 residual random variable 121 residual variance 122 return 1, 49 – expected 53 – including dividends 50 – logarithmic 34, 52 reversed butterﬂy 209 rho 197 risk 10, 91 – diversiﬁable 122 – market price of 212 – systematic 122 – undiversiﬁable 122 risk premium 119, 123 risk-neutral – expectation 60, 83 – market 60 – probability 60, 83, 250 scenario 47 security market line 123 self-ﬁnancing strategy 76, 88 short forward position 11, 125 short rate 235 short selling 5, 74, 77, 87 simple interest 22 spot rate 229 Standard and Poor Index 141 state 238 stochastic calculus 71, 185 stochastic diﬀerential equation 71 stock index 141 stock price 47 strategy 76, 87 – admissible 79, 88 – predictable 77, 88 – self-ﬁnancing 76, 88 – value of 76, 87 strike price 13, 147 stripped bond 230 swap 256 swaption 258 systematic risk 122 term structure 229 theta 197 time value of money 21 310 trinomial tree model Mathematics for Finance 64 underlying 85, 147 undiversiﬁable risk 122 unit bond 39 value at risk 202 value of a portfolio 2 value of a strategy 76, 87 VaR 202 variable interest 255 Vasiček model 260 vega 197 vega neutral portfolio volatility 71 weights in a portfolio Wiener process 69 yield 216 yield to maturity 229 zero-coupon bond 39 197 94

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The Mathematics of Banking and Finance by Dennis W. Cox, Michael A. A. Cox

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Further, a 2 is equally likely to appear on any subsequent roll of the dice. 4.3 ESTIMATION OF PROBABILITIES There are a number of ways in which you can arrive at an estimate of Prob(A) for the event A. Three possible approaches are: r A subjective approach, or ‘guess work’, which is used when an experiment cannot be easily r repeated, even conceptually. Typical examples of this include horse racing and Brownian motion. Brownian motion represents the random motion of small particles suspended in a gas or liquid and is seen, for example, in the random walk pattern of a drunken man. The classical approach, which is usually adopted if all sample points are equally likely (as is the case in the rolling of a dice as discussed above). The probability may be measured with certainty by analysing the event. Using the same mathematical notation, a mathematical definition of this is: Prob(A) = Number of events classifiable as A Total number of possible events A typical example of such a probability is a lottery.

pages: 261 words: 86,261

The Pleasure of Finding Things Out: The Best Short Works of Richard P. Feynman by Richard P. Feynman, Jeffrey Robbins

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I speak, of course, “in principle,” and I am not speaking about the actual manufacture of such devices. Let us therefore discuss what happens if we try to make the devices as small as possible. Reducing the Size FIGURE 5 So my third topic is the size of computing elements and now I speak entirely theoretically. The first thing that you would worry about when things get very small is Brownian motion*—everything is shaking about and nothing stays in place. How can you control the circuits then? Furthermore, if a circuit does work, doesn’t it now have a chance of accidentally jumping back? If we use two volts for the energy of this electric system, which is what we ordinarily use (Fig. 5), that is eighty times the thermal energy at room temperature (kT = 1/40 volt) and the chance that something jumps backward against 80 times thermal energy is e, the base of the natural logarithm, to the power minus eighty, or 10-43.

If the things flip back and then go forward later it is still all right. It’s very much the same as a particle in a gas which is bombarded by the atoms around it. Such a particle usually goes nowhere, but with just a little pull, a little prejudice that makes a chance to move one way a little higher than the other way, the thing will slowly drift forward and travel from one end to the other, in spite of the Brownian motion that it has made. So our computer will compute provided we apply a drift force to pull the thing across the calculation. Although it is not doing the calculation in a smooth way, nevertheless, calculating like this, forward and backward, it eventually finishes the job. As with the particle in the gas, if we pull it very slightly, we lose very little energy, but it takes a long time to get to one side from the other.

., 39, 40, 43 Bessel functions, 223 Beta decay, 192 Bethe, Hans, 11, 60–61, 64, 86, 190, 197–198, 235 Bets, 69 Big Bang, 199, 200 Biology, 99, 100, 101, 105, 123–124, 124–125, 126, 241. See also under Chemistry Black holes, 229(n) Bohr, Aage, 86, 87 Bohr, Niels, 86–88, 190, 203 Bongos, 191 Books of the world, 121–122 Bose-Einstein condensate, xvii Brains, 145, 194, 218, 222. See also Computers, analogy with brains Brass, 90 Brave New World (Huxley), 99 Bridgman, Percy, 118 British Museum Library, 121 Brownian motion, 38–39, 38(fig.), 42 Buddhism, 142 Cadmium, 74, 76 Calculus, 6–7, 195 California Institute of Technology (Caltech), 13, 191–192, 205–216, 226, 232–233 Caltech Cosmic Cube, 30 Cargo cults, 187, 242–243. See also Science, Cargo Cult Science Cathode ray oscilloscope, 120 Catholic Church, 7, 98, 111, 112–113 Censorship. See under Los Alamos Certainty. See Uncertainty Challenger. See Space Shuttle Challenger Chemistry and biology, 137, 138 chemical analysis/synthesis, 125, 137–138 chemical reactions, 131, 218 Chess, 48 chess game analogy, 13–14, 14–15 Chicago, 56–57 Children, 21–22, 145–146, 172 Christ, divinity of, 251, 254 Christie, Bob, 62, 73, 83 Communication, 113, 147 Communism, 251–252 Compton, 55, 56 Computers, 27–52, 126–129, 194 analogy with brains, 46–48 central processors, 29, 30–31 on Challenger Orbiter, 164–168 chips in, 28–29, 44 clock time vs. circuit time in, 35 debugging, 28 energy consumption of, 29, 32–37, 38, 42, 43, 44, 50–51 gates, reversible/irreversible, 39–40, 41(fig.), 43, 50.

pages: 445 words: 105,255

Radical Abundance: How a Revolution in Nanotechnology Will Change Civilization by K. Eric Drexler

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The idea of atoms, of course, had been around since antiquity. In Greece circa 400 BCE Democritus argued that matter must ultimately consist of indivisible particles—as indeed atoms are, barring nuclear reactions. In Rome circa 50 BCE Lucretius argued the same case in considerable depth and suggested that dust motes that could be seen dancing in sunbeams were, in fact, driven by what is now called “Brownian motion,” the effect of collisions with atoms (and for some of the motions he saw, he was right). Today, the most advanced forms of atomically precise fabrication rely on this Brownian dance to move molecules. After classical times, centuries passed before any further progress was achieved in understanding the atomic basis of the material world. Inquiry reached a landmark in England in the early 1800s when John Dalton observed that chemical reactions combined substances in fixed proportions and explained these proportions in terms of atoms.

PART 5 THE TRAJECTORY OF TECHNOLOGY CHAPTER 12 Today’s Technologies of Atomic Precision HUMAN TECHNOLOGY EVENTUALLY LED to machines, yet it began with wood, hide, stones, and hands—which is to say, with biopolymers (cellulose, collagen) and harder, inorganic materials used to make hand-held tools. AP nanotechnologies are following a similar path, but with AP control of biopolymers and inorganic materials using assembly driven by Brownian motion rather than hands. Once again, advances will lead to machines for making things. Just as an early blacksmith’s hammer and tongs differ from an automated machine in a watch-making factory, today’s early tools differ greatly from advanced APM systems. And just as blacksmith-level technology led to today’s machines, so today’s AP molecular technologies will lead to tomorrow’s nanomachines. Where do we stand today on the road to advanced atomically precise fabrication, the road that leads to APM?

To understand the next steps and the road ahead, it’s important to understand how self-assembly and stereotactic methods can be combined in complementary ways. Today’s Self-Assembly Methods Self-assembly has one great advantage: Because it employs thermal motion to move parts into place, assembling parts requires no nanoscale machinery. Researchers can use conventional biological or chemical means to make the parts, and if they’re properly designed, thermally driven Brownian motion can do the rest. Self-assembly, however, also brings with it a set of challenges and limitations. To produce complex, non-repetitive structures by self-assembly, each part must bind in just one specific place and position as the structure comes together, and this means that each part must have a unique shape, like a piece of a jigsaw puzzle. These molecular puzzle pieces can’t be too small, or they’d be too simple and too much alike; they can’t bind too strongly, or near-matches would end up in the wrong places and never let go; and like a finished jigsaw puzzle, the end product would be divided by many irregular seams.

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Erwin Schrodinger and the Quantum Revolution by John Gribbin

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He spent the winter of 1914–15 there, enjoying the beautiful mountain scenery while the conflict on the Western Front settled into the grim trench warfare for which the First World War is best remembered by Britain and the other Western participants. His next posting was to the equally peaceful, but less scenic, garrison town of Komárom, between Vienna and Budapest. There, Schrödinger wrote a paper about the behaviour of small particles being jostled in a fluid (gas or liquid) by the impact of molecules of the fluid. This is known as Brownian motion, after the Scottish physicist Robert Brown (1773–1858), who studied it in the 1820s. In 1905, Albert Einstein had proved that this erratic jittering can be explained statistically as caused by the constant but uneven bombardment that particles such as pollen grains receive from atoms and molecules, and thereby provided compelling evidence for the reality of atoms—just too late for this to be much comfort to Boltzmann.1 In a quite separate investigation, culminating in 1912, the American Robert Andrews Millikan (1868–1953)—who also, incidentally, coined the term “cosmic rays”—had managed to measure the charge on the electron by monitoring the way tiny electrically charged droplets of water or oil drift in an electric field.

In 1905, Albert Einstein had proved that this erratic jittering can be explained statistically as caused by the constant but uneven bombardment that particles such as pollen grains receive from atoms and molecules, and thereby provided compelling evidence for the reality of atoms—just too late for this to be much comfort to Boltzmann.1 In a quite separate investigation, culminating in 1912, the American Robert Andrews Millikan (1868–1953)—who also, incidentally, coined the term “cosmic rays”—had managed to measure the charge on the electron by monitoring the way tiny electrically charged droplets of water or oil drift in an electric field. These droplets are small enough to be affected by Brownian motion, and Schrödinger analysed statistically the importance of these effects in Millikan-type experiments. Nothing dramatic came out of the study, but it is important in the context of Schrödinger’s career because it was his first published foray into statistics, which would later loom large in his work. By the time this paper was published, the war, and Schrödinger, had both moved on. Italy was persuaded to join the Triple Entente with promises of large chunks of Austria, and declared war on 23 May 1915.

Investment: A History by Norton Reamer, Jesse Downing

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He became quite ﬂuent in ﬁnance as a result of this experience, and soon Bachelier found himself back in academia working under the polymath Henri Poincaré.1 He defended the ﬁrst portion of his thesis, entitled “Theory of Speculation,” in March 1900. In it, he showed how to value complicated French derivatives using advanced mathematics. In fact, his approach bore some similarity to that of Fischer Black and Myron Scholes many years later. Bachelier’s work was the ﬁrst use of formal models of randomness to describe and evaluate markets. In his paper, Bachelier used a form of what is called Brownian motion.2 Brownian motion was named after Robert Brown, who studied the random motions of pollen in water. Albert Einstein would describe this same phenomenon in one of his famous 1905 papers. The mathematical underpinnings of this description of randomness could be applied not only to the motions of small particles but also to the movements of markets. Bachelier’s work did not seem to have an immediate and profound inﬂuence on those markets, however.

Savage wrote postcards to a group of economists asking if any economists were familiar with Bachelier.16 And Samuelson did rethink many of the assumptions Bachelier made, such as noting that the expected return of the speculator should not be zero, as Bachelier suggested, but should rather be positive and commensurate with the risk the speculator is enduring. Otherwise, the investor would simply either not invest or own the risk-free security (short-dated Treasury debt). He also redeﬁned Bachelier’s equations to have the returns in lieu of the actual stock prices move in accordance with a slightly different form of Brownian motion because Bachelier’s form of Brownian motion implied that a stock could potentially have a negative price, which is not sensible, as the concept of limited liability for shareholders implies that the ﬂoor of value is zero.17 The Emergence of Investment Theory 235 Samuelson helped motivate the work on derivatives pricing with a 1965 paper on warrants and a 1969 paper with Robert Merton on the same subject—although he did, as he would later note, miss one crucial assumption that Black and Scholes were able to make in their formulation of options prices.18 Samuelson can be considered an intermediary in calling attention to the subﬁeld of derivatives pricing, even if the cornerstone of the most famous ﬁnal theory was not his own.

See also commercial banks; merchant banks Barbarians at the Gate, 276 Bardi bank, 43–44 Barings Bank, 170–72 behavioral ﬁnance, 251–54 bell curve, 239 Benartzi, Shlomo, 252 benchmarking, 328–30 Benedict XIV (pope), 37 Bent, Bruce, 143 Bentham, Jeremy, 36 Index 417 Bergen Tunnel construction project, 178 Berlin Wall, fall of, 96 Bernanke, Ben, 9, 197, 208, 226 beta, 243–45; alpha and, 248–49, 254, 308–9 Bible, 34, 239 Bierman, Harold, 204 bills of exchange, 83–84 Birds, The (Aristophanes), 24 Bismarck, Otto von, 108–9 Black, Fischer, 230, 235–36 BlackRock, 299 Black Thursday (October 24, 1929), 164 Blunt, John, 67–68 Bocchoris, 23 Boesky, Ivan, 147, 181, 184–86 Bogle, Jack, 284–85 bond index funds, 285 bonds: convertible, 178; fabrication of Italian, 163; government, 6, 135, 176; high-yield, 276; holding, 93; investment in, 257, 259, 297, 301; management of, 102 Boness, James, 236 bookkeeping, double-entry, 41 borrower, reputation of, 22–23 Borsa Italiana, 95 Boston, 100 Boston Consulting Group, 194 Boston Post, 157 bourses, 84 Breitowitz, Yitzhok, 150 Bristol-Myers Squibb, 188 Britain: beggar-thy-neighbor policies in, 202; colonial rule of India, 49–50, 61; supplies contract, after American Revolution, 175 British Bankers’ Association, 182 British East India Company, 66, 326 Brookings Institution, 91 Brown, Henry, 143 Brown, Robert, 230 Brownian motion, 230, 234 Brumberg, Richard, 121–22 Brush, Charles, 81 Bubble Act of 1720, 68, 87 bubbles: causes of, 5; housing bubble of 2004–2006, 213–14; South Sea Bubble, 68–69; technology (dot-com bubble of 1999-2000), 187, 213, 223–24, 246, 263, 276, 287 bubonic plague, 75 bucket shops, 90 Buddhist temples, 29–30 budget deﬁcit projections, 218 Buffett, Warren: American Express and, 169; earnings of, 305; on efficient market hypothesis, 250–51; ﬁnancial leverage and, 6; on real ownership, 4; resource allocation and, 7; as value manager, 140 bullet payments, 321 bull market: in 1920s, 91; of 1990s, 269, 285; after World War II, 92, 143 burghers, 42 Bush, George W., 218, 225 BusinessWeek, 143, 188 Buttonwood Agreement, 88, 97 Byzantines, 52 Cabot, Paul, 141 Cady, Roberts decision, 192 Caesar, 28 Calahan, Edward, 90 California Public Employees Retirement System (CalPERS), 129 418 Investment: A History call option: performance fee as, 310–11; sale of, 151 CalPERS.

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The Inner Lives of Markets: How People Shape Them—And They Shape Us by Tim Sullivan

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When we looked at that list of papers and thought about what we could do with the information, it occurred to us that these relatively esoteric academic papers had had, like their counterparts in physics, an outsized influence. That seemed worth exploring, not by reprinting the original papers but by examining how those ideas have lived in the world. This half-century’s worth of economic thought—often as incomprehensible to outsiders in its original formulations as Einstein’s investigations into the theory of Brownian motion is to non-physicists—has been used to make markets work better and, in an ever-widening set of applications, has helped them reach more deeply into our lives. The Inner Lives of Markets explores the intersection of those economic ideas and our lives. INTRODUCTION TERMS OF SERVICE At 109 Lincoln Street in Rutland, Vermont, stands a dilapidated yellow clapboard building. Rutland was incorporated in the late nineteenth century, flush with money from the marble quarries just outside town.

His thesis was the beginning of his lifelong project to bring “unification—and clarification—in mathematics” to the profession. Nobel prize winners are generally associated with a particular theory, insight or cohesive set of insights, or even a single specific paper. For Samuelson, rewriting economics in the new language of math was the contribution itself, often borrowing ideas already developed by physicists and mathematicians. He introduced, for example, the idea of Brownian motion (which he borrowed from physics) as a way of understanding financial markets, and a version of Henry-Louis Le Chatelier’s principle (developed by chemists in the nineteenth century) as a tool for understanding market equilibrium. Samuelson didn’t undertake this project alone but was responsible for many of its central contributions. We also see in Samuelson the sense that the discipline imposed by mathematics made economics no less relevant to understanding real-world problems.

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The Wisdom of Finance: Discovering Humanity in the World of Risk and Return by Mihir Desai

As he watched pollen emit particles in water, these particles seemed to move about randomly. Why and how were they moving? Soot particles did the same thing, making it clear that the pollen particles weren’t autonomously doing something. The conventional history of subsequent intellectual developments goes like this: in his annus mirabilis of 1905, when he produced four remarkable breakthroughs, Albert Einstein provided the first understanding of the mechanisms of so-called Brownian motion. He demonstrated that many processes that seem continuous (like the motion of dust or pollen) are in fact the product of many discrete particles moving about. In other words, the pollen particles were moving around in a continuous way because they were reacting to tiny water molecules that were bumping them at random. This foundational idea transformed physics by demonstrating the presence of atoms and also provided the machinery to mathematically describe all kinds of seemingly random processes, ultimately giving rise to quantum mechanics.

As such, the idea that finance, a study of markets and inherently social phenomena, lost its way by aping physics, a “hard” science of precision, is plain wrong. Instead, as philosopher and historian Jim Holt has described, “Here, then, is the correct chronology. A theory is proposed to explain a mysterious social institution (the Paris Bourse). It is then used to resolve a mid-level mystery in physics (Brownian motion). Finally, it clears up an even deeper mystery in physics (quantum behavior). The implication is plain: Market weirdness explains quantum weirdness, not the other way around. Think of it this way: If Isaac Newton had worked at Goldman Sachs instead of sitting under an apple tree, he would have discovered the Heisenberg uncertainty principle.” Aside from an interesting reversal of conventional wisdom, the story of Bachelier’s discovery is also the story of the two most important risk management strategies—options and diversification.

., 123 Berryman, John, 33 Bessemer Venture Partners, 147 beta. See capital asset pricing model Bewkes, Jeff, 108 Bhagavad Gita, 155 Big Short, The (film), 97 Book of Matthew, 60 Brooks, Mel, 8, 75, 91–96 Critic, The (film), 93 relationship with Anne Bancroft, 95–96 relationship with Gene Wilder, 94 relationship with mother, 93 relationship with Sid Caesar, 94 Brown, Robert, Brownian motion, 38–40 Buck v. Bell (1927), 21 Buffett, Warren, 22 Burroughs, John, 117 Byrne, David, 13–14 C capital asset pricing model, xi, 53–56 advertising company betas, 53–54 gold, 54–55 high-beta and low-beta assets, 53–56 insurance, 54–55 LinkedIn relationships vs. friendship vs. unconditional love, 55–56 negative betas, 54–55 Case, Steve, 108, 109, 112 Cather, Willa, 9, 170, 172, 174 Charles, Ray, 99 Chaucer, Geoffrey, 18, 74 choix du roi, 104 Coase, Ronald, 115 Code of Justinian, Lex Rhodia, 23 commitments, 6, 154–56, 159–60 Confusion de Confusiones (de la Vega), 5, 43–44 Consilience (Wilson), 177 Cook, Tim, 77, 79, 80, 83 corporate finance, 7–9 corporate governance misaligned incentives, 80, 81 stock-based compensation, 81 Corrêa da Serra, José, 140 Cosmopolis (DeLillo), 165 Costa, Dora, 30 Crews, Frederick, 94 Crossroads of Should and Must, The (Luna), 90–92 Curry, Stephen, 51 D Davies, Owen, 25 Dead Poets Society (film), 17 debt overhang, 132–35 “Defence of Usury” (Bentham), 121 de la Vega, Joseph.

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The Information: A History, a Theory, a Flood by James Gleick

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NORBERT WIENER (1956) (Illustration credit 8.1) He was short and rotund, with heavy glasses and a Mephistophelian goatee. Where Shannon’s fire-control work drilled down to the signal amid the noise, Wiener stayed with the noise: swarming fluctuations in the radar receiver, unpredictable deviations in flight paths. The noise behaved statistically, he understood, like Brownian motion, the “extremely lively and wholly haphazard movement” that van Leeuwenhoek had observed through his microscope in the seventeenth century. Wiener had undertaken a thoroughgoing mathematical treatment of Brownian motion in the 1920s. The very discontinuity appealed to him—not just the particle trajectories but the mathematical functions, too, seemed to misbehave. This was, as he wrote, discrete chaos, a term that would not be well understood for several generations. On the fire-control project, where Shannon made a modest contribution to the Bell Labs team, Wiener and his colleague Julian Bigelow produced a legendary 120-page monograph, classified and known to the several dozen people allowed to see it as the Yellow Peril because of the color of its binder and the difficulty of its treatment.

.… The night was noisier than the day, and at the ghostly hour of midnight, for what strange reasons no one knows, the babel was at its height.♦ But engineers could now see the noise on their oscilloscopes, interfering with and degrading their clean waveforms, and naturally they wanted to measure it, even if there was something quixotic about measuring a nuisance so random and ghostly. There was a way, in fact, and Albert Einstein had shown what it was. In 1905, his finest year, Einstein published a paper on Brownian motion, the random, jittery motion of tiny particles suspended in a fluid. Antony van Leeuwenhoek had discovered it with his early microscope, and the phenomenon was named after Robert Brown, the Scottish botanist who studied it carefully in 1827: first pollen in water, then soot and powdered rock. Brown convinced himself that these particles were not alive—they were not animalcules—yet they would not sit still.

This idea is carried still further in certain commercial codes where common words and phrases are represented by four- or five-letter code groups with a considerable saving in average time. The standardized greeting and anniversary telegrams now in use extend this to the point of encoding a sentence or two into a relatively short sequence of numbers.♦ To illuminate the structure of the message Shannon turned to some methodology and language from the physics of stochastic processes, from Brownian motion to stellar dynamics. (He cited a landmark 1943 paper by the astrophysicist Subrahmanyan Chandrasekhar in Reviews of Modern Physics.♦) A stochastic process is neither deterministic (the next event can be calculated with certainty) nor random (the next event is totally free). It is governed by a set of probabilities. Each event has a probability that depends on the state of the system and perhaps also on its previous history.

Adaptive Markets: Financial Evolution at the Speed of Thought by Andrew W. Lo

The unpredictable nature of this fair game will reveal itself in a precarious two-step dance, as you lurch back and forth like a drunk driver attempting to walk a straight line at a sobriety checkpoint. Any fair game like a martingale will produce wins and losses in a random pattern like a “drunkard’s walk”—and as Bachelier discovered, so do the prices in the stock market. Today, we call Bachelier’s discovery the Random Walk Model of stock prices. Bachelier’s analysis was decades ahead of its time. In fact, Bachelier anticipated Albert Einstein’s very similar work in physics on Brownian motion—the random motion of a tiny particle suspended in fluid, among other things—by five years.14 From an economist’s perspective, however, Bachelier did much more than Einstein.15 Bachelier had come up with a general theory of market behavior, and he did so by arguing that an investor could never profit from past price changes. Because the random price movements in a market were martingales, Bachelier concluded, “the mathematical expectation of the speculator was zero.”

The intellectual environment of economics was ripe with problems that could be solved with these ultramathematical techniques. What’s more, this borrowing from physics was also financially profitable. As we saw in chapter 1, the parallels between finance and physics can be very close. For example, the similarities between the movements of an asset’s price and the movements of a particle in Brownian motion led to the Random Walk Hypothesis. This means that financial economists can often use the same mathematics as the physicists: the Black-Scholes/ Merton option pricing formula also happens to be the solution to the heat equation in thermodynamics (heat is also the product of random motion). It shouldn’t be a surprise to learn that Samuelson was also instrumental in the birth of modern financial economics.

The rather supercilious references to him, as an unrigorous pioneer in stochastic processes and stimulator of work in that area by more rigorous mathematicians such as Kolmogorov, hardly does Bachelier justice. His methods can hold their own in rigor with the best scientific work of his time, and his fertility was outstanding. Einstein is properly revered for his basic, and independent, discovery of the theory of Brownian motion 5 years after Bachelier. But years ago when I compared the two texts, I formed the judgment (which I have not checked back on) that Bachelier’s methods dominated Einstein’s in every element of the vector. Thus the Einstein-Fokker-Planck Fourier equation for diff usion of probabilities is already in Bachelier, along with subtle uses of the nowstandard method of reflected images. 19. Kendall (1953). 20.

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Models. Behaving. Badly.: Why Confusing Illusion With Reality Can Lead to Disaster, on Wall Street and in Life by Emanuel Derman

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I showed that expected return is proportional to risk by using just two principles: (1) you should expect equal returns from equal risks, and (2) a stock’s risk is solely the volatility of the diffusion illustrated in Figure 5.3. The first principle is pure theory and hard to argue with: If two securities truly have the same risk, how could you not expect the same return from them? But that’s an expectation. In life, expectations aren’t necessarily fulfilled. The second assumption is pure model. The EMM’s picture of price movements goes by several names: a random walk, diffusion, and Brownian motion. One of its origins is in the description of the drift of pollen particles through a liquid as they collide with its molecules. Einstein used the diffusion model to successfully predict the square root of the average distance the pollen particles move through the liquid as a function of temperature and time, thus lending credence to the existence of hypothetical molecules and atoms too small to be seen.

See evil bailouts bare electrons Barfield, Owen Bedazzled (film) Begin, Menachem behavior, human: adequate knowledge and EMM as assumption about explanations for and humans as responsible for their actions and idolatry of models Law of One Price and laws of pragmamorphism and Ben-Gurion, David Bernoulli, Daniel Bernstein, Jeremy beta: CAPM and Betar (Brit Yosef Trumpeldor) binocular diplopia birds Black, Fischer Black-Scholes Model Merton and Blake, William Bnei Akiva (Sons of Akiva) Bnei Zion (Sons of Zion) body-mind relationship Bohr, Aage Bohr, Niels bonds: financial models and See also type of bond Boyle’s Law Brahe, Tycho brain Brave New World (Huxley) Brownian motion bundling of complex products cage: moth in perfect calibration Cape Flats Development Association (South Africa) Capital Asset Pricing Model (CAPM) capitalism caricatures: models as cash. See currency/cash Cauchy, Augustin-Louis causes Cepheid variable stars chalazion chaver (comrade) Chekhov, Anton chemistry: electromagnetic theory and Chesterton, G. K. Chinese choice chromatography Churchill, Winston Coetzee, J.

The Singularity Is Near: When Humans Transcend Biology by Ray Kurzweil

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.: Landes Bioscience, 1999), pp. 309–12, http://www.nanomedicine.com/NMI/9.4.2.5.htm. 154. George Whitesides, "Nanoinspiration: The Once and Future Nanomachine," Scientific American 285.3 (September 16,2001): 78–83. 155. "According to Einstein's approximation for Brownian motion, after 1 second has elapsed at room temperature a fluidic water molecule has, on average, diffused a distance of ~50 microns (~400,000 molecular diameters) whereas a l-rnicron nanorobot immersed in that same fluid has displaced by only ~0.7 microns (only ~0.7 device diameter) during the same time period. Thus Brownian motion is at most a minor source of navigational error for motile medical nanorobots," See K. Eric Drexler et al., "Many Future Nanomachines: A Rebuttal to Whitesides' Assertion That Mechanical Molecular Assemblers Are Not Workable and Not a Concern," a Debate about Assemblers, Institute for Molecular Manufacturing, 2001, http://www.imm.org/SciAmDebate2/whitesides.html. 156.

George Whitesides complained in Scientific American that "for nanoscale objects, even if one could fabricate a propeller, a new and serious problem would emerge: random jarring by water molecules. These water molecules would be smaller than a nanosubmarine but not much smaller."154 Whitesides's analysis is based on misconceptions. All medical nanobot designs, including those of Freitas, are at least ten thousand times larger than a water molecule. Analyses by Freitas and others show the impact of the Brownian motion of adjacent molecules to be insignificant. Indeed, nanoscale medical robots will be thousands of times more stable and precise than blood cells or bacteria.155 It should also be pointed out that medical nanobots will not require much of the extensive overhead biological cells need to maintain metabolic processes such as digestion and respiration. Nor do they need to support biological reproductive systems.

It's simple enough, but consider the diverse and beautiful ways it manifests itself: the endlessly varying patterns as it cascades past rocks in a stream, then surges chaotically down a waterfall (all viewable from my office window, incidentally); the billowing patterns of clouds in the sky; the arrangement of snow on a mountain; the satisfying design of a single snowflake. Or consider Einstein's description of the entangled order and disorder in a glass of water (that is, his thesis on Brownian motion). Or elsewhere in the biological world, consider the intricate dance of spirals of DNA during mitosis. How about the loveliness of a tree as it bends in the wind and its leaves churn in a tangled dance? Or the bustling world we see in a microscope? There's transcendence everywhere. A comment on the word "transcendence" is in order here. "To transcend" means "to go beyond," but this need not compel us to adopt an ornate dualist view that regards transcendent levels of reality (such as the spiritual level) to be not of this world.

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Average Is Over: Powering America Beyond the Age of the Great Stagnation by Tyler Cowen

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Larry Kaufman, who developed the evaluation function for the Rybka program, and who is the mastermind of the Komodo program, graduated from MIT with an undergraduate degree in economics in 1968. He went to work on Wall Street as a broker and soon started developing his own form of options-pricing theory, working independently of Fischer Black and Myron Scholes; Scholes later won a Nobel Prize for that contribution. Kaufman’s theory was based on ideas of Brownian motion and the logistic function, the latter of which he took from formulas for calculating chess ratings. In the 1970s he made money by applying his options-pricing work through a trading firm and stopped when the profits went away, and he has since dedicated his life to chess and computer chess, including his work on Rybka and Komodo. He lives in a fine house in one of the nicest parts of suburban Maryland, with his beautiful wife and young daughter.

., 37, 164 Babbage, Charles, 6 Banerjee, Abhijit, 222 BBC, 144 Becker, Gary, 226–27 behavioral economics, 75–76, 99, 105, 110, 149, 227 Belle (chess program), 46 benefit costs, 36, 59, 113 Benjamin, Joel, 47 Berlin, Germany, 246 Berra, Yogi, 229 biases, cognitive, 99–100 Bierce, Ambrose, 134 “Big Data,” 185, 221 Black, Fischer, 203 blogs, 180–81 Bonaparte, Napoleon, 148 Borjas, George, 162 “bots,” 144–45 “brain emulation,” 137–38. See also artificial intelligence (AI) branes, 214 Brazil, 20 Breedlove, Philip M., 20 Bresnahan, Timothy F., 33 Brookings Institution, 53 Brooklyn, New York, 172, 240 Brownian motion, 203 Brynjolfsson, Erik, 6, 33 Burks, John, 62 business cycles, 45 business negotiations, 73, 158 California, 8, 241 Campbell, Howard, 246 Canada, 20, 171, 177 Candidates Match, 156 Capablanca, Jose Raoul, 150 capital flows, 166 capitalism, 258 careers, 41–44, 119–25, 126, 202 Carlsen, Magnus, 104, 156, 189 Carr, Nicholas, 153–54 Caterpillar, 38 cell phone service, 118 CEOs, 100 Chen, Yingheng, 79 chess and cheating, 146–51 Chess Olympiad, 147, 189 computer’s influence on quality of play, 106–8 and decision making, 98–99, 101–2, 104–5, 129 early computer chess, 7, 46–47, 67–70 and face-to-face instruction, 195 and gender issues, 31, 106–8 and globalization of competition, 168 and intuition, 68–70, 72, 97, 99, 101, 105–6, 109–10, 114–15 machine and human styles contrasted, 75–76, 77–86 machine vs. machine matches, 70–75 as model for education, 185–88, 191–92, 202–3 and opening books, 83–85, 86–87, 107, 135, 203 and player ratings, 120 simplicity of rules, 48–49 spectator interest in, 156–57 See also Freestyle chess Chess Tiger (chess program), 78 children and wealth inequality, 249 China chess players from, 108, 189 and demographic trends, 230 and geographic trends, 177 and global competition, 171 and labor competition, 5, 163–64, 167, 169–70 and political trends, 252 and scientific specialization, 216 choice.

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The Mathematics of Love: Patterns, Proofs, and the Search for the Ultimate Equation by Hannah Fry

To me, these failings reflect a false impression of mathematics that’s as grave an error as a mistrust of it all together. But beyond its limitations, for me, mathematics has a beauty that encapsulates the realistic, the idiosyncratic and the abstract. And I’ll never get tired of finding more hidden patterns and counter-intuitive results in the real world, regardless of the assumptions it took to get there. Endnotes 1. Breaking the problem down makes the estimate like Brownian motion. An estimate with steps would have an error that diffused like √n. 2. And this is my book. 3. Just so we’re clear, we’re talking about reflection symmetry here. Rotational symmetry in a face is generally considered a bad thing. 4. See, for example, In Your Face by David Perrett for a well-written and comprehensive overview. 5. For n questions, the formula becomes: . 6. This is all true. 7.

Einstein's Dreams by Alan Lightman

For one, perhaps The Old One is not interested in getting close to his creations, intelligent or not. For another, it is not obvious that knowledge is closeness. For yet another, this time project could be too big for a twenty-six-year-old. On the other hand, Besso thinks that his friend might be capable of anything. Already this year, Einstein has completed his Ph.D. thesis, finished one paper on photons and another on Brownian motion. The current project actually began as an investigation of electricity and magnetism, which, Einstein suddenly announced one day, would require a reconception of time. Besso is dazzled by Einstein’s ambition. For a while, Besso leaves Einstein alone with his thoughts. He wonders what Anna has cooked for dinner and looks down a side street where a silver boat on the Aare glints in the low sun.

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From eternity to here: the quest for the ultimate theory of time by Sean M. Carroll

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This was problematic, as the Temperance movement was strong in America at the time, and Berkeley in particular was completely dry; a recurring theme in Boltzmann’s account is his attempts to smuggle wine into various forbidden places.193 We will probably never know what mixture of failing health, depression, and scientific controversy contributed to his ultimate act. On the question of the existence of atoms and their utility in understanding the properties of macroscopic objects, any lingering doubts that Boltzmann was right were rapidly dissipating when he died. One of Albert Einstein’s papers in his “miraculous year” of 1905 was an explanation of Brownian motion (the seemingly random motion of small particles suspended in air) in terms of collisions with individual atoms; most remaining skepticism on the part of physicists was soon swept away. Questions about the nature of entropy and the Second Law remain with us, of course. When it comes to explaining the low entropy of our early universe, we won’t ever be able to say, “Boltzmann was right,” because he suggested a number of different possibilities without ever settling on one in particular.

TIME IS PERSONAL 53 On the other hand, the achievements for which Paris Hilton is famous are also pretty mysterious. 54 Einstein’s “miraculous year” was 1905, when he published a handful of papers that individually would have capped the career of almost any other scientist: the definitive formulation of special relativity, the explanation of the photoelectric effect (implying the existence of photons and laying the groundwork for quantum mechanics), proposing a theory of Brownian motion in terms of random collisions at the atomic level, and uncovering the equivalence between mass and energy. For most of the next decade he concentrated on the theory of gravity; his ultimate answer, the general theory of relativity, was completed in 1915, when Einstein was thirty-six years old. He died in 1955 at the age of seventy-six. 55 We should also mention Dutch physicist Hendrik Antoon Lorentz, who beginning in 1892 developed the idea that times and distances were affected when objects moved near the speed of light, and derived the “Lorentz transformations,” relating measurements obtained by observers moving with respect to each other.

See also event horizons; singularities and arrow of time and baby universes model and closed timelike curves and entropy evaporation of and growth of structure and Hawking radiation and holographic principle and information and Laplace and particle accelerators and quantum tunneling and quasars and the real world and redshift and spacetime and string theory thermodynamic analogy and uncertainty principle uniformity of The Black Hole Wars (Susskind) block time/block universe perspective Bohr, Niels Boltzmann, Emma Boltzmann, Ludwig and anthropic principle and arrow of time and atomic theory and black holes death and de Sitter space and entropy and the H-Theorem and initial conditions of the universe and kinetic theory and Loschmidt’s reversibility objection and Past Hypothesis and Principle of Indifference and recurrence theorem and the Second Law of Thermodynamics and statistical mechanics Boltzmann brains Boltzmann-Lucretius scenario Bondi, Hermann boost. bosons bouncing-universe cosmology boundary conditions and cause and effect described and initial conditions of the universe and irreversibility and Maxwell’s Demon and recurrence theorem and time symmetry Bousso, Raphael Brahe, Tycho branes Brillouinéon brown dwarfs Brownian motion Bruno, Giordano bubbles of vacuum Buddhism Bureau of Longitude Callender, Craig Callisto caloric Calvin, John Calvino, Italo Carnot, Lazare Carnot, Nicolas Léonard Sadi Carrey, Jim Carroll, Lewis cause and effect celestial mechanics cellular automata CERN C-field Chandrasekhar Limit chaotic dynamics The Character of Physical Law (Feynman) charge charge conjugation checkerboard world exercise and arrow of time background of and conservation of information and Hawking radiation and holographic principle and information loss and interaction effects and irreversibility and Principle of Indifference and symmetry and testing hypotheses chemistry Chen, Jennifer choice Chronology Protection Conjecture circles in time.

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A Short History of Nearly Everything by Bill Bryson

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Partly it was to do with the limitations of equipment—there were, for instance, no centrifuges until the second half of the century, severely restricting many kinds of experiments—and partly it was social. Chemistry was, generally speaking, a science for businesspeople, for those who worked with coal and potash and dyes, and not gentlemen, who tended to be drawn to geology, natural history, and physics. (This was slightly less true in continental Europe than in Britain, but only slightly.) It is perhaps telling that one of the most important observations of the century, Brownian motion, which established the active nature of molecules, was made not by a chemist but by a Scottish botanist, Robert Brown. (What Brown noticed, in 1827, was that tiny grains of pollen suspended in water remained indefinitely in motion no matter how long he gave them to settle. The cause of this perpetual motion—namely the actions of invisible molecules—was long a mystery.) Things might have been worse had it not been for a splendidly improbable character named Count von Rumford, who, despite the grandeur of his title, began life in Woburn, Massachusetts, in 1753 as plain Benjamin Thompson.

(An application to be promoted to technical examiner second class had recently been rejected.) His name was Albert Einstein, and in that one eventful year he submitted to Annalen der Physik five papers, of which three, according to C. P. Snow, “were among the greatest in the history of physics”—one examining the photoelectric effect by means of Planck's new quantum theory, one on the behavior of small particles in suspension (what is known as Brownian motion), and one outlining a special theory of relativity. The first won its author a Nobel Prize and explained the nature of light (and also helped to make television possible, among other things).*17 The second provided proof that atoms do indeed exist—a fact that had, surprisingly, been in some dispute. The third merely changed the world. Einstein was born in Ulm, in southern Germany, in 1879, but grew up in Munich.

“Atoms cannot be perceived by the senses . . . they are things of thought,” he wrote. The existence of atoms was so doubtfully held in the German-speaking world in particular that it was said to have played a part in the suicide of the great theoretical physicist, and atomic enthusiast, Ludwig Boltzmann in 1906. It was Einstein who provided the first incontrovertible evidence of atoms' existence with his paper on Brownian motion in 1905, but this attracted little attention and in any case Einstein was soon to become consumed with his work on general relativity. So the first real hero of the atomic age, if not the first personage on the scene, was Ernest Rutherford. Rutherford was born in 1871 in the “back blocks” of New Zealand to parents who had emigrated from Scotland to raise a little flax and a lot of children (to paraphrase Steven Weinberg).

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Fortune's Formula: The Untold Story of the Scientific Betting System That Beat the Casinos and Wall Street by William Poundstone

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As we’ve already seen, the fluctuations of a bettor’s bankroll in a game of chance constitute a random walk (a one-dimensional random walk, since wealth can only move up or down). With time, the gambler’s wealth strays further and further from its original value, and this eventually leads to ruin. At about the time Bachelier was writing, Albert Einstein was puzzling over Brownian motion, the random jitter of microscopic particles suspended in a fluid. The explanation, Einstein surmised, was that the particles were being hit on all sides by invisible molecules. These random collisions cause the visible motion. The mathematical treatment of Brownian motion that Einstein published in 1905 was similar to, but less advanced than, the one that Bachelier had already derived for stock prices. Einstein, like practically everyone else, had never heard of Bachelier. The Random Walk Cosa Nostra SAMUELSON ADOPTED Bachelier’s ideas into his own thinking.

pages: 315 words: 92,151

Ten Billion Tomorrows: How Science Fiction Technology Became Reality and Shapes the Future by Brian Clegg

The researchers at the Australian National Laboratory in Canberra made small hollow glass spheres heat up at particular points around their surface with the laser. Where the surface was heated, air molecules that come into contact with the glass gain extra energy. As air molecules push away from the surface, the recoil moves the sphere in the opposite direction. In effect, this is controlled Brownian motion. This is the mechanism that causes small particles like pollen grains to dance around in water, as if they are alive. Albert Einstein explained the effect as being caused by impact from the unseen water molecules on the tiny grains. The interesting aspect of the laser “tractor beam” is that the position heated on the surface of the glass spheres can be modified by changing the polarization of the laser light, so that the effect can be used to move the spheres in any desired direction.

See also particle gun; phaser; ray guns Archimedes as inventor of light and mirrors in projectile compared to SF use of “Beep” (Blish) Bell, Alexander Graham Bell, Jocelyn Bell Aircraft Corporation Bell Labs laser development by speech recognition and de Bergerac, Cyrano Berkeley Lower Extremity Exoskeleton (BLEEX) The Bionic Woman Blade Runner BLEEX. See Berkeley Lower Extremity Exoskeleton Blish, James AI governments by instantaneous transmitter by virtual learning and Borg (fictional characters) Bose-Einstein condensate Bostrom, Nick The Boys from Brazil (Levin) Bradbury, Ray brain to brain link to computer link electrodes in implants interface to programming BrainGate Brownian motion Brunner, John Bulwer-Lytton, Edward Campbell, John W. Campbell, Murray S. A Canticle for Leibowitz (Miller) Čapek, Karel carbon aliens and artificial muscles and Moon’s lack of nanotubes Carmack, John Carrington event (1859) Casimir effect Cayley, George cell phones Cernan, Eugene chaos theory chess computerized invention of mechanical Chiao, Raymond The Chrysalids (Wyndham) Cities in Flight series (Blish) Clarke, Arthur C.

pages: 153 words: 12,501

Mathematics for Economics and Finance by Michael Harrison, Patrick Waldron

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It follows very naturally from the stuff in Section 5.4. 7.2.2 The Black-Scholes option pricing model Fischer Black died in 1995. In 1997, Myron Scholes and Robert Merton were awarded the Nobel Prize in Economics ‘for a new method to determine the value of derivatives.’ See http://www.nobel.se/announcement-97/economy97.html Black and Scholes considered a world in which there are three assets: a stock, whose price, S̃t , follows the stochastic differential equation: dS̃t = µS̃t dt + σ S̃t dz̃t , where {z̃t }Tt=0 is a Brownian motion process; a bond, whose price, Bt , follows the differential equation: dBt = rBt dt; and a call option on the stock with strike price X and maturity date T . Revised: December 2, 1998 138 7.2. ARBITRAGE AND PRICING DERIVATIVE SECURITIES They showed how to construct an instantaneously riskless portfolio of stocks and options, and hence, assuming that the principle of no arbitrage holds, derived the Black-Scholes partial differential equation which must be satisfied by the option price.

pages: 119 words: 10,356

Topics in Market Microstructure by Ilija I. Zovko

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Models of spread, starting withDemsetz (1968); Tinic (1972); Stoll (1978); Amihud and Mendelson (1980); Ho and Stoll (1981), have examined the possible determinants of spreads as a result of rational, utilitymaximizing problem faced by the market makers. Models providing insight into the utility-maximizing response of agents to other various measures of market conditions such as volatility are for example Lo (2002) who investigate a simple model in which the log stock price is modeled as a Brownian motion diffusion process. Provided agents prefer a lower expected execution time, their model predicts a positive relationship between volatility and limit order placement. Copeland and Galai (1983); Glosten and Milgrom (1985); Easley and O’Hara (1987); Glosten (1995); Foucault (1999); Easley et al. (2001) examine asymmetric information effects on order placement. Andersen (1996) modifies the Glosten and Milgrom (1985) model with the stochastic volatility and information flow perspective.

pages: 1,079 words: 321,718

Surfaces and Essences by Douglas Hofstadter, Emmanuel Sander

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One other factor that might have contributed to Einstein’s faith in his analogy between the physics of the ideal gas and that of the black body (not just between the mathematical formulas for their spectra) was the fact that only a few months earlier, he had found and deeply exploited an analogy between an ideal gas and another physical system — namely, a liquid containing colloidal particles whose nonstop, apparently random hopping-about could be observed through a microscope. This analogy had allowed him to argue persuasively for the existence of extremely tiny invisible molecules that were incessantly pelting the far larger colloidal particles (like thousands of gnats bashing randomly into hanging lamps) and giving them their mysterious hops, known as “Brownian motion”. It is thus probable that two distinct forces in Einstein’s mind — the mathematical similarity of the formulas and also his recent Brownian-motion analogy — gave him great trust in his analogy between a black body and an ideal gas. In any case, building on the bedrock of his latest analogy, Einstein undertook a series of computations, all based on thermodynamics, the branch of physics that he thought of as the deepest and most reliable of all. First he calculated the entropy of each of the systems and then he transformed the two entropy formulas so that they would look as similar as possible to each other; in fact, at the end of his ingenious manipulations, they wound up exactly identical except for the algebraic form of one simple exponent.

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I Am a Strange Loop by Douglas R. Hofstadter

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Think of how the water in a glass sitting on a table seems completely still to us. If our eyes could shift levels (think of the twist that zooms binoculars in or out) and allow us to peer at the water at the micro-level, we would realize that it is not peaceful at all, but a crazy tumult of bashings of water molecules. In fact, if colloidal particles are added to a glass of water, then it becomes a locus of Brownian motion, which is an incessant random jiggling of the colloidal particles, due to a myriad of imperceptible collisions with the water molecules, which are far tinier. (The colloidal particles here play the role of simmballs, and the water molecules play the role of simms.) The effect, which is visible under a microscope, was explained in great detail in 1905 by Albert Einstein using the theory of molecules, which at the time were only hypothetical entities, but Einstein’s explanation was so far-reaching (and, most crucially, consistent with experimental data) that it became one of the most important confirmations that molecules do exist.

On a skiing vacation in the Sierra Nevada, far away from home, my children and I took advantage of the “doggie cam” at the Bloomington kennel where we had boarded our golden retriever Ollie, and thanks to the World Wide Web, we were treated to a jerky sequence of stills of a couple of dozen dogs meandering haphazardly in a fenced-in play area outdoors, looking a bit like particles undergoing random Brownian motion, and although each pooch was rendered by a pretty small array of pixels, we could often recognize our Ollie by subtle features such as the angle of his tail. For some reason, the kids and I found this act of visual eavesdropping on Ollie quite hilarious, and although we could easily describe this droll scene to our human friends, and although I would bet a considerable sum that these few lines of text have vividly evoked in your mind both the canine scene at the kennel and the human scene at the ski resort, we all realized that there was not a hope in hell that we could ever explain to Ollie himself that we had been “spying” on him from thousands of miles away.

pages: 349 words: 134,041

Traders, Guns & Money: Knowns and Unknowns in the Dazzling World of Derivatives by Satyajit Das

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The daily price change is scaled to an annual volatility by multiplying the daily price changes by the square root of time; that is, 1% per day translates into an annual volatility of 15.81% (1% × √250 days in the year). In the world of precise high finance the business year is almost always assumed to be roughly 250 days (52 weeks × 5 days –, say, 10 public holidays). This is the root mean square rule, a common statistical trick, based on Geometric Brownian Motion (GBM). GBM describes how something like the stock price moves randomly over time from its current price in such a way that the daily price changes are distributed normally. The average price change is proportional to the square root of the elapsed time. GBM derives from the work of a botanist, Robert Brown. Brown wrote a paper entitled ‘A Brief Account of Microscopical Observations Made in the Months of June, July and August 1827, on the Particles Contained in the Pollen of Plants’.

pages: 532 words: 133,143

To Explain the World: The Discovery of Modern Science by Steven Weinberg

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This step toward unification was resisted by some physicists, including Pierre Duhem, who doubted the existence of atoms and held that the theory of heat, thermodynamics, was at least as fundamental as Newton’s mechanics and Maxwell’s electrodynamics. But soon after the beginning of the twentieth century several new experiments convinced almost everyone that atoms are real. One series of experiments, by J. J. Thomson, Robert Millikan, and others, showed that electric charges are gained and lost only as multiples of a fundamental charge: the charge of the electron, a particle that had been discovered by Thomson in 1897. The random “Brownian” motion of small particles on the surface of liquids was interpreted by Albert Einstein in 1905 as due to collisions of these particles with individual molecules of the liquid, an interpretation confirmed by experiments of Jean Perrin. Responding to the experiments of Thomson and Perrin, the chemist Wilhelm Ostwald, who earlier had been skeptical about atoms, expressed his change of mind in 1908, in a statement that implicitly looked all the way back to Democritus and Leucippus: “I am now convinced that we have recently become possessed of experimental evidence of the discrete or grained nature of matter, which the atomic hypothesis sought in vain for hundreds and thousands of years.”4 But what are atoms?

See also equant black holes, 267 blood, circulation of, 118 Boethius of Dacia, 124–25, 128 Bohr, Niels, 261 Bokhara, sultan of, 111 Bologna, University of, 127, 147 Boltzmann, Ludwig, 259–60, 267 Bonaventure, Saint, 129 Book of the Fixed Stars (al-Sufi), 108 Born, Max, 261–62 bosons, 263, 264 Boyle, Robert, 194, 199–200, 202, 213, 217, 265 Boyle’s law, 200 Bradwardine, Thomas, 138 Brahe, Tycho. See Tycho Brahe Broglie, Louis de, 248, 261 Brownian motion, 260 Bruno, Giordano, 157, 181, 188 Bullialdus, Ismael, 226 Buridan, Jean, 71, 132–35, 137, 161, 212 Burning Sphere, The (al-Haitam), 110 Butterfield, Herbert, 145 Byzantine Empire, 103–4, 116 Caesar, Julius, 31, 50, 60 calculus, 15, 195, 223–26, 231–32, 236, 315, 327 differential, 223–25 integral, 39, 223–25 limits in, 236 calendars Antikythera Mechanism for, 71n Arabs and, 109, 116, 118 Greeks and, 56, 58–61 Gregorian, 61, 158 Julian, 61 Khayyam and, 109 Moon vs.

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Big Bang by Simon Singh

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Albert and Mileva were married in 1903, and their first son, Hans Albert, was born the next year. In 1905, while juggling the responsibilities of fatherhood and his obligations as a patent clerk, Einstein finally managed to crystallise his thoughts about the universe. His theoretical research climaxed in a burst of scientific papers which appeared in the journal Annalen der Physik. In one paper, he analysed a phenomenon known as Brownian motion and thereby presented a brilliant argument to support the theory that matter is composed of atoms and molecules. In another paper, he showed that a well-established phenomenon called the photoelectric effect could be fully explained using the newly developed theory of quantum physics. Not surprisingly, this paper went on to win Einstein a Nobel prize. The third paper, however, was even more brilliant.

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Handbook of Asset and Liability Management, Volume I, Zenios and Ziemba (eds.). Elsevier 2006. Available at: www.EdwardOThorp.com. Toepke, Jerry. 2004. “Fill ‘Er Up! Benefit from Seasonal Price Patterns in Energy Futures.” Stocks, Futures and Options Magazine. March 3(3). Available at: www.sfomag.com/issuedetail.asp?MonthNameID=March& yearID=2004. Uhlenbeck, George, and Leonard Ornstein. 1930. “On the Theory of Brownian Motion.” Physical Review 36: 823–841. Van Norden, Simon, and Huntley Schaller. 1997. “Regime Switching in Stock Market Returns.” Applied Financial Economics 7: 177–191. P1: JYS bib JWBK321-Chan September 24, 2008 15:5 172 Printer: Yet to come P1: JYS ata JWBK321-Chan August 27, 2008 10:58 Printer: Yet to come About the Author rnest P. Chan is the founder of E. P. Chan & Associates (www.epchan.com), a consulting firm focusing on trading strategy and software development for money managers.

pages: 108 words: 63,808

The State of the Art by Iain M. Banks

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It was a TV production I'd seen on the BBC while I was in London ... or maybe the ship had repeated it. I couldn't recall. What I did recall was the plot and the setting, both of which seemed so apposite to my little scene with Linter that I started to wonder whether the beast upstairs was watching all this. Probably was, come to think of it. And not much point in looking for anything; the ship could produce bugs so small the main problem with camera stability was Brownian motion. Was The Ambassadors a sign from it then? Whatever; the play was replaced by a commercial for Odor-Eaters. 'I've told you,' Linter brought me back from my musings, speaking quietly, 'I'm prepared to take my chances. Do you think I haven't thought it all through before, many times? This isn't sudden, Sma; I felt like this my first day here, but I waited for months before I said anything, so I'd be sure.

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A Brief History of Time by Stephen Hawking

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For centuries the argument continued without any real evidence on either side, but in 1803 the British chemist and physicist John Dalton pointed out that the fact that chemical compounds always combined in certain proportions could be explained by the grouping together of atoms to form units called molecules. However, the argument between the two schools of thought was not finally settled in favor of the atomists until the early years of this century. One of the important pieces of physical evidence was provided by Einstein. In a paper written in 1905, a few weeks before the famous paper on special relativity, Einstein pointed out that what was called Brownian motion—the irregular, random motion of small particles of dust suspended in a liquid—could be explained as the effect of atoms of the liquid colliding with the dust particles. By this time there were already suspicions that these atoms were not, after all, indivisible. Several years previously a fellow of Trinity College, Cambridge, J. J. Thomson, had demonstrated the existence of a particle of matter, called the electron, that had a mass less than one thousandth of that of the lightest atom.

What Kind of Creatures Are We? (Columbia Themes in Philosophy) by Noam Chomsky

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Henri Poincaré went so far as to say that we adopt the molecular theory of gases only because we are familiar with the game of billiards. Ludwig Boltzmann’s scientific biographer speculates that he committed suicide because of his failure to convince the scientific community to regard his theoretical account of these matters as more than a calculating system—ironically, shortly after Albert Einstein’s work on Brownian motion and broader issues had convinced physicists of the reality of the entities he postulated. Niels Bohr’s model of the atom was also regarded as lacking “physical reality” by eminent scientists. In the 1920s, America’s first Nobel Prize–winning chemist dismissed talk about the real nature of chemical bonds as metaphysical “twaddle”: they are nothing more than “a very crude method of representing certain known facts about chemical reactions, a mode of representation” only, because the concept could not be reduced to physics.

pages: 239 words: 56,531

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pages: 295 words: 66,824

A Mathematician Plays the Stock Market by John Allen Paulos

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Louis Bachelier, whom I mentioned in chapter 4, also devised a formula for options more than one hundred years ago. Bachelier’s formula was developed in connection with his famous 1900 doctoral dissertation in which he was the first to conceive of the stock market as a chance process in which price movements up and down were normally distributed. His work, which utilized the mathematical theory of Brownian motion, was way ahead of its time and hence was largely ignored. His options formula was also prescient, but ultimately misleading. (One reason for its failure is that Bachelier didn’t take account of the effect of compounding on stock returns. Over time this leads to what is called a “lognormal” distribution rather than a normal one.) The Black-Scholes options formula depends on five parameters: the present price of the stock, the length of time until the option expires, the interest rate, the strike price of the option, and the volatility of the underlying stock.

pages: 232

Planet of Slums by Mike Davis

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Likewise in Bangalore, the urban fringe is where entrepreneurs can most profitably mine cheap labor with minimal oversight by the state.96 Millions of temporary workers and desperate peasants also hover around the edges of such world capitals of super-exploitation as Surat and Shenzhen. These labor nomads lack secure footing in either city or countryside, and often spend their lifetimes in a kind of desperate Brownian motion 93 See Seabrook, In the Cities of the South, p. 187. 94 Mohamadou Abdoul, "The Production of the City and Urban Informalities," in Enwezor et al. Under Siege, p. 342 95 Guy Thuillier, "Gated Communities in the Metropolitan Area of Buenos Aires," Housing Studies 20:2 (March 2005), p. 255. 96 Hans Schenk, "Urban Fringes in Asia: Markets versus Plans," in I. S. A. Baud and J. Post (eds), Realigning Actors in an Urbanising World: Governance and Institutions from a Development Perspective, Aldershot 2002, pp. 121-22, 131.

pages: 667 words: 186,968

The Great Influenza by John M. Barry

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I have never known him to engage in purposeless rivalries or competitive research. But often have I seen him sit calmly, lost in thought, while all around him others with great show of activity were flitting about like particles in Brownian motion; then, I have watched him rouse himself, smilingly saunter to his desk, assemble a few pipettes, borrow a few tubes of media, perhaps a jar of ice, and then do a simple experiment which answered the question.” But now, in the midst of a killing epidemic, everything and everyone around him—including even the pressure from Welch—shouldered thought aside, shouldered perspective and preparation aside, substituting for it what Avery so disdained: Brownian motion—the random movement of particles in a fluid. Others hated influenza for the death it caused. Avery hated it for that, too, but for a more personal assault as well, an assault upon his integrity.

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A Beautiful Mind by Sylvia Nasar

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Einstein had used to complain around the Institute that “Birkhoff is one of the world’s great academic anti-Semites.” Whether or not this was true, Birkhoff’s bias had prevented him from taking advantage of the emigration of the brilliant Jewish mathematicians from Nazi Germany.27 Indeed, Harvard also had ignored Norbert Wiener, the most brilliant American-born mathematician of his generation, the father of cybernetics and inventor of the rigorous mathematics of Brownian motion. Wiener happened to be a Jew and, like Paul Samuelson, the future Nobel Laureate in economics, he sought refuge at the far end of Cambridge at MIT, then little more than an engineering school on a par with the Carnegie Institute of Technology.28 William James, the preeminent American philosopher and older brother of the novelist Henry James, once wrote of a critical mass of geniuses causing a whole civilization to “vibrate and shake.”29 But the man in the street didn’t feel the tremors emanating from Princeton until World War II was practically over and these odd men with their funny accents, peculiar dress, and passion for obscure scientific theories became national heroes.

., 261 Brandeis University, 314–22 Brauer, Fred, 146 Brenner, Joseph, 239, 258 Brezhnev, Leonid, 332 Bricker, Jacob Leon, 144, 223, 321 Alicia Larde and, 200–201 Eleanor Stier and, 177, 178, 181, 182, 206–7 Nash’s delusions about, 326 Nash’s relationship with, 180–83, 204, 206–7 bridge, 142 Brieskorn, Egbert, 318 Brod, Max, 278 Brode, Wallace, 279 Bronx High School, 142 Brouwer’s fixed point theorem, 45, 128, 362 Browder, Earl, 153 Browder, Eva, 233–34, 380 Browder, Felix, 73, 142, 154, 157, 229, 244, 246–47 Nashes’ British trip and, 233–34 Nashes’ socializing with, 380, 386 on Nash’s defection effort, 281 Nash’s McLean commitment and, 257 Browder, William, 309, 335 Brown, Douglas, 126, 310, 312 Brownian motion, 55 Buchanan, James, 364 Buchwald, Art, 271 Bulletin de la Société Mathématique de France, 298 Bunker Hill Community College, 344 Burr, Stefan, 299 Bush, Vannevar, 137 Calabi, Eugenio, 64, 68, 72, 232, 244–45 Calabi, Giuliana, 245 calculus, tensor, 380 California Institute of Technology, 375 Camus, Albert, 271 Cantorian set theory, 52 Cappell, Sylvain, 99 Carl XVI Gustav, king of Sweden, 379–80 Carleson, Lennart, 223–24, 226, 227 Carnegie Institute of Technology, 35, 39–45, 129, 362 description of, 40 Carrier Clinic, 304, 305–8, 312–13, 43, 344 Cartan, Elie-Joseph, 157 Cartwright, Mary, 57 Casals, Pablo, 193 Castle, The (Kafka), 273, 278 Cauchy problem, 297–98 Cauvin, Jean-Pierre, 284, 298, 308 Cauvin, Louisa, 308 Central Bank of Sweden Prize in Economic Science in Memory of Alfred Nobel, see Nobel Prize in economics Central Intelligence Agency (CIA), 134 Centre de la Recherches Nationale Scientifiques, 298 Chamberlain, Gary, 354 Charles, Ray, 255 Chern, Shiing-shen, 72, 236, 279 Chiang Kai-shek, 153 Chicago, University of, 45, 132, 236, 237, 244 China, 153 Choate, Hall & Steward, 153 Chung, Kai Lai, 66 Church, Alonzo, 63, 64, 93 CIA (Central Intelligence Agency), 134 City College, 142, 144, 180 John Bates Clark medal for economics, 369 Clark University, 59 C.

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The Discovery of France by Graham Robb

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On the high road, for the twelve miles between Le Bouchet-St-Nicolas and Pradelles, the only other travellers he saw were ‘a cavalcade of stride-legged ladies and a pair of post-runners’, but he also saw some of the million tendrils of the other network that carried most of the traffic: The little green and stony cattle-tracks wandered in and out of one another, split into three or four, died away in marshy hollows, and began again sporadically on hillsides or at the borders of a wood. There was no direct road to Cheylard, and it was no easy affair to make a passage in this uneven country and through this intermittent labyrinth of tracks. This labyrinth is the reason why the towns and villages of France were both cut off and connected. Wares and produce travelled through the system of tracks and tiny roads by something akin to Brownian motion, changing hands slowly over great distances. When the main roads were improved and railways were built, trade was drained from the capillary network, links were broken, and a large part of the population suddenly found itself more isolated than before. Many regions today are experiencing the same effect because of the TGV railway system. * IT WOULD TAKE a thousand separate maps to show the movements of the migrant population through this labyrinth of tracks, but an excellent overall view can be gained from any large–scale relief map or satellite photograph.

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The Age of Radiance: The Epic Rise and Dramatic Fall of the Atomic Era by Craig Nelson

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After they helped her, she danced for them privately at their home and introduced them to sculptor Auguste Rodin; the four became regular friends and perhaps the only two people in the world the Curies saw regularly who weren’t scientists or blood relatives. Their closest friends remained the next-door neighbors at boulevard Kellermann, Jean and Henriette Perrin; he was a physics professor at the Sorbonne who verified Einstein’s explanation of Brownian motion, correctly estimated the size of water molecules and atoms, and established cathode rays as negatively charged particles—electrons. Pierre presented his and Marie’s scientific findings to France’s Academy of Sciences on March 16, 1903, and the Swedish Academy of Sciences then awarded them and Becquerel the Nobel Prize. Behind the scenes, four members of the French Académie had recommended that Becquerel and Pierre alone share the Nobel, leaving out Marie’s work entirely.

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Bad Samaritans: The Myth of Free Trade and the Secret History of Capitalism by Ha-Joon Chang

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Before the Convention went into effect in July 1884, Britain, Ecuador and Tunisia signed up, bringing the number of original member countries to 14. Subsequently, Ecuador, El Salvador and Guatemala denounced the Convention, and did not re-join it until the 1990s. The information is from the WIPO (World Intellectual Property Organization) website: http://www.wipo.int/about-ip/en/iprm/pdf/ch5.pdf#paris. 22 They were on the Brownian motion, the photoelectric effect and, most importantly, special relativity. 23 It was only in 1911, six years after he finished his Ph.D., that he was made a professor of physics in the University of Zürich. 24 For further details on the history of Swiss patent system, see Schiff (1971), Industrialisation without National Patents – the Netherlands, 1869–1912 and Switzerland, 1850–1907 (Princeton University Press, Princeton). 25 Moreover, the 1817 Dutch patent law was rather deficient even by the standards of the time.

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Is God a Mathematician? by Mario Livio

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The Black-Scholes model won its originators (Myron Scholes and Robert Carhart Merton; Fischer Black passed away before the prize was awarded) the Nobel Memorial Prize in economics. The key equation in the model enables the understanding of stock option pricing (options are financial instruments that allow bidders to buy or sell stocks at a future point in time, at agreed-upon prices). Here, however, comes a surprising fact. At the heart of this model lies a phenomenon that had been studied by physicists for decades—Brownian motion, the state of agitated motion exhibited by tiny particles such as pollen suspended in water or smoke particles in the air. Then, as if that were not enough, the same equation also applies to the motion of hundreds of thousands of stars in star clusters. Isn’t this, in the language of Alice in Wonderland, “curiouser and curiouser”? After all, whatever the cosmos may be doing, business and finance are definitely worlds created by the human mind.

pages: 313 words: 101,403

My Life as a Quant: Reflections on Physics and Finance by Emanuel Derman

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After my experiences in physics, I never again worried too much about sharing credit with collaborators. It almost never did you harm. 1RMS is a common mnemonic for both Risk Management System and Root Mean Square.Volatility-a crucial measure of risk that is defined as the square root of the mean of the squares of the stock's daily returns, or "root mean square" in common statistical parlance. Root mean square is also suggestive of Brownian motion, the process by which a randomly moving stock price diffuses from its initial value in such a way that the average price change is proportional to the square root of the elapsed time. 'A few years later a lady who cut my hair asked for my title at work. When I said I was a Vice President at Goldman Sachs, she congratulated me on having only one person above me. She didn't understand that I was one of probably 3,000 VPs.

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Sun in a Bottle: The Strange History of Fusion and the Science of Wishful Thinking by Charles Seife

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Physicists soon joined the chemists in their support of atomic theory; they began to provide evidence for the existence of tiny atomic particles. Theorists like Ludwig Boltzmann realized that you could explain the properties of gases simply by imagining matter as a collection of atoms madly bouncing around. Observers even saw the random motion of atoms indirectly: the jostling of water molecules makes a tiny pollen grain swim erratically about. (Albert Einstein helped explain this phenomenon—Brownian motion—in 1905.) Though a few stubborn holdouts absolutely refused to believe in atomic theory,14 by the beginning of the twentieth century the scientific community was convinced. Matter was made of invisible atoms of various kinds: hydrogen atoms, oxygen atoms, carbon atoms, iron atoms, gold atoms, uranium atoms, and a few dozen others. But, as scientists were soon to find out, atoms are not quite as uncuttable as the ancient Greeks thought.

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High-Frequency Trading: A Practical Guide to Algorithmic Strategies and Trading Systems by Irene Aldridge

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In mathematical terms, if a linear model can be expressed as shown in equation (8.37), reprinted here for convenience, then nonlinear models are best expressed as shown in equation (8.38) which follows: yt = α + ∞ βi xt−i + εt (8.37) i=0 yt = f (xt , xt−1 , xt−2 , · · ·) (8.38) where {yt } is the time series of random variables that are to be forecasted, {xt } is a factor significant in forecasting {yt }, and α and β are coefficients to be estimated. The one-step-ahead nonlinear forecast conditional on the information available in the previous period is usually specified using a Brownian motion formulation, as shown in equation (8.39): yt+1 = µt+1 + σt+1 ξt+1 (8.39) forecast of the mean of the where µt+1 = Et [yt+1 ] is the one-period-ahead variable being forecasted, σt+1 = vart [xt+1 ] is the one-period-ahead forecast of the volatility of the variable being forecasted, and ξt+1 is an identically and independently distributed random variable with mean 0 and variance 1. The term ξt+1 is often referred to as a standardized shock or innovation.

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The Investopedia Guide to Wall Speak: The Terms You Need to Know to Talk Like Cramer, Think Like Soros, and Buy Like Buffett by Jack (edited By) Guinan

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In contrast, less liquid assets such as a small-cap stock will have wider spreads, sometimes as high as 1 to 2% of the asset’s value. Related Terms: • Ask • Market Maker • New York Stock Exchange—NYSE • Bid • Pink Sheets Black Scholes Model What Does Black Scholes Model Mean? A model of price variation over time in financial instruments such as stocks that often is used to calculate the price of a European call option. The model assumes that the price of heavily traded assets follows a geometric Brownian motion with constant drift and volatility. When applied to a stock option, the model incorporates the constant price variation of the stock, the time value of money, the option’s strike price, and the time to the option’s expiration. Also known as the Black-Scholes-Merton Model. Investopedia explains Black Scholes Model The Black Scholes Model is one of the most important concepts in modern financial theory.

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Bad Samaritans: The Guilty Secrets of Rich Nations and the Threat to Global Prosperity by Ha-Joon Chang

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Before the Convention went into effect in July 1884, Britain, Ecuador and Tunisia signed up, bringing the number of original member countries to 14. Subsequently, Ecuador, El Salvador and Guatemala denounced the Convention, and did not re-join it until the 1990s. The information is from the WIPO (World Intellectual Property Organization) website: http://www.wipo.int/aboutip/en/iprm/pdf/ch5.pdf#paris. 22 They were on the Brownian motion, the photoelectric effect and, most importantly, special relativity. 23 It was only in 1911, six years after he finished his Ph.D., that he was made a professor of physics in the University of Zürich. 24 For further details on the history of Swiss patent system, see Schiff (1971), Industrialisation without National Patents – the Netherlands, 1869–1912 and Switzerland, 1850–1907 (Princeton University Press, Princeton). 25 Moreover, the 1817 Dutch patent law was rather deficient even by the standards of the time.

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How Not to Network a Nation: The Uneasy History of the Soviet Internet (Information Policy) by Benjamin Peters

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Aleksandr Bogdanov—old Bolshevik revolutionary, right-hand man to Vladimir Lenin, and philosopher—developed a wholesale theory that analogized between society and political economy, which he published in 1913 as Tektology: A Universal Organizational Science, a proto-cybernetics minus the mathematics, whose work Wiener may have seen in translation in the 1920s or 1930s.39 Stefan Odobleja was a largely ignored Romanian whose pre–World War II work prefaced cybernetic thought.40 John von Neumann, the architect of the modern computer, a founding game theorist, and a Macy Conference participant, was a Hungarian émigré. Szolem Mandelbrojt, a Jewish Polish scientist and uncle of fractal founder Benoit Mandelbrot, organized Wiener’s collaboration on harmonic analysis and Brownian motion in 1950 in Nancy, France. Roman Jakobson, the aforementioned structural linguist, a collaborator in the Macy Conferences, and a Russian émigré, held the chair in Slavic studies at Harvard founded by Norbert Wiener’s father. And finally, Wiener’s own domineering and brilliant father, Leo Wiener, was a self-made polymath, the preeminent translator of Tolstoy into English in the twentieth-century, the founder of Slavic studies in America, an émigré from a Belarusian shtetl, and like his son, a humanist committed to uncovering methods for nearly universal communication.41 Although summarizing the intellectual and international sources for the consolidation of cybernetics as a midcentury science for self-governing systems is beyond the scope of this project, the following statement is probably not too far of a stretch.

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Market Sense and Nonsense by Jack D. Schwager

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The efficient market hypothesis is inextricably linked to an underlying assumption that market price changes follow a random walk process (that is, price changes are normally distributed7). The assumption of a normal distribution allows one to calculate the probability of different-size price moves. Mark Rubinstein, an economist, colorfully described the improbability of the October 1987 stock market crash: Adherents of geometric Brownian motion or lognormally distributed stock returns (one of the foundation blocks of modern finance) must ever after face a disturbing fact: assuming the hypothesis that stock index returns are lognormally distributed with about a 20% annualized volatility (the historical average since 1928), the probability that the stock market could fall 29% in a single day is 10−160. So improbable is such an event that it would not be anticipated to occur even if the stock market were to last for 20 billion years, the upper end of the currently estimated duration of the universe.

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Endless Forms Most Beautiful: The New Science of Evo Devo by Sean B. Carroll

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There are some excellent tutorials and animations on these authors’ Web site concerning the generation of periodic and spacing patterns: www.eb.tuebingen.mpg.de/dept4/meinhardt/home.html. François Jacob’s quotation of Jean Perrin appears in his essay “Evolution and Tinkering,” Science 196 (1977): 1161–66. Jean Perrin was a Nobel laureate in Physics (1926) who was cited for his work on colloids and Brownian motion. He wrote a very popular book, Les Atomes (1913), from which the quotation is taken. 5. The Dark Matter of the Genome: Operating Instructions for the Tool Kit I first encountered “dark matter” in Brian Greene’s The Elegant Universe (New York: W. W. Norton, 1999), a very engaging book about the structure of the universe from the very smallest to the largest scale, and in Martin Rees’s excellent Just Six Numbers: The Deep Forces That Shape the Universe (New York: Basic Books, 2001).

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It was, thought Raf, like looking into a deep well and not even knowing if there was water at the bottom. 'Send him over,' said Hu San. 'But tell him to lose that cigar first ...' Around the edge of the room, on black leather banquettes, slouched Seattle's wealthy. Tall and blond or dark, handsome and unfortunately not tall at all, elegantly dressed or expensively dishevelled, both women and men talked intently or stood to shake hands and air-kiss briefly. The Brownian motion of money. The woman with the fox fur was repeating her story of meeting a horrible delivery boy on the way in. She was telling it for the third time and her partner was still pretending to be shocked. Only a few of those in the room showed their age in a surgical tightness around the eyes, the regrettable side effects of having reached middle age before the start of nanetíc surgery. The rest had that youthful permanence which came from being able to afford faces that were constantly rebuilt from the inside.

The Art of Computer Programming by Donald Ervin Knuth

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Brent, Richard Peirce, 8, 28, 40, 130, 136, 139, 141, 241, 279, 280, 313, 348, 352-353, 355, 356, 382, 386, 403, 501, 529-534, 539-540, 556, 590, 600, 643, 644, 646, 657, 695, 719-721. Brezinski, Claude, 357, 721. Brillhart, John David, 29, 394, 396, 400, 660. Brockett, Roger Ware, 712. Brocot, Achille, 655. Bronte, Emily Jane, 292. Brooks, Frederick Phillips, Jr., 226. Brouwer, Luitzen Egbertus Jan, 179. Brown, David, see Spencer Brown. Brown, George William, 135. Brown, Mark Robbin, 712. Brown, Robert, see Brownian motion. Brown, William Stanley, 419, 428, 438, 454, 686. Brownian motion, 559. Bruijn, Nicolaas Govert de, 181, 212, 568, 653, 664, 686, 694. cycle, 38-40. Brute force, 642. Bshouty, Nader Hanna (^j-i. l^ jjL>), 700. Buchholz, Werner, 202, 2~26. Bunch, James Raymond, 500. Buneman, Oscar, 706. Biirgisser, Peter, 515. Burks, Arthur Walter, 202. Burrus, Charles Sidney, 701. Butler, James Preston, 77. Butler, Michael Charles Richard, 442.

The probability that X, < x is F(x), so we have the binomial distribution discussed in Section 1.2.10: Fn(x) = s/n with probability (") F{x)s(l - F(x))n~s; the mean is F(x); the standard deviation is y/F(x)(l - F(x))/n. [See Eq. 1.2.10-(i9). This suggests that a slightly better statistic would be to define max {Fn(x) - F( — oo<x<oo see exercise 22. We can calculate the mean and standard deviation of Fn(y) — Fn(x), for x < y, and obtain the covariance of Fn(x) and Fn(y). Using these facts, it can be shown that for large values of n the function Fn(x) behaves as a "Brownian motion," and techniques from this branch of probability theory may be used to study it. The situation is exploited in articles by J. L. Doob and M. D. Donsker, Annals Math. Stat. 20 A949), 393-403 and 23 A952), 277-281; their approach is generally regarded as the most enlightening way to study the KS tests.] 7. Set j = n in Eq. A3) to see that K? is never negative, and that it can get as high as ^/n- Similarly, set j = 1 to make the same observations about K~. 8.

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Physics of the Future: How Science Will Shape Human Destiny and Our Daily Lives by the Year 2100 by Michio Kaku

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Effects that we can ignore, such as van der Waals forces, surface tension, the uncertainty principle, the Pauli exclusion principle, etc., become dominant in the nanoworld. To appreciate this problem, imagine that the atom is the size of a marble and that you have a swimming pool full of these atoms. If you fell into the swimming pool, it would be quite different from falling into a swimming pool of water. These “marbles” would be constantly vibrating and hitting you from all directions, because of Brownian motion. Trying to swim in this pool would be almost impossible, since it would be like trying to swim in molasses. Every time you tried to grab one of the marbles, it would either move away from you or stick to your fingers, due to a complex combination of forces. In the end, both scientists agreed to disagree. Although Smalley was unable to throw a knockout punch against the molecular replicator, several things became clear after the dust settled.

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The End of Wall Street by Roger Lowenstein

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Wall Street adopted quantitative strategies because they afforded more precision than old-fashioned judgment—they seemed to convert financial gambles into hard science. Investment banks stocked risk departments with PhDs. The problem was that homeowners weren’t molecules, and finance wasn’t physics. Merrill hired John Breit, a particle theorist, as a risk manager, and Breit tried to explain to his peers that the laws of Brownian motion didn’t truly describe finance—this wasn’t science, it was pseudoscience. The models said a diversified portfolio of municipal bonds would lose money once every 10,000 years, but as Breit pointed out, such a portfolio had been devastated merely 150 years ago, during the Civil War. With regard to Merrill’s portfolio of CDOs, the firm judged its potential loss to be “\$71.3 million.”6 This was absurd—not because the number was high or low, but because of the arrogance and self-delusion embedded in such fine, decimal-point precision.

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Future Shock by Alvin Toffler

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And by attracting community and parental participation—businessmen, trade unionists, scientists, and others—the movement could build broad political support for the super-industrial revolution in education. It would be a mistake to assume that the present-day educational system is unchanging. On the contrary, it is undergoing rapid change. But much of this change is no more than an attempt to refine the existent machinery, making it ever more efficient in pursuit of obsolete goals. The rest is a kind of Brownian motion, self-canceling, incoherent, directionless. What has been lacking is a consistent direction and a logical starting point. The council movement could supply both. The direction is super-industrialism. The starting point: the future. THE ORGANIZATIONAL ATTACK Such a movement will have to pursue three objectives—to transform the organizational structure of our educational system, to revolutionize its curriculum, and to encourage a more future-focused orientation.

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The Master Algorithm: How the Quest for the Ultimate Learning Machine Will Remake Our World by Pedro Domingos

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Humans do it all the time: an executive can move from, say, a media company to a consumer-products one without starting from scratch because many of the same management skills still apply. Wall Street hires lots of physicists because physical and financial problems, although superficially very different, often have a similar mathematical structure. Yet all the learners we’ve seen so far would fall flat if we, say, trained them to predict Brownian motion and then asked them to predict the stock market. Stock prices and the velocities of particles suspended in a fluid are just different variables, so the learner wouldn’t even know where to start. But analogizers can do this using structure mapping, an algorithm invented by Dedre Gentner, a psychologist at Northwestern University. Structure mapping takes two descriptions, finds a coherent correspondence between some of their parts and relations, and then, based on that correspondence, transfers further properties from one structure to the other.

pages: 442 words: 39,064

Why Stock Markets Crash: Critical Events in Complex Financial Systems by Didier Sornette

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In other words, we have in mind the process of the emergence of intelligent behaviors at a macroscopic scale that individuals at the microscopic scale cannot perceive. This process has been discussed in biology, for instance in animal populations such as ant colonies or in connection with the emergence of conciousness [8, 198] Let us mention another realization of this concept, which is found in the information contained in options prices on the ﬂuctuations of their underlying assets. Despite the fact that the prices do not follow geometrical Brownian motion, whose existence is a prerequisite for most options pricing models, traders have apparently adapted to empirically incorporating subtle information in the correlation of price distributions with fat tails [337]. In this case and in contrast to the crashes, the traders 280 chapter 7 have had time to adapt. The reason is probably that traders have been exposed for decades to options trading in which the characteristic time scale for option lifetime is in the range of month to years at most.

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The God Delusion by Richard Dawkins

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The very nerve impulses with which we do our thinking and our imagining depend upon activities in Micro World. But no action that our wild ancestors ever had to perform, no decision that they ever had to take, would have been assisted by an understanding of Micro World. If we were bacteria, constantly buffeted by thermal movements of molecules, it would be different. But we Middle Worlders are too cumbersomely massive to notice Brownian motion. Similarly, our lives are dominated by gravity but are almost oblivious to the delicate force of surface tension. A small insect would reverse that priority and would find surface tension anything but delicate. Steve Grand, in Creation: Life and How to Make It, is almost scathing about our preoccupation with matter itself. We have this tendency to think that only solid, material ‘things’ are ‘really’ things at all.

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Efficiently Inefficient: How Smart Money Invests and Market Prices Are Determined by Lasse Heje Pedersen

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In its most extreme form, it says not only that markets discount the future, but they accurately reflect all fundamentals of economies and everything that’s known about companies and so on and so forth, and they synthesize this information into completely perfect prices. This theory would be laughable if it wasn’t so widely believed in. It’s come out of valuing options and modeling diffusion processes: price movement as a diffusion using the Brownian motion and the heat equation. That is a good approximation for modeling short-term options, but to extend that to the idea that there is this perfect matrix of prices that reflects everything perfectly is putting too much weight on a small base of evidence, as they say in science. LHP: So how do your models exploit that markets are not perfectly efficient? DWH: Markets are social institutions and reflect all sorts of phenomena that you’d expect such social institutions to reflect.

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Rise of the Machines: A Cybernetic History by Thomas Rid

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It didn’t mention the unsuccessful experiments in the little MIT lab or any mechanical implementation of his theory. The paper mentioned the antiaircraft problem only two times, buried in a forest of mathematical formulas on page 76. Neither the title nor the introduction or index contained a single reference to the problem that had motivated the project’s funding. Instead, Wiener offered a mind-numbing alphabet soup of abstruse mathematics: Brownian motion, Cesàro partial sum, Fourier integral, Hermitian form, Lebesgue measure, Parseval’s theorem, Poisson distribution, Schwarz inequality, Stieltjes integral, Weyl’s lemma, and many more. When Weaver received the paper, he had it classified and bound in an orange cover. Engineers nicknamed the document “yellow peril,” a joking reference to the paper’s impenetrable theory and lack of practical relevance.

How I Became a Quant: Insights From 25 of Wall Street's Elite by Richard R. Lindsey, Barry Schachter

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I, on the other hand, muddled along, testing model after model to little effect, all the while trying to make sense of my surroundings. I was to discover that graduate school had prepared me in a decidedly tangential manner for what I was to encounter. The foreign exchange markets, which I had studied extensively in the context of purchasing power parity, uncovered interest rate parity, and geometric JWPR007-Lindsey April 30, 2007 18:3 Andrew B. Weisman 189 Brownian motion, were almost unrecognizable to me. I frequently felt lost in a world of rapid-fire economic and political developments, blaring broker boxes, and arcane market nomenclature. It was as if I had embarked on a career as a professional pugilist armed with an extensive knowledge of anatomy and the basic instruction set that I must strike my opponent firmly about the head and abdomen while avoiding same.

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Fool Me Twice: Fighting the Assault on Science in America by Shawn Lawrence Otto

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They do not extend in sudden and dramatic paradigm shifts, and they didn’t in Einstein’s day, either. In fact, many of the ideas Einstein developed were done collaboratively, with considerable debate, a prime example being the cosmological constant. His early papers were extensions of the work of Max Planck, the Austrian physicist Ludwig Boltzmann, and others, and his revolutionary findings on Brownian motion were independently discovered by Polish physicist Marian von Smoluchowski, who was also building on Boltzmann’s work. Hubble’s revolutionary discovery of the expansion of the universe also extended from ideas that were talked about for years. The redshift was first noted by American astronomer Vesto Slipher in 1912—nearly two decades before Hubble’s discovery. Galileo’s revolution was an extension of Copernicus’s writings of some seventy years before, which were widely discussed.

The Science of Language by Noam Chomsky

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Even great scientists, such as, say, Poincaré – one of the twentieth century's greatest scientists – just laughed at it. [Those who laughed] were very much under Machian [Ernst Mach's] influence: if you can't see it, touch it . . . [you can't take it seriously]; so you just have a way of calculating. Boltzmann actually committed suicide – in part, apparently, because of his inability to get anyone to take him seriously. By a horrible irony, he did it in 1905, the year that Einstein's Brownian motion paper came out, and everyone began to take it seriously. And it goes on. I've been interested in the history of chemistry. Into the 1920s, when I was born – so it isn't that far back – leading scientists would have just ridiculed the idea of taking any of this seriously, including Nobel prizewinning chemists. They thought of [atoms and other such ‘devices’] as ways of calculating the results of experiments.

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A Man for All Markets by Edward O. Thorp

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He wrote down equations that correctly predicted the statistical properties of the random motion of the particles. Until that time no one had ever seen a molecule or an atom (molecules are groups of atoms of various types bound together by electrical forces), and their existence had been disputed. Here was the final proof that atoms and molecules were real. This article became one of the most widely cited in all of physics. Unknown to Einstein, his equations describing the Brownian motion of pollen particles were essentially the same as the equations that Bachelier had used for his thesis five years earlier to describe a very different phenomenon, the ceaseless, irregular motion of stock prices. Bachelier employed the equations to deduce the “fair” prices for options on the underlying stocks. Unlike Einstein’s work, Bachelier’s remained generally unknown until future Nobel laureate (1970) Paul Samuelson came across it in a Paris library in the 1950s and had it translated into English.

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Stamping Butterflies by Jon Courtenay Grimwood

It merely put them into the hands of our dangerous lunatics as opposed to their dangerous lunatics... She sighed. "We'd need your agreement," Professor Mayer said. "And you needn't feel guilty about the man who takes your place. He's going to die anyway... Why would we do this?" she asked, watching smoke trickle towards the ceiling. A whole world of rigid rules covering temperature, convection and Brownian motion all busily pretending to be truly chaotic. No wonder she loved smoking so much. "Because we've got a situation. And mostly it's that we now know who you really are..." Not a flicker from Prisoner Zero. "...and according to your family you're already dead." Professor Mayer glanced round the prisoner's new room with its neat bed, built-in shower and view of the Mediterranean. Dr. Petrov could be forgiven her verdict of emotional autism, naïve though it was, because the man seemed totally oblivious to anything going on around him.

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Nexus by Ramez Naam

"But there are ways to increase the range." Pieces clicked together for Sam. The "no transmitters" rule. The "repeaters" Rangan had mentioned. These kids had found a way to extend Nexus transmissions. Dear god. "Sounds great," she replied. "I'll take your lead." Her pulse was quick now. Her stomach was a knot. Ilya popped the top of the vial. Sam caught a glimpse of a metallic liquid swirling through the glass. Brownian motion mixed tendrils of grey and silver. For an instant she had the impression of the drug as a living thing, aware, alert, purposeful. The moment passed. Ilya handed her the vial, followed closely by a glass of juice from the table. Sam downed the drug. The liquid tasted strongly metallic, slightly bitter. It felt heavy on her tongue, oily as it flowed down her throat. She sipped the juice.

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American Gods by Neil Gaiman

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Shadow had never seen the Nile, but there was a blinding afternoon sun burning on the wide brown river that made him think of the muddy expanse of the Nile: not the Nile as it is now, but as it was long ago, flowing like an artery through the papyrus marshes, home to cobra and jackal and wild cow... A road sign pointed to Thebes. The road was built up about twelve feet, so he was driving above the marshes. Clumps and clusters of birds in flight were questing back and forth, black dots against the blue sky, moving in some desperate Brownian motion. In the late afternoon the sun began to lower, gilding the world in elf-light, a thick warm custardy light that made the world feel unearthly and more than real, and it was in this light that Shadow passed the sign telling him he was Now Entering Historical Cairo. He drove under a bridge and found himself in a small port town. The imposing structures of the Cairo courthouse and the even more imposing customs house looked like enormous freshly baked cookies in the syrupy gold of the light at the end of the day.

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The Wealth of Networks: How Social Production Transforms Markets and Freedom by Yochai Benkler

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This stickiness could be the efficacy of a cluster of connections in pursuit of a goal one cares about, as in the case of the newly emerging peer-production enterprises. It could be the ways in which the internal social interaction has combined social norms with platform design to offer relatively stable relations with others who share common interests. Users do not amble around in a social equivalent of Brownian motion. They tend to cluster in new social relations, albeit looser and for more limited purposes than the traditional pillars of community. 667 The conceptual answer has been that the image of "community" that seeks a facsimile of a distant pastoral village is simply the wrong image of how we interact as social beings. We are a networked society now--networked individuals connected with each other in a mesh of loosely knit, overlapping, flat connections.

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Never Let a Serious Crisis Go to Waste: How Neoliberalism Survived the Financial Meltdown by Philip Mirowski

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Let us look more closely at the practical mechanics of orthodox contemporary “economics imperialism.” While gleefully encroaching upon the spheres of interest of other disciplines, orthodox economics has also freely appropriated formalisms and methods from those other disciplines: think of the advent of “experimental economics” or the embrace of magnetic resonance imaging, or attempts to absorb chaos theory or nonstandard analysis or Brownian motion through the Ito calculus. Indeed, if there has been any conceptual constant throughout the history of neoclassical theory since the 1870s, it has been slavish attempts to slake its physics envy through gorging on half-digested imitations of physical models. A social science so promiscuous in its avidity to mimic the tools and techniques of other disciplines has no principled discrimination about what constitutes just and proper argumentation within its own sphere; and this has only become aggravated in the decades since 1980.

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The Fabric of the Cosmos by Brian Greene

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As we will see in Chapter 13, recent work in string theory has suggested that strings may be much larger than the Planck length, and this has a number of potentially critical implications—including the possibility of making the theory experimentally testable. 13. The existence of atoms was initially argued through indirect means (as an explanation of the particular ratios in which various chemical substances would combine, and later, through Brownian motion); the existence of the first black holes was confirmed (to many physicists’ satisfaction) by seeing their effect on gas that falls toward them from nearby stars, instead of “seeing” them directly. 14. Since even a placidly vibrating string has some amount of energy, you might wonder how it’s possible for a string vibrational pattern to yield a massless particle. The answer, once again, has to do with quantum uncertainty.

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Behave: The Biology of Humans at Our Best and Worst by Robert M. Sapolsky

.* The main point here is that transposons occur in the brain.11 In humans transpositional events occur in stem cells in the brain when they are becoming neurons, making the brain a mosaic of neurons with different DNA sequences. In other words, when you make neurons, that boring DNA sequence you inherited isn’t good enough. Remarkably, transpositional events occur in neurons that form memories in fruit flies. Even flies evolved such that their neurons are freed from the strict genetic marching orders they inherit. Chance Finally, chance lessens genes as the Code of Codes. Chance, driven by Brownian motion—the random movement of particles in a fluid—has big effects on tiny things like molecules floating in cells, including molecules regulating gene transcription.12 This influences how quickly an activated TF reaches the DNA, splicing enzymes bump into target stretches of RNA, and an enzyme synthesizing something grabs the two precursor molecules needed for the synthesis. I’ll stop here; otherwise, I’ll go on for hours.

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The Snowball: Warren Buffett and the Business of Life by Alice Schroeder

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“Billy Rogers Died of Drug Overdose,” Omaha World-Herald, April 2, 1987; “Cause Is Sought in Death of Jazz Guitarist Rogers,” Omaha World-Herald, February 21, 1987. 35. Interview with Arjay Miller. 36. Interviews with Verne McKenzie, Malcolm “Kim” Chace III, Don Wurster, Dick and Mary Holland. 37. Interview with George Brumley. 38. Louis Jean-Baptiste Alphonse Bachelier, Theory of Speculation, 1900. Bachelier applied the scientific theory of “Brownian motion” to the market, probably the first of many attempts to bring the rigor and prestige of hard science to the soft science of economics. 39. Charles Ellis, Investment Policy: How to Win the Loser’s Game. Illinois: Dow-Jones-Irwin, 1985, which is based on his article “Winning the Loser’s Game” in the July/August 1975 issue of the Financial Analysts Journal. 40. The modern-day equivalents of Tweedy Browne’s Jamaica Water warrants still exist, for example. 41.