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

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For a model that takes into account the impact of a large investor’s behaviour, (ρm, Sm) is affected by {γk}k<m. 3.14 Conclusions • A discrete time market model is the most generic one, and it covers any market with time series of prices. Strategies developed for this model can be implemented directly. The discrete time model does not require the theory of stochastic integrals. • Unfortunately, discrete time models are difficult for theoretical investigations, and their role in mathematical finance is limited. A discrete time market model is complete only for the very special case of a two-point distribution (for the Cox-Ross-Rubinstein model and for a model from Remark 3.39). Therefore, pricing is difficult for the general case. Some useful theorems from continuous time setting are not valid for the general discrete time model. Many problems are still unsolved for discrete time market models (including pricing problems and optimal portfolio selection problems)

We will first demonstrate how to apply the binomial trees for European options, and then this method will be extended for American options. 6.1 The binomial tree for stock prices 6.1.1 General description Binomial trees are used to approximate the distributions of continuous time random processes of the stock price S(t) via discrete time processes. It is suggested to replace the price process by a random process with the following properties: changes only at discrete times t0=0, t1=∆t,…, tk=k∆, tN=T, where • The process ∆=T/N, T is terminal time, and ∆ denotes the one time step. • If the price of the underlying asset is possible values, Figure 6.1). • The probability p of probability q=1−p of at time tk, then it may take only one of two at time tk+1, where moving up to and u>1 (see is known, as well as the moving down to The dynamics of the process can be visualized via a graph called a binomial tree. If the risk-free interest rate r is non-random and constant, then the corresponding market model is equivalent to the Cox-Ross-Rubinstein discrete time market model for the discrete time prices The parameters p, u, and d are chosen to match the stock price expectation and volatility

© 2007 Nikolai Dokuchaev 8 Review of statistical estimation In this chapter, we collect some core facts from mathematical statistics and statistical inference that will be used later to estimate parameters for continuous time market models. 8.1 Some basic facts about discrete time random processes In this section, several additional definitions and facts about discrete time stochastic processes are given. Definition 8.1 A process ξt is said to be stationary (or strict-sense stationary), if the does not depend on m for any N>0, t1,…, distribution of the vector tN. and Definition 8.2 A (vector) process ξt is said to be wide-sense stationary, if does not depend on time shift m for all t, θ, and m. It can happen that a process is wide-sense stationary, but it is not stationary. In fact, stationarity in a wide sense is sufficient for many applications. Definition 8.3 Let ξt, t=0, 1, 2,…, be a discrete time random process such that ξt are mutually independent and have the same distribution, and Eξt≡0. Then the process ξt is said to be a discrete time white noise.

Market Risk Analysis, Quantitative Methods in Finance by Carol Alexander

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The model (I.3.141) is called arithmetic Brownian motion. Arithmetic Brownian motion is the continuous time version of a random walk. To see this, we first note that the discrete time equivalent of the Brownian increment dZt is a standard normal independent process: i.e. as we move from continuous to discrete time, dZ t → Zt ∼ NID0 1 Probability and Statistics 137 Also the increment dX t in continuous time becomes a first difference Xt = Xt − Xt−1 in discrete time.56 Using these discrete time equivalents gives the following discretization of (I.3.141): Xt = + \$t where \$t = Zt so \$t ∼ NID 0 2 But this is the same as Xt = + Xt−1 + \$t , which is the discrete time random walk model (I.3.140). So (I.3.141) is non-stationary. However a stationary continuous time process can be defined by introducing a mean reversion mechanism to the drift term.

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. I.3.7.4 Jumps and the Poisson Process A Poisson process, introduced in Section I.3.3.2, is a stochastic process governing the occurrences of events through time.

They are usually written at a higher mathematical level than the present text but have fewer numerical and empirical examples. • Those which focus on discrete time mathematics, including statistics, linear algebra and linear regression. Among these books are Watsham and Parramore (1996) and Teall and Hasan (2002), which are written at a lower mathematical level and are less comprehensive than the present text. Continuous time finance and discrete time finance are subjects that have evolved separately, even though they approach similar problems. As a result two different types of notation are used for the same object and the same model is expressed in two different ways. One of the features that makes this book so different from many others is that I focus on both continuous and discrete time finance, and explain how the two areas meet. Although the four volumes of Market Risk Analysis are very much interlinked, each book is self-contained.

Derivatives Markets by David Goldenberg

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However they aren’t very far from martingales. 524 OPTIONS Definition of a Sub (Super) Martingale 1. A discrete-time stochastic process (Xn())n=0,1,2,3,… is called a sub-martingale if E(Xn)<∞, E(Xn2 )<∞ and E(Xn+1()|Xn)>Xn for all n=0,1,2,3,… 2. A discrete-time stochastic process (Xn())n=0,1,2,3,… is called a super-martingale if E(Xn)<∞, E(Xn2)<∞, and E(Xn+1()|Xn )<Xn for all n=0,1,2,3,… We expect stock prices to be sub-martingales, not martingales, for two separate and different reasons: 1. All assets, risky or not, have to provide a reward for time and waiting. This reward is the risk-free rate. 2. Risky assets have to also reward investors for non-diversiﬁable risks. 15.4.1 The Equivalent Martingale Representation of Stock Prices Not all hope is lost for the martingale representation of stock prices under risk neutrality. We start with deﬁning a bond price process in discrete time, which in this context is a numeraire.

INTRODUCTION TO FINANCIAL FUTURES CONTRACTS 211 PART 2 Trading Structures Based on Forward Contracts 8. STRUCTURED PRODUCTS, INTEREST-RATE SWAPS 271 273 vi CONTENTS PART 3 Options 321 9. INTRODUCTION TO OPTIONS MARKETS 323 10. OPTION TRADING STRATEGIES, PART 1 345 11. RATIONAL OPTION PRICING 369 12. OPTION TRADING STRATEGIES, PART 2 415 13. MODEL-BASED OPTION PRICING IN DISCRETE TIME, PART 1: THE BINOMIAL OPTION PRICING MODEL (BOPM, N=1) 435 14. OPTION PRICING IN DISCRETE TIME, PART 2: DYNAMIC HEDGING AND THE MULTI-PERIOD BINOMIAL OPTION PRICING MODEL, N >1 473 15. EQUIVALENT MARTINGALE MEASURES: A MODERN APPROACH TO OPTION PRICING 507 16. OPTION PRICING IN CONTINUOUS TIME 539 17. RISK-NEUTRAL VALUATION, EMMS, THE BOPM, AND BLACK–SCHOLES 595 Index 637 DETAILED CONTENTS List of figures List of tables Preface Acknowledgments xxiii xxvii xxxi xxxvii PART 1 Forward Contracts and Futures Contracts CHAPTER 1 SPOT, FORWARD, AND FUTURES CONTRACTING 1 3 1.1 Three Ways to Buy and Sell Commodities 5 1.2 Spot Market Contracting (Motivation and Examples) 5 1.3 Forward Market Contracting (Motivation and Examples) 7 1.4 Problems with Forward Markets 11 1.5 Futures Contracts as a Solution to Forward Market Problems (Motivation and Examples) 13 1.6 Futures Market Contracting 17 1.7 Mapping Out Spot, Forward, and Futures Prices 20 1.7.1 Present and Future Spot Prices 20 1.7.2 Forward Prices 24 1.7.3 Futures Prices 25 CHAPTER 2 HEDGING WITH FORWARD CONTRACTS 33 2.1 Motivation for Hedging 33 2.2 Payoff to a Long Forward Position 37 2.3 Payoff to a Short Forward Position 39 viii DETAILED CONTENTS 2.4 Hedging with Forward Contracts 43 2.5 Profits to a Naked (Unhedged) Long Spot Position 45 2.6 Profits to a Fully Hedged Current Long Spot Position 47 2.7 Adding Profit Tables to Determine Profits from a Fully Hedged Position 50 Combining Charts to See Profits from the Hedged Position 54 2.8 CHAPTER 3 3.1 3.2 VALUATION OF FORWARD CONTRACTS ON ASSETS WITHOUT A DIVIDEND YIELD 65 Comparing the Payoffs from a Naked Long Spot Position to the Payoffs from a Naked Long Forward Position 66 Pricing Zero-Coupon, Unit Discount Bonds in Continuous Time 69 3.2.1 3.2.2 Continuous Compounding and Continuous Discounting 69 Pricing Zero-Coupon Bonds 71 3.3 Price vs.

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.

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

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With a larger number of stocks comprising the index the transaction costs would have been too high to make such a construction practicable. This page intentionally left blank 7 Options: General Properties In Chapters 1 and 4 we have seen simple examples of call and put options in a one-step discrete-time setting. Here we shall establish some fundamental properties of options, looking at them from a wider perspective and using continuous time. Nevertheless, many conclusions will also be valid in discrete time. Chapter 8 will be devoted to pricing and hedging options. 7.1 Deﬁnitions A European call option is a contract giving the holder the right to buy an asset, called the underlying, for a price X ﬁxed in advance, known as the exercise price or strike price, at a speciﬁed future time T , called the exercise or expiry time.

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.

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. Here we shall accept this formula without proof, by analogy with the discrete time result. (The proof is based on an arbitrage argument combined with a bit of Stochastic Calculus, the latter beyond the scope of this book.) What is the risk-neutral probability P∗ , then? A necessary condition is that the expectation of the discounted stock prices e−rt S(t) should be constant (independent of t), just like in the discrete time case, see (3.5). Let us compute this expectation using the real market probability P . Since W (t) is normally distributed with mean 0 and variance t, it has density x2 √ 1 e− 2t 2πt under probability P . As a result, E e−rt S(t) = S(0)E eσW (t)+(m−r)t \$ ∞ 1 − x2 e 2t dx eσx+(m−r)t √ = S(0) 2πt −∞ \$ ∞ 1 2 1 − (x−σt)2 2t √ e = S(0)e(m−r+ 2 σ )t dx 2πt −∞ \$ ∞ 1 2 1 − y2 √ e 2t dy = S(0)e(m−r+ 2 σ )t 2πt −∞ 1 = S(0)e(m−r+ 2 σ 2 )t .

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

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In Eq. 1.7, FV = PVezctt holds as well with t = 3 months as with t = 3 years, for example. Further on, the discount factors in continuous time become: (1.8) that is, the continuous time equivalent of Eq. 1.5 in discrete time. Coming back to the previous example of zd = 5%, t = 4 years, PV = 1, where Dt was =1/1.054 = 0.8227 in discrete time, corresponding to FV = 1.2155, we have now, with the same 5% as a zc rate: and But in fact we must take into account that if zd = 5% was a discrete rate, its corresponding continuous value is giving and that is, the same results as in discrete time. 1.4 FORWARD RATES Let's have the following set of spot rates z1, z2, …, zt, whatever the corresponding time periods t = 1, 2, …, t are (e.g., years), and define ft, t+1 the forward zero-coupon rate between time t and time t + 1.

In the case of financial processes, the stochastic calculus is essentially developed within the framework of diffusion processes, considered as conveniently describing their behavior. 8.2 THE STANDARD WIENER PROCESS, OR BROWNIAN MOTION The simplest diffusion process is a random process whose values of a random variable in function of the time t follow a probabilistic distribution proportional to t. On a discrete time interval Δt, this process may be described by (8.1) (The presence of “” will be explained later.) Also, for t = 0, (t) is such as (0) = 0. For the sake of mathematical tractability, most financial diffusion processes assume that their random nature is fairly described by a Gaussian (or normal) probability distribution (or bell curve), fully determined by its mean μ and its variance σ2. Hence where the « » sign means that (t) “follows” a certain distribution probability law, and (.) is the Gaussian distribution of probabilities. In the case of the process, [y(t)] actually follows a « unit normal distribution », noted (0, 1), of mean E = 0 and variance V = 1 (hence, a standard deviation STD = √V = 1 as well). In discrete time, Eq. 8.1 means therefore that the change of (t) during Δt is following a Gaussian distribution with parameters E = 0 (because 0 × Δt = 0), STD = Δt (because 1 × Δt = Δt) and V = Δt.

To obtain this limit, let us use the classic algebraic formula defining the “e” number (= 2.71828…): By making x = n/z and raising each side to the power z we get and in Eq. 1.6, by making n → ∞, we get giving FV(n → ∞ = 100 e0.08 = 108.3287)…(not much more than FV(n = 365)). We therefore have the corresponding relationships for t = 1 year: where zc means the continuous (zero) rate while zd is a discrete (zero) rate. It results from the previous table that the relationship between zc and zd is: (1.6bis) Note that one also speaks of continuous time versus discrete time to refer to continuous or discrete compounding. In practice, one shall consider that z without subscript means zd, and if there is a risk of confusion one must specify zd or zc. The correspondence may be generalized on t years, and with zero-coupon rates zct and zdt respectively, as follows: (1.7) In particular, due to its very essence of implying an instantaneous compounding, it appears that the “continuous” formula no longer needs a different formulation whether t is inferior or superior of 1 year (or 0.5 year) as with the “discrete” form.

Tools for Computational Finance by Rüdiger Seydel

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For illustration see the right-hand grid in Figure 1.7, and Figure 1.10. A Discrete Model We begin with discretizing the continuous time t, replacing t by equidistant time instances ti . Let us use the notations M : number of time steps T ∆t := M ti := i · ∆t, i = 0, ..., M Si := S(ti ) So far the domain of the (S, t) half strip is semi-discretized in that it is replaced by parallel straight lines with distance ∆t apart, leading to a discrete-time model. The next step of discretization replaces the continuous values Si along the parallel t = ti by discrete values Sji , for all i and appropriate j. (Here the indices j, i in Sji mean a matrix-like notation.) For a better understanding of the S-discretization compare Figure 1.8. This ﬁgure shows a mesh of the grid, namely the transition from t to t + ∆t, or from ti to ti+1 . t t+ ∆ t t Sd Su S i+1 i+1 1-p t t p S i S i S Fig. 1.8.

(1.21a) (1.21b) The relations (1.21a,b) can be derived from Deﬁnition 1.7 (−→ Exercise 1.9). The relation (1.21b) is also known as 1.6 Stochastic Processes E((∆Wt )2 ) = ∆t . 27 (1.21c) The independence of the increments according to Deﬁnition 1.7(c) implies for tj+1 > tj the independence of Wtj and (Wtj+1 − Wtj ), but not of Wtj+1 and (Wtj+1 − Wtj ). Wiener processes are examples of martingales — there is no drift. Discrete-Time Model Let ∆t > 0 be a constant time increment. For the discrete instances tj := j∆t the value Wt can be written as a sum of increments ∆Wk , Wj∆t = j Wk∆t − W(k−1)∆t . k=1 =:∆Wk The ∆Wk are independent and because of (1.21) normally distributed with Var(∆Wk ) = ∆t. Increments ∆W with such a distribution can be calculated from standard normally distributed random numbers Z. The implication √ Z ∼ N (0, 1) =⇒ Z · ∆t ∼ N (0, ∆t) leads to the discrete model of a Wiener process √ ∆Wk = Z ∆t for Z ∼ N (0, 1) for each k

. √ Because of relation (1.21b) the standard deviation of the numerator is ∆t. Hence for ∆t → 0 the normal distribution of the diﬀerence quotient disperses and no convergence can be expected. 1.6.2 Stochastic Integral Let us suppose that the price development of an asset is described by a Wiener process Wt . Let b(t) be the number of units of the asset held in a portfolio at time t. We start with the simplifying assumption that trading is only possible at discrete time instances tj , which deﬁne a partition of the interval 0 ≤ t ≤ T . Then the trading strategy b is piecewise constant, b(t) = b(tj−1 ) for tj−1 ≤ t < tj and 0 = t0 < t1 < . . . < tN = T . (1.23) Such a function b(t) is called step function. The trading gain for the subinterval tj−1 ≤ t < tj is given by b(tj−1 )(Wtj − Wtj−1 ), and N j=1 b(tj−1 )(Wtj − Wtj−1 ) (1.24) 1.6 Stochastic Processes 29 represents the trading gain over the time period 0 ≤ t ≤ T .

Mathematics for Economics and Finance by Michael Harrison, Patrick Waldron

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Note that corresponding to each Pareto efficient allocation there is at least one: 1. set of non-negative weights defining (a) the objective function in 4. and (b) the representative agent in 5. and 2. initial allocation leading to the competitive equilibrium in 2. 4.9 Multi-period General Equilibrium In Section 4.2, it was pointed out that the objects of choice can be differentiated not only by their physical characteristics, but also both by the time at which they are consumed and by the state of nature in which they are consumed. These distinctions were suppressed in the intervening sections but are considered again in this section and in Section 5.4 respectively. The multi-period model should probably be introduced at the end of Chapter 4 but could also be left until Chapter 7. For the moment this brief introduction is duplicated in both chapters. Discrete time multi-period investment problems serve as a stepping stone from the single period case to the continuous time case. The main point to be gotten across is the derivation of interest rates from equilibrium prices: spot rates, forward rates, term structure, etc. This is covered in one of the problems, which illustrates the link between prices and interest rates in a multiperiod model. Revised: December 2, 1998 CHAPTER 5.

It can also be thought of as a vector-valued function on the sample space Ω. A stochastic process is a collection of random variables or random vectors indexed by time, e.g. {x̃t : t ∈ T } or just {x̃t } if the time interval is clear from the context. For the purposes of this part of the course, we will assume that the index set consists of just a finite number of times i.e. that we are dealing with discrete time stochastic processes. Then a stochastic process whose elements are N -dimensional random vectors is equivalent to an N |T |-dimensional random vector. The (joint) c.d.f. of a random vector or stochastic process is the natural extension of the one-dimensional concept. Random variables can be discrete, continuous or mixed. The expectation (mean, average) of a discrete r.v., x̃, with possible values x1 , x2 , x3 , . . . is given by E [x̃] ≡ ∞ X xi P r (x̃ = xi ) .

Note that the expected utility axioms are neither necessary nor sufficient to guarantee that the Taylor approximation to n moments is a valid representation of the utility function. Some counterexamples of both types are probably called for here, or maybe can be left as exercises. Extracts from my PhD thesis can be used to talk about signing the first three coefficients in the Taylor expansion, and to speculate about further extensions to higher moments. 5.9 The Kelly Strategy In a multi-period, discrete time, investment framework, investors will be concerned with both growth (return) and security (risk). There will be a trade-off between the two, and investors will be concerned with finding the optimal tradeoff. This, of course, depends on preferences, but some useful benchmarks exist. There are three ways of measuring growth: 1. the expected wealth at time t Revised: December 2, 1998 104 5.10.

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

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Long-memory in ﬁnancial datasets has been observed in practice in the past, long before the use of long-range dependent stochastic volatility models. For example, Ding et al. (1993), De Lima and Crato (1994), and Breidt et al. (1998), among others, observed that the squared returns of market indexes have the long-memory property, which intuitively means that observations that are far apart are highly correlated. Harvey (1998) and Breidt et al. (1998) independently introduced a discrete time model under which the log-volatility is modeled as a fractional ARIMA(p,d,q) process, while at the same time Comte and Renault (1998) introduced a continuous-time long-range dependent volatility model. 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].

In Section 8.3, we use simulated data to test the preformance of the described methodologies, while in Section 8.4, we apply them in real data. In the last section, we summarize our results. 222 CHAPTER 8 Estimation and calibration for LMSV 8.2 Statistical Inference Under the LMSV Model The main goal of this section is to present the most popular methods for statistical inference under the LMSV model (Eq. 8.1). The model is in continuous-time and the volatility process is not observed, but we only have access to discrete time observations of historical stock prices. However, we assume that we are able to obtain high frequency (intraday) data, for example, tick-by-tick observations. 8.2.1 LOG-PERIODOGRAM REGRESSION HURST PARAMETER ESTIMATOR A common practice in the literature is the use of the absolute or log squared returns in order to estimate the long-memory parameter semiparametrically. For example, the reader can refer to Breidt et al. (1998) and Andersen and Bollerslev (1997).

For example, the reader can refer to Breidt et al. (1998) and Andersen and Bollerslev (1997). The estimator used in these cases is the well-known GPH estimator that was initially introduced by Geweke and Porter-Hudak (1983) and is based on the log-periodogram regression. The asymptotic behavior of the GPH estimator in the case of Gaussian observations has been studied by Robinson (1995a,b) and Hurvich, Deo and Brodsky (1998). However, the log squared returns in the discrete time LMSV model are not Gaussian and asymptotic properties of the estimator in this case have been established by Deo and Hurvich (2001). Our model is in continuous time, thus, we are going to discretize it ﬁrst before applying the log-periodogram method. This is a common approach when we deal with continuous-time models and is also suggested by Comte and Renault (1998). REMARK 8.1 For convenience in the illustration of the method, we drop the drift term.

Monte Carlo Simulation and Finance by Don L. McLeish

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In general, for more complex features of the derivative such as the distribution of return, important for considerations such as the Value at Risk, we need to obtain a solution {Xt , 0 < t < T }to an equation of the above form which is a stochastic process. Typically this can only be done by simulation. One of the simplest methods of simulating such a process is motivated through a crude interpretation of the above equation in terms of discrete time steps, that is that a small increment Xt+h − Xt in the process is approximately normally distributed with mean given by a(Xt , t)hand variance given by σ 2 (Xt , t)h. We generate these increments sequentially, beginning with an assumed value for X0 , and then adding to obtain an approximation to the value of the process at discrete times t = 0, h, 2h, 3h, . . .. Between these discrete points, we can linearly interpolate the values. Approximating the process by assuming that the conditional distribution of Xt+h − Xt is N (a(Xt , t)h, σ 2 (Xt , t)h) is called Euler’s method by analogy to a simple method by the same name for solving ordinary diﬀerential equations.

Much of the simulation literature concerns discrete event simulations (DES), simulations of systems that are assumed to change instantaneously in response to sudden or discrete events. These are the most common in operations research and examples are simulations of processes such as networks or queues. Simulation models in which the process is characterized by a state, with changes only at discrete time points are DES. In modeling an inventory system, for example, the arrival of a batch of raw materials can be considered as an event which precipitates a sudden change in the state of the system, followed by a demand some discrete time later when the state of the system changes again. A system driven by diﬀerential equations in continuous time is an example of a DES because the changes occur continuously in time. One approach to DES is future event 203 204 CHAPTER 4. VARIANCE REDUCTION TECHNIQUES simulation which schedules one or more future events at a time, choosing the event in the future event set which has minimum time, updating the state of the system and the clock accordingly, and then repeating this whole procedure.

BASIC MONTE CARLO METHODS and l > 0, Z ∞ 1 xp−1 exp{−(ln(x) − µ)2 /2σ2 }dx E[X I(X > l)] = √ σ 2π l Z ∞ 1 = √ ezp exp{−(z − µ)2 /2σ2 }dz σ 2π ln(l) pµ+p2 σ 2 /2 Z ∞ 1 exp{−(z − ξ)2 /2σ 2 }dz where ξ = µ + σ 2 p = √ e σ 2π ln(l) 2 2 ξ − ln(l) = epµ+p σ /2 Φ( ) σ 2 σ 1 (3.11) = η p exp{− p(1 − p)}Φ(σ−1 ln(η/l) + σ(p − )) 2 2 p where Φ is the standard normal cumulative distribution function. Application: A Discrete Time Black-Scholes Model Suppose that a stock price St , t = 1, 2, 3, ... is generated from an independent sequence of returns Z1 , Z2 over non-overlapping time intervals. If the value of the stock at the end of day t = 0 is S0 , and the return on day 1 is Z1 then the value of the stock at the end of day 1 is S1 = S0 eZ1 . There is some justice in the use of the term “return” for Z1 since for small values Z1 , S0 eZ1 ' S0 (1 + Z1 ) and so Z1 is roughly S1 −S0 S1 .

Frequently Asked Questions in Quantitative Finance by Paul Wilmott

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Volatility is a required input for all classical option-pricing models, it is also an input for many asset-allocation problems and risk estimation, such as Value at Risk. Therefore it is very important to have a method for forecasting future volatility. There is one slight problem with these econometric models, however. The econometrician develops his volatility models in discrete time, whereas the option-pricing quant would ideally like a continuous-time stochastic differential equation model. Fortunately, in many cases the discrete-time model can be reinterpreted as a continuous-time model (there is weak convergence as the time step gets smaller), and so both the econometrician and the quant are happy. Still, of course, the econometric models, being based on real stock price data, result in a model for the real and not the risk-neutral volatility process.

Even MBAs could now join in the fun. See Cox, Ross and Rubinstein (1979). Figure 1-3: The branching structure of the binomial model. 1979- 81 Harrison, Kreps, Pliska Until these three came onto the scene quantitative finance was the domain of either economists or applied mathematicians. Mike Harrison and David Kreps, in 1979, showed the relationship between option prices and advanced probability theory, originally in discrete time. Harrison and Stan Pliska in 1981 used the same ideas but in continuous time. From that moment until the mid 1990s applied mathematicians hardly got a look in. Theorem, proof everywhere you looked. See Harrison and Kreps (1979) and Harrison and Pliska (1981). 1986 Ho and Lee One of the problems with the Vasicek framework for interest rate derivative products was that it didn’t give very good prices for bonds, the simplest of fixed income products.

It is justifiable to rely on sensitivities of prices to variables, but usually not sensitivity to parameters. To get around this problem it is possible to independently model volatility, etc., as variables themselves. In such a way it is possible to build up a consistent theory. Static hedging There are quite a few problems with delta hedging, on both the practical and the theoretical side. In practice, hedging must be done at discrete times and is costly. Sometimes one has to buy or sell a prohibitively large number of the underlying in order to follow the theory. This is a problem with barrier options and options with discontinuous payoff. On the theoretical side, the model for the underlying is not perfect, at the very least we do not know parameter values accurately. Delta hedging alone leaves us very exposed to the model, this is model risk.

Empirical Market Microstructure: The Institutions, Economics and Econometrics of Securities Trading by Joel Hasbrouck

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Dealer markets, on the other hand, often have no publicly visible bids or offers, nor any trade reporting, and are therefore usually considered opaque. 1.4 Econometric Issues Microstructure data are distinctive. Most microstructure series consist of discrete events randomly arranged in continuous time. Within the timeseries taxonomy, they are formally classified as point processes. Point process characterizations are becomingly increasingly important, but for many purposes it suffices to treat observations as continuous variables realized at regular discrete times. Microstructure data are often well ordered. The sequence of observations in the data set closely corresponds to the sequence in which the economic events actually happened. In contrast, most macroeconomic data are time-aggregated. This gives rise to simultaneity and uncertainty about the directions of causal effects. The fine temporal resolution, sometimes described as ultra-high frequency, often supports stronger conclusions about causality (at least in the post hoc ergo propter hoc sense).

With high µ, all days will tend to have one-sided order flow, a preponderance of buys or sells, depending on the outcome of the value draw. Although this model could be estimated by maximum likelihood, actual applications are based on a modified version, described in the next section. 6.2 Event Uncertainty and Poisson Arrivals This model is a variation of the sequential trade model with event uncertainty (section 5.4.4). The principal difference is that agents are not sequentially drawn in discrete time but arrive randomly in continuous time. These events are modeled as a Poisson arrival process. Specifically, suppose that the traders arrive randomly in time such that the probability of an arrival in a time interval of length t is λt where λ is the arrival intensity per unit time, and the probability of two traders arriving in the same interval goes to zero in the limit as t → 0. Then, • The number of trades in any finite interval of length t is Poisson with parameter θ = λt.

Although these sorts of analyses are sensible first steps, they are based on samples that are generated with a selection bias. The source of this bias lies in the decisions of the traders who submitted the orders. Corrections to deal with this bias have been implemented by Madhavan and Cheng (1997) and Bessembinder (2004). 15 Prospective Trading Costs and Execution Strategies In this chapter we discuss minimization of expected implementation cost in two stylized dynamic trading problems. Both analyses are set in discrete time, and in each instance a trader must achieve a purchase by a deadline. The first problem concerns long-term order splitting and timing. A large quantity is to be purchased over a horizon that spans multiple days. Strategic choice involves quantities to be sent to the market at each time, but order choice is not modeled. The second problem involves purchase of a single unit over a shorter horizon, typically several hours.

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

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The MIT era for Merton was highly productive. He produced both a large quantity and a high average quality of papers within the MIT 142 The Rise of the Quants tradition of elegant and simple continuous-time stochastic models that often employed the representative agent approach. These models differentiated themselves from the less sophisticated discrete-time models in vogue elsewhere. Such a continuous-time approach depended crucially on solutions to differential equations, while the discrete-time analog took a far less elegant approach using algebra, some calculus, and seemingly myriad special cases. While more mathematically difficult, the continuous-time approach of the MIT School was much more elegant and powerful. Because of its elegance and generality, the continuous-time approach could, in turn, take on more difficult problems and tease out more general and generalizable results.

He wrote a prominent textbook, called Investment, now in its sixth edition, with Gordon Alexander and Jeffrey Bailey, and Fundamentals of Investments, also with Gordon Alexander and Jeffrey Bailey, and now in its third edition. Sharpe also began to study pensions in the post-CAPM portion of his career. In his research, he continued to look into ways in which theoretical concepts can be reduced to methodologies that can be applied by practitioners. For instance, he produced a discrete-time binomial option pricing procedure that offered a readily applicable procedure for BlackScholes securities pricing, which will be covered in the next part of this book. He also developed the Sharpe ratio, a measure of the risk of a mutual or index fund versus its reward. Sharpe continued to work to make financial concepts more democratic and more accessible. He helped develop Financial Engines, an Internetbased application to deliver investment advice online. 78 The Rise of the Quants Ever concerned about the practitioner’s side of finance, Sharpe began to consult with investment houses, first Merrill Lynch and then Wells Fargo.

In his Nobel speech, he placed his work as resting somewhere between the simple, elegant, and insightful two period models that had been the hallmark of MIT and Cambridge research on the East Coast for almost two decades and the more abstract general equilibrium models, produced on the other coast, that were the hallmark of Berkeley at the University of California. Merton’s innovation, and his research signature, is the powerful tool of continuous-time modeling. As opposed to all the special cases and indeterminate solutions that discrete-time models provide once even two or three time intervals are connected, the continuous-time models often provided surprisingly simple and analytic solutions, despite the more sophisticated tools necessary to solve them. Because of the simplicity of the solutions and the relatively small numbers of variables he chose to include, his models were also more amenable to econometric verification. Merton’s research agenda in the period leading up to his influential contribution to the Black-Scholes equation produced ten papers in five years.

Analysis of Financial Time Series by Ruey S. Tsay

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There are two types of stochastic process for modeling the price of an asset. The first type is called the discrete-time stochastic process, in which the price changes at discrete time points. All the processes discussed in the previous chapters belong to this category. For example, the daily closing price of IBM stock on the New York Stock Exchange forms a discrete-time stochastic process. Here the price changes only at the closing of a trading day. Price movements within a trading day are not necessarily relevant to the observed daily price. The second type of stochastic process is the continuous-time process, in which the price changes continuously, even though the price is only observed at discrete time points. One can think of the price as the “true value” of the stock that always exists and is time varying.

A continuous price can assume any positive real number, whereas a discrete price can only assume a countable number of possible values. Assume that the price of an asset is a continuous-time stochastic process. If the price is a continuous random variable, then we have a continuous-time continuous process. If the price itself is discrete, then we have a continuous-time discrete process. Similar classifications apply to discretetime processes. The series of price change in Chapter 5 is an example of discrete-time discrete process. In this chapter, we treat the price of an asset as a continuous-time continuous stochastic process. Our goal is to introduce the statistical theory and tools needed to model financial assets and to price options. We begin the chapter with some terminologies of stock options used in the chapter. In Section 6.2, we provide a brief introduction of Brownian motion, which is also known as a Wiener process.

For a given η, {x(η, t)} is a time series with values depending on the time t. For simplicity, we 223 STOCHASTIC PROCESSES write a continuous-time stochastic process as {xt } with the understanding that, for a given t, xt is a random variable. In the literature, some authors use x(t) instead of xt to emphasize that t is continuous. However, we use the same notation xt , but call it a continuous-time stochastic process. 6.2.1 The Wiener Process In a discrete-time econometric model, we assume that the shocks form a white noise process, which is not predictable. What is the counterpart of shocks in a continuoustime model? The answer is the increments of a Wiener process, which is also known as a standard Brownian motion. There are many ways to define a Wiener process {wt }. We use a simple approach that focuses on the small change wt = wt+ t − wt associated with a small increment t in time.

Data Mining in Time Series Databases by Mark Last, Abraham Kandel, Horst Bunke

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Proceedings of the Sixteenth International Conference on Data Engineering, pp. 33–42. 53. Policker, S., and Geva, A.B. (2000). Non-Stationary Time Series Analysis by Temporal Clustering. IEEE Transactions on Systems, Man, and Cybernetics, Part B, 30(2), 339–343. 54. Pratt, K.B. and Fink, E. (2002). Search for Patterns in Compressed Time Series. International Journal of Image and Graphics, 2(1), 89–106. 55. Pratt, K.B. (2001). Locating patterns in discrete time series. Master’s thesis, Computer Science and Engineering, University of South Florida. 56. Qu, Y., Wang, C., and Wang, X.S. (1998). Supporting Fast Search in Time Series for Movement Patterns in Multiple Scales. Proceedings of the Seventh International Conference on Information and Knowledge Management, pp. 251–258. 57. Sahoo, P.K., Soltani, S., Wong, A.K.C., and Chen, Y.C. (1988). A Survey of Thresholding Techniques.

Extending the LCSS Model Having seen that there exists an eﬃcient way to compute the LCSS between two sequences, we extend this notion in order to deﬁne a new, more ﬂexible, similarity measure. The LCSS model matches exact values, however in our model we want to allow more ﬂexible matching between two sequences, when the values are within certain range. Moreover, in certain applications, the stretching that is being provided by the LCSS algorithm needs only to be within a certain range, too. We assume that the measurements of the time-series are at ﬁxed and discrete time intervals. If this is not the case then we can use interpolation [23,34]. Deﬁnition 3. Given an integer δ and a real positive number ε, we deﬁne the LCSSδ,ε (A, B) as follows:   0 if A or B is empty       1 + LCSSδ,ε (Head(A), Head(B))  LCSSδ,ε (A, B) = if |an − bn | < ε and |n − m| ≤ δ      max(LCSSδ,ε (Head(A), B), LCSSδ,ε (A, Head(B)))     otherwise Indexing Time-Series under Conditions of Noise 77 Fig. 5.

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

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The arbitrage models involve the compilation, and where possible the resolution, of an equation with partial derivatives for the price Pt (s, v1 , v2 , . . .) considered as a function of t, v1 , v2 , . . . (s ﬁxed), using: 18 Ho T. and Lee S., Term structure movement and pricing interest rate contingent claims, Journal of Finance, Vol. 41, No. 5., 1986, pp. 1011–29. 19 Heath D., Jarrow R. and Morton A., Bond Pricing and the Term Structure of Interest Rates: a New Methodology, Cornell University, 1987. Heath D., Jarrow R. and Morton A., Bond pricing and the term structure of interest rates: discrete time approximation, Journal of Financial and Quantitative Analysis, Vol. 25, 1990, pp. 419–40. Bonds 139 • the absence of arbitrage opportunity; • hypotheses relating to stochastic processes that govern the evolutions in the state variables v1 , v2 etc. The commonest of the models with just one state variable are the Merton model,20 the Vasicek model21 and the Cox, Ingersoll and Ross model;22 all these use the instant term rate r(t) as the state variable.

The ARMA model is a classical model; it considers that the volumes observed are produced by a random stable process, that is, the statistical properties do not change over the course of time. The variables in the process (that is, mathematical anticipation, valuation–valuation) are independent of time and follow a Gaussian distribution. The variation must also be ﬁnished. Volumes will be observed at equidistant moments (case of process in discrete time). We will take as an example the ﬂoating-demand savings accounts in LUF/BEF Techniques for Measuring Structural Risks in Balance Sheets 319 from 1996 to 1999, observed monthly (data on CD-ROM). The form given in the model is that of the recurrence system, Volum t = a0 + p ai Volum t−i + εt i=1 where a0 + a1 Volum t−1 + . . . + aP Volum t−p represents the autoregressive model that is ideal or perfectly adjusted to the chronological series, thus being devoid of uncertainty, and εt is a mobile average process. εt = q bi ut−i i=0 The ut−I values constitute ‘white noise’ (following the non-autocorrelated and centred normal random variables with average 0 and standard deviation equal to 1). εt is therefore a centred random variable with constant variance.

., A theory of the term structure of interest rates, Econometrica, Vol. 53, No. 2, 1985, pp. 385–406. Fabozzi J. F., Bond Markets, Analysis and Strategies, Prentice-Hall, 2000. Bibliography 385 Heath D., Jarrow R., and Morton A., Bond Pricing and the Term Structure of Interest Rates: a New Methodology, Cornell University, 1987. Heath D., Jarrow R., and Morton A., Bond pricing and the term structure of interest rates: discrete time approximation, Journal of Financial and Quantitative Analysis, Vol. 25, 1990, pp. 419–40. Ho T. and Lee S., Term structure movement and pricing interest rate contingent claims, Journal of Finance, Vol. 41, No. 5, 1986, pp. 1011–29. Macauley F., Some Theoretical Problems Suggested by the Movements of Interest Rates, Bond Yields and Stock Prices in the United States since 1856, New York, National Bureau of Economic Research, 1938, pp. 44–53.

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

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The faster-than-exponential growth of the return and of the crash hazard rate correspond to nonconstant growth rates, which increase with the return and with the hazard rate. The following reasoning allows us to understand intuitively the origin of the appearance of an inﬁnite slope or inﬁnite value in a ﬁnite time at tc , called a ﬁnite-time singularity. Suppose, for instance, that the growth rate of the hazard rate doubles when the hazard rate doubles. For simplicity, we consider discrete-time intervals as follows. Starting with a hazard rate of 1 per unit time, we assume it grows at a constant rate of 1% per day until it doubles. We estimate the doubling time as proportional to the inverse of the growth rate, that is, approximately 1/1% = 1/001 = 100 days. There is a multiplicative correction term equal to ln 2 = 069 such that the doubling time is ln 2/1% = 69 days. But we drop this proportionality model ing bubbles a n d c r a s h e s 161 factor ln 2 = 069 for the sake of pedagogy and simplicity.

Similarly, the economic and ﬁnancial development of the United States and Europe and of other parts of the world are interdependent due to the existence of several coupling mechanisms (exchanges of goods, services, transfer of research and development, immigration, etc.) The faster-than-exponential growths observed in Figures 10.1 and 10.2 correspond to nonconstant growth rates, which increase with population or with the size of economic factors. Suppose, for instance, that the growth rate of the population doubles when the population doubles. For simplicity, we consider discrete time 2 05 0: the end of t h e g r o w t h e r a? 365 intervals as follows. Starting with a population of 1,000, we assume it grows at a constant rate of 1% per year until it doubles. We estimate the doubling time as proportional to the inverse of the growth rate, that is, approximately 1/1% = 1/0.01 = 100 years. Actually, there is a multiplicative correction term equal to ln 2 = 069 such that the doubling time is ln 2/1% = 69 years.

Financial Modelling in Python by Shayne Fletcher, Christopher Gardner

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As already mentioned, in the interest of brevity we stay in the terminal measure. In particular this means that we can re-use the fil component already discussed for the Monte-Carlo model. t In the terminal measure there is only one state variable that needs to be evolved: namely, 0 C(s)dW (s). In a typical application, the evolve step will be peformed on a discrete set of contiguous times. In other words, suppose we have the discrete times {T1 , T2 , . . . , Ti } with the T time today denoted by T0 , then the simulation of 0 i C(s)dW (s) is carried out as the discrete sum shown below Ti i Tk+1 C(s)dW (s) = C 2 (s)ds Z K (8.2) 0 k=0 Tk with Z 1 , Z 2 , . . . independent, identical distributed normal variates with distribution N (0, 1). The evolve component of the Hull–White model is implemented in the ppf.model. hull white.monte carlo.evolve module as shown below. class evolve: def init (self, ccy, seed = 1234, antithetic = True): self. ccy = ccy self. seed = seed self. antithetic = antithetic def evolve(self, t, T, state, req, env): The Hull–White Model 113 if t > T: raise RuntimeError, ’attempting to evolve backwards’ if t == T: return variates = state.get variates() num sims = variates.shape if self. antithetic: raise RuntimeError, \ ’expected number of simulations to be even with’ \ ’antithetic’ num sims /= 2 volt = req.term vol(t, self. ccy, env) volT = req.term vol(T, self. ccy, env) vartT = volT*volT-volt*volt if vartT < 0: raise RuntimeError, ’negative incremental variance’ voltT = math.sqrt(vartT) generator = random.Random(self. seed) for i in range(num sims): z = generator.gauss(0, 1.0) variates[i] = variates[i]+voltT*z if self. antithetic: variates[num sims+i] = variates[num sims+i]-voltT*z state.set variates(variates) self. seed = self. seed+1 The evolve component is constructed by passing in the currency, the start seed for the random generator and a boolean to control whether we wish to have antithetic variates.

Think Complexity by Allen B. Downey

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is written with a capital letter because it is a complex number, which you can think of as a vector with a magnitude, , and an angle. The power spectral density is related to the Fourier transform by the following relation: Depending on the application, we may not care about the difference between f and . In that case, we would use the one-sided power spectral density: So far we have assumed that is a continuous function, but often it is a series of values at discrete times. In that case, we can replace the continuous Fourier transform with the discrete Fourier transform (DFT). Suppose that we have N values hk with k in the range from 0 to . The DFT is written Hn, where n is an index related to frequency: 9.1 Each element of this sequence corresponds to a particular frequency. If the elements of hk are equally spaced in time, with time step d, the frequency that corresponds to Hn is: To get the one-sided power spectral density, you can compute Hn with n in the range to , and: To avoid negative indices, it is conventional to compute Hn with n in the range 0 to , and use the relation to convert.

Turing's Vision: The Birth of Computer Science by Chris Bernhardt

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One-Dimensional Cellular Automata A one-dimensional cellular automaton consists of an infinite tape divided into cells. Each cell can have one of a number of states. We will only look at cases where there are just two states, which we will denote by white and black. This means that the tape will be divided into an infinite number of cells, which we depict as squares, each of which is colored either black or white. The computation takes place at discrete time intervals. The initial tape is given to us at time 0. It first gets updated at time 1, then at time 2 and so on. Even though it is really one tape that is evolving at each time interval, it is easier to describe each instance as a separate tape. Given a tape, the subsequent tape, one unit of time later, is given by an updating rule that for each cell looks at the state of that cell and the states of some of its neighbors on the initial tape and gives the cell’s new state on the updated tape.

Elements of Mathematics for Economics and Finance by Vassilis C. Mavron, Timothy N. Phillips

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The divergence is uniform (see Fig. 12.5). 270 Elements of Mathematics for Economics and Finance X 30 20 10 0 1 2 3 4 5 t -10 Figure 12.4 divergence. Graph of the solution of Example 12.3.4 showing oscillatory 12.5 The Cobweb Model The Cobweb model is an economic model for analysing periodic fluctuations in price, supply, and demand that oscillate towards equilibrium. It is assumed that the quantities involved change only at discrete time intervals and that there is a time lag in the response of suppliers to price changes. For instance, the supply this year of a particular agricultural product depends on the price obtained from the previous year’s harvest. The demand for the produce will depend of course on this year’s price. Another example is that of package holidays. The holiday company’s supply of holidays for this season will depend on the prices obtained for last season’s.

Extreme Money: Masters of the Universe and the Cult of Risk by Satyajit Das

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By adjusting the ratio of options to the shares, you can construct a portfolio where the changes in the value of the options and shares exactly offset, at least, for small movements in the stock price. Working as research assistant to Paul Samuelson, Robert Merton was also working on option pricing. Merton introduced an idea—continuous time mathematics. Black and Scholes assumed that the portfolio would be rebalanced to keep it free of risk by changing the number of shares held at discrete time intervals. Merton forced the time intervals into infinitely small fragments of time, effectively allowing continuous and instantaneous rebalancing. Although unrealistic and practically impossible, this allowed a mathematical solution using Ito’s (pronounced “Eto”) Lemma to solve the equation. Ito, an eccentric Japanese mathematician, later did not remember deriving the eponymous technique. The Black-Scholes-Merton (BSM) option pricing model relied on the CAPM, itself reliant on the EMH, and arbitrage.

See also exotic products AIG, 230-234 arbitrage, 242 central banks, 281-282 deconstruction, 235-236 first-to-default (FtD) swaps, 220-221 Harvard case studies, 214-215 hedging, 216-217 Italy, 215-216 Jerome Kerviel, 226-230 managing risk, 124 markets, 235, 334 municipal bonds, 211-214 price movements, 210-211 risk, 218-219 design of, 225 Fiat, 222-223 Greece, 223, 225 sale of to ordinary investors, 332-333 sovereign debt, 236-238 TARDIS trades, 217-218 TOBs (tender option bonds), 222 Derman, Emanuel, 309 Derrida, Jacques, 236 Descartes, René, 228 Detroit, 42 Deutsche Bank, 79, 195, 272, 312 Devaney, John, 255 di Lampedusa, Giuseppe, 353 digitals, 211 Dillion Read, 201 Dimon, Jamie, 283, 290 dinars, 21 Diners Club, 71 Dirac, Paul, 104 dirty bombs, 26 disaster capitalism, 342 discrete time intervals, 121 dismal science, 102-104 dispersion swaps, 255 distressed debt trading, 242 distributions, normal, 126 diversification, 122-124 dividends, 119 Dixon, Geoff, 156 documentation requirements, 181 DOG (debt overburdened group), 161 dollars American, 21-22, 28, 87 aussies, 21 kiwis, 21 Zimbabwe, 23 domain knowledge, 64 domestic corporate profits, United States, 276 Dominion Bond Rating Service, 283 doomsday clock, 34 Dorgan, Bryan, 67 Douglas, Michael, 167, 310 Dow 36,000, 99 Dow 40,000: Strategies for Profiting from the Greatest Bull Market in History, 97 Dow 100,000, 97 Dow Jones Industrial Average (DJIA), 89, 97, 126 Dr.

Chaos by James Gleick

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Population biology learned quite a bit about the history of life, how predators interact with their prey, how a change in a country’s population density affects the spread of disease. If a certain mathematical model surged ahead, or reached equilibrium, or died out, ecologists could guess something about the circumstances in which a real population or epidemic would do the same. One helpful simplification was to model the world in terms of discrete time intervals, like a watch hand that jerks forward second by second instead of gliding continuously. Differential equations describe processes that change smoothly over time, but differential equations are hard to compute. Simpler equations—“difference equations”—can be used for processes that jump from state to state. Fortunately, many animal populations do what they do in neat one-year intervals.

A Primer for the Mathematics of Financial Engineering by Dan Stefanica

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rt 10 ° B'(T) B(T) dT = In(B(T))I~:~ Therefore, 13(t) = 13(0) exp Bond Pricing. Yield of a Bond. Bond Duration and Bond Convexity. The term exp ( - ° The zero rate r(O, t) between time and time t is the rate of return of a cash deposit made at time and maturing at time t. If specified for all values of t, then r(O, t) is called the zero rate curve 2 and is a continuous function of t. Interest can be compounded at discrete time intervals, e.g., annually, semiannually, monthly, etc., or can be compounded continuously. Unless otherwise specified, we assume that interest is compounded continuously. For continuously compounded interest, the value at time t of B(O) currency units (e.g., U.S. dollars) is where exp(x) = eX. The value at time of B(t) currency units at time t is r(t) = lim ~ . B(t + dt) - B(t) = B'(t). dt-70 dt B(t) B(t) ----------------------2We note, and further explain this in section 2.7.1, that r(O,t) is the yield of a zero~ coupon bond with maturity t.

Scala in Depth by Tom Kleenex, Joshua Suereth

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To understand this issue, we’ll look at how to write a good equality method. 2.5.1. Example: A timeline library We’d like to construct a time line, or calendar, widget. This widget needs to display dates, times, and time ranges as well as associated events with each day. The fundamental concept in this library is going to be an InstantaneousTime. InstantaneousTime is a class that represents a particular discrete time within the time series. We could use the java.util.Date class, but we’d prefer something that’s immutable, as we’ve just learned how this can help simplify writing good equals and hashCode methods. In an effort to keep things simple, let’s have our underlying time storage be an integer of seconds since midnight, January 1, 1970, Greenwich Mean Time on a Gregorian calendar. We’ll assume that all other times can be formatted into this representation and that time zones are an orthogonal concern to representation.

The Data Revolution: Big Data, Open Data, Data Infrastructures and Their Consequences by Rob Kitchin

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The interlinking of data in Obama’s campaign created what Crampton et al. (2012) term an ‘information amplifier effect’, wherein the sum of data is more than the parts. Velocity A fundamental difference between small and big data is the dynamic nature of data generation. Small data usually consist of studies that are freeze-framed at a particular time and space. Even in longitudinal studies, the data are captured at discrete times (e.g., every few months or years). For example, censuses are generally conducted every five or ten years. In contrast, big data are generated on a much more continuous basis, in many cases in real-time or near to real-time. Rather than a sporadic trickle of data, laboriously harvested or processed, data are flowing at speed. Therefore, there is a move from dealing with batch processing to streaming data (Zikopoulos et al. 2012).

The End of College: Creating the Future of Learning and the University of Everywhere by Kevin Carey

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A host of other elite research universities from around the world soon followed. By 2014, edX was offering hundreds of free online courses in subjects including the Poetry of Walt Whitman, the History of Early Christianity, Computational Neuroscience, Flight Vehicle Aerodynamics, Shakespeare, Dante’s Divine Comedy, Bioethics, Contemporary India, Historical Relic Treasures and Cultural China, Linear Algebra, Autonomous Mobile Robots, Electricity and Magnetism, Discrete Time Signals and Systems, Introduction to Global Sociology, Behavioral Economics, Fundamentals of Immunology, Computational Thinking and Data Science, and an astrophysics course titled Greatest Unsolved Mysteries of the Universe. Doing this seemed to contradict five hundred years of higher-education economics in which the wealthiest and most sought-after colleges enforced a rigid scarcity over their products and services.

The Art of R Programming by Norman Matloff

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Table 15-1: Common GDB Commands Command Description l b r n s p c h q List code lines Set breakpoint Run/rerun Step to next statement Step into function call Print variable or expression Continue Help Quit 15.1.5 Extended Example: Prediction of Discrete-Valued Time Series Recall our example in Section 2.5.2 where we observed 0- and 1-valued data, one per time period, and attempted to predict the value in any period from the previous k values, using majority rule. We developed two competing functions for the job, preda() and predb(), as follows: # prediction in discrete time series; 0s and 1s; use k consecutive # observations to predict the next, using majority rule; calculate the # error rate preda <- function(x,k) { n <- length(x) k2 <- k/2 # the vector pred will contain our predicted values pred <- vector(length=n-k) Interfacing R to Other Languages 327 for (i in 1:(n-k)) { if (sum(x[i:(i+(k-1))]) >= k2) pred[i] <- 1 else pred[i] <- 0 } return(mean(abs(pred-x[(k+1):n]))) } predb <- function(x,k) { n <- length(x) k2 <- k/2 pred <- vector(length=n-k) sm <- sum(x[1:k]) if (sm >= k2) pred <- 1 else pred <- 0 if (n-k >= 2) { for (i in 2:(n-k)) { sm <- sm + x[i+k-1] - x[i-1] if (sm >= k2) pred[i] <- 1 else pred[i] <- 0 } } return(mean(abs(pred-x[(k+1):n]))) } Since the latter avoids duplicate computation, we speculated it would be faster.

Them And Us: Politics, Greed And Inequality - Why We Need A Fair Society by Will Hutton

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But he and some of the partners in his unit complained that they were not being rewarded as well as their counterparts at hedge funds and private-equity firms, so they quit. Seventy million dollars was deemed unfair because it was too low. There were numerous similar stories in both New York and London. Finance is also extraordinarily short term. Executives expect top pay for that year’s outcome, even though it may take years to see whether the deals they struck have truly worked or the profits are anything more than transient. Next year is another discrete time period; profits and thus bonuses are decided year by year. This approach has been cemented by the ‘mark-to-market’ accounting convention in which asset values and profits have to be recognised each year, tracking market movements. As economist John Kay puts it, the trader or dealer ‘not only eats what he kills but also takes credit for the expected cull as soon as the hunters’ guns are primed’.6 Year One’s profits may turn to dust in Year Two, but by then who cares?

What to Think About Machines That Think: Today's Leading Thinkers on the Age of Machine Intelligence by John Brockman

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KAUFFMAN Pioneer of biocomplexity research; affiliate, Institute for Systems Biology, Seattle; author, Reinventing the Sacred: A New View of Science, Reason, and Religion The advent of quantum biology, light-harvesting molecules, bird navigation, perhaps smell, suggests that sticking to classical physics in biology may turn out to be simply stubborn. Now Turing Machines are discrete state (0,1), discrete time (T, T+1) subsets of classical physics. We all know they, like Shannon information, are merely syntactic. Wonderful mathematical results such as Gregory Chaitin’s omega—the probability that a program will halt, which is totally non-computable and nonalgorithmic—tell us that the human mind, as Roger Penrose also argued, cannot be merely algorithmic. Mathematics is creative. So is the human mind.

Commodity Trading Advisors: Risk, Performance Analysis, and Selection by Greg N. Gregoriou, Vassilios Karavas, François-Serge Lhabitant, Fabrice Douglas Rouah

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This should help prevent deleting observations where returns were truly zero. The return data were converted to log changes,1 so they can be interpreted as percentage changes in continuous time. The mean returns presented in Table 3.1 show CTA returns are higher than those of public or private returns. This result is consistent with those formula used was rit = ln (1 + dit /100) × 100, where, dit is the discrete time return. The adjustment factor of 100 is used since the data are measured as percentages. 1The 33 Performance of Managed Futures in previous literature. The conventional wisdom as to why CTAs have higher returns is that they incur lower costs. However, CTA returns may be higher because of selectivity or reporting biases. Selectivity bias is not a major concern here, because the comparison is among CTAs, not between CTAs and some other investment.

Principles of Protocol Design by Robin Sharp

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After a number of attempts, the message will presumably get through, unless of course the system is overloaded or one of the senders has a defect and never stops transmitting3 . Contention protocols offer a form of what is known as statistical multiplexing, where the capacity of the multiplexed service is divided out among the senders in a non-deterministic manner. Their analysis (see, for example, Chapter 4 in ) is usually based on the theory of discrete-time Markov chains, which we shall not consider in detail here. In the case of unrestricted contention protocols, this analysis yields the intuitively obvious result that unless the generated traffic (number of new messages generated per unit time) is very moderate, then unrestricted contention is not a very effective method, leading to many collisions and long delays before a given message in fact gets through. The Blockchain Alternative: Rethinking Macroeconomic Policy and Economic Theory by Kariappa Bheemaiah

Consider our previous stock market example - it is easier to think about the behavioural traits of an individual stock broking agent rather than think about how news of that agent winning the lottery will affect throes of agents who are related to this agent. In fact, the modeler does not even have to go into excruciating detail regarding the individual agents’ traits. Starting with some initial hypotheses, the modeller can generate a model that represents these hypotheses. As the dynamics of the system evolve over discrete time steps, the results can be tested for validity and if they are representative of real world phenomenon, a proof of concept is formed. The advantage of this approach is that it can be employed to study general properties of a system which are not sensitive to the initial conditions, or to study the dynamics of a specific system with fairly well- known initial conditions, e. g. the impact of the baby boomers’ retirement on the US stock market (Bandini et al., 2012).

Antifragile: Things That Gain From Disorder by Nassim Nicholas Taleb

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Many people blew up on misunderstanding the effect. Example of detection and mapping of convexity bias (ωA), from author’s doctoral thesis: The method is to find what needs dynamic hedging and dynamic revisions. Among the members of the class of instruments considered that are not options stricto-sensu but require dynamic hedging can be rapidly mentioned a broad class of convex instruments: (1) Low coupon long dated bonds. Assume a discrete time framework. Take B(r,T,C) the bond maturing period T, paying a coupon C where rt = ∫rs ds. We have the convexity д2B/дr2 increasing with T and decreasing with C. (2) Contracts where the financing is extremely correlated with the price of the Future. (3) Baskets with a geometric feature in its computation. (4) A largely neglected class of assets is the “quanto-defined” contracts (in which the payoff is not in the native currency of the contract), such as the Japanese NIKEI Future where the payoff is in U.S. currency.

The Data Warehouse Toolkit: The Definitive Guide to Dimensional Modeling by Ralph Kimball, Margy Ross

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Because the date dimension is likely the most frequently constrained dimension in a schema, it should be kept as small and manageable as possible. If you want to ﬁlter or roll up time periods based on summarized day part groupings, such as activity during 15-minute intervals, hours, shifts, lunch hour, or prime time, time-of-day would be treated as a full-ﬂedged dimension table with one row per discrete time period, such as one row per minute within a 24-hour period resulting in a dimension with 1,440 rows. If there’s no need to roll up or ﬁlter on time-of-day groupings, time-of-day should be handled as a simple date/time fact in the fact table. By the way, business users are often more interested in time lags, such as the transaction’s duration, rather than discreet start and stop times. Time lags can easily be computed by taking the difference between date/time stamps.

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

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With this calculus we introduce the geometric Brownian motion model of stock price evolution and deduce the Black-Scholes equation. We then show how the BlackScholes equation can be reduced to the heat equation. This yields a derivation of the Black-Scholes formula. In Chapter 6, we step up another mathematical gear and this is the most mathematically demanding chapter. We introduce the concept of a martingale in both continuous and discrete time, and use martingales to examine the concept of riskneutral pricing. We commence by showing that option prices determine synthetic probabilities in the context of a single time horizon model. We then move on to study discrete pricing in martingale terms. Having motivated the definitions using the discrete case, we move on to the continuous case, and show how martingales can be used to develop arbitrage-free prices in the continuous framework.

Data Mining: Concepts and Techniques: Concepts and Techniques by Jiawei Han, Micheline Kamber, Jian Pei

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Pattern analysis is useful in the analysis of spatiotemporal data, time-series data, image data, video data, and multimedia data. An area of spatiotemporal data analysis is the discovery of colocation patterns. These, for example, can help determine if a certain disease is geographically colocated with certain objects like a well, a hospital, or a river. In time-series data analysis, researchers have discretized time-series values into multiple intervals (or levels) so that tiny fluctuations and value differences can be ignored. The data can then be summarized into sequential patterns, which can be indexed to facilitate similarity search or comparative analysis. In image analysis and pattern recognition, researchers have also identified frequently occurring visual fragments as “visual words,” which can be used for effective clustering, classification, and comparative analysis.

The Art of Computer Programming: Fundamental Algorithms by Donald E. Knuth

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By contrast, a "continuous simulation" would be simulation of activities that are under continuous changes, such as traffic moving on a highway, spaceships traveling to other planets, etc. Continuous simulation can often be satisfactorily approximated by discrete simulation with very small time intervals between steps; however, in such a case we usually have "synchronous" discrete simulation, in which many parts of the system are slightly altered at each discrete time interval, and such an application generally calls for a somewhat different type of program organization than the kind considered here. The program developed below simulates the elevator system in the Mathe- Mathematics building of the California Institute of Technology. The results of such a simulation will perhaps be of use only to people who make reasonably frequent visits to Caltech; and even for them, it may be simpler just to try using the elevator several times instead of writing a computer program.