# martingale

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Tools for Computational Finance by Rüdiger Seydel

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t0 t0 Hence the integrals bn (s)dWs form a Cauchy sequence with respect to convergence in the mean. Accordingly the Itô integral of f is deﬁned as t t f (s)dWs := l.i.m.n→∞ bn (s)dWs , t0 t0 for simple processes bn deﬁned by (1.30). The value of the integral is independent of the choice of the bn in (1.30). The Itô integral as function in t is a stochastic process with the martingale property. If an integrand a(x, t) depends on a stochastic process Xt , the function f is given by f (t) = a(Xt , t). For the simplest case of a constant integrand a(Xt , t) = a0 the Itô integral can be reduced via (1.29) to t dWs = Wt − Wt0 . t0 For the “ﬁrst” nontrivial Itô integral consider Xt = Wt and a(Wt , t) = Wt . Its solution will be presented in Section 3.2. 1.7 Stochastic Diﬀerential Equations 1.7.1 Itô Process Many phenomena in nature, technology and economy are modeled by means d x = a(x, t).

Generally for 0 ≤ s < t the property Wt −Ws ∼ N (0, t−s) holds, in particular E(Wt − Ws ) = 0 , Var(Wt − Ws ) = E((Wt − Ws )2 ) = t − s. (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.

Our e−r∆t or e−rT is consistent with the approach of Black, Merton and Scholes. For references on risk-neutral valuation we mention [Hull00], [MR97], [Kwok98] and [Shr04]. on Section 1.6: Introductions into stochastic processes and further hints on advanced literature may be found in [Doob53], [Fr71], [Ar74], [Bi79], [RY91], [KP92], [Shi99], [Sato99]. The requirement (a) of Deﬁnition 1.7 (W0 = 0) is merely a convention of technical relevance; it serves as normalization. This Brownian motion ist called standard Brownian motion. For a proof of the nondiﬀerentiability of Wiener processes, see [HuK00]. For more hints on martingales, see Appendix B2. In contrast to the results for Wiener processes, diﬀerentiable functions Wt satisfy for δN → 0 (Wtj − Wtj−1 )2 −→ 0 . |Wtj − Wtj−1 | −→ |Ws |ds , The Itô integral and the alternative Stratonovich integral are explained in [Doob53], [Ar74], [CW83], [RY91], [KS91], [KP92], [Mik98], [Øk98], [Sc80], [Shr04].

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

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If yes, give an example; if no, prove it. such that x and y are rational Problem 1.60 Let Q2 be the set of all pairs 2 numbers. We consider a random direct line L in R such that with probability 1, and that the angle between L and the vector (1, 0) has the uniform distribution on [0, π). Find the probability that the set © 2007 Nikolai Dokuchaev is finite. 2 Basics of stochastic processes In this chapter, some basic facts and definitions from the theory of stochastic (random) processes are given, including filtrations, martingales, Markov times, and Markov processes. 2.1 Definitions of stochastic processes Sometimes it is necessary to consider random variables or vectors that depend on time. Definition 2.1 A sequence of random variables ξt, t=0, 1, 2,…, is said to be a discrete time stochastic (or random) process. be given. A mapping ξ:[0,T]×Ω→R is said to be a Definition 2.2 Let continuous time stochastic (random) process if ξ(t,ω) is a random variable for a.e.

Definition 2.6 Let ξt be a discrete time white noise, and let t=0, 1, 2,…. Then the process ηt is said to be a random walk. The theory of stochastic processes studies their pathwise properties (or properties of trajectories ξ(t, ω) for given ω, as well as the evolution of the probability distributions. © 2007 Nikolai Dokuchaev Mathematical Finance 18 Definition 2.7 A continuous time process ξ(t)=ξ(t, ω) is said to be continuous (or pathwise continuous), if trajectories ξ(t, ω) are continuous in t a.s. (i.e., with probability 1, or for a.e. ω). It can happen that a continuous time process is not continuous (for instance, a process with jumps). 2.2 Filtrations, independent processes and martingales In this section, we shall assume that either or t=0, 1, 2,…. Filtrations In addition to evolving random variables, we shall use evolving σ-algebras. is called a filtration if for s<t.

., the sequence (Note that is not ‘non-decreasing’), therefore to make the sequence non-decreasing, we must replace is not a filtration; by Independent processes Definition 2.13 Random processes ξ(·) and η(·) are said to be independent iff the events and are independent for all m,n, and all times (t1,…, tn) and (τ1,…, τm), and all sets In fact, processes are independent iff all events from the filtrations generated by them are mutually independent. Martingales Definition 2.14 Let ξ(t) be a process such that E|ξ(t)|2<+∞ for all t, and let if filtration. We say that ξ(t) is a martingale with respect to be a Note that we require that E|ξ(t)|2<+∞ because, for simplicity, we have defined the conditional expectation only for this case. In the literature, the martingales are often defined under the condition E|ξ(t)|<+∞, which is less restrictive. Sometimes the term ‘martingale’ is used without mentioning the filtration. Definition 2.15 Let ξ(t) be a process, and let be the filtration generated by this process. We say that ξ(t) is a martingale if ξ(t) is a martingale with respect to the filtration Problem 2.16 Prove that any discrete time random walk is a martingale. © 2007 Nikolai Dokuchaev Mathematical Finance 20 Problem 2.17 Let ζ be a random variable such that E|ζ|2<+∞, and let be a filtration.

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

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Haug, The Complete Guide to Option Pricing Formulas, Irwin, 1997. [67] M.B. Haugh, L. Kogan, Pricing American Options: A Duality Approach, Operations Research 52(2), 2004, 258-270. References 529 [68] J.M. Harrison, D.M. Kreps, Martingales and arbitrage in multi-period securities markets, Journal of Economic Theory 20, 1979, 381-408. [69] J.M. Harrison, S.R. Pliska, Martingales and stochastic integration in the theory of continuous trading, Stochastic Processes and Applications 11, 1981, 215-60. [70] J.M. Harrison, S.R. Pliska, Martingales and stochastic integration in the theory of continuous trading, Stochastic Processes and Applications 13, 1983, 313-16. [71] D. Heath, R. Jarrow, A. Morton, Bond pricing and the term structure of interest rates: a new methodology for contingent claims valuation, Econometrica 60, 1992, 77-105. [72] S.

Exercise 6.20 Find the Black-Scholes price of an option paying (ST - K)+ at time T. 180 Risk neutrality and martingale measures Exercise 6.21 Find the Black-Scholes price of an option paying (K - ST)+ at time T. Exercise 6.22 Let Wt be a Brownian motion, and let Ft be its filtration. Compute the following when t > s: lE(Wt 1 mss); E(Wr IY'); ]E(W41y). What happens if s < t? Exercise 6.23 A derivative pays (log ST)3 at time T. Develop a price in the BlackScholes world. Exercise 6.24 If Wt is a Brownian motion, is Wt a martingale? Justify your answer. Exercise 6.25 Give an example of a continuous time stochastic process, Xt, such that IE(Xt) = 0, and X t is not a martingale. Exercise 6.26 If S and B follow Black-Scholes assumptions, what is the drift of S in the martingale measure associated to taking S + B as numeraire? The practical pricing of a European option 7.1 Introduction We have developed several techniques for pricing an option: trees, PDEs, riskneutral valuation and replication plus variants of each.

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Derivatives Markets by David Goldenberg

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First, note that under the risk-neutral probability measure E r, the risk-neutralized stock price (which we will continue to call S0 ) is clearly not a martingale, unless riskless interest rates actually are equal to zero. The second equation above says that, E r(S1()|S0)=(1+r′)S0>S0 unless r′=0. Even under risk neutrality (which doesn’t mean zero interest rates), the martingale requirement that E r(S1()|S0)=S0 is clearly violated. Stock prices under risk neutrality are not martingales. 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.

The ﬁnal EQUIVALENT MARTINGALE MEASURES 521 equality is due to the fact that E(X2())=0, because it is a fair game by assumption. We can also prove that E(W1()|W0)=W0. n CONCEPT CHECK 3 a. Prove that E(W1()|W0)=W0, where we are standing at time t=0. We now have two results, E(W1()|W0)=W0 and, E(W2()|W1)=W1. 15.3.2 Definition of a Discrete-Time Martingale A discrete-time stochastic process (Xn())n=0,1,2,3,.. is called a martingale if, 1. E(Xn)<∞ and E(Xn2)<∞ for all n and, 2. E(Xn+1()|Xn)=Xn for all n=0,1,2,3,… Note that our martingales have ﬁnite ﬁrst and second moments. As we have seen in example 2, martingales are constructed from independent fair games. If we add independent fair games to a given starting wealth process, we will end up with a wealth process that is a martingale. Conversely, if the wealth process is a martingale with ﬁnite means and variances, then it must have been generated in this way, by adding uncorrelated fair games to an initial wealth process and proceeding in this way.

Instead of going that route, we will examine some actual continuous time price processes that are consistent with the EMH, and for which we can price options and therefore, for which there is at least one EMM for the discounted price process (FTAP1). We will also look for martingale components in the actual price processes. Chapter 16 begins this program. 15.7 APPENDIX: ESSENTIAL MARTINGALE PROPERTIES Here we collect a few of the many properties of martingales that are used in proving results that make martingales useful in applied ﬁnance. We restrict attention to discrete-time martingales and sometimes even choose N=2. No attempt at mathematical rigor is claimed. The intuition behind these results is the primary concern. We start with a discrete-time stochastic process (Xn()n=0,1,2,3,… with ﬁnite ﬁrst and second moments E(Xn )<∞ and E(Xn2 )<∞ for all n=0,1,2,3,… and the martingale property, E(Xn+1()|Xn)=Xn for all n=0,1,2,3,… (MP1) 1. Tower Property (TP) E(X2|X0)=E{E(X2|X1)|X0} (TP) This is a general property of conditional expectations and doesn’t require a martingale.

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

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The iterative method we will use for this problem was developed by Chadam and Yin in Ref. 22 to study a similar partial integro-differential problem. 13.3.1 STATEMENT OF THE PROBLEM As pointed out in Ref. 17, when modeling high frequency data in applications, a Lévy-like stochastic process appears to be the best ﬁt. When using these models, option prices are found by solving the resulting PIDE. For example, integrodifferential equations appear in exponential Lévy models, where the market price of an asset is represented as the exponential of a Lévy stochastic process. These models have been discussed in several published works such as Refs 17 and 23. 365 13.3 Another Iterative Method In this section, we consider the following integro-differential model for a European call option ∂C σ 2S2 ∂ 2C ∂C (S, t) − rC(S, t) (S, t) + rS (S, t) + ∂t ∂S 2 ∂S 2 ∂C y y + ν(dy) C(Se , t) − C(S, t) − S(e − 1) (S, t) = 0, ∂S (13.28) where the market price of an asset is represented as the exponential of a Lévy stochastic process (see Chapter 12 of Ref. 17).

Physica A 2003;318:279–292 [Proceedings of International Statistical Physics Conference, Kolkata]. 19. Mantegna RN, Stanley HE. Stochastic process with ultra-slow convergence to a Gaussian: the truncated Levy ﬂight. Phys Rev Lett 1994;73:2946–2949. 20. Peng CK, Mietus J, Hausdorff JM, Havlin S, Stanley HE, Goldberger AL. Longrange anticorrelations and non-Gaussian behavior of the heartbeat. Phys Rev Lett 1993;70:1343–1346. 21. Peng CK, Buldyrev SV, Havlin S, Simons M, Stanley HE, Goldberger AL. Mosaic organization of DNA nucleotides. Phys Rev E 1994;49:1685–1689. 22. Levy P. Calcul des probabilités. Paris: Gauthier-Villars; 1925. 23. Khintchine AYa, Levy P. Sur les lois stables. C R Acad Sci Paris 1936;202:374–376. 24. Koponen I. Analytic approach to the problem of convergence of truncated Levy ﬂights towards the Gaussian stochastic process. Phys Rev E 1995;52:1197–1199. 25. Podobnik B, Ivanov PCh, Lee Y, Stanley HE.

Stable non-Gaussian random processes: stochastic models with inﬁnite variance. New York: Chapman and Hall; 1994. 6. Levy P. Calcul des probabilités. Paris: Gauthier-Villars; 1925. 7. Khintchine AYa, Levy P. Sur les lois stables. C R Acad Sci Paris;1936;202:374. 8. Mantegna RN, Stanley HE. Stochastic process with ultra-slow convergence to a Gaussian: the truncated Levy ﬂight. Phys Rev Lett;1994;73:2946– 2949. 9. Koponen I. Analytic approach to the problem of convergence of truncated Levy ﬂights towards the Gaussian stochastic process. Phys Rev E;1995;52:1197–1199. 10. Weron R. Levy-stable distributions revisited: tail index> 2 does not exclude the Levy-stable regime. Int J Mod Phys C; 2001;12:209–223. Chapter Thirteen Solutions to Integro-Differential Parabolic Problem Arising on Financial Mathematics MARIA C.

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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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In particular, since our process is Markovian (cf. Sections 7.1 and 7.2), neither does S(T) depend on earlier values of S at previous times t. With this respect, the geometric Wiener process under Q, of Eq. 8.16, using the risk neutral probability measure, is called a semimartingale, That is, a variant of a “martingale”. A martingale is a Markovian (memory-less) stochastic process such as, at t, the conditional expected value of St+1 is St. In our case, we talk of a semimartingale, that is, a martingale completed by a finite variation, of the ert form here. Indeed, in our case, E(St + 1|St, …, S1) does not equal St, but the forward value Ft + 1Q = St er(t + 1). These notions will play a major role in the option pricing theory, see Chapter 10, in particular Section 10.2.4. Finally, let us come back to the relationship 8.17, valuing a forward or future under Q, the risk neutral probability measure: as a consequence, the geometric general Wiener process (Eq. 8.16) under Q, applied to a forward or a future, comes down to (8.18) ANNEX 8.1: PROOFS OF THE PROPERTIES OF dZ(t) (see Section 2) These proofs are given for information purpose only, the calculations bringing no useful concept or more insight about stochastic calculus.

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

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When µ = 0, pt cannot be forecast beyond its most recent value: E[pt+1 | pt , pt−1 , . . .] = pt . A process with this property is generally described as a martingale. One definition of a martingale is a discrete stochastic process {xt } where E|xt | < ∞ for all t, and E(xt+1 | xt , xt−1 , . . . ) = xt (see Karlin and Taylor (1975) or Ross (1996)). Martingale behavior of asset prices is a classic result arising in many economic models with individual optimization, absence of arbitrage, or security market equilibrium (Cochrane (2005)). The result is generally contingent, however, on assumptions of frictionless trading opportunities, which are not appropriate in most microstructure applications. The martingale nevertheless retains a prominent role. To develop this idea, note that expectations in the last paragraph are conditioned on lagged pt or xt , that is, the history of the process.

One of the basic goals of microstructure analysis is a detailed and realistic view of how informational efficiency arises, that is, the process by which new information comes to be impounded or reflected in prices. In microstructure analyses, transaction prices are usually not martingales. Sometimes it is not even the case that the public information includes the history of transaction prices. (In dealer markets, trades are often not reported.) By imposing economic or statistical structure, though, it is often possible to identify a martingale component of the price (with respect to a particular information set). Later chapters will indicate how this can be accomplished. A random-walk is a process constructed as the sum of independently and identically distributed (i.i.d.) zero-mean random variables (Ross (1996), p. 328). It is a special case of a martingale. The price in Equation 3.1, for example, cumulates the ut . Because the ut are i.i.d., the price process is time-homogenous, that is, it exhibits the same behavior whenever in time we sample it.

To develop this idea, note that expectations in the last paragraph are conditioned on lagged pt or xt , that is, the history of the process. A more general definition involves conditioning on broader information sets. The process {xt } is a martingale with respect to another (possibly multidimensional) process {zt } if E|xt | < ∞ for all t and E(xt+1 | zt , zt−1 , . . .) = xt (Karlin and Taylor (1975), definition 1.2, p. 241). In particular, suppose that at some terminal time the cash value or payoff of a security is a random variable v . Traders form a sequence of beliefs based on a sequence of information sets 1 , 2 , . . . This sequence does not contract: Something known at time t is known at time τ > t. Then the conditional expectation 25 26 EMPIRICAL MARKET MICROSTRUCTURE xt = E[v |t ] is a martingale with respect to the sequence of information sets {k }. When the conditioning information is “all public information,” the conditional expectation is sometimes called the fundamental value or (with a nod to the asset pricing literature) the efficient price of the security.

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Frequently Asked Questions in Quantitative Finance by Paul Wilmott

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A 269 383-386 Hagan, P, Kumar, D, Lesniewski, A & Woodward, D 2002 Managing smile risk. Wilmott magazine, September Halton, JH 1960 On the efficiency of certain quasi-random sequences of points in evaluating multi-dimensional integrals. Num. Maths. 2 84-90 Hammersley, JM & Handscomb, DC 1964 Monte Carlo Methods. Methuen, London Harrison, JM & Kreps, D 1979 Martingales and arbitrage in multiperiod securities markets. Journal of Economic Theory 20 381-408 Harrison, JM & Pliska, SR 1981 Martingales and stochastic integrals in the theory of continuous trading. Stochastic Processes and their Applications 11 215-260 Haselgrove, CB 1961 A method for numerical integration. Mathematics of Computation 15 323-337 Heath, D, Jarrow, R & Morton, A 1992 Bond pricing and the term structure of interest rates: a new methodology. Econometrica 60 77-105 Ho, T & Lee, S 1986 Term structure movements and pricing interest rate contingent claims.

Scientist 23 18-40 Black, F & Scholes, M 1973 The pricing of options and corporate liabilities. Journal of Political Economy 81 637-59 Cox, J & Rubinstein, M 1985 Options Markets. Prentice-Hall Derman, E & Kani, I 1994 Riding on a smile. Risk magazine 7 (2) 32-39 Dupire, B 1994 Pricing with a smile. Risk magazine 7 (1) 18-20 Harrison, JM & Kreps, D 1979 Martingales and arbitrage in multiperiod securities markets. Journal of Economic Theory 20 381-408 Harrison, JM & Pliska, SR 1981 Martingales and stochastic integrals in the theory of continuous trading. Stochastic Processes and their Applications 11 215-260 Joshi, M 2003 The Concepts and Practice of Mathematical Finance. CUP Rubinstein, M 1976 The valuation of uncertain income streams and the pricing of options. Bell J. Econ. 7 407-425 Rubinstein, M 1994 Implied binomial trees. Journal of Finance 69 771-818 Wilmott, P 2006 Paul Wilmott On Quantitative Finance, second edition.

Because zero probability sets don’t change, a portfolio is an arbitrage under one measure if and only if it is one under all equivalent measures. Therefore a price is non-arbitrageable in the real world if and only if it is non-arbitrageable in the risk-neutral world. The risk-neutral price is always non-arbitrageable. If everything has a discounted asset price process which is a martingale then there can be no arbitrage. So if we change to a measure in which all the fundamental assets, for example the stock and bond, are martingales after discounting, and then define the option price to be the discounted expectation making it into a martingale too, we have that everything is a martingale in the risk-neutral world. Therefore there is no arbitrage in the real world. Explanation 4: If we have calls with a continuous distribution of strikes from zero to infinity then we can synthesize arbitrarily well any payoff with the same expiration.

Monte Carlo Simulation and Finance by Don L. McLeish

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Then the process Zs is the analogue of the Radon-Nikodym derivative dQ dP of the processes restricted to the time interval 0 · t · s. For a more formal definition, as well as an explanation of how we should interpret the integral, see the appendix. This process Zs is, both in discrete and continuous time, a martingale. MODELS IN CONTINUOUS TIME 67 Wiener Process 3 2.5 2 W(t) 1.5 1 0.5 0 -0.5 -1 0 1 2 3 4 5 t 6 7 8 9 Figure 2.6: A sample path of the Wiener process Models in Continuous Time We begin with some oversimplified rules of stochastic calculus which can be omitted by those with a background in Brownian motion and diﬀusion. First, we define a stochastic process Wt called the standard Brownian motion or Wiener process having the following properties; 1. For each h > 0, the increment W (t+h)−W (t) has a N (0, h) distribution and is independent of all preceding increments W (u) − W (v), t > u > v > 0. 2.

Under the risk-neutral measure, the discounted price Yt = St /Bt forms a martingale. A martingale is a process Yt for which the expectation of a future value given the present is equal to the present i.e. E(Yt+1 |Ht ) = Yt .for all t. (2.7) Properties of a martingale are given in the appendix and it is easy to show that for such a process, when T > t, E(YT |Ht ) = E[...E[E(YT |HT −1 )|HT −2 ]...|Ht ] = Yt . (2.8) A martingale is a fair game in a world with no inflation, no need to consume and no mortality. Your future fortune if you play the game is a random variable whose expectation, given everything you know at present, is your present fortune. Thus, under a risk-neutral measure Q in a complete market, all marketable securities discounted to the present form martingales. For this reason, we often refer to the risk-neutral measure as a martingale measure.

Now let us return to the constraints on the vector of stock prices. In order that the discounted stock price forms a martingale under the Q measure, we require that EQ [S(t + 1)|Ht ] = (1 + r(t))S(t). This is achieved if we define Q such that for any event A ∈ Ht , Q(A) = Z Zt dP where A Zs = kt exp( s X ηt0 (St+1 − St )) (2.22) t=1 where kt are Ht measurable random variables chosen so that Zt forms a martingale E(Zt+1 |Ht ) = Zt . 66 CHAPTER 2. SOME BASIC THEORY OF FINANCE Theorem 9 shows that this exponentially tilted distribution has the property of being the closest to the original measure P while satisfying the condition that the normalized sequence of stock prices forms a martingale. There is a considerable literature exploring the links between entropy and risk-neutral valuation of derivatives.

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

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The proof of the Fundamental Theorem of Asset Pricing is quite technical and will be omitted. Deﬁnition 4.5 A sequence of random variables X(0), X(1), X(2), . . . such that E∗ (X(n + 1)|S(n)) = X(n) for each n = 0, 1, 2, . . . is said to be a martingale with respect to P∗ . Condition (4.3) can be expressed by saying that the discounted stock prices S j (0), S j (1), S j (2), . . . form a martingale with respect to P∗ . The latter is called a risk-neutral or martingale probability on the set of scenarios Ω. Moreover, E∗ is called a risk-neutral or martingale expectation. 84 Mathematics for Finance Example 4.5 Let A(0) = 100, A(1) = 110, A(2) = 121 and suppose that stock prices can follow four possible scenarios: Scenario ω1 ω2 ω3 ω4 S(0) 90 90 90 90 S(1) 100 100 80 80 S(2) 112 106 90 80 The tree of stock prices is shown in Figure 4.2.

This results in the following bond prices at time 1: 101.14531 in the up state and 100.9999 in the down state. (The latter is the same as for the par bond.) Expectation with respect to the risk-neutral probability gives the initial bond price 100.05489, so the ﬂoor is worth 0.05489. Bibliography Background Reading: Probability and Stochastic Processes Ash, R. B. (1970), Basic Probability Theory, John Wiley & Sons, New York. Brzeźniak, Z. and Zastawniak, T. (1999), Basic Stochastic Processes, Springer Undergraduate Mathematics Series, Springer-Verlag, London. Capiński, M. and Kopp, P. E. (1999), Measure, Integral and Probability, Springer Undergraduate Mathematics Series, Springer-Verlag, London. Capiński, M. and Zastawniak, T. (2001), Probability Through Problems, Springer-Verlag, New York. Chung, K. L. (1974), A Course in Probability Theory, Academic Press, New York.

But p∗ (1 + u) + (1 − p∗ )(1 + d) = 1 + r by (3.4), which implies that E∗ (S(n + 1)|S(n) = x) = x(1 + r) for any possible value x of S(n), completing the proof. Dividing both sides of the equality in Proposition 3.5 by (1 + r)n+1 , we obtain the following important result for the discounted stock prices S(n) = −n S(n) (1 + r) . Corollary 3.6 (Martingale Property) For any n = 0, 1, 2, . . . + 1)|S(n)) = S(n). E∗ (S(n We say that the discounted stock prices S(n) form a martingale under the risk-neutral probability p∗ . The probability p∗ itself is also referred to as the martingale probability. Exercise 3.19 Let r = 0.2. Find the risk-neutral conditional expectation of S(3) given that S(2) = 110 dollars. 3.3 Other Models This section may be skipped at ﬁrst reading because the main ideas to follow later do not depend on the models presented here. 64 Mathematics for Finance 3.3.1 Trinomial Tree Model A natural generalisation of the binomial tree model extends the range of possible values of the one-step returns K(n) to three.

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

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Kurtosis – a statistical measure of the distribution of observations about the expected mean as a deviation from that predicted by the normal distribution. Life cycle – the characterization of a process from its birth to death. Life Cycle Model – a model of household consumption behavior from the beginning of its earning capacity to the end of the household. Markov process – a stochastic process with the memorylessness property for which the present state, future state, and past observations are independent. Markowitz bullet – the upper boundary of the efficient frontier of various portfolios when graphed according to risk and return. Martingale – a model of a process for which past events cannot predict future outcomes. Mean – a mathematical technique that can be calculated based on a number of alternative weightings to produce an average for a set of numbers. MIT School – an approach to economic and financial studies that favors dynamic (time-variant) modeling and simple, elegant, but predictively powerful theories.

Black saw the description and prediction of interest rates to be a multi-faceted and challenging problem. While he had demonstrated that an options price depends on the underlying stock price mean and volatility, and the risk-free interest rate, the overall market for interest rates is much more multi-dimensional. The interest rate yield curve, which graphs rates against maturities, depends on many markets and instruments, each of which is subject to stochastic processes. His interest and collaboration with Emanuel Derman and Bill Toy resulted in a model of interest rates that was first used profitably by Goldman Sachs through the 1980s, but eventually entered the public domain when they published their work in the Financial Analysts Journal in 1990.2 Their model provided reasonable estimates for both the prices and volatilities of treasury bonds, and is still used today.

Black-Scholes model – a model that can determine the price of a European call option based on the assumption that the underlying security follows a geometric Brownian motion with constant drift and volatility. Bond – a financial instrument that provides periodic (typically semi-annual) interest payments and the return of the paid-in capital upon maturity in exchange for a fixed price. Brownian motion – the simplest of the class of continuous-time stochastic processes that describes the random motion of a particle or a security that is buffeted by forces that are normally distributed in strength. Calculus of variations – a mathematical technique that can determine the optimal path of a variable, like savings or consumption, over time. Call – an option to purchase a specified security at a specified future time and price. Capital allocation line – a line drawn on the graph of all possible combinations of risky and risk-free assets that shows the best risk–reward horizon.

Analysis of Financial Time Series by Ruey S. Tsay

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ISBN: 0-471-41544-8 CHAPTER 6 Continuous-Time Models and Their Applications Price of a financial asset evolves over time and forms a stochastic process, which is a statistical term used to describe the evolution of a random variable over time. The observed prices are a realization of the underlying stochastic process. The theory of stochastic process is the basis on which the observed prices are analyzed and statistical inference is made. 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.

For more description on options, see Hull (1997). 6.2 SOME CONTINUOUS-TIME STOCHASTIC PROCESSES In mathematical statistics, a continuous-time continuous stochastic process is defined on a probability space (, F, P), where is a nonempty space, F is a σ -field consisting of subsets of , and P is a probability measure; see Chapter 1 of Billingsley (1986). The process can be written as {x(η, t)}, where t denotes time and is continuous in [0, ∞). For a given t, x(η, t) is a real-valued continuous random variable (i.e., a mapping from to the real line), and η is an element of . For the price of an asset at time t, the range of x(η, t) is the set of non-negative real numbers. 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.

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. This means that the rate of change in expectation is zero and the rate of change in variance is 1. In practice, the mean and variance of a stochastic process can evolve over time in a more complicated manner. Hence, further generalization of stochastic process is needed. To this end, we consider the generalized Wiener process in which the expectation has a drift rate µ and the rate of variance change is σ 2 . Denote such a process by xt and use the notation dy for a small change in the variable y.

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

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Market Efficiency Tests Based on the Martingale Hypothesis A classic definition of market efficiency in terms of security returns is due to Market Ineﬃciency and Proﬁt Opportunities at Diﬀerent Frequencies 87 Samuelson (1965), who showed that properly anticipated prices fluctuate randomly in an efficient market. In other words, if all of the news is incorporated instantaneously into the price of a given financial security, the expected price of the security given current information is always the current price of the security itself. This relationship is known as a martingale. Formally, a stochastic price process {Pt } is a martingale within information set It if the best forecast of Pt+1 based on current information It is equal to Pt : E[Pt+1 |It ] = Pt (7.13) Applying the martingale hypothesis to changes in price levels, we can express “abnormal,” or returns in excess of expected returns given current information, as follows: Zt+1 = Pt+1 − E[Pt+1 |It ] (7.14) A market in a particular financial security or a portfolio of financial securities is then said to be efficient when abnormal return Zt+1 is a “fair game”—that is, E[Zt+1 |It ] = 0 (7.15) LeRoy (1989) provides an extensive summary of the literature on the subject.

In the Garman (1976) model, the market has one monopolistic market maker (dealer). The market maker is responsible for deciding on and then setting bid and ask prices, receiving all orders, and clearing trades. The market maker’s objective is to maximize profits while avoiding bankruptcy or failure. The latter arise whenever the market maker has no inventory or cash. Both buy and sell orders arrive as independent stochastic processes. The model solution for optimal bid and ask prices lies in the estimation of the rates at which a unit of cash (e.g., a dollar or a “clip” of 10 million in FX) “arrives” to the market maker when a customer comes in to buy securities (pays money to the dealer) and “departs” the market maker when a customer comes in to sell (the dealer pays the customer). Suppose the probability of an arrival, a customer order to buy a security at the market ask price pa is denoted λa .

Fama (1991) also suggested that the efficient markets hypothesis is difficult to test for the following reason: the idea of a market fully reflecting all available information contains a joint hypothesis. On the one hand, expected values of returns are a function of information. On the other hand, differences of realized returns from their expected values are random. Incorporating both issues in the same test is difficult. Nevertheless, martingale-based tests for market efficiencies exist. Froot and Thaler (1990), for example, derive a specification for a test of market efficiency of a foreign exchange rate. In equilibrium, foreign exchange markets follow the uncovered interest rate parity hypothesis that formulates the price of a foreign exchange rate as a function of interest rates in countries on either side of the interest rate. Under the uncovered interest rate parity, an expected change in the equilibrium spot foreign exchange rate S, given that the information set It is a function of the interest 88 HIGH-FREQUENCY TRADING rate differential between domestic and foreign interest rates, rt − rtd and risk premium ξt of the exchange rate: E[St+1 |It ] = rt − rtd + ξt (7.16) where the risk premium ξt is zero for risk-neutral investors and is diversifiable to zero for others.

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

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Like many mathematicians and physicists, I found the mathematics of the Black-Scholes options pricing formula incredibly interesting. For starters, after years of specializing in pure mathematics, I was starting from scratch in a totally new area. It allowed me to start to learn basic mathematics instead of delving deeper and deeper into advanced subjects. I literally had to start from scratch and learn probability theory and then the basics of stochastic processes, things I knew nothing at all about. Not to mention I knew nothing about financial markets, derivatives, or JWPR007-Lindsey 122 May 7, 2007 16:55 h ow i b e cam e a quant anything at all to do with finance. It was exciting to learn so much from scratch. In the midst of reading about Black-Scholes, I was also deeply involved with writing the book with Victor Ginzburg from the University of Chicago.

Richard Grinold, who was my prethesis advisor, gave me a copy of the HJM paper a couple of weeks before the seminar and told me to dig into it. This represents some of the best academic advice I have ever received since I am not sure that I would have immediately realized the model’s importance and potential for further work by myself. The rest, in some sense, is history. I really enjoyed the paper because I was struggling to understand some of the rather abstract questions in stochastic process theory that it dealt with, and I quickly decided to work on the HJM model for my dissertation. Broadly speaking, the HJM paradigm still represents the state of the art in interest rate derivatives pricing, so having been working with it from the very beginning is definitely high on my list of success factors later in life. In my five years at Berkeley, I met a few other people of critical importance to my career path, and life in general.

At Columbia College, I decided to enroll in its three-two program, which meant that I spent three years studying the contemporary civilization and humanities core curriculum, as well as the hard sciences, and then two years at the Columbia School of Engineering. There, I found a home in operations research, which allowed me to study computer science and applied mathematics, including differential equations, stochastic processes, statistical quality control, and mathematical programming. While studying for my master’s in operations research at Columbia, I had the opportunity to work at the Rand Institute, where math and computer science were applied to real-world problems. There I was involved in developing a large-scale simulation model designed to optimize response times for the New York City Fire Department. My interest in applied math led me to Carnegie-Mellon’s Graduate School of Industrial Administration, which had a strong operations research faculty.

pages: 338 words: 106,936

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

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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

The Code-Breakers: The Comprehensive History of Secret Communication From Ancient Times to the Internet. New York: Scribner. Kaplan, Ian. 2002. “The Predictors by Thomas A. Bass: A Retrospective.” This is a comment on The Predictors by a former employee of the Prediction Company. Available at http://www.bearcave.com/bookrev/predictors2.html. Karlin, Samuel, and Howard M. Taylor. 1975. A First Course in Stochastic Processes. 2nd ed. San Diego, CA: Academic Press. — — — . 1981. A Second Course in Stochastic Processes. San Diego, CA: Academic Press. Katzmann, Robert A. 2008. Daniel Patrick Moynihan: The Intellectual in Public Life. Washington, DC: Woodrow Wilson Center Press. Kelly, J., Jr. 1956. “A New Interpretation of Information Rate.” IRE Transactions on Information Theory 2 (3, September): 185–89. Kelly, Kevin. 1994a. “Cracking Wall Street.”

“Consumer Prices, the Consumer Price Index, and the Cost of Living.” Journal of Economic Perspectives 12 (1, Winter): 3–26. Bosworth, Barry P. 1997. “The Politics of Immaculate Conception.” The Brookings Review, June, 43–44. Bouchaud, Jean-Philippe, and Didier Sornette. 1994. “The Black-Scholes Option Pricing Problem in Mathematical Finance: Generalization and Extensions for a Large Class of Stochastic Processes.” Journal de Physique 4 (6): 863–81. Bower, Tom. 1984. Klaus Barbie, Butcher of Lyons. London: M. Joseph. Bowman, D. D., G. Ouillion, C. G. Sammis, A. Sornette, and D. Sornette. 1998. “An Observational Test of the Critical Earthquake Concept.” Journal of Geophysical Research 103: 24359–72. Broad, William J. 1992. “Defining the New Plowshares Those Old Swords Will Make.” The New York Times, February 5.

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

Cardano offered some very wise advice on speculation that we would all do well to follow, even today: “The most fundamental principle of all in gambling is simply equal conditions, e.g., of opponents, of bystanders, of money, of situation, of the dice box, and of the die itself. To the extent to which you depart from that equality, if it is in your opponent’s favour, you are a fool, and if in your own, you are unjust.”11 This notion of a “fair game”—one that doesn’t favor you or your opponent—came to be known as a martingale.12 Few of us want to be unjust, and no one wants to be a fool. The martingale is a very subtle idea, at the heart of many concepts in mathematics and physics, but the important takeaway here is surprisingly simple. In a fair game, your winnings or losses can’t be forecast by looking at your past performance. If they could, then the game isn’t fair, because you could increase your bet when the forecast is positive, and decrease your bet when it’s negative.

Let’s return to Cardano’s fair game, the martingale. The game could be something as simple as a coin flip. In a fair game, past performance is no guarantee of future outcomes. After each turn, you’ll either win some money (heads) or lose some money (tails). Now imagine playing Are We All Homo economicus Now? • 19 this fair game repeatedly, but with a twist. Visualize your winnings and losses physically by taking a step forward or backward with every flip of the coin. (You might need to do this on a sidewalk, or in a hallway.) 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.

In a 1973 article on the mathematical underpinnings of financial speculation, Samuelson included a wonderful tribute to Bachelier: Notes to Chapter 1 • 423 Since illustrious French geometers almost never die, it is possible that Bachelier still survives in Paris supplementing his professorial retirement pension by judicious arbitrage in puts and calls. But my widespread lecturing on him over the last 20 years has not elicited any information on the subject. How much Poincaré, to whom he dedicates the thesis, contributed to it, I have no knowledge. Finally, as Bachelier’s cited works suggest, he seems to have had something of a one-track mind. But what a track! 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.

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Algorithms to Live By: The Computer Science of Human Decisions by Brian Christian, Tom Griffiths

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Like the famous Heisenberg uncertainty principle of particle physics, which says that the more you know about a particle’s momentum the less you know about its position, the so-called bias-variance tradeoff expresses a deep and fundamental bound on how good a model can be—on what it’s possible to know and to predict. This notion is found in various places in the machine-learning literature. See, for instance, Geman, Bienenstock, and Doursat, “Neural Networks and the Bias/Variance Dilemma,” and Grenander, “On Empirical Spectral Analysis of Stochastic Processes.” in the Book of Kings: The bronze snake, known as Nehushtan, gets destroyed in 2 Kings 18:4. “pay good money to remove the tattoos”: Gilbert, Stumbling on Happiness. duels less than fifty years ago: If you’re not too fainthearted, you can watch video of a duel fought in 1967 at http://passerelle-production.u-bourgogne.fr/web/atip_insulte/Video/archive_duel_france.swf. as athletes overfit their tactics: For an interesting example of very deliberately overfitting fencing, see Harmenberg, Epee 2.0.

Nature 363 (1993): 315–319. Gould, Stephen Jay. “The Median Isn’t the Message.” Discover 6, no. 6 (1985): 40–42. Graham, Ronald L., Eugene L. Lawler, Jan Karel Lenstra, and Alexander H. G. Rinnooy Kan. “Optimization and Approximation in Deterministic Sequencing and Scheduling: A Survey.” Annals of Discrete Mathematics 5 (1979): 287–326. Grenander, Ulf. “On Empirical Spectral Analysis of Stochastic Processes.” Arkiv för Matematik 1, no. 6 (1952): 503–531. Gridgeman, T. “Geometric Probability and the Number π.” Scripta Mathematika 25, no. 3 (1960): 183–195. Griffiths, Thomas L., Charles Kemp, and Joshua B. Tenenbaum. “Bayesian Models of Cognition.” In The Cambridge Handbook of Computational Cognitive Modeling. Edited by Ron Sun. Cambridge, UK: Cambridge University Press, 2008. Griffiths, Thomas L., Falk Lieder, and Noah D.

Peter Todd, a cognitive scientist at Indiana University, has explored this complexity (and how to simplify it) in detail. See Todd and Miller, “From Pride and Prejudice to Persuasion Satisficing in Mate Search,” and Todd, “Coevolved Cognitive Mechanisms in Mate Search.” Selling a house is similar: The house-selling problem is analyzed in Sakaguchi, “Dynamic Programming of Some Sequential Sampling Design”; Chow and Robbins, “A Martingale System Theorem and Applications”; and Chow and Robbins, “On Optimal Stopping Rules.” We focus on the case where there are potentially infinitely many offers, but these authors also provide optimal strategies when the number of potential offers is known and finite (which are less conservative—you should have a lower threshold if you only have finitely many opportunities). In the infinite case, you should set a threshold based on the expected value of waiting for another offer, and take the first offer that exceeds that threshold.

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

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Paul Cootner, one of the leading finance scholars of the 1960s, once delivered this accolade: “So outstanding is his work that we can say that the study of speculative prices has its moment of glory at its moment of conception.”1 Bachelier laid the groundwork on which later mathematicians constructed a full-fledged theory of probability. He derived a formula that anticipated Einstein’s research into the behavior of particles subject to random shocks in space. And he developed the now universally used concept of stochastic processes, the analysis of random movements among statistical variables. Moreover, he made the first theoretical attempt to value such financial instruments as options and futures, which had active markets even in 1900. And he did all this in an effort to explain why prices in capital markets are impossible to predict! Bachelier’s opening paragraphs contain observations about “fluctuations on the Exchange” that could have been written today.

(LOR) Leland-Rubinstein Associates Leverage Leveraged buyouts Liquidity management market money Preference theory stock “Liquidity Preference as Behavior Toward Risk” (Tobin) Linear programming Loading charges: see Brokerage commissions London School of Economics (LSE) London Stock Exchange Macroeconomics Management Science Marginal utility concept “Market and Industry Factors in Stock Price Performance” (King) Market theories (general discussion). See also specific theories and types of securities competitive disaster avoidance invisible hand linear regression/econometric seasonal fluctuations stochastic process Mathematical economics Mathematical Theory of Non-Uniform Gases, The Maximum expected return concept McCormick Harvester Mean-Variance Analysis Mean-Variance Analysis in Portfolio Choice and Capital Markets (Markowitz) “Measuring the Investment Performance of Pension Funds,” report Mellon Bank Merck Merrill Lynch Minnesota Mining MIT MM Theory “Modern Portfolio Theory. How the New Investment Technology Evolved” Money Managers, The (“Adam Smith”) Money market funds Mortgages government-guaranteed prepaid rates on “‘Motionless’ Motion of Swift’s Flying Island, The” (Merton) Multiple manager risk analysis (MULMAN) Mutual funds individual investment in performance analysis of portfolio management and Value Line National Bureau of Economic Research National General Naval Research Logistics Quarterly New School for Social Research New York Stock Exchange volume of trading New York Times averages “Noise” (Black) Noise trading asset prices and inefficiency of October, 1987, crash OPEC countries Operations Research Optimal capital structure Optimal investment strategy: see Diversification; Portfolio(s), optimal “Optimization of a Quadratic Function Subject to Linear Constraints, The” (Markowitz) Optimization theory Options call contracts expected return on implicit out-of-the-money/in-the-money pricing formulas put valuation Options markets over-the-counter Pacific Stock Exchange Paul A.

“Diversification of Planning.” Trusts and Estates, Vol. 80 (May), pp. 469–473. Leibowitz, Martin. 1990. Speech in Honor of William F. Sharpe. October 17. Leland, Hayne E. and Mark Rubinstein. 1988. “The Evolution of Portfolio Insurance.” Portfolio Insurance: A Guide to Dynamic Hedging, Donald Luskin, ed. New York: John Wiley & Sons, pp. 3–10. Leroy, Stephen F. 1989. “Efficient Capital Markets and Martingales.” Journal of Economic Literature, Vol. XXVII (December), pp. 1583–1621. This article contains an extensive bibliography. Lichtenberg, Frank. 1990. “Industrial Diversification and Its Consequences for Productivity.” Cambridge, MA: National Bureau of Economic Research, Working Paper #3231. Lichtenberg, Frank and Donald Siegel. 1989. “The Effects of Leveraged Buyouts on Productivity and Related Aspects of Firm Behavior.”