# stochastic process

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Mathematical Finance: Theory, Modeling, Implementation by Christian Fries

.: Conditional Expectation: Let the σ-algebra C be generated by the sets C1 = {ω1 , ω2 , ω3 }, C2 = {ω4 , ω5 , ω6 }, C3 = {ω7 , . . . , ω10 }. In this sense C may be interpreted as an information set and X|C as a filtered version of X. If it is only possible to make statements upon events in C then one may only make statements about X which could also be made about X|C . C| 2.2. Stochastic Processes q Definition 17 (Stochastic Process): A family X = {Xt | 0 ≤ t < ∞} of random variables Xt : (Ω, F ) → (S , S) is called (time continuous) stochastic process. If (S , S) = (Rd , B(Rd )), we say that X is a d-dimensional stochastic process. The family X may also be interpreted as a X : [0, ∞) × Ω → S : X(t, ω) := Xt (ω) ∀ (t, ω) ∈ [0, ∞) × Ω. y If the range (S , S) is not given explicitly we assume (S , S) = 35 This work is licensed under a Creative Commons License. http://creativecommons.org/licenses/by-nc-nd/2.5/deed.en (Rd , B(Rd )).

Random Variables: Z • Z(ω) dP(ω) – Lebesgue integral. Integral of a random variable Z with respect to a measure P (cf. expectation). Stochastic Processes: Z • X(t1 , ω) dP(ω) – Lebesgue integral. Integral of a random variable X(t1 ) with respect to a measure P. X(t) dt – Lebesgue Integral or Riemann integral. The (pathwise) integral of the stochastic process X with respect to t. X(t) dW(t) – Itô integral. The (pathwise) integral of the stochastic process X with respect to a Brownian motion W. t1 Ω Ω t2 Z • t1 Z • t2 t1 Z Ω X(t,ω) X(t1 , ω) dP(ω) ω1 0 t1 T Z T 0 X(t)dW (t)[ω1 ] t Figure 2.10.: Integration of stochastic processes The notion of a stochastic integral may be extended to more general integrands and/or more general integrators. For completeness we mention: 57 This work is licensed under a Creative Commons License. http://creativecommons.org/licenses/by-nc-nd/2.5/deed.en Comments welcome. ©2004, 2005, 2006 Christian Fries Version 1.3.19 [build 20061210]- 13th December 2006 http://www.christian-fries.de/finmath/ CHAPTER 2.

A Family of σ-algebras {Ft | t ≥ 0}, where F s ⊆ Ft ⊆ F q for 0 ≤ s ≤ t, is called a Filtration on (Ω, F ). y Definition 20 (Generated Filtration): Let X denote a stochastic process on (Ω, F ). We define q FtX := σ(X s ; 0 ≤ s ≤ t) := the smallest σ-algebra with respect to which X s is measurable ∀ s ∈ [0, t]. y Definition 21 (Adapted Process): q Let X denote a stochastic process on (Ω, F ) and {Ft } a filtration on (Ω, F ). The process X is called {Ft }-adapted, if Xt is Ft -measurable for all t ≥ 0. y Interpretation: In Figure 2.4 we depict a filtration of four σ-algebras with increasing refinement (left to right). The black borders surround the generators of the corresponding σ-algebra. If a stochastic process maps a gray value to each elementary event (or path) ωi of Ω (left), then we have 36 This work is licensed under a Creative Commons License. http://creativecommons.org/licenses/by-nc-nd/2.5/deed.en Comments welcome. ©2004, 2005, 2006 Christian Fries Version 1.3.19 [build 20061210]- 13th December 2006 http://www.christian-fries.de/finmath/ 2.4.

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

This is a distribution symmetrical with respect to 0, which corresponds to a normal distribution for n = 2 and gives rise to a leptokurtic distribution (resp. negative kurtosis distribution) for n < 2 (n > 2). 2.3 STOCHASTIC PROCESSES 2.3.1 General considerations The term stochastic process is applied to a random variable that is a function of the time variable: {Xt : t ∈ T }. 354 Asset and Risk Management f (x) v=1 v=2 v=3 x 0 Figure A2.15 Generalised error distribution If the set T of times is discrete, the stochastic process is simply a sequence of random variables. However, in a number of ﬁnancial applications such as Black and Scholes’ model, it will be necessary to consider stochastic processes in continuous time. For each possible result ω ∈ , the function of Xt (ω) of the variable t is known as the path of the stochastic process. A stochastic process is said to have independent increments when, regardless of the times t1 < t2 < . . . < tn , the r.v.s Xt1 , Xt2 − Xt1 , Xt3 − Xt2 , . . . are independent.

It is, however, possible to extend the deﬁnition to a concept of stochastic differential, through the theory of stochastic integral calculus.8 As the stochastic process zt is deﬁned within the interval [a; b], the stochastic integral of zt is deﬁned within [a; b] with respect to the standard Brownian motion wt by: a 7 8 b zt dwt = lim n→∞ δ→0 n−1 ztk (wtk+1 − wtk ) k=0 The root function presents a vertical tangent at the origin. The full development of this theory is outside the scope of this work. Probabilistic Concepts where, we have: 357 a = t0 < t1 < . . . < tn = b δ = max (tk − tk−1 ) k=1,...,n Let us now consider a stochastic process Zt (for which we wish to deﬁne the stochastic differential)and a standard Brownian motion wt . If there is a stochastic process zt such that t Zt = Z0 + 0 zs dws , then it is said that Zt admits the stochastic differential dZt = zt dwt .

8.1.2 The data in the example 8.2 Calculations 8.2.1 Treasury portfolio case 8.2.2 Bond portfolio case 8.3 The normality hypothesis PART IV FROM RISK MANAGEMENT TO ASSET MANAGEMENT Introduction 9 224 224 230 234 235 238 241 243 243 243 244 244 244 250 252 255 256 Portfolio Risk Management 9.1 General principles 9.2 Portfolio risk management method 9.2.1 Investment strategy 9.2.2 Risk framework 257 257 257 258 258 10 Optimising the Global Portfolio via VaR 10.1 Taking account of VaR in Sharpe’s simple index method 10.1.1 The problem of minimisation 10.1.2 Adapting the critical line algorithm to VaR 10.1.3 Comparison of the two methods 10.2 Taking account of VaR in the EGP method 10.2.1 Maximising the risk premium 10.2.2 Adapting the EGP method algorithm to VaR 10.2.3 Comparison of the two methods 10.2.4 Conclusion 10.3 Optimising a global portfolio via VaR 10.3.1 Generalisation of the asset model 10.3.2 Construction of an optimal global portfolio 10.3.3 Method of optimisation of global portfolio 265 266 266 267 269 269 269 270 271 272 274 275 277 278 11 Institutional Management: APT Applied to Investment Funds 11.1 Absolute global risk 11.2 Relative global risk/tracking error 11.3 Relative fund risk vs. benchmark abacus 11.4 Allocation of systematic risk 285 285 285 287 288 x Contents 2.2 Theoretical distributions 2.2.1 Normal distribution and associated ones 2.2.2 Other theoretical distributions 2.3 Stochastic processes 2.3.1 General considerations 2.3.2 Particular stochastic processes 2.3.3 Stochastic differential equations 347 347 350 353 353 354 356 Appendix 3 Statistical Concepts 3.1 Inferential statistics 3.1.1 Sampling 3.1.2 Two problems of inferential statistics 3.2 Regressions 3.2.1 Simple regression 3.2.2 Multiple regression 3.2.3 Nonlinear regression 359 359 359 360 362 362 363 364 Appendix 4 Extreme Value Theory 4.1 Exact result 4.2 Asymptotic results 4.2.1 Extreme value theorem 4.2.2 Attraction domains 4.2.3 Generalisation 365 365 365 365 366 367 Appendix 5 Canonical Correlations 5.1 Geometric presentation of the method 5.2 Search for canonical characters 369 369 369 Appendix 6 371 Algebraic Presentation of Logistic Regression Appendix 7 Time Series Models: ARCH-GARCH and EGARCH 7.1 ARCH-GARCH models 7.2 EGARCH models 373 373 373 Appendix 8 Numerical Methods for Solving Nonlinear Equations 8.1 General principles for iterative methods 8.1.1 Convergence 8.1.2 Order of convergence 8.1.3 Stop criteria 8.2 Principal methods 8.2.1 First order methods 8.2.2 Newton–Raphson method 8.2.3 Bisection method 375 375 375 376 376 377 377 379 380 Contents 8.3 Nonlinear equation systems 8.3.1 General theory of n-dimensional iteration 8.3.2 Principal methods xi 380 381 381 Bibliography 383 Index 389 Collaborators Christian Berbé, Civil engineer from Université libre de Bruxelles and ABAF ﬁnancial analyst.

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Market Risk Analysis, Quantitative Methods in Finance by Carol Alexander

Readers interested in estimating the parameters of a GARCH model when they come to Chapter II.4 will need to understand maximum likelihood estimation. Section I.3.7 shows how to model the evolution of financial asset prices and returns using a stochastic process in both discrete and continuous time. The translation between discrete and continuous time, and the relationship between the continuous time representation and the discrete time representation of a stochastic process, is very important indeed. The theory of finance requires an understanding of both discrete time and continuous time stochastic processes. Section I.3.8 summarizes and concludes. Some prior knowledge of basic calculus and elementary linear algebra is required to understand this chapter. Specifically, an understanding of Sections I.1.3 and I.2.4 is assumed.

But we do 134 Quantitative Methods in Finance not know and so we need to estimate the variance using the maximum likelihood estimator ˆ 2 given by (I.3.132). Then, using ˆ in place of we have ˆ (I.3.135) estse X = √ n and ˆ2 (I.3.136) estse ˆ 2 = √ 2n I.3.7 STOCHASTIC PROCESSES IN DISCRETE AND CONTINUOUS TIME A stochastic process is a sequence of identically distributed random variables. For most of our purposes random variables are continuous, indeed they are often assumed to be normal, but the sequence may be over continuous or discrete time. That is, we consider continuous state processes in both continuous and discrete time. • The study of discrete time stochastic processes is called time series analysis. In the time domain the simplest time series models are based on regression analysis, which is introduced in the next chapter. A simple example of a time series model is the first order autoregression, and this is defined below along with a basic test for stationarity.

I.3.7.3 Stochastic Models for Asset Prices and Returns Time series of asset prices behave quite differently from time series of returns. In efficient markets a time series of prices or log prices will follow a random walk. More generally, even in the presence of market frictions and inefficiencies, prices and log prices of tradable assets are integrated stochastic processes. These are fundamentally different from the associated returns, which are generated by stationary stochastic processes. Figures I.3.28 and I.3.29 illustrate the fact that prices and returns are generated by very different types of stochastic process. Figure I.3.28 shows time series of daily prices (lefthand scale) and log prices (right-hand scale) of the Dow Jones Industrial Average (DJIA) DJIA 12000 9.4 Log DJIA 9.3 11000 9.2 10000 9.1 9000 9 8000 8.9 Sep-01 May-01 Jan-01 Sep-00 May-00 Jan-00 Sep-99 May-99 Jan-99 Sep-98 May-98 8.8 Jan-98 7000 Figure I.3.28 Daily prices and log prices of DJIA index 56 This is not the only possible discretization of a continuous increment.

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Tools for Computational Finance by Rüdiger Seydel

The easiest way to consider stochastic movements is via an additive term, 32 Chapter 1 Modeling Tools for Financial Options dx = a(x, t) + b(x, t)ξt . dt Here we use the notations a: deterministic part, bξt : stochastic part, ξt denotes a generalized stochastic process. An example of a generalized stochastic process is white noise. For a brief deﬁnition of white noise we note that to each stochastic process a generalized version can be assigned [Ar74]. For generalized stochastic processes derivatives of any order can be deﬁned. Suppose that Wt is the generalized version of a Wiener process, then Wt can be diﬀerentiated. Then white noise ξt is d Wt , or vice versa, deﬁned as ξt = Ẇt = dt t Wt = ξs ds. 0 That is, a Wiener process is obtained by smoothing the white noise. The smoother integral version dispenses with using generalized stochastic processes. Hence the integrated form of ẋ = a(x, t) + b(x, t)ξt is studied, t t x(t) = x0 + a(x(s), s)ds + b(x(s), s)ξs ds, t0 t0 and we replace ξs ds = dWs .

Here we consider the continuoustime situation. That is, t ∈ IR varies continuously in a time interval I, which typically represents 0 ≤ t ≤ T . A more complete notation for a stochastic process is {Xt , t ∈ I}, or (Xt )0≤t≤T . Let the chance play for all t in the interval 0 ≤ t ≤ T , then the resulting function Xt is called realization or path of the stochastic process. Special properties of stochastic processes have lead to the following names: Gaussian process: All ﬁnite-dimensional distributions (Xt1 , . . . , Xtk ) are Gaussian. Hence speciﬁcally Xt is distributed normally for all t. Markov process: Only the present value of Xt is relevant for its future motion. That is, the past history is fully reﬂected in the present value.4 An example of a process that is both Gaussian and Markov, is the Wiener process. 4 This assumption together with the assumption of an immediate reaction of the market to arriving informations are called hypothesis of the eﬃcient market [Bo98]. 26 Chapter 1 Modeling Tools for Financial Options 11500 11000 10500 10000 9500 9000 8500 8000 7500 7000 0 50 100 150 200 250 300 350 400 450 500 Fig. 1.14.

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

Analysis of Financial Time Series by Ruey S. Tsay

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

F forward price, or future price (depends on the context) FV future value -ibor generic for LIBOR, EURIBOR, or any other inter-bank market rate K strike price of an option κ kurtosis M month or million, depending on context MD modified duration MtM “Marked to Market” (= valued to the observed current market price) μ drift of a stochastic process N total number of a series (integer number), or nominal (notional) amount (depends on the context) (.) Gaussian (normal) density distribution function N(.) Gaussian (normal) cumulative distribution function P put price P{.} probability of {.} PV present value (.) Poisson density distribution function r generic symbol for a rate of return rf risk-free return ρ(.) correlation of (.) skew skewness S spot price of an asset (equity, currency, etc.), as specified by the context STD(.) standard deviation of (.) σ volatility of a stochastic process t current time, or time in general (depends on the context) t0 initial time T maturity time τ tenor, that is, time interval between current time t and maturity T V(.) variance of (.) (.) stochastic process of (.)

Provided F(x) is continuously differentiable, we can determine the corresponding density function f(x) associated to the random variable X as Stochastic Processes A stochastic process can be defined as a collection of random variables defined on the same probability space (Ω, , P) and “indexed” by a set of parameter T, that is, {Xt, t ∈ T}. Within the framework of our chapter, t is the time. For a given outcome or sample ω, Xt(ω) for t ∈ T is called a sample path, realization or trajectory of the process. The space containing all possible values of Xt is called the state space. Further in this chapter, we will only consider a one-dimension state space, namely the set of real numbers , that refers to T, and random variables Xt involved in stochastic processes {Xt, t ∈ T} will be denoted by where “∼” indicates its random nature over time t; these random variables will be such as a price, a rate or a return.

., NEFTCI in the further reading at the end of the chapter). 9 Other financial models: from ARMA to the GARCH family The previous chapter dealt with stochastic processes, which consist of (returns) models involving a mixture of deterministic and stochastic components. By contrast, the models developed here present three major differences: These models are deterministic; since they are aiming to model a non-deterministic variable such as a return, the difference between the model output and the actual observed value is a probabilistic error term. By contrast with stochastic processes described by differential equations, these models are built in discrete time, in practice, the periodicity of the modeled return (daily, for example). By contrast with usual Markovian stochastic processes, these models incorporate in the general case a limited number of previous return values, so that they are not Markovian.

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

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 + ν(dy) C(Sey , t) − C(S, t) − S(ey − 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.

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

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.

It suffices to apply Theorem 4.42 with f≡0, b≡1, then yx,s(t)=w(t)−w(s)+x, and the corresponding operator is In Example 4.49, representation (4.16) is said to be the probabilistic representation of the solution. In particular, it follows that where is the probability density function for N(x, T−s). Note that this function is also well known in the theory of parabolic equations: it is the so-called fundamental solution of the heat equation. The representation of functions of the stochastic processes via solution of parabolic partial differential equations (PDEs) helps to study stochastic processes: one can use numerical methods developed for PDEs (i.e., finite differences, fundamental solutions, etc.). On the other hand, the probabilistic representation of a solution of parabolic PDEs can also help to study PDEs. For instance, one can use Monte Carlo simulation for numerical solution of PDEs. Some theoretical results can also be proved easier with probabilistic representation (for example, the so-called maximum principle for parabolic equations follows from this representation: if φ≥0 and Ψ≥0 in (4.15), then V≥0).

British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library Library of Congress Cataloging in Publication Data A catalog record for this book has been requested ISBN 0-203-96472-1 Master e-book ISBN ISBN10: 0-415-41447-4 (hbk) ISBN10: 0-415-41448-2 (pbk) ISBN10: 0-203-96472-1 (Print Edition) (ebk) ISBN13: 978-0-415-41447-0 (hbk) ISBN13: 978-0-415-41448-7 (pbk) ISBN13: 978-0-203-96472-9 (Print Edition) (ebk) © 2007 Nikolai Dokuchaev Contents Preface vi 1 Review of probability theory 1 2 Basics of stochastic processes 17 3 Discrete time market models 23 4 Basics of Ito calculus and stochastic analysis 49 5 Continuous time market models 75 6 American options and binomial trees 110 7 Implied and historical volatility 132 8 Review of statistical estimation 139 9 Estimation of models for stock prices 168 Legend of notations and abbreviations 182 Selected answers and key figures 183 Bibliography 184 © 2007 Nikolai Dokuchaev Preface Dedicated to Natalia, Lidia, and Mikhail This book gives a systematic, self-sufficient, and yet short presentation of the mainstream topics of Mathematical Finance and related part of Stochastic Analysis and Statistical Finance that covers typical university programs.

Risk Management in Trading by Davis Edwards

RANDOM NUMBERS Another use of variables is to represent a random quantity. Randomness, in finance, is typically described using notation from probability. Probability is the branch of mathematics that studies how likely or unlikely something is to occur. The probability that an event will occur is represented as a value 64 RISK MANAGEMENT IN TRADING KEY CONCEPT: STOCHASTIC PROCESSES Stochastic is a term that describes a type of random process that evolves over time. In a stochastic process, prices might be modeled as a series whose next value depends on the current value plus a random component. This is slightly different than a completely random process (like the series of numbers obtained by rolling a pair of dice). between a 0 percent chance of occurrence (something will not occur) and a 100 percent chance of occurrence (something will definitely occur).

between a 0 percent chance of occurrence (something will not occur) and a 100 percent chance of occurrence (something will definitely occur). In finance, the term stochastic is often used as a synonym for random. Stochastic describes a type of random sequence that evolves over time. In this type of sequence, the value of the next item in the sequence depends on the value of the previous item plus or minus a random value. In finance, stochastic processes are particularly important. This is because prices are often modeled as stochastic processes, and prices are a fundamental input into trading decisions. Common examples of random numbers are the results of throwing dice or flipping a coin. Each roll of the dice or flip of a coin generates a realization of a defined process. The probability of the coin landing on either a head or a tail is 50 percent and the probability of any single number on a regular, six‐sided die is 1/6 (assuming a fair dice roll and fair coin flip).

The time that has passed 74 RISK MANAGEMENT IN TRADING 0.1% 0.2% 50/50 chance of +1 or −1 Cumulative Result 10 9 8 7 6 5 4 3 2 1 0 −1 −2 −3 −4 −5 −6 −7 −8 −9 −10 1.0% 0.4% 1.8% 0.8% 1.6% 3.1% 25.0% 50.0% 100.0% 15.6% 25.0% 37.5% 37.5% 25.0% 16.4% 21.9% 23.4% 6.3% 24.6% 27.3% 15.6% 16.4% 20.5% 16.4% 11.7% 10.9% 9.4% 3.1% 24.6% 27.3% 31.3% 25.0% 12.5% 20.5% 24.6% 27.3% 31.3% 11.7% 16.4% 21.9% 23.4% 31.3% 37.5% 50.0% 50.0% 4.4% 7.0% 10.9% 9.4% 6.3% 12.5% 3.1% 5.5% 7.0% 5.5% 3.1% 1.6% 0.8% 4.4% 1.8% 1.0% 0.4% 0.2% 0.1% 0 1 2 3 4 5 6 7 8 9 10 Time FIGURE 3.9 Dispersion in a Random Series For financial mathematics, the Wiener process is often generalized to include a constant drift term that pushes prices upward. The constant drift term is due to risk‐free inflation (and described later in the chapter in the “time value of money” discussion). Continuous time versions of this process are called Generalized Wiener Process or the Ito Process. (See Equation 3.8, A Stochastic Process.) A stochastic process with discrete time steps can be described as: ΔSt = μΔt + σΔWt St or ΔSt = μSt Δt + σSt ΔWt where ΔSt Change in Price. The change in price that will occur St Price. The price of an asset at time t (3.8) Financial Mathematics μ Drift. The drift term that pushes prices upwards. Commonly, this is a constant, but can be generalized to vary over time Δt Change in Time. Typically, finance uses convention that Δt = 1.0 is a one year passage of time.

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After Hosking's paper, the literature on this subject has been surprisingly scarce, adding up to eight journal articles written by only nine authors: Hosking, Johansen, Nielsen, MacKinnon, Jensen, Jones, Popiel, Cavaliere, and Taylor. See the references for details. Most of those papers relate to technical matters, such as fast algorithms for the calculation of fractional differentiation in continuous stochastic processes (e.g., Jensen and Nielsen [2014]). Differentiating the stochastic process is a computationally expensive operation. In this chapter we will take a practical, alternative, and novel approach to recover stationarity: We will generalize the difference operator to non-integer steps. 5.4 The Method Consider the backshift operator, B, applied to a matrix of real-valued features {Xt}, where BkXt = Xt − k for any integer k ≥ 0.

While assessing the probability of backtest overfitting is a useful tool to discard superfluous investment strategies, it would be better to avoid the risk of overfitting, at least in the context of calibrating a trading rule. In theory this could be accomplished by deriving the optimal parameters for the trading rule directly from the stochastic process that generates the data, rather than engaging in historical simulations. This is the approach we take in this chapter. Using the entire historical sample, we will characterize the stochastic process that generates the observed stream of returns, and derive the optimal values for the trading rule's parameters without requiring a historical simulation. 13.3 The Problem Suppose an investment strategy S invests in i = 1, …I opportunities or bets. At each opportunity i, S takes a position of mi units of security X, where mi ∈ ( − ∞, ∞).

Chapters 10 and 16 are dedicated to this station, with the understanding that it would be unreasonable for a book to reveal specific investment strategies. 1.3.1.4 Backtesters This station assesses the profitability of an investment strategy under various scenarios. One of the scenarios of interest is how the strategy would perform if history repeated itself. However, the historical path is merely one of the possible outcomes of a stochastic process, and not necessarily the most likely going forward. Alternative scenarios must be evaluated, consistent with the knowledge of the weaknesses and strengths of a proposed strategy. Team members are data scientists with a deep understanding of empirical and experimental techniques. A good backtester incorporates in his analysis meta-information regarding how the strategy came about. In particular, his analysis must evaluate the probability of backtest overfitting by taking into account the number of trials it took to distill the strategy.

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Mathematics for Economics and Finance by Michael Harrison, Patrick Waldron

A random vector is just a vector of random variables. 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 ) .

This framework is sufficient to illustrate the similarities and differences between the most popular approaches. When we consider consumer choice under uncertainty, consumption plans will have to specify a fixed consumption vector for each possible state of nature or state of the world. This just means that each consumption plan is a random vector. Let us review the associated concepts from basic probability theory: probability space; random variables and vectors; and stochastic processes. Let Ω denote the set of all possible states of the world, called the sample space. A collection of states of the world, A ⊆ Ω, is called an event. Let A be a collection of events in Ω. The function P : A → [0, 1] is a probability function if 1. (a) Ω ∈ A (b) A ∈ A ⇒ Ω − A ∈ A (c) Ai ∈ A for i = 1, . . . , ∞ ⇒ S∞ i=1 Ai ∈ A (i.e. A is a sigma-algebra of events) Revised: December 2, 1998 86 5.2.

The prices of this, and the other elementary claims, must, by no arbitrage, equal the prices of the corresponding replicating portfolios. 5.5 The Expected Utility Paradigm 5.5.1 Further axioms The objects of choice with which we are concerned in a world with uncertainty could still be called consumption plans, but we will acknowledge the additional structure now described by terming them lotteries. If there are k physical commodities, a consumption plan must specify a k-dimensional vector, x ∈ <k , for each time and state of the world. We assume a finite number of times, denoted by the set T . The possible states of the world are denoted by the set Ω. So a consumption plan or lottery is just a collection of |T | k-dimensional random vectors, i.e. a stochastic process. Again to distinguish the certainty and uncertainty cases, we let L denote the collection of lotteries under consideration; X will now denote the set of possible values of the lotteries in L. Revised: December 2, 1998 94 5.5. THE EXPECTED UTILITY PARADIGM Preferences are now described by a relation on L. We will continue to assume that preference relations are complete, reflexive, transitive, and continuous.

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

We shall say that the family X of random variables Xt satisfies the stochastic differential equation, dXt = µ(t, Xt)dt + a(t, Xt)dWt, (5.8) The Ito calculus 106 if for any t, we have that Xr+h - Xt - h s(t, Xt) - a(t, Xr)(Wt+h - Wt) is a random variable with mean and variance which are o(h). We shall call such a family of random variables an Ito process or sometimes just a stochastic process. Note that if a is identically zero, we have that Xt+h - Xt - h s(t, Xt) (5.9) is of mean and variance o(h). We have thus essentially recovered the differential equation dXt (5.10) µ(t, Xt). dt The essential aspect of this definition is that if we know X0 and that Xt satisfies the stochastic differential equation, (5.8), then Xt is fully determined. In other terms, the stochastic differential equation has a unique solution. An important corollary of this is that µ and a together with Xo are the only quantities we need to know in order to define a stochastic process. Equally important is the issue of existence - it is not immediately obvious that a family Xt satisfying a given stochastic differential equation exists.

Rather surprisingly, this leads to the Black-Scholes price. We therefore have a very powerful alternative method for pricing options. Justifying this procedure requires an excursion into some deep and powerful mathematics. 6.4 The concept of information 141 Before we can proceed to a better understanding of option pricing, we need a better understanding of the nature of stochastic processes. In particular, we need to think a little more deeply about what a stochastic process is. We have talked about a continuous family of processes, Xt, such that X, - XS has a certain distribution. As long as we only look at a finite number of values of t and s this is conceptually fairly clear, but once we start looking at all values at once it as a lot less obvious what these statements mean. One way out is to take the view that each random variable Xt displays some aspect of a single more fundamental variable.

The argument we gave above still works; if a portfolio is of zero value and can be positive with positive probability tomorrow then to get the expectation to be zero, there must be a positive probability of negative value tomorrow. Hence, as before arbitrage is impossible. This is still not particularly useful however, as we know that a risky asset will in general grow faster than a riskless bond on average due to the risk aversion of market participants. To get round this problem, we ask what the rate of growth means for a stochastic process. The stochastic process is determined by a probability measure on the sample space which is the space of paths. However, the definition of an arbitrage barely mentions the probability measure. All it says is that it is impossible to set up a portfolio with zero value today which has a positive probability of being of positive value in the future, and a zero probability of being of negative value. The actual magnitude of the positive probability is not mentioned.

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Derivatives Markets by David Goldenberg

How do we take derivatives of smooth functions of stochastic processes, say F(Xt,t), such as (GBM SDE) where the process is the solution of a stochastic differential equation dXt=μXtdt+σXtdWt with initial value X0? We start with the observation that we can expect to end up with another stochastic process that is also the solution to another stochastic differential equation. This new stochastic differential equation for the total differential of F(Xt,t) will have a new set of drift and diffusion coefficients. The question is what exactly are the drift and diffusion coefficients of dF(Xt,t)? This is one of the problems that K. Itô solved in his famous formula called Itô’s lemma. To understand Itô’s lemma, keep in mind that there are two stochastic processes involved. The first is the underlying process (think of it as the stock).

The second equation above says that, Er(S1(ω)|S0)=(1+r′)S0>S0 unless r′=0. Even under risk neutrality (which doesn’t mean zero interest rates), the martingale requirement that Er(S1(ω)|S0)=S0 is clearly violated. Stock prices under risk neutrality are not martingales. However they aren’t very far from martingales. Definition of a Sub (Super) Martingale 1. A discrete-time stochastic process (Xn(ω))n=0,1,2,3,… is called a sub-martingale if E(Xn)<∞, and E(Xn+1(ω)|Xn)>Xn for all n=0,1,2,3,… 2. A discrete-time stochastic process (Xn(ω))n=0,1,2,3,… is called a super-martingale if E(Xn)<∞, , and E(Xn+1(ω)|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.

We will begin with the prototype of all continuous time models, and that is arithmetic Brownian motion (ABM). ABM is the most basic and important stochastic process in continuous time and continuous space, and it has many desirable properties including the strong Markov property, the martingale property, independent increments, normality, and continuous sample paths. Of course, here we want to focus on options pricing rather than the pure mathematical theory. The idea here is to partially prepare you for courses in mathematical finance. The details we have to leave out are usually covered in such courses. 16.1 ARITHMETIC BROWNIAN MOTION (ABM) ABM is a stochastic process {Wt(ω)}t≥0 defined on a sample space (Ω,ℑW,℘W). We won’t go into all the details as to exactly what (Ω,ℑW,℘W) represents but you can think of the probability measure, ℘W, which is called Wiener measure, to be defined in terms of the transition density function p(T,y;t,x) for τ =T–t, Norbert Wiener gave the first rigorous mathematical construction (existence proof) for ABM and, because of this, it is sometimes called the Wiener process.

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Frequently Asked Questions in Quantitative Finance by Paul Wilmott

Short Answer Brownian Motion is a stochastic process with stationary independent normally distributed increments and which also has continuous sample paths. It is the most common stochastic building block for random walks in finance. Example Pollen in water, smoke in a room, pollution in a river, are all examples of Brownian motion. And this is the common model for stock prices as well. Long Answer Brownian motion (BM) is named after the Scottish botanist who first described the random motions of pollen grains suspended in water. The mathematics of this process were formalized by Bachelier, in an option-pricing context, and by Einstein. The mathematics of BM is also that of heat conduction and diffusion. Mathematically, BM is a continuous, stationary, stochastic process with independent normally distributed increments.

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. Journal of Finance 42 1129-1142 Itô, K 1951 On stochastic differential equations.

Journal of Financial Economics 3 167-79 Haug, EG 2003 Know your weapon, Parts 1 and 2. Wilmott magazine, May and July Haug, EG 2006 The complete Guide to Option Pricing Formulas. McGraw-Hill Lewis, A 2000 Option Valuation under Stochastic Volatility. Finance Press What are the Forward and Backward Equations? Short Answer Forward and backward equations usually refer to the differential equations governing the transition probability density function for a stochastic process. They are diffusion equations and must therefore be solved in the appropriate direction in time, hence the names. Example An exchange rate is currently 1.88. What is the probability that it will be over 2 by this time next year? If you have a stochastic differential equation model for this exchange rate then this question can be answered using the equations for the transition probability density function.

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

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.

Keynesian model – a model developed by John Maynard Keynes that demonstrates savings may not necessarily be balanced with new investment and the gross domestic product may differ from that which would result in full employment. 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.

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

Price discreteness, for example, reflects a tick size (minimum pricing increment) that is generally set in level units. For reasons that will be discussed shortly, the drift can be dropped in most microstructure analyses. 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.

Placing the price change first is simply an expositional simplification and carries no implications that this variable is first in any causal sense. The chapter treats the general case but uses a particular structural model for purposes of illustration. The structural model is a bivariate model of price changes and trade directions: yt = [pt qt ]′ . 9.1 Modeling Vector Time Series The basic descriptive statistics of a vector stochastic process { yt } are the process mean µ = E[yt ] and the vector autocovariances. The vector autocovariances are defined as the matrices 78 MULTIVARIATE LINEAR MICROSTRUCTURE MODELS Ŵk = E( yt − E [yt ])(yt−k − E [yt ])′ for k = . . . −2, −1, 0, +1, +2, . . . (9.1) In suppressing the dependence of µ and Ŵk on t, we have implicitly invoked an assumption of covariance stationarity. Note that although a univariate autocorrelation has the property that γk = γ−k , the corresponding property in the multivariate case is Ŵk = Ŵ′−k .

The buy limit price is denoted Lt . If at time t, pt ≥ Lt , then the agent has effectively submitted a marketable limit order, which achieves immediate execution. A limit order priced at Lt < pt will be executed during period t if pτ ≤ Lt for any time t < τ < t + 1. The situation is depicted in figure 15.2. A limit order priced at Lt executes if the stock price follows path B but not path A. This is a standard problem in stochastic processes, and many exact results are available. The diffusion-barrier notion of execution is at best a first approximation. In many markets, a buy limit order might be executed by a market (or marketable) sell order while the best ask is still well above the limit price. We will subsequently generalize the execution mechanism to allow this. For the moment, though, it might be noted that the present situation is not without precedent.

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

.♦ To illuminate the structure of the message Shannon turned to some methodology and language from the physics of stochastic processes, from Brownian motion to stellar dynamics. (He cited a landmark 1943 paper by the astrophysicist Subrahmanyan Chandrasekhar in Reviews of Modern Physics.♦) A stochastic process is neither deterministic (the next event can be calculated with certainty) nor random (the next event is totally free). It is governed by a set of probabilities. Each event has a probability that depends on the state of the system and perhaps also on its previous history. If for event we substitute symbol, then a natural written language like English or Chinese is a stochastic process. So is digitized speech; so is a television signal. Looking more deeply, Shannon examined statistical structure in terms of how much of a message influences the probability of the next symbol.

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

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.

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

Mandelbrot was a student of Paul Lévy’s—the son of the man who gave Bachelier bad marks at his examination at the École Polytechnique in 1900. Lévy’s research focused on “stochastic processes”: mathematical models that describe the behavior of some variable through time. For example, we saw in Chapter 15 that Jules Regnault proposed and tested a stochastic process that varied randomly, which resulted in a rule about risk increasing with the square root of time. Likewise, Louis Bachelier more formally developed a random-walk stochastic process. Paul Lévy formalized these prior random walk models into a very general family of stochastic processes referred to as Lévy processes. Brownian motion was just one process in the family of Lévy processes—and perhaps the best behaved of them. Other stochastic processes have such things as discontinuous jumps and unusually large shocks (which might, for example, explain the crash of 1987, when the US stock market lost 22.6% of its value in a single day).

One of his major contributions to the literature on finance (published in 1966) was a proof that an efficient market implies that stock prices may not follow a random walk, but that they must be unpredictable. It was a nice refinement of Regnault’s hypothesis articulated almost precisely a century prior. Although Mandelbrot ultimately developed a fractal-based option-pricing model with two of his students that allowed for extreme events and a more general stochastic process, for various reasons Mandelbrot never saw it adopted in practice to any great extent. I suspect that this is because the solution, while potentially useful, is complicated and contradicts most other tools that quantitative financiers use. With Mandelbrot’s models, it is all or nothing. You have to take a leap beyond the world of Brownian motion and throw out old friends like Bernoulli’s law of large numbers.

Benoit Mandelbrot believed he had discovered a deep structure to the world in general and financial markets in particular. His insights, however, can be traced directly back to the special tradition of mathematical inquiry that has its roots in the Enlightenment. I think this is what most excited him about his work—thinking of it in historical context as a culmination of applications of probability to markets. Although not all quants are aware of it, when they use a stochastic process (like Brownian motion) to price a security or figure out a hedge, they are drawing from a very deep well of mathematical knowledge that would not have existed but for the emergence of financial markets in Europe. Yes, the models that modern quants have applied to markets can go wrong. Models are crude attempts to characterize a reality that is complex and continually evolving. Despite the crashes—or perhaps because of them—financial markets have continually challenged the best and brightest minds with puzzles that hold the promise of intellectual and pecuniary rewards.

., 2012, “High Frequency Trading and Volatility”, SSRN Working Paper. Brunnermeier, M., and L. H. Pedersen, 2005, “Predatory Trading”, Journal of Finance 40(4), pp. 1825–63. Carlin, B., M. Sousa Lobo and S. Viswanathan, 2007, “Episodic Liquidity Crises: Cooperative and Predatory Trading”, Journal of Finance 42(5), pp. 2235–74. Clark, P. K., 1970, “A Subordinated Stochastic Process Model of Cotton Futures Prices”, PhD Dissertation, Harvard University. Clark, P. K., 1973, “A Subordinated Stochastic Process Model with Finite Variance for Speculative Prices”, Econometrica 41(1), pp. 135–55. Donefer, B. S., 2010, “Algos Gone Wild: Risk in the World of Automated Trading Strategies”, The Journal of Trading 5, pp. 31–4. Easley, D., N. Kiefer, M. O’Hara and J. Paperman, 1996, “Liquidity, Information, and Infrequently Traded Stocks”, Journal of Finance 51, pp. 1405–36.

John Wiley and Sons, Chichester. 18 i i i i i i “Easley” — 2013/10/8 — 11:31 — page 19 — #39 i i THE VOLUME CLOCK: INSIGHTS INTO THE HIGH-FREQUENCY PARADIGM Linton, O., and M. O’Hara, 2012, “The Impact of Computer Trading on Liquidity, Price Efficiency/Discovery and Transactions Costs”, in Foresight: The Future of Computer Trading in Financial Markets. An International Perspective, Final Project Report. The Government Office for Science, London. Mandelbrot, B., 1973, “Comments on ‘A Subordinated Stochastic Process Model with Finite Variance for Speculative Prices by Peter K. Clark’ ”, Econometrica 41(1), pp. 157–59. Mandelbrot, B., and M. Taylor, 1967, “On the Distribution of Stock Price Differences”, Operations Research 15(6), pp. 1057–62. NANEX, 2010, “Analysis of the ‘Flash Crash’ ”, June 18. URL: http://www.nanex.net/ 20100506/FlashCrashAnalysis_CompleteText.html. NANEX, 2011, “Strange Days June 8’th, 2011 – NatGas Algo”.

Algorithmic approaches to execution problems are fairly well studied, and often apply methods from the stochastic control literature (Bertsimas and Lo 1998; Bouchaud et al 2002; Cont and Kukanov 2013; Guéant et al 2012; Kharroubi and Pham 2010). The aforementioned papers seek to solve problems similar to ours, ie, to execute a certain number of shares over some fixed period as cheaply as possible, but approach it from another direction. They typically start with an assumption that the underlying “true” stock price is generated by some known stochastic process. There is also a known impact function that specifies how arriving liquidity demand pushes market prices away from this true value. Having this information, as well as time and volume constraints, it is then possible to compute the optimal strategy explicitly. This can be done either in closed form or numerically (often using dynamic programming, the basis of reinforcement learning). There are also interesting game-theoretic variants of execution problems in the presence of an arbitrageur (Moallemi et al 2012), and examinations of the tension between exploration and exploitation (Park and van Roy 2012).

Monte Carlo Simulation and Finance by Don L. McLeish

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. W (0 ) = 0 . [FIGURE 2.6 ABOUT HERE] The fact that such a process exists is by no means easy to see. It has been an important part of the literature in Physics, Probability and Finance at least since the papers of Bachelier and Einstein, about 100 years ago.

And when the drift term a(Xt , t ) is linear in Xt , the solution of an ordinary diﬀerential equation will allow the calculation of the expected value of the process and this is the first and most basic description of its behaviour. The MODELS IN CONTINUOUS TIME 77 appendix provides an elementary review of techniques for solving partial and ordinary diﬀerential equations. However, that the information about a stochastic process obtained from a deterministic object such as a ordinary or partial diﬀerential equation is necessarily limited. For example, while we can sometimes obtain the marginal distribution of the process at time t it is more diﬃcult to obtain quantities such as the joint distribution of variables which depending on the path of the process, and these are important in valuing certain types of exotic options such as lookback and barrier options.

Solving deterministic diﬀerential equations can sometimes provide a solution to a specific problem such as finding the arbitrage-free price of a derivative. 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, . . ..

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

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.

Erdmann Oxford University L.C.G. Rogers University of Cambridge E. Süli Oxford University J.F. Toland University of Bath Other books in this series A First Course in Discrete Mathematics I. Anderson Analytic Methods for Partial Differential Equations G. Evans, J. Blackledge, P. Yardley Applied Geometry for Computer Graphics and CAD D. Marsh Basic Linear Algebra, Second Edition T.S. Blyth and E.F. Robertson Basic Stochastic Processes Z. Brzeźniak and T. Zastawniak Elementary Differential Geometry A. Pressley Elementary Number Theory G.A. Jones and J.M. Jones Elements of Abstract Analysis M. Ó Searcóid Elements of Logic via Numbers and Sets D.L. Johnson Essential Mathematical Biology N.F. Britton Fields, Flows and Waves: An Introduction to Continuum Models D.F. Parker Further Linear Algebra T.S. Blyth and E.F. Robertson Geometry R.

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The Man Who Solved the Market: How Jim Simons Launched the Quant Revolution by Gregory Zuckerman

Members of Axcom’s team viewed investing through a math prism and understood financial markets to be complicated and evolving, with behavior that is difficult to predict, at least over long stretches—just like a stochastic process. It’s easy to see why they saw similarities between stochastic processes and investing. For one thing, Simons, Ax, and Straus didn’t believe the market was truly a “random walk,” or entirely unpredictable, as some academics and others argued. Though it clearly had elements of randomness, much like the weather, mathematicians like Simons and Ax would argue that a probability distribution could capture futures prices as well as any other stochastic process. That’s why Ax thought employing such a mathematical representation could be helpful to their trading models. Perhaps by hiring Carmona, they could develop a model that would produce a range of likely outcomes for their investments, helping to improve their performance.

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Building Habitats on the Moon: Engineering Approaches to Lunar Settlements by Haym Benaroya

Each is a random variable, signifying that there are uncertainties about both, and here they are modeled using normal probability density functions. When the stress exceeds the strength, the system has failed. A measure of the reliability (probability of failure) is given by the overlapped area, shown hatched. The random variable is a static property – the shape of the density function does not change with time. Where the density function is time-dependent, the variable is called a random, or stochastic, process. Before examining some commonly used densities, we define an averaging procedure known as the mathematical expectation for probabilistic variables. 10.3 Mathematical Expectation The single most important descriptor of a random variable is its mean or expected value. This defines the most likely value of a variable. However, random variables may have the same mean, but their spread of possible values, or their variance, can be considerably different.

As we know, and as we will discuss in more detail subsequently, the complete structure can behave in ways that are unpredictable, if based only on the behavior of its components. Our reliability estimates are guesses about the future, not extrapolations from past data. More on this later. Now that we have an understanding of the autocorrelation, we proceed to study its Fourier transform, the spectral density . 10.6 Power Spectrum A measure of the ‘energy’ of the stochastic process X(t) is given by its power spectrum , or spectral density , S XX (ω), which is the Fourier transform of its autocorrelation function: and thus: (10.14) These equations are known as the Wiener-Khintchine formulas. Since R XX (−τ) = R XX (τ), S XX (ω) is not a complex function but a real and even function. For τ = 0: where S XX (ω) ≥ 0 since, as a measure of energy, it must be positive semi-definite.

Happel (1993): Indigenous materials for lunar construction. Applied Mechanics Reviews, 46(6), pp.313–325. Footnotes 1From the Greek we also have the stochastic (στoκoς) process. 2An axiom is a rule that is assumed to be true, and upon which further rules and facts are deduced. For engineering, the deduced facts must conform to reality. An excellent book on the basics of probabilistic modeling is Probability, Random Variables, and Stochastic Processes, A. Papoulis, McGraw-Hill, 1965. © Springer International Publishing AG 2018 Haym BenaroyaBuilding Habitats on the MoonSpringer Praxis Bookshttps://doi.org/10.1007/978-3-319-68244-0_11 11. Reliability and damage Haym Benaroya1 (1)Professor of Mechanical & Aerospace Engineering, Rutgers University, New Brunswick, New Jersey, USA “We need to make sure it survives for a while.”

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

Faff and Hallahan (2001) argue that survivorship bias is more likely to cause performance reversals than performance persistence. The data used show considerable kurtosis (see Table 3.1). However, this kurtosis may be caused by heteroskedasticity (returns of some funds are more variable than others). REGRESSION TEST OF PERFORMANCE PERSISTENCE To measure performance persistence, a model of the stochastic process that generates returns is required. The process considered is: rit = αi + βi rt + εit , ε it ~ N(0, σ i2 ) i = 1, K , n and t = 1, K , T (3.1) where rit = return of fund (or CTA) i in month t rt = average fund returns in month t slope parameter bi = differences in leverage. The model allows each fund to have a different variance, which is consistent with past research. We also considered models that assumed that bi is zero, with either fixed effects (dummy variables) for time or random effects instead.

This demonstrates that most of the nonnormality shown in Table 3.1 is due to heteroskedasticity. MONTE CARLO STUDY In their method, EGR ranked funds by their mean return or modified Sharpe ratio in a first period, and then determined whether the funds that ranked high in the first period also ranked high in the second period. We use Monte Carlo simulation to determine the power and size of hypothesis tests with EGR’s method when data follow the stochastic process given in equation 3.1. Data were generated by specifying values of α, β, and σ. The simulation used 1,000 replications and 120 simulated funds. The mean return over all funds, r̄t, is derived from the values of α and β as: Σα i Σε it + n n rt = Σβ i 1− n where all sums are from i = 1 to n. A constant value of α simulates no performance persistence. For the data sets generated with persistence present, α was generated randomly based on the mean and variance of β’s in each of the three data sets.

Chicago Mercantile Exchange. (1999) “Question and Answer Report: Managed Futures Accounts.” Report No. M584/10M/1299. www.cve.com. Christoffersen, P. (2003) Elements of Financial Risk Management. San Diego, CA: Academic Press. Chung, S. Y. (1999) “Portfolio Risk Measurement: A Review of Value at Risk.” Journal of Alternative Investments, Vol. 2, No. 1, pp. 34–42. Clark, P. K. (1973) “A Subordinated Stochastic Process Model with Finite Variance for Speculative Prices.” Econometrica, Vol. 41, No. 1, pp. 135–155. Clayton, U. (2003) A Guide to the Law of Securitisation in Australia. Sydney, Australia: Clayton Company. Cooley, P. L., R. L. Roenfeldt, and N. K. Modani. (1977) “Interdependence of Market Risk Measures.” Journal of Business, Vol. 50, No. 3, pp. 356–363. Cootner, P. (1967) “Speculation and Hedging.”

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

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.

Bootstrapping: Douglas Engelbart, Coevolution, and the Origins of Personal Computing (Writing Science) by Thierry Bardini

In the conceptual world, both the transmission and the trans- formation of what Whorf called "culturally ordained forms and categories" IS the process by which people learn. The crucial point in Bateson's synthesis lay in the characterization of all such processes as "stochastic": Both genetic change and the process called learnIng (including the somatic changes induced by the envIronment) are stochastic processes. In each case there is, I believe, a stream of events that is random in certain aspects and in each case there is a nonrandom selective process which causes certain of the random com- ponents to "surVIve" longer than others. Without the random, there can be no new thIng. . . . We face, then, two great stochastic systems that are partly in interaction and partly isolated from each other. One system IS withIn the individual and is called learnIng; the other is immanent In heredIty and in populations and IS called evolutIon.

In all three of these features, David stresses "software over hardware," or "the touch typist's memory of a particular arrangement of the keys" over this particular arrangement of the keys, and concludes "this, then, was a situation in which the precise details of timing in the developmental se- quence had made it profitable in the short run to adapt machines to the habit of men (or to women, as was increasingly the case) rather than the other way around. And things have been this way ever since" (ibid., 336).6 Thus, it was by institutionalization as an incorporating practice that the QWERTY standard became established. The establishment of a commercial education network favoring the QWERTY was the decisive factor, the source of the" historical accident" that governed the stochastic process that secured forever the supremacy of the QWERTY. It is indeed because of such an "acci- dent" that the six or seven years during which Remington enjoyed the early advantage of being the sole owner of the typewriter patent also saw its selling agents establish profitable and durable business associations with the com- mercial education business. These early business ties soon gave place to an or- ganized and institutional network of associations that secured Remington's position in the typewriter business.

See also Hypertext Atari, 103 Atlas computer, 126, 25 2n6 Augmentation Research Center (ARC), 275 276 Index 145-47,157; staff, 121-22; spa- tial organization of laboratory, 122- 23; Framework ActivIty (FRAMAC), 194-95,211, 259nI6; Personal and Organizational Development Activ- ity (PODAC), 194-201, 259nnI6- 18; Line Activity (LINAC), 194,211, 259nI6; and est, 201-8, 260nnI9- 20; as "breakthrough lab," 211-13; Engelbart's eulogy, 214 -as NIC, see under ARPA: Network Information Center Augmented knowledge workshop, I 16, 21 9 AutomatIon, 18 - I 9, 24 0n 5 Automobile, 18, 3 I Baby boomers, 125 Bandwidth, see under Information Baran, Paul, 18 4, 257n4 Bass, Walter, I98ff; and est, 202, 204, 260nI9 Batch processing, 4 Bates, Roger, 1°9,120,123,156 Bateson, Gregory, 17, 26, 52, 56, 102, 135, 228- 2 9, 23 6nI 3, 24 0n 3, 242nI8; on coevolution, 56, 24 2 - 43 n 24; on stochastIc process, 56, 24 2n2 4 Baudot, Maurice-Emile, 65, 67 f , 79 BBN (Bolt, Beranek and Newman), 30, 12 4, 19 1 , 247 nI ,25 8n 7 BCC (Berkeley Computer CorporatIon), 155 f , 25 6n 9 Beam pen, 89. See also LIght pen "Behavior, Purpose and Teleology," 25 Bell Laboratories, 24 7 n 5 Benedict, Henry H., 78 Bergson, Henri-Louis, 48 Berkeley Computer CorporatIon (BCC), I55 f , 25 6n 9 Berman, Melvyn, 1°9-10 Bewley, William, 177 Bigelow, Julian, 25 Bliss, James C., 61-62, 222-23, 244nI Boaz, Franz, 24 0n 3 .

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The Collapse of Western Civilization: A View From the Future by Naomi Oreskes, Erik M. Conway

T h e F r e n z y o F F o s s i l F u e l s 17 This was consistent with the expectation—based on physical theory—that warmer sea surface temperatures in regions of cyclogenesis could, and likely would, drive either more hurricanes or more intense ones. However, they backed away from this conclusion under pressure from their scientific colleagues. Much of the argument surrounded the concept of statistical significance. Given what we now know about the dominance of nonlinear systems and the distribution of stochastic processes, the then-dominant notion of a 95 percent confidence limit is hard to fathom. Yet overwhelming evidence suggests that twentieth-century scientists believed that a claim could be accepted only if, by the standards of Fisherian statistics, the possibility that an observed event could have happened by chance was less than 1 in 20. Many phenomena whose causal mechanisms were physically, chemically, or biologically linked to warmer temperatures were dis-missed as “unproven” because they did not adhere to this standard of demonstration.

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Skin in the Game: Hidden Asymmetries in Daily Life by Nassim Nicholas Taleb

Adaptation of Theorem 1 to Brownian Motion The implications of simplified discussion do not change whether one uses richer models, such as a full stochastic process subjected to an absorbing barrier. And of course in a natural setting the eradication of all previous life can happen (i.e., Xt can take extreme negative value), not just a stopping condition. The Peters and Gell-Mann argument also cancels the so-called equity premium puzzle if you add fat tails (hence outcomes vastly more severe pushing some level equivalent to ruin) and absence of the fungibility of temporal and ensemble. There is no puzzle. The problem is invariant in real life if one uses a Brownian-motion-style stochastic process subjected to an absorbing barrier. In place of the simplified representation we would have, for an process subjected to L, an absorbing barrier from below, in the arithmetic version: or, for a geometric process: where Z is a random variable.

He has built and deployed extremely low latency, high throughput automated trading systems for trading exchanges around the world, across multiple asset classes. He specializes in statistical arbitrage market-making, and pairs trading strategies for the most liquid global futures contracts. He works as a Senior Quantitative Developer at a trading firm in Chicago. He holds a Masters in Computer Science from the University of Southern California. His areas of interest include Computer Architecture, FinTech, Probability Theory and Stochastic Processes, Statistical Learning and Inference Methods, and Natural Language Processing. About the reviewers Nataraj Dasgupta is the VP of Advanced Analytics at RxDataScience Inc. He has been in the IT industry for more than 19 years and has worked in the technical & analytics divisions of Philip Morris, IBM, UBS Investment Bank, and Purdue Pharma. He led the Data Science team at Purdue, where he developed the company's award-winning Big Data and Machine Learning platform.

In the next chapter, we will review and implement some simple regression and classification methods and understand the advantages of applying supervised statistical learning methods to trading. Predicting the Markets with Basic Machine Learning In the last chapter, we learned how to design trading strategies, create trading signals, and implement advanced concepts, such as seasonality in trading instruments. Understanding those concepts in greater detail is a vast field comprising stochastic processes, random walks, martingales, and time series analysis, which we leave to you to explore at your own pace. So what's next? Let's look at an even more advanced method of prediction and forecasting: statistical inference and prediction. This is known as machine learning, the fundamentals of which were developed in the 1800s and early 1900s and have been worked on ever since. Recently, there has been a resurgence in interest in machine learning algorithms and applications owing to the availability of extremely cost-effective processing power and the easy availability of large datasets.

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Fooled by Randomness: The Hidden Role of Chance in Life and in the Markets by Nassim Nicholas Taleb

The Tools The notion of alternative histories discussed in the last chapter can be extended considerably and subjected to all manner of technical refinement. This brings us to the tools used in my profession to toy with uncertainty. I will outline them next. Monte Carlo methods, in brief, consist of creating artificial history using the following concepts. First, consider the sample path. The invisible histories have a scientific name, alternative sample paths, a name borrowed from the field of mathematics of probability called stochastic processes. The notion of path, as opposed to outcome, indicates that it is not a mere MBA-style scenario analysis, but the examination of a sequence of scenarios along the course of time. We are not just concerned with where a bird can end up tomorrow night, but rather with all the various places it can possibly visit during the time interval. We are not concerned with what the investor’s worth would be in, say, a year, but rather of the heart-wrenching rides he may experience during that period.

Starting at \$100, in one scenario it can end up at \$20 having seen a high of \$220; in another it can end up at \$145 having seen a low of \$10. Another example is the evolution of your wealth during an evening at a casino. You start with \$1,000 in your pocket, and measure it every fifteen minutes. In one sample path you have \$2,200 at midnight; in another you barely have \$20 left for a cab fare. Stochastic processes refer to the dynamics of events unfolding with the course of time. Stochastic is a fancy Greek name for random. This branch of probability concerns itself with the study of the evolution of successive random events—one could call it the mathematics of history. The key about a process is that it has time in it. What is a Monte Carlo generator? Imagine that you can replicate a perfect roulette wheel in your attic without having recourse to a carpenter.

Fifty Challenging Problems in Probability With Solutions by Frederick Mosteller

As is well known, no strategy can give him a higher probability of achieving his goal, and the probability is this high if and only if he makes sure either to lose x or win y eventually. The Leeser Paradise The Lesser Paradise resembles the Golden Paradise with the imoortant difference that before leaving the haH the gambler must pay an income tax ·First pUblished, 1965 Reprinted by Dover Publications. Inc in 1976 under the title Inequalities for stochastic processes 56 of t 100% (0 < t < 1) on any net positive income that he has won there. It is therefore no harder or easier for him to win y dollars with an initial fortune of x than it is for his brother in the Golden Paradise to win y/(I - t) dollars. The greatest probability with which he can achieve his goal is therefore (I - t)x (I) (1 - t)x +y The Paradise Lost Here, the croupier collects the tax of !

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

During World War II, Wiener researched ways to integrate human gunner and analog computer agency in antiaircraft artillery fire-control systems, vaulting his wartime research on the feedback processes among humans and machines into a general science of communication and control, with the gun and gunner ensemble (the man and the antiaircraft gun cockpit) as the original image of the cyborg.5 To designate this new science of control and feedback mechanisms, Wiener coined the neologism cybernetics from the Greek word for steersman, which is a predecessor to the English term governor (there is a common consonant-vowel structure between cybern- and govern—k/g + vowel + b/v + ern). Wiener’s popular masterworks ranged further still, commingling complex mathematical analysis (especially noise and stochastic processes), exposition on the promise and threat associated with automated information technology, and various speculations of social, political, and religious natures.6 For Wiener, cybernetics was a working out of the implications of “the theory of messages” and the ways that information systems organized life, the world, and the cosmos. He found parallel structures in the communication and control systems operating in animal neural pathways, electromechanical circuits, and information flows in larger social systems.7 The fact that his work speaks in general mathematical terms also sped his work’s reception and eventual embrace by a wide range of readers, including Soviet philosopher-critics, as examined later.

Because the coauthors were sensitive to how language, especially foreign terms, packs in questions of international competition, the coauthors attempted to keep their language as technical and abstract as possible, reminding the reader that the cybernetic mind-machine analogy was central to the emerging science but should be understood only “from a functional point of view,” not a philosophical one.76 The technical and abstract mathematical language of Wiener’s cybernetics thus served as a political defense against Soviet philosopher-critics and as ballast for generalizing the coauthors’ ambitions for scientists in other fields. They employed a full toolbox of cybernetic terminology, including signal words such as homeostasis, feedback, entropy, reflex, and the binary digit. They also repeated Wiener and Shannon’s emphases on probabilistic, stochastic processes as the preferred mathematical medium for scripting behavioral patterns onto abstract logical systems, including a whole section that elaborated on the mind-machine analogy with special emphasis on the central processor as capable of memory, responsiveness, and learning.77Wiener’s call for cyberneticists with “Leibnizian catholicity” of scientific interests was tempered into its negative form—a warning against disciplinary isolationism.78 On the last page of the article, the coauthors smoothed over the adoption of Wiener, an American, as foreign founder of Soviet cybernetics by summarizing and stylizing Wiener’s “sharp critique of capitalist society,” his pseudo-Marxist prediction of a “new industrial revolution” that would arise out of the “chaotic conditions of the capitalist market,” and his widely publicized postwar fear of “the replacement of common workers with mechanical robots.”79 A word play in Russian animates this last phrase: the Russian word for worker, or rabotnik, differs only by a vowel transformation from robot, the nearly universal term coined in 1927 by the playwright Karel Capek from the Czech word for “forced labor.”80 The first industrial revolution replaced the hand with the machine, or the rabotnik with the robot, and Wiener’s science, the coauthors dreamed, would help usher in a “second industrial revolution” in which the labor of the human mind could be carried out by intelligent machines, thus freeing, as Marx had intimated a century earlier, the mind to higher pursuits.

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Chaos: Making a New Science by James Gleick

It seems to have been the issue on which many different fields of science were stuck—they were stuck on this aspect of the nonlinear behavior of systems. Now, nobody would have thought that the right background for this problem was to know particle physics, to know something about quantum field theory, and to know that in quantum field theory you have these structures known as the renormalization group. Nobody knew that you would need to understand the general theory of stochastic processes, and also fractal structures. “Mitchell had the right background. He did the right thing at the right time, and he did it very well. Nothing partial. He cleaned out the whole problem.” Feigenbaum brought to Los Alamos a conviction that his science had failed to understand hard problems—nonlinear problems. Although he had produced almost nothing as a physicist, he had accumulated an unusual intellectual background.

Astute readers, though, could tell that I preferred Joe Ford’s more freewheeling “cornucopia” style of definition—“Dynamics freed at last from the shackles of order and predictability…”—and still do. But everything evolves in the direction of specialization, and strictly speaking, “chaos” is now a very particular thing. When Yaneer Bar-Yam wrote a kilopage textbook, Dynamics of Complex Systems, in 2003, he took care of chaos proper in the first section of the first chapter. (“The first chapter, I have to admit, is 300 pages, okay?” he says.) Then came Stochastic Processes, Modeling Simulation, Cellular Automata, Computation Theory and Information Theory, Scaling, Renormalization, and Fractals, Neural Networks, Attractor Networks, Homogenous Systems, Inhomogenous Systems, and so on. Bar-Yam, the son of a high-energy physicist, had studied condensed matter physics and become an engineering professor at Boston University, but he left in 1997 to found the New England Complex Systems Institute.

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Topics in Market Microstructure by Ilija I. Zovko

Quantitative Finance, 2:346–353, 2002. 100 BIBLIOGRAPHY W. S. Choi, S. B. Lee, and P. I. Yu. Estimating the permanent and transitory components of the bid/ask spread. In C.-F. e. Lee, editor, Advances in investment analysis and portfolio management. Volume 5. Elsevier, 1998. T. Chordia and B. Swaminathan. Trading volume and crossautocorrelations in stock returns. Journal of Finance, LV(2), April 2000. P. K. Clark. Subordinated stochastic process model with finite variance for speculative prices. Econometrica, 41(1):135–155, 1973. K. J. Cohen, S. F. Maier, R. A. Schwartz, and D. K. Whitcomb. Transaction costs, order placement strategy, and existence of the bid-ask spread. Journal of Political Economy, 89(2):287–305, 1981. K. J. Cohen, R. M. Conroy, and S. F. Maier. Order flow and the quality of the market. In Y. Amihud, T. Ho, and R.

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

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.

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Think Complexity by Allen B. Downey

, Stanley Milgram sorting, Analysis of Basic Python Operations, Analysis of Basic Python Operations source node, Dijkstra spaceships, Structures, Life Patterns spanning cluster, Percolation special creation, Falsifiability spectral density, Spectral Density spherical cow, The Axes of Scientific Models square, Fractals stable sort, Analysis of Basic Python Operations Stanford Large Network Dataset Collection, Zipf, Pareto, and Power Laws state, Cellular Automata, Stephen Wolfram, Sand Piles stochastic process, The Axes of Scientific Models stock market, SOC, Causation, and Prediction StopIteration, Iterators __str__, Representing Graphs, Representing Graphs strategy, Prisoner’s Dilemma string concatenation, Analysis of Basic Python Operations string methods, Analysis of Basic Python Operations Strogatz, Steven, Paradigm Shift?, Watts and Strogatz The Structure of Scientific Revolutions, Paradigm Shift?

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

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.

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Money: 5,000 Years of Debt and Power by Michel Aglietta

This mimetic model’s strength is that it reveals the emergence, from amid this general confusion, of a polarisation around one single object of desire recognised by all (see Box 1.1).21 Box 1.1 Theorem of mimetic convergence In a population of N agents (i = 1, N), on date t each person has a belief ui(t) regarding the debt that represents absolute liquidity. i chooses his belief in t+1 by copying an agent j at random, with the probability pij for j = 1, N. So we have Pr{ui (t+1) = uj(t)} = pij with Σpij = 1 for each i. The mimetic interdependency is formalised as a Markovian stochastic process defined by the matrix Such that the dynamic process is written U (t+1) = PU(t) The theorem shows that -If the graph associated with P is strongly correspondent (matrix P does not break down into independent sub-matrices); -and aperiodic (the process of revising beliefs is not cyclical); -the mimetic contagion converges towards unanimity around a belief, which can be any of the initial beliefs.

These rules are ways of describing a discretionary monetary policy, which is constrained by situations of uncertainty. These are systems of constrained discretion, in which the rule is used as a safeguard. Box 6.1 Interest rate rules 1) The Wicksellian norm which the Rilskbank used to break out of inflation in the 1930s set a target for price levels and not inflation rates. It is associated with the rates rule it = īt + φpt in which pt = the log of the price index that is be stabilised. īt follows a stochastic process that is independent of price movements but is correlated to the exogenous fluctuations in the natural rate rt. The relationship defining the equilibrium nominal rate is it = rt + Etpt + 1 – pt. Eliminating it we get: If we separate out the processes followed by rt and īt then pt has a single solution: It follows that prices fluctuate around a long-term level: The long-term value of the general level of prices is independent from demand for money.

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

General proof that properly anticipated prices are random. Samuelson has proved a general theorem showing that the concept that prices are unpredictable can actually be deduced rigorously [357] from a model that hypothesizes that a stock’s present price pt is set at the expected discounted value of its future dividends dt dt+1 dt+2 (which are supposed to be random variables generated according to any general (but known) stochastic process): pt = dt + 1 dt+1 + 1 2 dt+2 + 1 2 3 dt+3 + · · · (3) where the factors i = 1 − r < 1, which can ﬂuctuate from one time period to the next, account for the depreciation of a future price calculated at present due to the nonzero consumption price index r. We see that pt = dt + 1 pt+1 , and thus the expectation Ept+1 of pt+1 conditioned on the knowledge of the present price pt is Ept+1 = pt − dt (4) 1 This shows that, barring the drift due to the inﬂation and the dividend, the price increment does not have a systematic component or memory of the past and is thus random.

Inductive reasoning and bounded rationality (The El Farol Problem), American Economic Review (Papers and Proceedings) 84. 18. Arthur, W., Lane, D., and Durlauf, S., Editors (1997). The economy as an evolving complex system II (Addison-Wesley, Redwood City). 19. Arthur, W. B. (1987). Self-reinforcing mechanisms in economics, Center for Economic Policy Research 111, 1–20. 20. Arthur, W. B., Ermoliev, Y. M., and Kaniovsky, Y. M. (1984). Strong laws for a class of path-dependent stochastic processes with applications, in Proceedings of the International Conference on Stochastic Optimization, A. Shiryaev and R. Wets, editors (Springer-Verlag, New York), pp. 287–300. 21. Arthur, W. B., Holland, J. H., LeBaron, B., Palmer, R., and Taylor, P. (1997). Asset pricing under endogenous expectations in an artiﬁcial stock market, in The Economy as an Evolving Complex System II, W. Arthur, D. Lane, and S.

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Antifragile: Things That Gain From Disorder by Nassim Nicholas Taleb

Next we turn to a central distinction between the things that like stress and other things that don’t. 1 Cato was the statesman who, three books ago (Fooled by Randomness), expelled all philosophers from Rome. 2 This little bit of effort seems to activate the switch between two distinct mental systems, one intuitive and the other analytical, what psychologists call “system 1” and “system 2.” 3 There is nothing particularly “white” in white noise; it is simply random noise that follows a Normal Distribution. 4 The obvious has not been tested empirically: Can the occurrence of extreme events be predicted from past history? Alas, according to a simple test: no, sorry. 5 Set a simple filtering rule: all members of a species need to have a neck forty centimeters long in order to survive. After a few generations, the surviving population would have, on average, a neck longer than forty centimeters. (More technically, a stochastic process subjected to an absorbing barrier will have an observed mean higher than the barrier.) 6 The French have a long series of authors who owe part of their status to their criminal record—which includes the poet Ronsard, the writer Jean Genet, and many others. CHAPTER 3 The Cat and the Washing Machine Stress is knowledge (and knowledge is stress)—The organic and the mechanical—No translator needed, for now—Waking up the animal in us, after two hundred years of modernity The bold conjecture made here is that everything that has life in it is to some extent antifragile (but not the reverse).

My dream—the solution—is that we would have a National Entrepreneur Day, with the following message: Most of you will fail, disrespected, impoverished, but we are grateful for the risks you are taking and the sacrifices you are making for the sake of the economic growth of the planet and pulling others out of poverty. You are at the source of our antifragility. Our nation thanks you. 1 A technical comment on why the adaptability criterion is innocent of probability (the nontechnical reader should skip the rest of this note). The property in a stochastic process of not seeing at any time period t what would happen in time after t, that is, any period higher than t, hence reacting with a lag, an incompressible lag, is called nonanticipative strategy, a requirement of stochastic integration. The incompressibility of the lag is central and unavoidable. Organisms can only have nonanticipative strategies—hence nature can only be nonpredictive. This point is not trivial at all, and has even confused probabilists such as the Russian School represented by Stratonovich and the users of his method of integration, who fell into the common mental distortion of thinking that the future sends some signal detectable by us.

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Seeing Like a State: How Certain Schemes to Improve the Human Condition Have Failed by James C. Scott

.",, Kollontay's point of departure, like Luxemburg's, is an assumption about what kinds of tasks are the making of revolutions and the creating of new forms of production. For both of them, such tasks are voyages in uncharted waters. There may be some rules of thumb, but there can be no blueprints or battle plans drawn up in advance; the numerous unknowns in the equation make a one-step solution inconceivable. In more technical language, such goals can be approached only by a stochastic process of successive approximations, trial and error, experiment, and learning through experience. The kind of knowledge required in such endeavors is not deductive knowledge from first principles but rather what Greeks of the classical period called nietis, a concept to which we shall return. Usually translated, inadequately, as "cunning," metis is better understood as the kind of knowledge that can be acquired only by long practice at similar but rarely identical tasks, which requires constant adaptation to changing circumstances.

Metis is not merely the specification of local values (such as the local mean temperature and rainfall) made in order to successfully apply a generic formula to a local case. Taking language as a parallel, I believe that the rule of thumb is akin to formal grammar, whereas metis is more like actual speech. Metis is no more derivative of general rules than speech is derivative of grammar. Speech develops from the cradle by imitation, use, trial and error. Learning a mother tongue is a stochastic process-a process of successive, selfcorrecting approximations. We do not begin by learning the alphabet, individual words, parts of speech, and rules of grammar and then trying to use them all in order to produce a grammatically correct sentence. Moreover, as Oakeshott indicates, a knowledge of the rules of speech by themselves is compatible with a complete inability to speak intelligible sentences.

Longevity: To the Limits and Beyond (Research and Perspectives in Longevity) by Jean-Marie Robine, James W. Vaupel, Bernard Jeune, Michel Allard

Finch' Summary In this essay, I inquire about little explored sources of non-genetic factors in individual life spans that are displayed between individuals with identical genotypes in controlled laboratory environments. The numbers of oocytes found in the ovaries of inbred mice, for example, show a > 5-fold range between individuals. Smaller, but still extensive variations are also indicated for hippocampal neurons. These variations in cell number can be attributed to stochastic processes during organogenesis, i.e. "developmental noise in cell fate determination." They may be of general importance to functional changes during aging, as argued for reproductive senescence in females which is strongly linked to the time of oocyte depletion. More generally, I hypothesize that variations in cell numbers during development result in individual differences in the reserve cell numbers which, in turn, set critical thresholds for dysfunctions and sources of morbidity during aging.

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Finding Alphas: A Quantitative Approach to Building Trading Strategies by Igor Tulchinsky

The disadvantage is that each of these approaches presumes some specific data model. Trend analysis is an example of applications of statistical models in alpha research. In particular, a hidden Markov model is frequently utilized for that purpose, based on the belief that price movements of the stock market are not totally random. In a statistics framework, the hidden Markov model is a composition of two or more stochastic processes: a hidden Markov chain, which accounts for the temporal variability, and an observable process, which accounts for the spectral variability. In this approach, the pattern of the stock market behavior is determined based on these probability values at a particular time. The goal is to figure out the hidden state sequence given the observation sequence, extract the long-term probability distribution, and identify the current trend relative to that distribution.

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Epigenetics Revolution: How Modern Biology Is Rewriting Our Understanding of Genetics, Disease and Inheritance by Nessa Carey

But over decades all these mild abnormalities in gene expression, resulting from a slightly inappropriate set of chromatin modifications, may lead to a gradually increasing functional impairment. Clinically, we don’t recognise this until it passes some invisible threshold and the patient begins to show symptoms. The epigenetic variation that occurs in developmental programming is at heart a predominantly random process, normally referred to as ‘stochastic’. This stochastic process may account for a significant amount of the variability that develops between the MZ twins who opened this chapter. Random fluctuations in epigenetic modifications during early development lead to non-identical patterns of gene expression. These become epigenetically set and exaggerated over the years, until eventually the genetically identical twins become phenotypically different, sometimes in the most dramatic of ways.

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Dark Pools: The Rise of the Machine Traders and the Rigging of the U.S. Stock Market by Scott Patterson

The following ad for Getco, for instance, appeared in January 2012: CHICAGO, IL: Work with inter-disciplinary teams of traders & technologists & use trading models to trade profitably on major electronic exchanges; use statistical & mathematical approaches & develop new models to leverage trading capabilities. Must have Master’s in Math, Statistics, Physical Science, Computer Science, or Engineering w/min GPA of 3.4/4.0. Must have proven graduate level coursework in 2 or more of the following: Stochastic Processes, Statistical Methods, Mathematical Finance, Applied Numerical Methods, Machine Learning. Then, in the summer of 2011, a new contender for the high-frequency crown had emerged. Virtu Financial, the computer trading outfit that counted former Island attorney and Nasdaq executive Chris Concannon as a partner, merged with EWT, a California speed-trading operation that operated on exchanges around the world.

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The Irrational Economist: Making Decisions in a Dangerous World by Erwann Michel-Kerjan, Paul Slovic

The existing literature is based on a completely standard expected utility modelling, whereby the welfare of each future generation is evaluated by computing its expected utility based on a probability distribution for the GDP per capita that it will enjoy. A major difficulty, however, is that these probability distributions are ambiguous, in the sense that they are not based on scientific arguments, or on a database large enough to make them completely objective. Indeed, more than one stochastic process is compatible with existing methods for describing economic growth. The Ellsberg paradox tells us that most human beings are averse to ambiguity, which means that they tend to overestimate the probability of the worst-case scenario when computing their subjective expected utility. This suggests that agents systematically violate Savage’s “Sure Thing Principle” (Savage, 1954). More precisely, it seems that the way we evaluate uncertain prospects depends on how precise our information about the underlying probabilities is.

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

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 .

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Rage Inside the Machine: The Prejudice of Algorithms, and How to Stop the Internet Making Bigots of Us All by Robert Elliott Smith

v=GGgiGtJk7MA 2The Pew Research Center, 2017, Public Trust in Government: 1958–2017, www.people-press.org/2017/12/14/public-trust-in-government-1958-2017/ 3Gallup, 2018, Confidence in Institutions, https://news.gallup.com/poll/1597/confidence-institutions.aspx 4This in fact led to a protracted conversation on the difference between UK, European and American methods of presenting odds, which led to a wasted afternoon of my graduate studies, a sleepless night working all the relationships out, an inferior mid-term exam score in my Stochastic Processes course and hard work to get an A in the end. So I have omitted this for the reader’s benefit. 5Colin E. Beech, 2008, The Grail and the Golem: The Sociology of Aleatory Artifacts. PhD dissertation. Rensselaer Polytechnic Institute, Troy, NY. Advisor(s) Sal Restivo. AAI3342844. https://dl.acm.org/citation.cfm?id=1627267 6Prakash Gorroochurn, 2012, Some Laws and Problems of Classical Probability and How Cardano Anticipated Them.

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

A straightforward adaptation of a clustering method for outlier detection can be very costly, and thus does not scale up well for large data sets. Clustering-based outlier detection methods are discussed in detail in Section 12.5. 12.3. Statistical Approaches As with statistical methods for clustering, statistical methods for outlier detection make assumptions about data normality. They assume that the normal objects in a data set are generated by a stochastic process (a generative model). Consequently, normal objects occur in regions of high probability for the stochastic model, and objects in the regions of low probability are outliers. The general idea behind statistical methods for outlier detection is to learn a generative model fitting the given data set, and then identify those objects in low-probability regions of the model as outliers. However, there are many different ways to learn generative models.

The kernel density approximation of the probability density function is(12.9) where K() is a kernel and h is the bandwidth serving as a smoothing parameter. Once the probability density function of a data set is approximated through kernel density estimation, we can use the estimated density function to detect outliers. For an object, o, gives the estimated probability that the object is generated by the stochastic process. If is high, then the object is likely normal. Otherwise, o is likely an outlier. This step is often similar to the corresponding step in parametric methods. In summary, statistical methods for outlier detection learn models from data to distinguish normal data objects from outliers. An advantage of using statistical methods is that the outlier detection may be statistically justifiable.

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Indianapolis, IN: Wiley. â•‡ [2] Peterson, E. 2004. Web Analytics Demystified: A Marketer’s Guide to Understanding How Your Web Site Affects Your Business. New York: Celilo Group Media. â•‡ [3] Pedrick, J. H. and Zufryden, F. S. 1991. “Evaluating the Impact of Advertising Media Plans: A Model of Consumer Purchase Dynamics Using Single Source Data.” Marketing Science, vol. 10(2), pp. 111–130. â•‡ [4] Penniman, W. D. 1975. “A Stochastic Process Analysis of Online User Behavior.” In The Annual Meeting of the American Society for Information Science, Washington, DC, pp. 147–148. â•‡ [5] Meister, D. and Sullivan, D. 1967. “Evaluation of User Reactions to a Prototype On-Line Information Retrieval System: Report to NASA by the Bunker-Ramo Corporation. Report Number NASA CR-918.” Bunker-Ramo Corporation, Oak Brook, IL. â•‡ [6] Directors, A.

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Doing Data Science: Straight Talk From the Frontline by Cathy O'Neil, Rachel Schutt

The degrees themselves aren’t giving us a real understanding of how interconnected a given node is, though, so in the next iteration, add the degrees of all the neighbors of a given node, again scaled. Keep iterating on this, adding degrees of neighbors one further step out each time. In the limit as this iterative process goes on forever, we’ll get the eigenvalue centrality vector. A First Example of Random Graphs: The Erdos-Renyi Model Let’s work out a simple example where a network can be viewed as a single realization of an underlying stochastic process. Namely, where the existence of a given edge follows a probability distribution, and all the edges are considered independently. Say we start with nodes. Then there are pairs of nodes, or dyads, which can either be connected by an (undirected) edge or not. Then there are possible observed networks. The simplest underlying distribution one can place on the individual edges is called the Erdos-Renyi model, which assumes that for every pair of nodes , an edge exists between the two nodes with probability .

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The Theory That Would Not Die: How Bayes' Rule Cracked the Enigma Code, Hunted Down Russian Submarines, and Emerged Triumphant From Two Centuries of Controversy by Sharon Bertsch McGrayne

Venter, Gary G. (fall 1987) Credibility. CAS Forum 81–147. Chapter 7. From Tool to Theology Armitage P. (1994) Dennis Lindley: The first 70 years. In Aspects of Uncertainty: A Tribute to D. V. Lindley, eds., PR Freeman and AFM Smith. John Wiley and Sons. Banks, David L. (1996) A Conversation with I. J. Good. Statistical Science (11) 1–19. Dubins LE, Savage LJ. (1976) Inequalities for Stochastic Processes (How to Gamble If You Must). Dover. Box, George EP, et al. (2006) Improving Almost Anything. Wiley. Box GEP, Tiao GC. (1973) Bayesian Inference in Statistical Analysis. Addison-Wesley. Cramér, H. (1976). Half of a century of probability theory: Some personal recollections. Annals of Probability (4) 509–46. D’Agostini, Giulio. (2005) The Fermi’s Bayes theorem. Bulletin of the International Society of Bayesian Analysis (1) 1–4.

The Trade Lifecycle: Behind the Scenes of the Trading Process (The Wiley Finance Series) by Robert P. Baker

The market risk calculation is in theory attempting to replicate every possible combination of market data. Some simplifications have to be made because each piece of market data is technically a random variable and its connection (or correlation) to other market data is very hard, if not impossible, to determine. 3. Decide the calculation methodology There are two basic approaches – stochastic or historical. Stochastic processes If a piece of market data is assumed to be a normal distribution then we can ascribe different probabilities to different values. For example: 1% probability of 140, 5% probability of 155, 50% probability of 182 and so on. This removes the need for a large amount of data but ignores correlation between different market data. Historical data We go back over a certain period of market data, apply every day-on-day change in all market data to the set of trades under examination and for each day we get a different total value.

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Narrative Economics: How Stories Go Viral and Drive Major Economic Events by Robert J. Shiller

., 2007. 7. Long et al., 2008. 8. JSTOR catalogs over nine million scholarly articles and books in all fields, and 7% of these are in business or economics, but 25% of the articles with “ARIMA,” “ARMA,” or “autoregressive” are in business or economics. 9. Moving average models are sometimes justified by reference to the Wold decomposition theorem (1954), which shows that any covariance stationary stochastic process can be modeled as a moving average of noise terms plus a deterministic component. But there is no justification for assuming that simple variants of ARIMA models are so general. We may be better able to do economic forecasting in some cases if we represent these error terms or driving variables as the result of co-epidemics of narratives about which we have some information. 10. See Nsoesie et al., 2013. 11.

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Networks, Crowds, and Markets: Reasoning About a Highly Connected World by David Easley, Jon Kleinberg

How vicious are cycles of intransitive choice? Theory and Decision, 24:119–145, 1988. [41] Albert-László Barabási and Réka Albert. Emergence of scaling in random networks. Science, 286:509–512, 1999. [42] Albert-László Barabási and Zoltan Oltvai. Network biology: Understanding the cell’s functional organization. Nature Reviews Genetics, 5:101–113, 2004. [43] A. D. Barbour and D. Mollison. Epidemics and random graphs. In Stochastic Processes in Epidemic Theory, volume 86 of Lecture Notes in Biomathematics, pages 86–89. Springer, 1990. [44] John A. Barnes. Social Networks. Number 26 in Modules in Anthropology. Addison Wesley, 1972. [45] Chris Barrett and E. Mutambatsere. Agricultural markets in developing countries. In Lawrence E. Blume and Steven N. Durlauf, editors, The New Palgrave Dictionary of Economics. Oxford University Press, second edition, 2008

Mathematics of Operations Research, 28(2):294–308, 2003. [239] Peter D. Killworth and H. Russell Bernard. Reverse small world experiment. Social Networks, 1:159–192, 1978. [240] Peter D. Killworth, Eugene C. Johnsen, H. Russell Bernard, Gene Ann Shelley, and Christopher McCarty. Estimating the size of personal networks. Social Networks, 12(4):289–312, December 1990. [241] John F. C. Kingman. The coalescent. Stochastic Processes and their Applications, 13:235–248, 1982. [242] Aniket Kittur and Robert E. Kraut. Harnessing the wisdom of crowds in Wikipedia: Quality through coordination. In Proc. CSCW’08: ACM Conference on Computer-Supported Cooperative Work, 2008. BIBLIOGRAPHY 817 [243] Jon Kleinberg. Authoritative sources in a hyperlinked environment. Journal of the ACM, 46(5):604–632, 1999. A preliminary version appears in the Proceedings of the 9th ACM-SIAM Symposium on Discrete Algorithms, Jan. 1998

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The Age of Spiritual Machines: When Computers Exceed Human Intelligence by Ray Kurzweil

Engines of Change: The American Industrial Revolution, 1790-1860. Washington, D.C.: Smithsonian Institution Press, 1986. Hoage, R. J. and Larry Goldman. Animal Intelligence: Insights into the Animal Mind. Washington, D.C.: Smithsonian Institution Press, 1986. Hodges, Andrew. Alan Turing: The Enigma. New York: Simon and Schuster, 1983. Hoel, Paul G., Sidney C. Port, and Charles J. Stone. Introduction to Stochastic Processes. Boston: Houghton-Mifflin, 1972. Hofstadter, Douglas R. Gödel, Escher, Bach: An Eternal Golden Braid. New York: Basic Books, 1979. _________. Metamagical Themas: Questing for the Essence of Mind and Pattern. New York: Basic Books, 1985. Hofstadter, Douglas R. and Daniel C. Dennett. The Mind’s I: Fantasies and Reflections on Self and Soul. New York: Basic Books, 1981. Hofstadter, Douglas R., Gray Clossman, and Marsha Meredith.

Principles of Protocol Design by Robin Sharp

For example, you might like to pursue all the references to the Alternating Bit Protocol in the literature, starting with the ones given in connection with Protocol 5. This will lead you into the area of other proof techniques for protocols, as well as illustrating how new mechanisms develop as time goes by. Finally, you might like to investigate quantitative properties of some protocols, such as their throughput and delay in the presence of varying loads of traffic. Generally speaking, this requires a knowledge of queueing theory and the theory of stochastic processes. This is not a subject which we pay more than passing attention to in this book. However, some protocols, especially multiplexing protocols, have been the subject of intensive investigation from this point of view. Good discussions of the general theory required are found in [73], while [11] relates the theory more explicitly to the analysis of network protocols. 118 4 Basic Protocol Mechanisms Exercises 4.1.

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Hands-On Machine Learning With Scikit-Learn and TensorFlow: Concepts, Tools, and Techniques to Build Intelligent Systems by Aurélien Géron

Whereas PG algorithms directly try to optimize the policy to increase rewards, the algorithms we will look at now are less direct: the agent learns to estimate the expected sum of discounted future rewards for each state, or the expected sum of discounted future rewards for each action in each state, then uses this knowledge to decide how to act. To understand these algorithms, we must first introduce Markov decision processes (MDP). Markov Decision Processes In the early 20th century, the mathematician Andrey Markov studied stochastic processes with no memory, called Markov chains. Such a process has a fixed number of states, and it randomly evolves from one state to another at each step. The probability for it to evolve from a state s to a state s′ is fixed, and it depends only on the pair (s,s′), not on past states (the system has no memory). Figure 16-7 shows an example of a Markov chain with four states. Suppose that the process starts in state s0, and there is a 70% chance that it will remain in that state at the next step.

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

That kind of wave function, concentrated entirely on a single possible observational outcome, is known as an “eigenstate.” Once the system is in that eigenstate, you can keep making the same kind of observation, and you’ll keep getting the same answer (unless something kicks the system out of the eigenstate into another superposition). We can’t say with certainty which eigenstate the system will fall into when an observation is made; it’s an inherently stochastic process, and the best we can do is assign a probability to different outcomes. We can apply this idea to the story of Miss Kitty. According to the Copenhagen interpretation, our choice to observe whether she stopped by the food bowl or the scratching post had a dramatic effect on her wave function, no matter how sneaky we were about it. When we didn’t look, she was in a superposition of the two possibilities, with equal amplitude; when she then moved on to the sofa or the table, we added up the contributions from each of the intermediate steps, and found there was interference.

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Cooking for Geeks by Jeff Potter

It’s possible to break up the collagen chemically, too: lysosomal enzymes will attack the structure and "break the covalent bonds" in chem-speak, but this isn’t so useful to know in the kitchen. Note For fun, try marinating a chunk of meat in papaya, which contains an enzyme, papain, that acts as a meat tenderizer by hydrolyzing collagen. One piece of information that is critical to understand in the kitchen, however, is that hydrolysis takes time. The structure has to literally untwist and break up, and due to the amount of energy needed to break the bonds and the stochastic processes involved, this reaction takes longer than simply denaturing the protein. Hydrolyzing collagen not only breaks down the rubbery texture of the denatured structure, but also converts a portion of it to gelatin. When the collagen hydrolyzes, it breaks into variously sized pieces, the smaller of which are able to dissolve into the surrounding liquid, creating gelatin. It’s this gelatin that gives dishes such as braised ox tail, slow-cooked short ribs, and duck confit their distinctive mouthfeel.

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

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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Debunking Economics - Revised, Expanded and Integrated Edition: The Naked Emperor Dethroned? by Steve Keen

The impact of this power inversion can be seen in the physicist Joe McCauley’s observations about the need to reform economics education: The real problem with my proposal for the future of economics departments is that current economics and finance students typically do not know enough mathematics to understand (a) what econophysicists are doing, or (b) to evaluate the neo-classical model (known in the trade as ‘The Citadel’) critically enough to see, as Alan Kirman put it, that ‘No amount of attention to the walls will prevent The Citadel from being empty.’ I therefore suggest that the economists revise their curriculum and require that the following topics be taught: calculus through the advanced level, ordinary differential equations (including advanced), partial differential equations (including Green functions), classical mechanics through modern nonlinear dynamics, statistical physics, stochastic processes (including solving Smoluchowski–Fokker–Planck equations), computer programming (C, Pascal, etc.) and, for complexity, cell biology. Time for such classes can be obtained in part by eliminating micro- and macro-economics classes from the curriculum. The students will then face a much harder curriculum, and those who survive will come out ahead. So might society as a whole. (McCauley 2006: 607–8) This amplifies a point that, as a critic of economics with a reasonable grounding in mathematics myself, has long set me apart from most other critics: neoclassical economics is not bad because it is mathematical per se, but because it is bad mathematics. 16 | DON’T SHOOT ME, I’M ONLY THE PIANO Why mathematics is not the problem Many critics of economics have laid the blame for its manifest failures at the feet of mathematics.

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The Transhumanist Reader by Max More, Natasha Vita-More

If we estimate about 102 bytes of information to encode these details (which may be low), we have 1016 bytes, considerably more than the 109 bytes that you mentioned. One might ask: How do we get from 107 bytes that specify the brain in the genome to 1016 bytes in the mature brain? This is not hard to understand, since we do this type of meaningful data expansion routinely in our self-organizing software paradigms. For example, a genetic ­algorithm can be efficiently coded, but in turn creates data far greater in size than itself using a stochastic process, which in turn self-organizes in response to a complex environment (the problem space). The result of this process is meaningful information far greater than the original program. We know that this is exactly how the creation of the brain works. The genome specifies initially semi-random interneuronal connection wiring patterns in specific regions of the brain (random within certain constraints and rules), and these patterns (along with the ­neurotransmitter-concentration levels) then undergo their own internal evolutionary process to self-organize to reflect the interactions of that person with their experiences and environment.

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Unelected Power: The Quest for Legitimacy in Central Banking and the Regulatory State by Paul Tucker

FRAMING A STANDARD FOR SYSTEM RESILIENCE: POLITICS, TRADE-OFFS, AND PUBLIC DEBATE If the public policy purpose of a central banking stability mandate should be continuity of services from the system as a whole, thus avoiding the worst costs of “bust,” the core of the regime must be a monitorable standard of resilience. That much is entailed by the first Design Precept, cast as a revived “nondelegation doctrine” in part III (chapter 14). The big questions are what it means in principle and in practice. Roughly speaking, policy makers need to determine the severity of shock that the system should be able to withstand. In principle, that would be driven by three things: A view of the underlying (stochastic) process generating the first-round losses from end borrowers that hit the system A picture (or model) of the structure of the financial system through which those losses and other shocks are transmitted around the system A tolerance for systemic crisis The first and second are properly objects of scientific inquiry by technocrats and researchers. The third is different. Whereas the central belief of monetary economics relevant to the design of policy institutions is that there is no long-run trade-off to speak of between economic activity and inflation, we do not yet know enough to judge whether prosperity would be damaged by totally eliminating the risk-taking structures that can threaten periodic bouts of instability.6 As I recollect former UK Treasury secretary George Osborne putting it, no one wants the stability of the graveyard.

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The Elements of Statistical Learning (Springer Series in Statistics) by Trevor Hastie, Robert Tibshirani, Jerome Friedman

Bayesian Data Analysis, CRC Press, Boca Raton, FL. Geman, S. and Geman, D. (1984). Stochastic relaxation, Gibbs distributions and the Bayesian restoration of images, IEEE Transactions on Pattern Analysis and Machine Intelligence 6: 721–741. Genkin, A., Lewis, D. and Madigan, D. (2007). Large-scale Bayesian logistic regression for text categorization, Technometrics 49(3): 291–304. Genovese, C. and Wasserman, L. (2004). A stochastic process approach to false discovery rates, Annals of Statistics 32(3): 1035–1061. Gersho, A. and Gray, R. (1992). Vector Quantization and Signal Compression, Kluwer Academic Publishers, Boston, MA. Girosi, F., Jones, M. and Poggio, T. (1995). Regularization theory and neural network architectures, Neural Computation 7: 219–269. Golub, G. and Van Loan, C. (1983). Matrix Computations, Johns Hopkins University Press, Baltimore.

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

Suppose each arc e of G has been assigned a probability p(e), where the probabilities satisfy the conditions 0 < p(e) < 1; ^ p(e) = 1 for 1 < j < n. init(e)=Vj Consider a random path, which starts at V\ and subsequently chooses branch e of G with probability p(e), until Vn is reached; the choice of branch taken at each step is to be independent of all previous choices. 2.3.4.2 ORIENTED TREES 381 For example, consider the graph of exercise 2.3.4.1-7, and assign the respective probabilities 1, \, \, |, 1, f, \, \, \ to arcs ei, e2,.. •, e9. Then the path "Start-A- B-C-A-D-B-C-Stop" is chosen with probability l-|-l-|-|-|-l-i = tIs- Such random paths are called Markov chains, after the Russian mathematician Andrei A. Markov, who first made extensive studies of stochastic processes of this kind. The situation serves as a model for certain algorithms, although our requirement that each choice must be independent of the others is a very strong assumption. The purpose of this exercise is to analyze the computation time for algorithms of this kind. The analysis is facilitated by considering the n x n matrix A — (aij), where aij = ^2p{e) summed over all arcs e that go from Vi to Vj.

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The Codebreakers: The Comprehensive History of Secret Communication From Ancient Times to the Internet by David Kahn

N.S.A. leads even such firms as I.B.M. and Remington Rand in important areas of computer development, such as time-sharing, and industry has adopted many N.S.A.-designed features. The second section, STED (for “Standard Technical Equipment Development”) conducts basic cryptographic research. It looks for new principles of encipherment. It ascertains whether new developments in technology, such as the transistor and the tunnel diode, have cryptographic applications. Using such esoteric tools as Galois field theory, stochastic processes, and group, matrix, and number theory, it will construct a mathematical model of a proposed cipher machine and will simulate its operation on a computer, thus producing the cipher without having to build the hardware. Rotor principles have often been tested for cryptographic strength in this way. It devises audio scramblers, from the ultra-secure types for high officials to the walkie-talkies of platoon commanders, as well as video scramblers for reconnaissance television and for facsimile.