martingale Concepts

Mathematical Finance: Core Theory, Problems and Statistical Algorithms

by Nikolai Dokuchaev · 24 Apr 2007

(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

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defined by the set of probability distributions of its components. Probability distributions on infinite dimensional spaces are commonly used in the studied in theory of stochastic processes. For example, the Wiener process Chapter 4 is a random (infinity-dimensional) vector with values at the space C(0,T) of continuous functions f

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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

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a discrete time white noise, and let t=0, 1, 2,…. Then the process ηt is said to be a random walk. The theory of stochastic processes studies their pathwise properties (or properties of trajectories ξ(t, ω) for given ω, as well as the evolution of the probability distributions. © 2007 Nikolai

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Example 2.12 Let Ω={ω1,ω2,ω3}, ξt, t=0, 1,… such that © 2007 Nikolai Dokuchaev Consider a discrete time random process Basics of Stochastic Processes Let us find again the filtration Let 19 generated by the process ξt, t=0, 1, 2. denote the σ-algebra generated by the random

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process ξ(t), it is true under some additional conditions; it suffices to require that ξ(t) is pathwise continuous). © 2007 Nikolai Dokuchaev Basics of Stochastic Processes 21 2.4 Markov processes Definition 2.23 Let ξ(t) be a process, and let say that ξ(t) is a Markov (Markovian) process

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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

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unlimited speed. This fact is linked with non-differentiability of y(t). © 2007 Nikolai Dokuchaev Basics of Ito Calculus and Stochastic Analysis 67 4.6 Martingale representation theorem is the filtration In this section, we assume that w(t) is an n-dimensional vector process, generated by w(t), and a

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continuous time market models. 8.1 Some basic facts about discrete time random processes In this section, several additional definitions and facts about discrete time stochastic processes are given. Definition 8.1 A process ξt is said to be stationary (or strict-sense stationary), if the does not depend on m for

Derivatives Markets

by David Goldenberg · 2 Mar 2016 · 819pp · 181,185 words

E(W1(ω)|W0)=W0 and, E(W2(ω)|W1)=W1. 15.3.2 Definition of a Discrete-Time Martingale A discrete-time stochastic process (Xn(ω))n=0,1,2,3,.. is called a martingale if, 1. E(Xn)<∞ and for all n and, 2. E(Xn+1(ω)|Xn)=Xn for all

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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

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all n=0,1,2,3,… 2. A discrete-time stochastic process (Xn(ω))n=0

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in applied finance. We restrict attention to discrete-time martingales and sometimes even choose N=2. No attempt at mathematical rigor is claimed. The intuition behind these results is the primary concern. We start with a discrete-time stochastic process (Xn(ω)n=0,1,2,3,… with finite first and second

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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

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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

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zero, because the stock price has effectively ceased to exist. In other words, zero is an absorbing boundary. However, ABM is still the most basic stochastic process of its kind, so not too much should initially be made of this defect. In particular, it is possible to generate an option pricing formula

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density function of XT(ω), given X0. 16.3.2 Transition Density Functions In general, the transition density function describes the probabilistic evolution of a stochastic process from a known position x=X0 assumed at time t to random positions y=yT(ω)=XT(ω) assumed at time T. Let τ denote

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f(x,t) with respect to t. So far so good. Now for a huge jump. 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

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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

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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 process is the derived process, which is a sufficiently smooth

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which assumes that the stock price process St resembles a standard GBM, with the twist that σt is not only time-dependent, but is a stochastic process itself. The risk-neutral form of the Heston model is, where the variance, is itself a process defined by, Note that Zt is a new

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the constant σ assumption, except to throw out Black–Scholes’ assumption of a stationary log-normal diffusion, and search for a viable (smile-consistent) underlying stochastic process among the vast set of alternatives, many of which will lead to incomplete markets. Black–Scholes and its modifications, however, still have tremendous appeal, especially

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differential equations (SDEs) 553, 559, 562–3, 564, 566, 567–8, 570, 571, 583; stochastic integral equations (SIEs) 559, 560, 561, 564, 565–6, 567; stochastic processes 540–1, 543, 562, 587, 588; transition density function for shifted arithmetic Brownian motion 545–6; Wiener measure (and process) 540–1 option sellers 328

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differential equations (SDEs) 553, 559, 562–3, 564, 566, 567–8, 570, 571, 583 stochastic integral equations (SIEs) 559, 560, 561, 564, 565–6, 567 stochastic processes 540–1, 543, 562, 587, 588 stock forwards when stock pays dividends 88–90 stock index futures 225–30; commentary 230; futures contracts, introduction of

Handbook of Modeling High-Frequency Data in Finance

by Frederi G. Viens, Maria C. Mariani and Ionut Florescu · 20 Dec 2011 · 443pp · 51,804 words

6.1 Introduction In recent years, a growing concern is the presence of long-term memory effects in ﬁnancial time series. The empirical characterization of stochastic processes usually requires the study of temporal correlations and the determination of asymptotic probability density function (pdfs). Major stock indices in developed countries have been previously

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consequence that the stable Levy processes have inﬁnite variance. To avoid the problems arising in the inﬁnite second moment, Mantegna and Stanley [19] considered a stochastic process with ﬁnite variance that follows scale relations called TLF . The TLF distribution is deﬁned by ⎧ ⎪ x>l ⎨0 (6.6) P(x) = cPL (x) −l

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physics and economic ﬂuctuations: do outliers exist? 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

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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. Scale-invariant truncated Levy process. Europhys Lett 2000;52

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models were introduced in order to model this observed random behavior of the market volatility. Under such a model, the volatility is described by a stochastic process. Among the ﬁrst models in the literature were these by Taylor (1986) and Hull and White (1987), under which the volatility dynamics are described by

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with application in continuous-time ﬁnancial models. J Appl Probab 2004;41:467–482. Gao J, Anh V, Heyde C, Tieng Q. Parameter estimation of stochastic processes with long-range dependence and intermittency. J Time Anal 2001;22:527–535. Geweke J, Porter-Hudak S. The estimation and application of long-memory

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). (10.39) W 1 is a Brownian motion on a ﬁltered probability space(, (Fs )s∈[0,T ] , P) and T σ is a continuous adapted stochastic process such that E[ 0 σ 4 (s)ds] < ∞. We will assume that σ and W 1 are independent processes. The one period, say [t − 1

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to ﬁnd an analytical solution for a class of stochastic differential equations (SDEs) that arises in population models. Consider the following nonlinear SDE for the stochastic process X = {Xt : t ∈ T }: dXt = rXt (k − Xtm )dt + βXt dBt (12.1) Handbook of Modeling High-Frequency Data in Finance, First Edition. Edited by

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processes: d(Xt Yt ) = Xt dYt + Yt dXt + dXt dYt . This formula is also called integration by parts, or derivative of a product for two stochastic processes. The proof of this identity is done by using the two dimensional Ito’s formula below, and (dXt )2 , (dYt )2 , dXt dYt are computed

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of ﬁnancial indices or asset prices when a market crash takes place. For these ﬁnancial data are more appropriated other models, like the Levy—like stochastic processes. In order to introduce these models, we ﬁrst present a brief introduction of Stable distributions. Consider the sum of n independent identically distributed (i.i

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=√ P2 [X (2t)] = √ . exp exp √ √ 8γ 8πγ 2π( 2σ ) 2( 2σ )2 That is, the variance is now σ22 = 2σ 2 . So, two stable stochastic processes exist: Lorentzian and Gaussian, and in both cases, their Fourier transform has the form ϕ(q) = exp(−γ |q|α ) with α = 1 for the

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Levy processes with α < 2 have inﬁnite variance. In order to avoid the problems arising in the inﬁnite second moment Mantegna and Stanley considered a stochastic process with ﬁnite variance that follows scale relations called Truncated Levy ﬂight (TLF) [8]. The TLF distribution is deﬁned by T (x) = cP(x)χ(−l

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behavior was compatible with a slow convergence to a Gaussian distribution but it was not possible to conclude that the Levy distribution was the appropriated stochastic process for explaining 346 CHAPTER 12 Stochastic Differential Equations and Levy Models the ﬁnancial indices evolution. The authors believe that the standardized Levy model that was

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is exactly the way in which a market crash is deﬁned: A market crash is an outlying point in the cumulative probability distribution of the stochastic process described by the Levy model. It should be noted that outlying points exactly reﬂect the crash of the corresponding ﬁnancial indices/stocks. 12.5.1

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: 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

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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

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allowed the volatility to be nonconstant or a stochastic variable. In this model, the underlying security S follows, as in the Black–Scholes model, a stochastic process dSt = μSt dt + σt St dZt , where Z is a standard Brownian motion. Unlike the classical model, the variance v(t) = σ 2 (t) also

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follows a stochastic process given by √ dvt = κ(θ − v(t))dt + γ vt dWt , where W is another standard Brownian motion. The correlation coefﬁcient between W and Z

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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

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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

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) − 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). Also, we assume the option has the ﬁnal payoff C(S, T ) = max(ST − K , 0), (13.29) where

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instance, in [10] a model with stochastic volatility is proposed. In this model the underlying security S follows, as in the Black–Scholes model, a stochastic process dS = μS dt + σ S dX1 , where X1 is a standard Brownian motion. Unlike the classical model, the variance v(t) = σ 2 (t) also

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follows a stochastic process given by dv = κ(θ − v(t)) dt + γ √ v dX2 , where X2 is another standard Brownian motion. The correlation coefﬁcient between X1 and X2

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, 336, 346 Lévy ﬂight, 125 Lévy ﬂight models, 336–340 Lévy ﬂight parameter estimating, 135 values of, 136, 138 Lévy-like stochastic process, 364 Lévy market, integro-differential equations in, 375–380 Lévy model(s), 4–5 for describing log returns, 22 log return process increments

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abnormal price movements, 45 Market-traded option prices, 219 Markov chain, stochastic volatility process with, 401 Markowitz-type optimization, 286 Martingale-difference process, 178. See also Continuous semimartingales; Equivalent martingale measure; Exponential martingale process Supermartingale Matlab, 14, 257 Matlab module, 125, 339 Maximum likelihood estimation (MLE), 13–14, 185 Index ﬁnite-sample performance

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–334 Stochastic differential equation solution, Lévy ﬂight parameter for, 340 Stochastic-Dirichlet problem, 317 Stochastic function of time, 245 Stochastic order ﬂow process, 237 Stochastic processes, 352, 400 empirical characterization of, 119 Lévy-like, 364 Stochastic recurrence equation (SRE), 179 Stochastic variable, 129 Stochastic volatility, 348, 354 ﬁnancial models with

The Concepts and Practice of Mathematical Finance

by Mark S. Joshi · 24 Dec 2003

8 3.9 4 4.9 5 6 Exercises The Ito calculus Introduction 5.1 Brownian motion 5.2 5.3 Quadratic variation 5.4 Stochastic processes 5.5 Ito's lemma 5.6 Applying Ito's lemma 5.7 An informal derivation of the Black-Scholes equation 5.8 Justifying

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therefore effectively T. This is vastly different from a differentiable function, and it expresses just how much more jagged Brownian motion paths are. 5.4 Stochastic processes Whilst Brownian motion is extremely interesting in its own right, it does not make the best model for stock movements since the probability of negative

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even depend on Xt. The simplest generalization one could try would be to let it and or be piecewise constant functions of time. 5.4 Stochastic processes 103 Suppose we want to know how Xt - X,s is distributed. Suppose the interval [s, t] is divided into [s, t1], [t1, t2], ...,

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). The important thing here is that the second and fourth terms are small as h goes to zero. To see this, note that 5.4 Stochastic processes 105 the second term is o(h) since e(t) = 0 and e is continuous because µ is. Moreover, we can rewrite g as 1

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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

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. 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. Fortunately,

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return to multiple time horizons and view stochastic processes as the slow revelation of a single path drawn in advance. The concept of information is examined in this context, and defined using filtrations. In the discrete setting, we define expectations conditioned on information. A martingale is defined to be a process such

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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

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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

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world. Exercise 6.24 If Wt is a Brownian motion, is Wt a martingale? Justify your answer. Exercise 6.25 Give an example of a continuous time stochastic process, Xt, such that IE(Xt) = 0, and X t is not a martingale. Exercise 6.26 If S and B follow Black-Scholes assumptions, what

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nodes. This is crucial in the tractability of the method in that it means we have to do N2 calculations. If the drift of our stochastic process was state-dependent, or the volatility was time-dependent, then an up-move followed by a down-move would not be the same as

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neutral measures? Invoking the multi-dimensional version of Gir- sanov's theorem, changing the measure will allow us to change the drifts of the two stochastic processes but will allow nothing else. If we are working in a deterministic interest-rate world with constant continuous compounding rate r then the risidess bond

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is really drawing a time-dependent function V (t) = a(t)2 for instantaneous variance. We are then drawing a path for S under the stochastic process dS S = rdt + a(t)dW. (16.8) We know that the terminal distribution for S under such a process is given by ST _

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References 529 [68] J.M. Harrison, D.M. Kreps, Martingales and arbitrage in multi-period securities markets, Journal of Economic Theory 20, 1979, 381-408. [69] J.M. Harrison, S.R. Pliska, Martingales and stochastic integration in the theory of continuous trading, Stochastic Processes and Applications 11, 1981, 215-60. [70] J.M. Harrison

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, S.R. Pliska, Martingales and stochastic integration in the theory of continuous trading, Stochastic Processes and Applications 13, 1983, 313-16. [71] D. Heath, R

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replication, see replication, static stepping methods for Monte Carlo, 439 stochastic, 433 stochastic calculus, 97 stochastic differential equation, 105 for square of Brownian motion, 107 stochastic process, 102-106, 141 stochastic volatility, 88, 389 and risk-neutral pricing, 390-393 implied, 400 pricing by Monte Carlo, 391-394 pricing by PDE and

Mathematical Finance: Theory, Modeling, Implementation

by Christian Fries · 9 Sep 2007

getProcessValue() getNumeraireValue() LogNormalProcess BrownianMotion getProcessValue(timeIndex, component) getInitialValue(component) getDrift(timeIndex, component) getVolatility(timeIndex, component) getBrownianIncrement(timeIndex, path, factor) Mathematical Finance Theory, Modeling and Implementation Stochastic Processes, Interest Rate Models, Hybrid Models, Numerical Methods, Object Oriented Implementation Christian P. Fries 13th December 2006 preliminary, uncorrected and incomplete version (public beta) 2 This

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. . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1. Abridged Versions . . . . . . . . . . . . . . . . . . . . . . . 1.1.2. Special Sections . . . . . . . . . . . . . . . . . . . . . . . . 21 22 22 23 I. 25 Foundations 2. Foundations 2.1. Probability Theory . . . . . . . . . . . . . . . . 2.2. Stochastic Processes . . . . . . . . . . . . . . . 2.3. Filtration . . . . . . . . . . . . . . . . . . . . . 2.4. Brownian Motion . . . . . . . . . . . . . . . . . 2.5. Wiener Measure, Canonical Setup . . . . . . . . 2.6. Itô Calculus . . . . . . . . . . . . . . . . . . . . 2.6.1. Itô Integral . . . . . . . . . . . . . . . . 2.6.2

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been keept as low as possible. Chapter 2 gives the foundations in the order of their dependence. The reader familiar with the concepts of stochastic processes and martingales may skip the chapter and use it as reference only. To get a feeling for the mathematical concepts one should read the special sections Interpretation

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. ©2004, 2005, 2006 Christian Fries Version 1.3.19 [build 20061210]- 13th December 2006 http://www.christian-fries.de/finmath/ Part I. Foundations: Probability Theory, Stochastic Processes and Risk Neutral Evaluation 25 This work is licensed under a Creative Commons License. http://creativecommons.org/licenses/by-nc-nd/2.5/deed.en

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.en Comments welcome. ©2004, 2005, 2006 Christian Fries Version 1.3.19 [build 20061210]- 13th December 2006 http://www.christian-fries.de/finmath/ 2.2. STOCHASTIC PROCESSES    if ω ∈ F E(X|F) • For C = {∅, F, Ω \ F, Ω} we have X|C (ω) =  ,  E(X|Ω \ F) if ω ∈ Ω \

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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

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in Figure 2.1 and 2.3. C| 2.4. Brownian Motion Definition 23 (Brownian Motion): q n Let W : [0, ∞) × Ω → R denote a stochastic process with the following properties: 37 This work is licensed under a Creative Commons License. http://creativecommons.org/licenses/by-nc-nd/2.5/deed.en

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. This theorem also gives a construction of the Brownian motion. Tip (time discrete realizations): In the following we will often consider the realizations of a stochastic process at discrete times 0 = T 0 < T 1 < . . . < T N only (e.g. this will be the case when we consider the implementation). If we

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the Brownian motion may be found in [12]. 2.6. Itô Calculus Motivation: The Brownian motion W is our first encounter with an important continuous stochastic process. The Brownian motion may be viewed as the limit of a scaled random walk.11 If we interpret the Brownian motion W in this sense

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): W(T i ) = i−1 X ∆W(T j ). j=0 Using the increments ∆W(T j ) we may define a whole family of discrete stochastic processes. We give an step by step introduction and use the illustrative interpretation of a particle movement: First we assume that the particle may lose energy

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) could depend on the paths, i.e. are assumed to be random variables. This might appear odd since then one could create any time discrete stochastic process.12 However it would make sense to allow that the parameters µ(T j ) and/or σ(T j ) (used in the increment from X(T

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1.3.19 [build 20061210]- 13th December 2006 http://www.christian-fries.de/finmath/ CHAPTER 2. FOUNDATIONS Definition 31 (Itô integral for elementary processes): A stochastic process φ is called elementary, if X φ(t, ω) = e j (ω) · 1(t j ,t j+1 ] (t), q j∈N∪{0} where {t

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Let W = (W1 , . . . , Wm ) denote an m-dimensional Brownian motion defined on (Ω, F , P). Let σi, j (i = 1, . . . , n, j = 1, . . . , m) denote stochastic processes belonging to the class of integrands of the Itô integral (compare Definition 34) with ! Z t 2 P σi, j (τ, ω) dτ < ∞ ∀t ≥ 0

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]- 13th December 2006 http://www.christian-fries.de/finmath/ 2.7. BROWNIAN MOTION WITH INSTANTANEOUS CORRELATION fi, j (i = 1, . . . , n, j = 1, . . . , m) denote stochastic processes belonging to the class of integrands of the Itô integral (see Definition 34 and Definition 37) with ! Z t 2 P fi, j (τ, ω

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set of factors to the relevant ones is discussed in Appendix A.4 and A.5. C| 2.8. Martingales Definition 46 (Martingale): q The stochastic process{X(t), Ft ; 0 ≤ t < ∞} is called a martingale with respect to the filtration {Ft } and the measure P if X s = E(X(t) | F s ) P

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(t))T denote an m-dimensional Brownian motion, FRt the corresponding filtration. Let M(t) denote a martingale with respect to Ft with Ω |M(t)|2 dP < ∞ (∀ t ≥ 0). Then there exists a stochastic process g (with g(t) belonging to the class of integrands of the Itô integral) with Z t

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to t. 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

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Let X denote a (real valued) stochastic process on (Ω, F ) and {Ft } a filtration on (Ω, F ). The process X is called {Ft }-previsible, if X is {Ft }-adapted and bounded with left continuous paths. y Definition 56 (Integral with respect to a semi-martingale as integrator22 ): Let Y denote a semi

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for more general integrators) [12], §4 and [18], §3. Further Reading: On stochastic processes: As introduction: [25]. For an in depth discussion: [18], [27] and [29]. C| 22 For a more detailed discussion and a definition of a semi-martingale see remark 57. 58 This work is licensed under a Creative Commons License

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The following list of symbols summarizes the most important notions from Chapter 2: Symbol Object Interpretation ω element of Ω State. In the context of stochastic processes: path. Ω set State space. X random variable Map which assigns an event / outcome (e.g. a number) to a state. Example: the payoff of

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a financial product (this may be interpreted as a snapshot of the financial product itself). X stochastic process Sequence (in time) of random variables (e.g. the evolution of a financial product (could be its payoffs but also its value)). X(t), Xt

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December 2006 http://www.christian-fries.de/finmath/ CHAPTER 8 Interest Rate Structures 8.1. Introduction In previous sections we have considered a one dimensional stochastic process as the underlying, representing a stock for example. We will now turn to the modeling of interest rates and later the pricing of interest rate

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/finmath/ CHAPTER 8. INTEREST RATE STRUCTURES T0 T1 T2 T3 T4 T5 T6 Figure 8.1.: Modeling an interest rate curve through a family of stochastic processes. 8.1.1. Fixing Times and Tenor Times If the interest rates F(T i ; t) for periods [t, T i ] for i = 1, . . . , k

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. The period structure {T 0 , T 1 , . . . , T k } is also called the tenor structure. 8.2. Definitions All of the following random variables and stochastic processes are assumed to be defined over the same probability space (Ω, F , P). As the building block of all interest rates we define the Bond

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as a volatility of some process considered at a future period in time. Remark 95 (Interest Rate Models): The various definitions of interest rates (as stochastic processes) are the starting point for the corresponding interest rate modeling. The corresponding models are 131 This work is licensed under a Creative Commons License. http

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general are generated for an approximating model, namely a time-discrete model. In Chapter 13 we start by consider the approximation of time-continuous stochastic processes through time discrete stochastic processes. We then consider the approximation of the random variables by a Monte-Carlo simulation or a discretization of the state space. 177 This

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://www.christian-fries.de/finmath/ CHAPTER 13 Discretization of time and state space In this chapter we consider methods for discretization and implementation of Itô stochastic processes. We give an integrated presentation of “path simulation” (MonteCarlo simulation) and “lattice methods” (e.g. trees). Finally, both methods may be combined, see Section 13

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. DISCRETIZATION OF TIME: THE EULER AND THE MILSTEIN SCHEME and a time discretization {ti | i = 0, . . . , n} with 0 = t0 < . . . < tn . Then the time-discrete stochastic process X̃ defined by X̃(ti+1 ) = X̃(ti ) + µ(ti , X̃(ti ))∆ti + σ(ti , X̃(ti ))∆W(ti ) Euler scheme of the process X (where

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1 µi := µ(τ)dτ, σi := σ2 (τ)dτ. ∆ti ti ∆ti ti 13.2. Discretization of Paths (Monte-Carlo Simulation) We consider the time-discrete stochastic process X(ti+1 ) = X(ti ) + µ(ti , X(ti ))∆ti + σ(ti , X(ti ))∆W(ti ). (13.6) We consider here an Euler scheme. The

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://www.christian-fries.de/finmath/ 13.4. PATH SIMULATION THROUGH A LATTICE: TWO LAYERS Further Reading: An in depth discussion of numerical methods to approximate stochastic processes may be found in [19]. C| 193 This work is licensed under a Creative Commons License. http://creativecommons.org/licenses/by-nc-nd/2.5

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this in the next section and conclude by giving Ũ(T ) a name: Definition 168 (Option Value upon Optimal Exercise): q Let Ũ be the stochastic process who’s time t value U(t) is the (Numéraire relative) 198 This work is licensed under a Creative Commons License. http://creativecommons.org

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;ωi ) N(T 2 ;ωi ) . The conditional expectation is a function of Z dividing the cloud of dots. Example: Consider a LIBOR Market Model with stochastic processes for the forward rates L1 , L2 , . . . Ln . In T 1 we wish to calculate the conditional expectation of a derivative with a Numéraire relative

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fix notation, let us restate Monte-Carlo pricing first: 15.2.1. Pricing using Monte Carlo Simulation Assume that our model is given as a stochastic process X, for example an Itô process dX = µdt + σ · dW(t) modeling our model primitives, like functions of the the underlyings (e.g. financial products

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∗ (ti ) i = 0, 1, 2, . . . time discretization scheme of X → target scheme X◦ ti 7→ X ◦ (ti ) i = 0, 1, 2, . . . any other time discrete stochastic process (assumed to be close to X ∗ ) → proxy scheme Let φY ◦ (y) denote the density of Y ◦ and φY ∗ (y) the density of Y ∗ . We require

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pure and never simple. Oscar Wilde The Importance of Being Earnest, [37]. Up to this point now we have considered Models of a single scalar stochastic process and options on it: The Black-Scholes model for a stock S , or the Black model for a forward rate L. The true challenge in

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to a maturity of 20 or 30 years, resulting in 80 or 120 interest rates to model. We denote the simulation time parameter of the stochastic process by t. 256 This work is licensed under a Creative Commons License. http://creativecommons.org/licenses/by-nc-nd/2.5/deed.en Comments welcome

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Li,0 := 6 PMarket (T i ) − PMarket (T i+1 ) , PMarket (T i+1 )(T i+1 − T i ) The parameters σi may well be stochastic processes. In this cases σi is called a stochastic volatility model. 265 This work is licensed under a Creative Commons License. http://creativecommons.org/licenses/by

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. Choice of the Volatilities Reproduction of Caplet Market Prices We assume here, that the σi ’s are deterministic functions (i.e. not random variables or stochastic processes). The forward rate Li follows the Itô process Q dLi (t) = µQ i (t)Li (t)dt + σi (t)Li (t)dWi (t) unter Q

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T ) = αQ (t, T )dt + σ(t, T ) · dW Q (t) f (0, T ) = f0 (T ) (19.3) Equation (19.3) represents a family of stochastic processes parametrized by T , which give a complete description of the interest rate curve: From Definition (90) ;t)) , i.e. we have f (t, T ) = − ∂ log

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short rate model (20.2) gives a complete description of the interest rate curve dynamic. Short rate models were and are popular, since the underlying stochastic process r is one-dimensional (i.e. scalar valued). Thus many techniques that are known from the modeling of (also one dimensional) stock price processes may

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consequence of a non-linear function of a stochastic process Integration of stochastic processes . . . . . . . . . . . . . . . . . . . 30 32 35 37 39 41 41 42 50 57 3.1. Replication: The two-times-two-states-two-assets example . . . . . . 3.2. Replication: Generalisation to multiple states . . . . . . . . . . . . . 3.3. Real Measure versus Martingale Measure . . . . . . . . . . . . . . . 65 67 70 6.1. 6

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-fries.de/finmath/ LIST OF FIGURES 7.5. Wert des Replikationsportfolios ohne Rehedging . . . . . . . . . . . 118 8.1. Modeling an interest rate curve through a family of stochastic processes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2. Cashflows for a forward bond. . . . . . . . . . . . . . . . . . . . . . 8.3. UML Diagram: The class DiscountFactors . . . . . . . . . . . 8.4. UML Diagram: The class DiscountFactors with bootstrapper . 128 129

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the LIBOR Market Model. Diploma Thesis. University of Karlsruhe, 2004. [22] M, M; R, M: Martingale methods in financial modelling: theory and applications. Springer-Verlag, 1997. ISBN 3-540-61477-X. [23] P, W; B J̈: Stochastic Processes. From Physics to Finance. Springer-Verlag, 2000. ISBN 3-540-66560-9. [24] P, A

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Company, 1976. [47] B, J-P; S, D: The Black-Scholes Option Pricing Problem in Mathematical Finance: Generalizations and Extensions for a Large Class of Stochastic Processes. Journal de Physique I 4, 863, 1994. [48] B, A; G, D; M, M: The Market Model of Interest Rate Dynamics. Mathematical Finance 7, p

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Spread . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 static (Java™ Schlüsselwort) . . . 366 statische Methode . . . . . . . . . . . . . . . . 365 Stochastic Integral – Itô Process as Integrator . . . . . . . . . . 46 stochastic integral . . . . . . . . . . . . . . . . . . 57 – semi-martingale as integrator . . . . . 58 – with Brownian motion as integrator 44 stochastic process . . . . . . . . . . . . . . . . . . 35 – lognormal . . . . . . . . . . . . . . . . . . . . . . . 49 – previsible . . . . . . . . . . . . . . . . . . . . . . . 58 – previsible process . . . . . . . . . . . . . . . . 58 Stochastische Differentialgleichung – Diskretisierung . . . . . . . . . . . . . . . . . 177 Stochastischer Prozess . . . . . . . . . . . . . 35 – Gestoppter Prozess . . . . . . . . . . . . . 222 – quadratische Variation

Mathematics for Finance: An Introduction to Financial Engineering

by Marek Capinski and Tomasz Zastawniak · 6 Jul 2003

. 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

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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

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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,

Monte Carlo Simulation and Finance

by Don L. McLeish · 1 Apr 2005

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

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tool in the construction of models in finance, and that is that a stochastic integral with respect to a Brownian motion process is always a martingale. To see this, note that in an approximating sum Z T h(t)dWt ≈ 0 n−1 X h(ti )(W (ti+1 ) − W (ti

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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

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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

Tools for Computational Finance

by Rüdiger Seydel · 2 Jan 2002 · 313pp · 34,042 words

1 Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Model of the Financial Market . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Numerical Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 The Binomial Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Risk-Neutral Valuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Stochastic Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6.1 Wiener Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6.2 Stochastic Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7 Stochastic Diﬀerential Equations . . . . . . . . . . . . . . . . . . . . . . . 1.7.1 Itô Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7.2 Application to

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“today,” without loss of generality. Then the interval 0 ≤ t ≤ T represents the remaining life time of the option. The price St is a stochastic process, compare Section 1.6. In real markets, the interest rate r and the volatility σ vary with time. To keep the models and the analysis

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say as solution of the partial diﬀerential equation (1.2). Second it is used as St with subscript t to emphasize its random character as stochastic process. When the subscript t is omitted, the current role of S becomes clear from the context. 1.3 Numerical Methods Applying numerical methods is inevitable

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In this sense the resulting value V0 applies to the real world. The risk-neutral valuation can be seen as a technical tool. 1.6 Stochastic Processes 25 The assumption of risk neutrality is just required to deﬁne and calculate a rational price or fair value of V0 . For this speciﬁc purpose

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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

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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.

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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

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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

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) The relations (1.21a,b) can be derived from Deﬁnition 1.7 (−→ Exercise 1.9). The relation (1.21b) is also known as 1.6 Stochastic Processes E((∆Wt )2 ) = ∆t . 27 (1.21c) The independence of the increments according to Deﬁnition 1.7(c) implies for tj+1 > tj the

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t < tj is given by b(tj−1 )(Wtj − Wtj−1 ), and N b(tj−1 )(Wtj − Wtj−1 ) j=1 (1.24) 1.6 Stochastic Processes 29 represents the trading gain over the time period 0 ≤ t ≤ T . The trading gain (possibly < 0) is determined by the strategy b(t) and

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independent of the choice of the bn in (1.30). The Itô integral as function in t is a stochastic process with the martingale property. If an integrand a(x, t) depends on a stochastic process Xt , the function f is given by f (t) = a(Xt , t). For the simplest case of a

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: 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

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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

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) . For the linear SDE of (1.33) the expectation E(St ) solves Ṡ = µS. Hence S0 eµ(t−t0 ) is the expectation of the stochastic process and µ is the expected continuously compounded return earned by an investor per year. The rate of return µ is also called growth rate. The function S0

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= β(σ − ζ)dt Here and sometimes later on, we suppress the subscript t, which may be done when the role of the variables as stochastic processes is clear from the context. The rate of return µ of S is zero; dW (1) and dW (2) may be 40 Chapter 1 Modeling

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1)-distributed random numbers (−→ Chapter 2) Integration methods for SDEs (−→ Chapter 3) 1.8 Itô Lemma and Implications Itô’s lemma is most fundamental for stochastic processes. It may help, for example, to derive solutions of SDEs (−→ Exercise 1.11). Itô’s lemma is the stochastic counterpart of the chain rule for

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! which is known as the Poisson distribution with parameter λ > 0 (−→ Appendix B1). This leads to the Poisson process. Deﬁnition 1.17 (Poisson process) The stochastic process {Jt , t ≥ 0} is called Poisson process if the following conditions hold: (a) J0 = 0 (b) Jt − Js are integer-valued for 0 ≤ s <

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approach of Black, Merton and Scholes. For references on risk-neutral valuation we mention [Hull00], [MR97], [Kwok98] and [Shr04]. on Section 1.6: Introductions into stochastic processes and further hints on advanced literature may be found in [Doob53], [Fr71], [Ar74], [Bi79], [RY91], [KP92], [Shi99], [Sato99]. The requirement (a) of Deﬁnition 1.

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. Chapter 3 Simulation with Stochastic Diﬀerential Equations This chapter provides an introduction into the numerical integration of stochastic diﬀerential equations (SDEs). Again Xt denotes a stochastic process and solution of an SDE, dXt = a(Xt , t)dt + b(Xt , t)dWt for 0 ≤ t ≤ T, where the driving process W is

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smooth interpolation is at variance with the stochastic nature of solutions of SDEs. When ∆t is small, a linear interpolation matches the appearance of a stochastic process, and is easy to be carried out. Such an interpolating continuous polygon was used for the Figures 1.15 and 1.16. Another easily

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.-I. Bischi, L. Gardini: From bi-stability to chaotic oscillations in a macroeconomic model. Chaos, Solitons and Fractals 12 (2001) 805-822. J.L. Doob: Stochastic Processes. John Wiley, New York (1953). K. Dowd: Beyond Value at Risk: The New Science of Risk Management. Wiley & Sons, Chichester (1998). D. Duﬃe: Dynamic

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. Handscomb: Monte Carlo Methods. Methuen, London (1964). G. Hämmerlin, K.-H. Hoﬀmann: Numerical Mathematics. Springer, Berlin (1991). J.M. Harrison, S.R. Pliska: Martingales and stochastic integrals in the theory of continuous trading. Stoch. Processes and their Applications 11 (1981) 215-260. P. Heider: A condition number for the

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Sci. Comput. 15 (1994) 1251–1279. W.J. Morokoﬀ: Generating quasi-random paths for stochastic processes. SIAM Review 40 (1998) 765–788. K.W. Morton: Numerical Solution of Convection-Diﬀusion Problems. Chapman & Hall, London (1996). M. Musiela, M. Rutkowski: Martingale Methods in Financial Modelling. (Second Edition 2005) Springer, Berlin (1997). S.N. Neftci

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51, 91–96, 104–107, 110–111, 115–116, 118–119, 121, 124, 215–216, 235, 244, 260–261 Stochastic integral 28–31, 39, 50 Stochastic process 6, 10, 25–32, 45, 50–52, 57, 91, 102, 213 Index Stochastic Taylor expansion 95–99 Stock 1, 33, 37, 41–42, 51, 58

Frequently Asked Questions in Quantitative Finance

by Paul Wilmott · 3 Jan 2007 · 345pp · 86,394 words

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

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1987 Theory of Financial Decision Making. Rowman & Littlefield What is Brownian Motion and What are its Uses in Finance? Short Answer Brownian Motion is a stochastic process with stationary independent normally distributed increments and which also has continuous sample paths. It is the most common stochastic building block for random walks in

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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. If Wt is the BM at time t then for every t, τ ≥ 0, Wt+τ − Wt is independent of

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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

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that other models, such as stochastic volatility, have difficulties in doing. References and Further Reading Cox, J & Ross, S 1976 Valuation of Options for Alternative Stochastic Processes. Journal of Financial Econometrics 3 Kingman, JFC 1995 Poisson Processes. Oxford Science Publications Lewis, A Series of articles in Wilmott magazine September 2002 to August

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models, in which volatility is a function of asset and time, σ(S, t), and stochastic volatility models, in which we represent volatility by another stochastic process. The latter models require a knowledge or specification of risk preferences since volatility risk cannot be hedged just with the underlying asset. If the variance

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ℚ-martingale and so for some process αt. Applying Itô’s lemma,dGt = (r + αη)Gtdt + αGt dWt. This stochastic differential equation can be rewritten as one

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arbitrage in multiperiod securities markets. Journal of Economic Theory 20 381-408 Harrison, JM & Pliska, SR 1981 Martingales and stochastic integrals in the theory of continuous trading. Stochastic Processes and their Applications 11 215-260 Joshi, M 2003 The Concepts and Practice of Mathematical Finance. CUP Rubinstein, M 1976 The valuation of uncertain

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

by Alain Ruttiens · 24 Apr 2013 · 447pp · 104,258 words

ON CURRENCIES 7.7 FUTURES ON (NON-FINANCIAL) COMMODITIES FURTHER READING Part II: The Probabilistic Environment Chapter 8: The basis of stochastic calculus 8.1 STOCHASTIC PROCESSES 8.2 THE STANDARD WIENER PROCESS, OR BROWNIAN MOTION 8.3 THE GENERAL WIENER PROCESS 8.4 THE ITÔ PROCESS 8.5 APPLICATION OF THE

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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

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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

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time t and maturity T V(.) variance of (.) (.) stochastic process of (.) stochastic variable zt “zero” or 0-coupon rate of maturity t Z standard Wiener process (Brownian motion, white noise) Introduction The world is the

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as it becomes convenient to buy forward at a reduced (market) cost. Part II The Probabilistic Environment 8 The basis of stochastic calculus 8.1 STOCHASTIC PROCESSES Stochastic is equivalent to random, hence the stochastic calculus develops rules of calculus to be applied if the problems to be handled are of a

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) = 0.7734. 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

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, 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

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, as follows: discrete parameter set: continuous parameter set: discrete state space: discrete parameter chain continuous parameter chain continuous state space: random sequence random function, or stochastic process In our time series, t may be considered as discrete or continuous. Most liquid financial instruments may look to be traded continuously in time, but

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a continuous time framework, the random variable of prices or returns will change continuously as well. So that the processes we will consider relate to “stochastic processes” properly said. Stationary or Non-Stationary Processes A random process may also be considered as stationary or non-stationary. Broadly speaking, a process is stationary

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what extent can we consider that these values represent the mean and variance of the distribution F(x) of the random variables of a given stochastic process? For sake of simplicity, let us further consider that it is the case. In particular, stationarity implies that the probability P that a random variable

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under Q, of Eq. 8.16, using the risk neutral probability measure, is called a semimartingale, That is, a variant of a “martingale”. A martingale is a Markovian (memory-less) stochastic process such as, at t, the conditional expected value of St+1 is St. In our case, we talk of a semimartingale, that

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.12. FURTHER READING Darrell DUFFIE, Security Markets: Stochastic Models, Academic Press Inc., 1988, 250 p. L.C.G. ROGERS, David WILLIAMS, Diffusions, Markov Processes and Martingales, vol. 1: Foundations, vol. 2: Itô Calculus, Cambridge University Press, 2nd ed., 2000, 406 and 494 p. A.G. MALLIARIS, W.A. BROCK, Stochastic Methods

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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

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-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

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usual Markovian stochastic processes, these models incorporate in the general case a limited number of previous return values, so that they are not Markovian. For a time series of

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moves appears generally more frequent than implied by the normal distribution, what is called a “fat tails” problem. Due to the poor performance of alternative stochastic processes developed with a non-normal distribution (cf. Chapter 15, Section 15.1), the market practice generally prefers to keep the Gaussian hypothesis but to adjust

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SABR models, let us also mention the one8 consisting – instead of starting from Eq. 12.3 – in considering the following relationship: that creates a third stochastic process Z3 that is independent (uncorrelated) with Z1. Provided some hypothesis can be reasonably made about ρ1, 2, presumably as a function of σt, the model

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4, Section 4.3.7). Finally, diffusion processes that involve jump discontinuities can be generalized by use of the more general class of continuous-time stochastic processes called Lévy processes, of which the Wiener, the Poisson and the gamma processes are particular cases (cf. Further Reading). 15.1.3 Other alternative processes

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spot instruments swaptions see also market prices; option pricing price of time, CAPM price-weighted indexes pricing sensitivities see sensitivities probability risk neutral see also stochastic processes Proportion of failures (POF) test putable bonds put options call-put parity see also options PV see present value quanto swaps randomness random numbers random

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standardized futures contracts standard Wiener process see also dZ; general Wiener process stationarity stationary Markovian processes stochastic processes basis of Brownian motion definition of process diffusion processes discrete/continuous variables general Wiener process Markovian processes martingales probability reminders risk neutral probability standard Wiener process stationary/non-stationary processes terminology stock indexes basket options