Wiener process Concepts

The Dream Machine: J.C.R. Licklider and the Revolution That Made Computing Personal

by M. Mitchell Waldrop · 14 Apr 2001

research at all. However, no one seems to have informed Wiener of that fact, and his mathematical output soon became legendary. The Wiener measure, the Wiener process, the Wiener- Hopf equations, the Paly-Wiener theorems, the Wiener extrapolation of linear times series, generalized harmonic analysis-he saw mathematics everywhere he looked. He

Mathematical Finance: Core Theory, Problems and Statistical Algorithms

by Nikolai Dokuchaev · 24 Apr 2007

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:[0, T]→R. It generates

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4 Basics of Ito calculus and stochastic analysis This chapter introduces the stochastic integral, stochastic differential equations, and core results of Ito calculus. 4.1 Wiener process (Brownian motion) Let T>0 be given, Definition 4.1 We say that a continuous time random process w(t) is a (onedimensional

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) Wiener process (or Brownian motion) if (i) w(0)=0; (ii) w(t) is Gaussian with Ew(t)=0, Ew(t)2=t, i.e., w(t)

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(N. Wiener). There exists a probability space such that there exists a pathwise continuous process with these properties. This is why we call it the Wiener process. The corresponding set Ω in Wiener’s proof of this theorem is the set C(0, T). Remember that C(0, T) denotes the set

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:[0, T]→R. Corollary 4.3 Let ∆t>0, Corollary 4.4 then Var ∆w=∆t. This can be interpreted as This means that a Wiener process cannot have pathwise differentiable trajectories. Its trajectories are very irregular (but they are still continuous a.s.). Let us list some basic properties of w

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We say that a continuous time process w(t) = (w1(t),…, wn(t)): [0, +∞)×Ω→Rn is a (standard) n-dimensional Wiener process if (i) wi(t) is a (one-dimensional) Wiener process for any i=1,…, n; (ii) the processes {wi(t)} are mutually independent. Remark 4.6 Let be a matrix such

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that Then the process is also said to be a Wiener process (but not a standard Wiener process, since it has correlated components). We shall omit

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the word ‘standard’ below; all Wiener processes in this book are assumed to be standard. For simplicity, one can assume for the first

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. After that, one can read this chapter again taking into account the general case. Proposition 4.7 A Wiener process is a Markov process. be the filtration Proof. We consider an n-dimensional Wiener process w(t). Let generated by w(t). We have that w(t+τ)=w(t+τ)−w(t)+w

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and F2:Rn→R. It follows that w(s) is a Markov process. Proposition 4.8 Let be a filtration such that an n-dimensional Wiener process w(t) and w(t+τ)−w(t) does not depend on Then w(t) is a martingale with is adapted to respect to Proof

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(t) does not depend on © 2007 Nikolai Dokuchaev Therefore, the martingale property holds. Basics of Ito Calculus and Stochastic Analysis 51 Corollary 4.9 A Wiener process w(t) is a martingale. (In other words, if is the ) filtration generated by w(t), then w(t) is a martingale with respect to

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Up to the end of this chapter, we assume that we are given an n-dimensional Wiener process w(t) and the filtration such as described in Proposition 4.8. One may assume that this filtration is generated by the process (w(t

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w(·). We assume also that terminal time. 4.2 Stochastic integral (Ito integral) Stochastic integral for step functions Let w(t) be a one-dimensional Wiener process. Repeat that described in Proposition 4.8. Notation: Let integer N>0, be me set of a set of times is a filtration such as

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construct it as a function of T for a fixed ω). Ito integral for a random time interval Let be the filtration generated by the Wiener process w(t), and let Then Markov time with respect to can define the Ito integral for a random time interval [0, τ] as Let τ

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time τ is such that Ew(τ)>0. In addition, Vector case be a filtration such as described Let w(t) be an n-dimensional Wiener process, and let in Proposition 4.8. Let f=(f1,…, fn):[0,T]×Ω→R1×n be a (vector row) process such that for all i

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Ito integral The right-hand part is well defined by the previous definitions. Ito processes Definition 4.23 Let w(t) be an n-dimensional Wiener process, for all i. Let Let a random process β=(β1,…, βn) take values in R1×n, and let © 2007 Nikolai Dokuchaev Basics of Ito Calculus

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(t)β2(t)dt. 3 Mathematische Annalen 104 (1931), 415–458. The vector case Let us assume first that w(t) is an n-dimensional Wiener process. Let random processes a=(a1,…, am) and β={βij} take values in Rm and Rm×w respectively, and let and for all i, j. Let

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(4.7) and (4.8) are distributed log-normally conditionally given ys. A generalization let Problem 4.40 Let w(t) be an n-dimensional Wiener process, 1×n σ(t)=(σ1(t),…, σn(t)) be a process with values in R such that and let some conditions on the growth for

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has continuous derivatives Theorem 4.42 and Proof. By the Ito formula, Then the proof follows. Vector case Let w(t) be an n-dimensional Wiener process. Let (non-random) functions f:Rm×[0, Let y(t) be a solution of the T]→Rm and b:Rm×[0, T]→Rm×n be

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equivalent formulation as the following martingale representation theorem. generated by a Theorem 4.52 Let ξ(t) be a martingale with respect to the filtration Wiener process w(t) such that Eξ(T)2<+∞. Then there exists a process f(t) with values in such that R1×n and with components from

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.58 Let the assumptions of Proposition 4.55 be satisfied, and let the measure P* be defined via the equation Then w*(t) is a Wiener process under P*. Proof of Theorem 4.58. We are going to prove only that (4,19) for all deterministic continuous functions f(·):[0, T]→R1

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Nikolai Dokuchaev for all Mathematical Finance 72 Example 4.61 Let us reconsider Example 4.57. We have that Hence (Remember that w* is a Wiener process with respect to P*, hence E*w*(T)=0.) One can verify that the integral in Example 4.57 has the value case of non

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by the following Ito equation: dS(t)=S(t)(a(t)dt+σ(t)dw(t)). (5.1) Here w(t) is a (one-dimensional) Wiener process, and a and σ are market parameters. Sometimes in the literature S(t) is called a geometric Brownian motion (for the case of non-random

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be a Brownian motion. Mathematicians prefer to use the term ‘Brownian motion’ for w(t) only (i.e., Brownian motion is the same as a Wiener process). Definition 5.1 In (5.1), a(t) is said to be the appreciation rate, σ(t) is said to be the volatility. Note that

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exists a random process η(t) that does not depend on w(·). This process describes additional random factors presented in the model besides the driving Wiener process w(t). be the filtration generated by (w(t),η(t)), and let Let by the process w(t) only. It follows that be the

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process (w(t), η(t)), where η(·) is a process independent from w(·) that describes additional random factors presented in the model besides the driving Wiener process (see also Remark 5.3). be a probability measure such that the Definition 5.20 Let process is a martingale with respect to the filtration

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5.23 Let Condition 5.22 be satisfied, and let the measure P* be defined by equation (5.10). Then (i) w*(t) is a Wiener process under P*; (ii) P* is an equivalent risk-neutral measure. Proof. Statement (i) follows from the Girsanov theorem (4.58), as well as the statement

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that P* is equivalent to P. Let us prove the rest of part (ii). We have that By Theorem 4.58, w*(t) is a Wiener process under P*. Then This completes the proof. Theorem 5.24 Let Condition 5.22 be satisfied, and let P* be the equivalent riskneutral measure defined

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respect to under P*. Continuous Time Market Models 83 where γ(t) is the number of shares. By Girsanov’s theorem, w*(t) is a Wiener process under P*. Then This completes the proof. 5.5 Replicating strategies Remember that T>0 is given. Let ψ be a random variable. Definition 5

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(T)=ψ iff © 2007 Nikolai Dokuchaev a.s. We have that Mathematical Finance 84 We have used here the fact that w*(t) is a Wiener process under P*, and ∫·dw* is an Ito integral under P*, so E* ∫·dw*=0. First application: the uniqueness of the replicating strategy Theorem 5.28

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(5.12) Hence (5.13) © 2007 Nikolai Dokuchaev Mathematical Finance 86 But since w* is a Wiener process under P*. This contradicts (5.13). We have used again the fact that w*(t) is a Wiener process under P*, and ∫·dw* is an Ito integral under P*, i.e., E* ∫·dw*=0. Problem 5

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assume that the filtration is generated by w(t) (or by S(t), or by Let w* and P* be defined as above, i.e., Wiener process with respect to P*, and ). Remember that w*(t) is a (5.14) Theorem 5.36 The Black-Scholes market is complete. be an arbitrary

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=σ(t, η), where η is a random process (or a random vector, or a random variable), independent from the driving Wiener process w(t) (for instance, η may represent another Wiener process). Clearly, any original probability measure P=Pη is a risk-neutral measure (note that for any η). Any probability measure is

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a process and S(·) is uniquely defined defined in Theorem 4.58. Therefore, the distribution of by the distribution of w*(·). Since w*(·) is a Wiener process under the corresponding measure P* for any η, then the distribution of S(·) is the same under all these P*. In other words, all these

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complete by adding new assets. For instance, if σ(t) is random and evolves as the solution of an Ito equation driven by a new Wiener process W(t) then the market can be made complete by allowing trading of any option on this stock (say, European call with given strike price

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is again based on Ito equations, which now can be written as Here w(t)=(w1(t),…, wn(t)) is a vector Wiener process; i.e., its components are scalar Wiener processes. Further, a(t)={ai(t)} is the vector of the appreciation rates, and σ(t)={σij(t)} is the volatility matrix. We

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is an equivalent risk-neutral measure. Set © 2007 Nikolai Dokuchaev Continuous Time Market Models 103 By the Girsanov theorem (4.58), w*(t) is a Wiener process under P*. Clearly, σ(t)dw*(t)=ã(t)dt+σ(t)dw(t). In addition, main diagonal where is a diagonal matrix with the

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to be random; (ii) the range for the discounted price processes is bounded; (iii) the number of securities is larger than the number of driving Wiener processes. © 2007 Nikolai Dokuchaev Mathematical Finance 104 The last feature (iii) has explicit economical sense: there are many different bonds (since bonds with different maturities represent

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). Feature (iii) can be expressed as the condition that σij(t)≡0 for all j>n,=1,…, N, where n is the number of driving Wiener processes, N is the number of bonds, N>>n. It follows that the matrix a is degenerate. This is a very essential feature of the bond

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given set of maturing times, Continuous Time Market Models 105 We consider the case where there is a driving n-dimensional Wiener process w(t). Let be a filtration generated by this Wiener process. We assume that the process r(t) is (To cover some special models, we do not assume that r(t

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known, and where the stock price evolves as dP(t)=adt+σ dw(t), where σ>0 is a given constant, w(t) is a Wiener process. Assume that the fair price for a call option is e−rTE* max(P(T)−K, 0), where K is the strike price, T is

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measure P*, as required by the Black-Scholes approach. This means that S(tk+1)=S(tk)Mk+1, where Here w*(t) is a Wiener process under P*, and ξk are i.i.d. (independent identically distributed) random variables with law N(0, 1) under P*. Proposition 6.1 Let γ

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an Ito stochastic differential equation dS(t)=a(t)S(t)dt+σ(t)S(t)dw(t). (7.1) Here w(t) is a Wiener process such that w(t)~N(0, t), i.e., the distribution law for w(t) is N(0, t). Remember that the coefficient σ(t

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)dt+σ(t)S(t)dw(t), (9.1) where a(t) is the appreciation rate, σ(t) is the volatility, w(t) is a Wiener process, w(t)~N(0, t). Remember that (9.2) Let us repeat the proof that was given above in a different form. Let f(x

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the stochastic integral, to define interval [s0, +∞), where time interval [0, +∞) Normally, a Wiener process is defined on the time is initial time; we had introduced Wiener processes for only. We can use the following definition: where is some standard Wiener process independent from w(·). Proof of Theorem 9.2. To proof convergency, one should notice

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stock prices: • The volatility is non-random, and the appreciation rate a(t) is an Ito process that evolves as where and where is some Wiener process; • (a(t), σ(t))=f(ξ(t)), where f is a deterministic function, ξ is a Markov chain process; • σ(t)=CS(t)p, where

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• the volatility σ(t) is an Ito process that evolves as where and where is some Wiener process. All these models (and many others) need statistical evaluation in implementation with real market data, and that is one of the mainstream research fields in

Tools for Computational Finance

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

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 the Stock Market . . . . . . . . . . . . . . . . . . 1.7

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, j, k, l, m, n, M, N, ν various variables: Xt , X, X(t) Wt y(x, τ ) w h ϕ ψ 1D random variable Wiener process, Brownian motion (Deﬁnition 1.7) solution of a partial diﬀerential equation for (x, τ ) approximation of y discretization grid size basis function (Chapter 5) test

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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

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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

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450 500 Fig. 1.14. The Dow at 500 trading days from September 8, 1997 through August 31, 1999 1.6.1 Wiener Process Deﬁnition 1.7 (Wiener process, Brownian motion) A Wiener process (or Brownian motion; notation Wt or W ) is a timecontinuous process with the properties (a) W0 = 0 (with probability one) (b) Wt

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1.7(c) implies for tj+1 > tj the independence of Wtj and (Wtj+1 − Wtj ), but not of Wtj+1 and (Wtj+1 − Wtj ). Wiener processes are examples of martingales — there is no drift. Discrete-Time Model Let ∆t > 0 be a constant time increment. For the discrete instances tj := j

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= Z ∆t for Z ∼ N (0, 1) for each k . (1.22) We summarize the numerical simulation of a Wiener process as follows: Algorithm 1.8 (simulation of a Wiener process) Start: t0 = 0, W0 = 0; ∆t loop j = 1, 2, ... : tj = tj−1 + ∆t draw Z ∼ N (0, 1) √ Wj = Wj−1 + Z

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are realizations of Wt at the discrete points tj . The Figure 1.15 shows a realization of a Wiener process; 5000 calculated points (tj , Wj ) are joined by linear interpolation. Almost all realizations of Wiener processes are nowhere diﬀerentiable. This becomes intuitively clear when the diﬀerence quotient ∆Wt Wt+∆t − Wt = ∆t ∆t 28

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0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Fig. 1.15. Realization of a Wiener process, with ∆t = 0.0002 is considered. √ Because of relation (1.21b) the standard deviation of the numerator is ∆t. Hence for ∆t → 0 the normal

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and no convergence can be expected. 1.6.2 Stochastic Integral Let us suppose that the price development of an asset is described by a Wiener process Wt . Let b(t) be the number of units of the asset held in a portfolio at time t. We start with the simplifying assumption

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limit exists and is called Riemann-Stieltjes integral T b(t)dWt . 0 In our situation this integral generally does not exist because almost all Wiener processes are not of bounded variation. That is, the ﬁrst variation of Wt , which is the limit of N |Wtj − Wtj−1 | , j=1 is unbounded

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derivation can be summarized to E((∆Wt )2 − ∆t) = 0 , Var((∆Wt )2 − ∆t) = 2(∆t)2 , hence (∆Wt )2 ≈ ∆t. This property of a Wiener process is symbolically written (1.28) (dWt )2 = dt It will be needed in subsequent sections. Now we know enough about the convergence of the left

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factor must be compensated by an unbounded growth of the other factor, so N |Wtj − Wtj−1 | → ∞ für δN → 0 . j=1 In summary, Wiener processes are not of bounded variation, and the integration with respect to Wt can not be deﬁned as an elementary limit of (1.24). The aim

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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

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. 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

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in this integral equation is an ordinary (Lebesgue- or Riemann-) integral. The second integral is an Itô integral to be taken with respect to the Wiener process Wt . The resulting stochastic diﬀerential equation (SDE) is named after Itô. Deﬁnition 1.10 (Itô stochastic diﬀerential equation) An Itô stochastic diﬀerential equation is dXt

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in (1.31) are named as follows: a(Xt , t): drift term or drift coeﬃcient b(Xt , t): diﬀusion term solution Xt : Itô process A Wiener process is a special case of an Itô process, because from Xt = Wt the trivial SDE dXt = dWt follows, hence the drift vanishes, a = 0, and

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discretized version of the Itô SDE ∆Xt = a(Xt , t)∆t + b(Xt , t)∆Wt (1.32) with the Algorithm 1.8 for approximating a Wiener process, using the same ∆t for both discretizations. The result is Algorithm 1.11 (Euler discretization of an SDE) Approximations yj to Xtj are calculated by

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calculate random numbers Z ∼ N (0, 1) (−→ Section 2.3). Solutions to the SDE or to its discretized version for a given realization of the Wiener process are called trajectories or paths. By simulation of the SDE we understand the calculation of one or more trajectories. For the purpose of visualization, the

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time increment ∆t = 1/300 the Figure 1.16 depicts a trajectory Xt of the SDE for 0 ≤ t ≤ 1. For another realization of a Wiener process Wt the solution looks diﬀerent. This is demonstrated for a similar SDE in Figure 1.17. 1.7.2 Application to the Stock Market Now

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written dSt = rSt dt + σSt [γdt + dWt ] . (1.37) It turns out (−→ Appendix B2) that there is another probability measure Q and —matching it— another Wiener process Wtγ depending on γ, such that dWtγ = γdt + dWt . (1.38) dSt = rSt dt + σSt dWtγ . (1.39) Then (1.37) becomes Comparing this SDE

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) written in the same notation. Then Xt = (Xt , . . . , Xt ) and a(Xt , t) are n-dimensional vectors. The Wiener processes of each component SDE need not be correlated. In the general situation, the Wiener process can be m(1) (m) dimensional, with components Wt , ..., Wt . Then b(Xt , t) is an (n × m)matrix

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be 40 Chapter 1 Modeling Tools for Financial Options correlated. The stochastic volatility σ follows the mean volatility ζ and is simultaneously perturbed by a Wiener process. Both σ und ζ provide mutual mean reversion, and stick together. The two SDEs for σ and ζ may be seen as a tandem controlling

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, t) be a C 2,1 -smooth function (continuous ∂x , ∂x 2, ∂t ). Then Yt := g(Xt , t) follows an Itô process with the same Wiener process Wt : ∂g ∂g 1 ∂ 2 g 2 ∂g a+ + b dWt b (1.43) dYt = dt + ∂x ∂t 2 ∂x2 ∂x where the derivatives of

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dWs Yt = Yt0 + (µ − σ 2 ) 2 t0 t0 (1.46) 1 2 = Yt0 + (µ − σ )(t − t0 ) + σ(Wt − Wt0 ) 2 From the properties of the Wiener process Wt we conclude that Yt is distributed normally. To write down the density function fˆ(Yt ) the mean µ̂ := E(Yt ) and the variance σ̂ are

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Section 1.7.2. Here we take the exponential of (1.46), or restart the derivation by again applying Itô’s lemma. For an arbitrary Wiener process Wt set Xt := Wt and σ2 Yt = g(Xt , t) := S0 exp µ− t + σXt . 2 From Xt = Wt follows the trivial SDE with coeﬃcients a

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process. We assume that qτ1 , qτ2 , ... are i.i.d. This process is called compound Poisson. Next we superimpose the jump process to the continuous Wiener process. The combined geometric Brownian und jump process is given by dSt = µSt dt + σSt dWt + (qt − 1)St dJt . (1.52) Here σ is the

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as normalization. This Brownian motion ist called standard Brownian motion. For a proof of the nondiﬀerentiability of Wiener processes, see [HuK00]. For more hints on martingales, see Appendix B2. In contrast to the results for Wiener processes, diﬀerentiable functions Wt satisfy for δN → 0 (Wtj − Wtj−1 )2 −→ 0 . |Wtj − Wtj−1 | −→ |Ws |ds

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Ft adapted and E f (s)2 ds < ∞. We assume that all integrals occuring in the text exist. The integrator Wt needs not be a Wiener process. The stochastic integral can be extended to semimartingales [HuK00]. Notes and Comments 51 on Sections 1.7, 1.8: The connection between white noise and

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Wiener processes is discussed in [Ar74]. White noise is a Gaussian process ξt with E(ξt ) = 0 and a spectral density that is constant on the entire

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, see Exercise 1.18.— Recently, many books on ﬁnancial markets have been published, see for instance [DaJ03], [ElK99], [Gem00], [MeVN02]. In view of their continuity, Wiener processes are not appropriate to model jumps, which are characteristic for the evolution of stock prices. The jumps lead to relatively heavy tails in the distribution

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and solution of an SDE, dXt = a(Xt , t)dt + b(Xt , t)dWt for 0 ≤ t ≤ T, where the driving process W is a Wiener process. The solution of a discrete version of the SDE is denoted yj . That is, yj should be an approximation to Xtj , or yt an approximation

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of the SDE a strong solution. In this sense the solution in (3.2) is a strong solution. If one is free to select a Wiener process, then a solution of the SDE is called weak solution. For a weak solution, only the distribution of X is of interest, not its path

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. Assuming an identical sample path of a Wiener process for the SDE and for the numerical approximation, a pathwise comparison of the trajectories 3.1 Approximation Error 93 Xt of (3.2) and y

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from (3.1) is possible for tj . For example, for tm = T the absolute error for a given Wiener process is |XT − yT |. Since the approximation yT also depends on the chosen step length h, we also write yTh . For another

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Wiener process the error is somewhat diﬀerent. We average the error over “all” sample paths of the Wiener process: Deﬁnition 3.1 (absolute error) The absolute error at T is (h) := E(|XT − yTh |). In practice

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we represent the set of all sample paths of a Wiener process by N diﬀerent simulations. Example 3.2 X0 = 50, α = 0.06, β = 0.3, T = 1. We want to investigate experimentally how the absolute

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,k , yT,k for k = 1, ..., N . Again: to obtain pairs of comparable trajectories, also the theoretical solution (3.2) is fed with the same Wiener process from (3.1). Then we calculate the estimate of the absolute error , (h) := N 1 h |XT,k − yT,k |. N k=1 Such an

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1.) y ∈ IRn , a, b ∈ IRn . Then, for instance, replace bb by ∂y the Jacobian matrix of all ﬁrst-order partial derivatives. 2.) For multiple Wiener processes the situation is more complicated, because then simple explicit integrals as in (3.9) do not exist. Only the Euler scheme remains simple: for m

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Wiener processes the Euler scheme is yj+1 = yj + a∆t + b(1) ∆W (1) + ... + b(m) ∆W (m) . The Figure 3.1 depicts two components of

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(1.49). For the chosen parameters we have S1 = 5 exp(0.015 + 0.3W1 ), which requires “the” value of the Wiener process at t = 1. Related values W1 of the Wiener process can be obtained from (1.22) with ∆t = T as 106 Chapter 3 Simulation with Stochastic Diﬀerential Equations 4.7 4

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Itô Integral in Equation (3.9) Let the interval 0 ≤ s ≤ t be partitioned into n subintervals, 0 = t1 < t2 < ... < tn+1 = t. For a Wiener process Wt assume Wt1 = 0. n n 1 2 1 Wtj+1 − Wtj a) Show Wtj Wtj+1 − Wtj = Wt2 − 2 2 j=1 j=1

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). Exercise 3.3 a) Show Integration by Parts for Itô Integrals t t sdWs = tWt − t0 Wt0 − t0 Ws ds t0 Hint: Start with the Wiener process Xt = Wt and apply the Itô Lemma with the transformation y = g(x, t) := tx. t s b) Denote ∆Y := t0 t0 dWz ds. Show

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order Show that the random variables ∆W O(∆t3 ) the same moments as ∆W and ∆Y . Exercises 121 Exercise 3.7 Brownian Bridge For a Wiener process Wt consider Xt := Wt − t WT T Calculate Var(Xt ) and show that ! t t 1− Z T for 0 ≤ t ≤ T. with Z ∼ N

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) ) = ρ dt, because E(dW (1) ) = E(dW (2) ) = 0. Compared to more general systems as in (1.41), the version (6.1a) with correlated Wiener processes has the advantage that each asset price has its own growth factor µ and volatility σ, which can be estimated from data. The correlation ρ is

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a1 + b1 b2 )dt + (Xb2 + Y b1 )dW . (B2.2) Application: dS = rSdt + σSdŴ ⇒ d(e−rt S) = e−rt σSdŴ (B2.3) for any Wiener process Ŵ . Filtration of a Brownian motion FtW := σ{Ws | 0 ≤ s ≤ t} (B2.4) Here σ{.} denotes the smallest σ-algebra containing the sets put

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unpredictable; Ms is the best forecast. The SDE of a martingale has no drift term. Examples: any Wiener process Wt , Wt2 − t for any Wiener process Wt , exp(λWt − 12 λ2 t) for any λ ∈ IR and any Wiener process Wt , Jt − λt for any Poisson process Jt with intensity λ. For martingales, see for instance

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– Marsaglias polar method 74 – Milstein integrator 99 – Monte Carlo simulation 105 – Projection SOR 156 – Quadratic approximation 171 – Radical–inverse function 83, 90 – Variance 53–54 – Wiener process 27 Analytic methods 124 Antithetic variates 86, 108–109, 111 Arbitrage 4–5, 9, 15, 23, 37, 52, 59, 124, 141, 143, 180, 212, 220

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, 121, 139–141, 167–168, 277–279 Box–Muller method 72–75, 85, 117 Bridge 102, 116, 118, 121, 236 Brownian motion 25–26, see Wiener process Bubnov 186 Business time 52 Calculus of variations 200 Calibration 38, 52, 54 Call, see Option Cancellation 54 Cauchy convergence 31, 275 Cauchy distribution 88

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–101, 109, 118 Weak derivative 273–274 Weak solution 92, 120, 195, 199–202 Weighted residuals 183–187 Weighting function 186 White noise 32, 51 Wiener process (Brownian motion) Wt 25–34, 36, 38–44, 47, 50–52, 56, 57, 91–93, 96–102, 105, 119–121, 212, 215, 257–261 Writer

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

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

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 excess of possible.1 The aim of this book is to present the quantitative aspects of

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of financial processes, the stochastic calculus is essentially developed within the framework of diffusion processes, considered as conveniently describing their behavior. 8.2 THE STANDARD WIENER PROCESS, OR BROWNIAN MOTION The simplest diffusion process is a random process whose values of a random variable in function of the time t follow a

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discrete time (or “finite”) intervals Δt to infinitely short, « infinitesimal » or « instantaneous » time intervals noted dt, Eq. 8.1 becomes (8.2) called a standard Wiener process, or a Brownian process or Brownian motion.1 This process is also called (although improperly2) white noise, by analogy with the very light but permanent

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expected value of the product of (t) at two different points of time t1 and t2, (8.5) Finally, the product of two different standard Wiener processes Z1 and Z2, is not random: (8.6) where ρ1, 2 is the correlation coefficient between the two processes. These relationships constitute the core

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So that the expected value and the variance of the Itô process are also functions of (t) and t. 8.5 APPLICATION OF THE GENERAL WIENER PROCESS General Wiener processes4 are widely used to describe the behavior of financial products. In practice, we actually model the returns (rather than the prices) of

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financial time series by a general Wiener process. Long time series can indeed present large prices variations, while returns are more stable over time, as can be viewed in the example next.

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, 0.000838 × √ (250 × 395/5) = 0.1177. Although 5 minutes is hardly an approximation for dt, discretizing dt by a Δt = 5', the general Wiener process (by discretizing Eq. 8.11 and by omitting “(t)” in S(t), to simplify the notations) is: where Δt is the unit time interval and

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a stock price S (Eq. 8.1), has been built by using a physical probability measure, given the μ drift, associated with the stochastic standard Wiener process dZ. By assuming μ = r, this equation can be rewritten with the risk neutral probability measure, called Q. Defining dZQ as we obtain (8

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.16) that is, a geometric Wiener process involving a standard Wiener process under Q8. Integrating Eq. 8.16 in the same manner as in Section 8.7, instead of obtaining (Eq. 8.14) we obtain

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. Sections 7.1 and 7.2), neither does S(T) depend on earlier values of S at previous times t. With this respect, the geometric Wiener process under Q, of Eq. 8.16, using the risk neutral probability measure, is called a semimartingale, That is, a variant of a “martingale”. A

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come back to the relationship 8.17, valuing a forward or future under Q, the risk neutral probability measure: as a consequence, the geometric general Wiener process (Eq. 8.16) under Q, applied to a forward or a future, comes down to (8.18) ANNEX 8.1: PROOFS OF THE PROPERTIES

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stochastic and is equal to its expected value. NB: for convenience, the “∼” symbol is omitted hereafter. Proof of Eq. 8.3 – Square of the Standard Wiener Process By definition of the variance, knowing that V[dZ(t)] = dt, since E[dZ(t)] = 0, And also, by definition of the variance, here,

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: V[dZ(t)] = dt second-order moments: negligible because they contain dtn with n > 2. Proof of Eq. 8.6 – Product of Two Standard Wiener Processes The expected value of the product of two stochastic variables is which introduces the covariance. Here, both E[dZi(t)] = 0, hence From the relationship

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hypotheses underlying the Black–Scholes formula are as follows: The underlying price is the only stochastic variable and is assumed to follow a geometric (general) Wiener process. This implies a constant drift and volatility of the underlying returns during the lifetime of the option. Financial markets are efficient; in other words, market

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updated in accordance with this market information. Efficiency also implies, practically speaking, enough market liquidity. Market prices are assumed to be continuous, like the Wiener process used to model the underlying (cf. Chapter 8, Sections 8.1 and 8.2). Market prices and interest rates are assumed to be traded at

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extended to European options on any kind of underlying offering a return ≠ 0, provided that its process can be reasonably modeled by a geometric Wiener process. This extension is valid if the underlying return can be considered as continuous in time.3 This will be the case of a LIBOR rate

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of: incorporating dividend payments (options on equities); American options; options on interest rates, volatility, or other underlyings, that do not fit with the geometric general Wiener process (see Chapters 11 and 12); second generation options (cf. Chapter 11); relaxing the Gaussian framework, for taking into account observed asymmetry and kurtosis in the

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this first step n times, for n → ∞, corresponding to an infinity of sub-periods dt, one replicates the Gaussian distribution, as used in the Wiener process for Black–Scholes for example. The CRR model is based on a set of n finite sub-periods Δt. Hence the following CRR algorithm for

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price a second-generation option in Chapter 11, Section 11.8). Let us start from an underlying spot price S, modeled by a geometric general Wiener process as used for the Black–Scholes formula (cf. Eq. 10.1): In discrete time, we have (cf. Eq. 8.1) where (t) is a

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due to Vasicek. The Vasicek model is a process governing a single rate r: (11.1) Its stochastic component is same as in the traditional Wiener process. The mean reversion logically applies to its deterministic component. The b coefficient is featuring the mean objective of r, on the long run: the

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considered in the next two sub-sections. Basically, the discretized variant of HJM is using n forward LIBOR rates that are modeled as n geometric Wiener processes, (11.2) where Li is the ith forward LIBOR rate, applying from maturity T to T + δ; δ is the LIBOR reference, in years,

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a strict mean-reversion term. Moreover, by doing so, the LMM model takes into account the correlation – via covariance terms – between each of the standard Wiener processes dZi (cf. Eq. 8.6): (11.3) To calibrate the model represented by Eq. 11.2 and Eq. 11.3, we must estimate the

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rates cannot go below zero. However, we may contest the validity of modeling the LIBOR rates involved in caps and floors by individual and independent Wiener processes, hence the need for a more adapted approach. The most common way is by using the LMM model, introduced at the end of the

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as developed in the previous chapter. Calling S1 and S2 the spot price of assets 1 and 2 respectively, both are modeled as a standard Wiener process and the correlation between dZ1 and dZ2 processes is ρ12, being considered as a constant, just as for σ1 and σ2. For a time T

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with the main application field for exchange options, that is, in MandA operations. Practically speaking, the hypotheses, namely μ and σ are constants, of both Wiener processes, and, more importantly, the constant correlation coefficient, express the limits of the valuation formula. Coming back to MandA operations, the above formula justifies – and

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a continuation of Chapter 10. As seen in Chapter 10, Section 10.1, the volatility, denoted σ, originates from processes such as the general Wiener process used to model an underlying, and appears as a key ingredient for pricing non-conditional derivatives such as options. As such, strictly speaking, options should

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SCt being the “open”, respectively the “close”, of day t; such as The Garman–Klass volatility is These formulae are based on a geometric Wiener process in prices, but they do not take into account the drift of the process. Rogers and Satchell proposed the following formula, that turns out to

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a volatility, since it is not a price or a rate of a financial instrument as such. However, even if usual models, like the Wiener process, consider the volatility of the instrument as a constant, in practice we are aware that volatility is changing over time, so why not try to

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Chapter 11, Section 11.3 (for interest rate processes). Considering the volatility σt of an asset of price S modeled by a geometric, general Wiener process as per Eq. 8.11b of Chapter 8, but where the constant volatility σ is replaced by the variable σt, a very simplistic volatility model

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for multivariate GARCH models11 or for a multivariate stochastic volatility model, generalizing the Heston model (cf. Section 12.2) in a matrix process of n Wiener processes, leading to a (complex) stochastic correlation model that still allows for analytic tractability.12 12.5 VOLATILITY AND VARIANCE SWAPS Volatility and variance swaps belong

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of unexpected market news, such as key economic statistical data. By nature, a process involving some Gaussian stochastic component, such as the usual geometric, general Wiener process, cannot involve prices jumps: starting from the discrete time form of this process (from Eq. 8.11b) if we make Δt → 0, changes in

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is combining a general Wiener with a Poisson process: where Q is a Poisson process, independent from Z. Note that the σ of the Wiener process refers to the standard deviation of the returns out of the occurrence of jumps! Further calculations lead to European option prices that differ from the

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with where σtot is the total volatility, including due to jumps: σtot = σ + σjumps γ = σjumps/σtot. As an example of combination between a general Wiener process and a Poisson process, let us start from one of the prices simulation of L'Oreal stock, used for the Monte Carlo simulation, in Chapter

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.9% and the Monte Carlo simulation is performed on successive 9000 intervals Δts of a 1/100th of a day, by simulating a usual general Wiener process. By adding, in Figure 15.2, a random generated Poisson process of four equal jump sizes of €2 (to make them appear clearly), we

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3, as if the market has showed several bullish shocks. Figure 15.2 Random generated Poisson process with our equal jump sizes Figure 15.3 Wiener process versus Wiener + Poisson process Note that the Poisson component of the jump-diffusion process refers to the probabilistic occurrence of an event, here a

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of a financial time series, so that they can better be viewed as forecasting models (at least, for risk management purpose), unlike the traditional Wiener process and the models presented in the previous sections of this chapter. Coming back to the starting point of our non-deterministic description of financial products

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, in Chapter 8, Section 8.2, we have defined the Brownian motion, or standard Wiener process (also called white noise) as per Eq. 8.2: where y(t) is distributed as (E = 0, V = 1), so that Z(t) is

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series is persistent or trending: two successive Δts are positively correlated. Hence the idea of modeling financial time series by a kind of “generalized” general Wiener process (generalizing Eq. 8.11) (15.1) that needs to significantly adapt the Itô lemma in particular, and the stochastic calculus in general (moreover, note

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non-linear feature. In its more general formulation, a stock, for example a regime-switching model, instead of determining a single process, like a general Wiener process leading to one determines k process regimes pt, t = 1, …, k, are possible during the next Δt: To simplify, by limiting k to 2,

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based on non-Markovian regime-switching processes. A similar step has been widely investigated from GARCH models (cf. Chapter 9, Section 9.6), instead of Wiener processes. Regime-switching models have been developed for most of the financial instruments (currencies, interest rates, stocks, etc.), but also on volatilities, to cope with

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proved to be non-negligible in the estimation of the parameters of usual diffusion processes (for example, the drift and the volatility of a general Wiener process), and affects these estimations more seriously than the “sampling discreteness” considered in the previous paragraph.13 15.2.3 Consequences for option pricing Option pricing

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/risk-free spot instruments zero-coupon bonds see also bond duration book value method bootstrap method Brinson’s BHB model Brownian motion see also standard Wiener process bullet bonds Bund (German T-bond) 10-year benchmark futures callable bonds call options call-put parity jump processes see also options Calmar ratio

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see bond duration duration D dVega/dTime dynamic replication see delta-Gamma neutral management dZ Black–Scholes formula fractional Brownian motion geometric Wiener process martingales properties of dZ(t) standard Wiener process economic capital ED see exposure at default effective duration, bonds efficient frontier efficient markets EGARCH see exponential GARCH process EONIA see Euro

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EWMA process I/E/MGARCH processes non-linear models regime-switching models variants volatility general Wiener process application fractional Brownian motion gamma processes geometric Wiener process Itô Lemma Itô process jump processes volatility modeling see also standard Wiener process geometric average geometric Wiener process German Bund see Bund (German T-Bond) global VaR Gordon–Shapiro method government bonds

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relative VaR return measures expected return performance Portfolio Theory in practice risk vs return ratios several stock positions single stock positions time periods returns general Wiener process instantaneous measures “reverse cash and carry” operations rho risk see individual types Risk at Value (RaV) “risk-free” bonds risk-free yield curve risk

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process variance swaps Vasicek model VDAX index vega VIX index volatility annualized basket options correlation modeling curves delta-gamma neutral management derivatives dVega/dTime general Wiener process historical implied intraday volatility modeling option pricing practical issues realized models smiles smirks variance swaps vega volga vomma VXN index weather White see Hull and

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White model white noise AR process see also Brownian motion; standard Wiener process Wiener see general Wiener process; standard Wiener process WTI Crude Oil futures Yang–Zang volatility yield, convenience yield curves capital markets components CRS pricing cubic splines method definition EONIA/OIS swaps

Analysis of Financial Time Series

by Ruey S. Tsay · 14 Oct 2001

of stock options used in the chapter. In Section 6.2, we provide a brief introduction of Brownian motion, which is also known as a Wiener process. We then discuss some diffusion equations and stochastic calculus, including the well-known Ito’s lemma. Most option pricing formulas are derived under the assumption

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to emphasize that t is continuous. However, we use the same notation xt , but call it a continuous-time stochastic process. 6.2.1 The Wiener Process In a discrete-time econometric model, we assume that the shocks form a white noise process, which is not predictable. What is the counterpart of

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shocks in a continuoustime model? The answer is the increments of a Wiener process, which is also known as a standard Brownian motion. There are many ways to define a Wiener process {wt }. We use a simple approach that focuses on the small change wt = wt+ t − wt associated

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with a small increment t in time. A continuous-time stochastic process {wt } is a Wiener process if it satisfies 1. 2. √ t, where is a standard normal random variable, and wt = wt is independent of w j for all j ≤ t

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time t is normally distributed with mean zero and variance t. To put it formally, for a Wiener process wt , we have that wt − w0 ∼ N (0, t). This says that the variance of a Wiener process increases linearly with the length of time interval. CONTINUOUS - TIME MODELS -0.4 w -1.5 -1

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.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 time time Figure 6.1. Four simulated Wiener processes. Figure 6.1 shows four simulated Wiener processes on the unit time interval [0, 1]. They are obtained by using a simple version of the Donsker’s Theorem in the

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, 1], let [nt] be the integer part of nt. Define wn,t = √1n i=1 z i . Then wn,t converges in distribution to a Wiener process wt on [0, 1] as n goes to infinity. The four plots start with w0 = 0, but drift apart as time increases, illustrating that the

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variance of a Wiener process increases with time. A simple time transformation from [0, 1) to [0, ∞) can be used to obtain simulated Wiener processes for t ∈ [0, ∞). Remark: A formal definition of a Brownian motion wt on a probability space

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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

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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

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for a small change in the variable y. Then the model for xt is d xt = µ dt + σ dwt , (6.1) where wt is a Wiener process. If we consider a discretized version of Eq. (6.1), then √ xt − x0 = µt + σ t for increment from 0 to t. Consequently, E(xt

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, µ and σ of Eq. (6.1) are referred to as the drift and volatility parameters of the generalized Wiener process xt . 6.2.3 Ito’s Processes The drift and volatility parameters of a generalized Wiener process are timeinvariant. If one further extends the model by allowing µ and σ to be functions 226 CONTINUOUS

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process xt is an Ito’s process if it satisfies d xt = µ(xt , t) dt + σ (xt , t) dwt , (6.2) where wt is a Wiener process. This process plays an important role in mathematical finance and can be written as t t xt = x0 + µ(xs , s) ds + σ (xs , s) dws

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(6.2) is referred to as a stochastic diffusion equation with µ(xt , t) and σ (xt , t) being the drift and diffusion functions, respectively. The Wiener process is a special Ito’s process because it satisfies Eq. (6.2) with µ(xt , t) = 0 and σ (xt , t) = 1. 6.3 ITO’S

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’s Lemma Assume that xt is a continuous-time stochastic process satisfying d xt = µ(xt , t) dt + σ (xt , t) dwt , where wt is a Wiener process. Furthermore, G(xt , t) is a differentiable function of xt and t. Then, dG = ∂G ∂G ∂G 1 ∂2G 2 σ (xt , t) dt + µ(xt

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) dwt . ∂x ∂t 2 ∂x2 ∂x (6.6) Example 6.1. As a simple illustration, consider the square function G(wt , t) = wt2 of the Wiener process. Here we have µ(wt , t) = 0, σ (wt , t) = 1 and ∂G = 2wt , ∂wt ∂G = 0, ∂t ∂2G = 2. ∂wt2 Therefore, 1 dwt2 = 2wt × 0

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dwt = µ − dt + σ dwt . Pt 2 Pt2 Pt 2 ITO ’ S LEMMA 229 This result shows that the logarithm of a price follows a generalized Wiener Process with drift rate µ − σ 2 /2 and variance rate σ 2 if the price is a geometric Brownian motion. Consequently, the change in logarithm of

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price of a stock follows the geometric Brownian motion d Pt = µPt dt + σ Pt dwt , then the logarithm of the price follows a generalized Wiener process σ2 d ln(Pt ) = µ − dt + σ dwt , 2 where Pt is the price of the stock at time t and wt is a

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Wiener process. Therefore, the change in log price from time t to T is normally distributed as σ2 2 ln(PT ) − ln(Pt ) ∼ N µ− (6.9) (T −

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= t for both Eqs. (6.11) and (6.12), one can construct a portfolio of the stock and the derivative that does not involve the Wiener process. The appropriate portfolio is short on derivative and long ∂∂GPtt shares of the stock. Denote the value of the portfolio by Vt . By construction, Vt

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of Black–Scholes differential equation in Section 6.5, (h + ) = ∂∂GPtt is the number of shares in the portfolio that does not involve uncertainty, the Wiener process. We know that ct = Pt (h + ) + Bt , where Bt is the dollar amount invested in risk-free bonds in the portfolio (or short on the

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xit is a continuous-time stochastic process satisfying d xit = µi (xt )dt + σi (xt ) dwit , i = 1, . . . , k, (6.21) where wit is a Wiener process. It is understood that the drift and volatility functions µi (xit ) and σi (xit ) are functions of time index t as well. We omit t

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from their arguments to simplify the notation. For i = j, the Wiener processes wit and w jt are different. We assume that the correlation between dwit and dw jt is ρi j . This means that ρi j is

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the opposite side of differentiation so that t d xs = xt − x0 0 continues to hold for a stochastic process xt . In particular, for the Wiener process wt , t t we have 0 dws = wt because w0 = 0. Next, consider the integration 0 ws dws . Using the prior result and taking integration

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stochastic differential equation nt d Pt = µdt + σ dwt + d (Ji − 1) , Pt i=1 (6.26) 245 JUMP DIFFUSION MODELS where wt is a Wiener process, n t is a Poisson process with rate λ, and {Ji } is a sequence of independent and identically distributed nonnegative random variables such that X

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table. EXERCISES 1. Assume that the log price pt = ln(Pt ) follows a stochastic differential equation dpt = γ dt + σ dwt , where wt is a Wiener process. Derive the stochastic equation for the price Pt . 2. Considering the forward price F of a nondividend-paying stock, we have Ft,T = Pt er

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ARMA model, 322 marginal models, 327 Vector MA model, 318 Volatility, 79 Volatility equation, 82 Volatility model, factor, 383 Volatility smile, 244 White noise, 26 Wiener process, 223 generalized, 225

Mathematics for Finance: An Introduction to Financial Engineering

by Marek Capinski and Tomasz Zastawniak · 6 Jul 2003

obtain a limit for all times t ≥ 0 simultaneously, but this is beyond the scope of this book. The limit W (t) is called the Wiener process (or Brownian motion). It inherits many of the properties of the random walk, for example: 1. W (0) = 0, which corresponds to wN (0) = 0

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resulting continuous time model the stock price is given by (8.5) S(t) = S(0)emt+σW (t) , where W (t) is the standard Wiener process (Brownian motion), see Section 3.3.2. This means, in particular, that S(t) has the log normal distribution. Consider a European option on the

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Pricing 187 a probability P∗ such that V (t) = W (t) + m − r + 12 σ 2 t/σ (rather than W (t) itself) becomes a Wiener process under P∗ , then the exponential factor 1 2 e(m−r+ 2 σ )t will be eliminated from the ﬁnal expression. (The existence of such

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−rT (S(T ) − X)+ . Let us compute this expectation. Because V (T ) = W (t) + m − r + 12 σ 2 σt for t ≥ 0 is a Wiener process under P∗ , the random variable V (T ) = W (T ) + 1 2 T m − r + 2 σ σ is normally distributed with mean 0 and variance

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)eσb+ru− 2 σ u = a and put V (t) = W (t)+ m − r + 12 σ 2 σt for any t ≥ 0, which is a Wiener process under P∗ . In particular, V (u) is normally distributed under P∗ with mean 0 and variance u. The right-hand side of (8.8) is

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σ u √ e− 2u dx 2πt −∞ $ b (x−σu)2 1 √ = S(0) e− 2u dx. 2πt −∞ Now observe that, since V (t) is a Wiener process under P∗ , the random variables V (u) and V (t) − V (u) are independent and normally distributed with mean 0 and variance u and t

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, futures contract value variance value at risk Greek parameter vega symmetric random walk; weights in a portfolio weights in a portfolio as a row matrix Wiener process, Brownian motion position in a risky security strike price position in a ﬁxed income (risk free) security; yield of a bond position in a derivative

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of a strategy 76, 87 VaR 202 variable interest 255 Vasiček model 260 vega 197 vega neutral portfolio volatility 71 weights in a portfolio Wiener process 69 yield 216 yield to maturity 229 zero-coupon bond 39 197 94

Frequently Asked Questions in Quantitative Finance

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

device for quantitative finance decades later. The starting point for almost all financial models, the first equation written down in most technical papers, includes the Wiener process as the representation for randomness in asset prices. See Wiener (1923). 1950s Samuelson The 1970 Nobel Laureate in Economics, Paul Samuelson, was responsible for setting

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. Itô’s lemma tells us the stochastic differential equation for the value of an option on that asset. In mathematical terms, if we have a Wiener process X with increments dX that are normally distributed with mean zero and variance dt then the increment of a function F(X) is given by

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stating the theorem. Given a random variable y satisfying the stochastic differential equationdy = a(y, t) dt + b(y, t) dX , where dX is a Wiener process, and a function f(y, t) that is differentiable with respect to t and twice differentiable with respect to y, then f satisfies the following

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, . . . , yn, t) of n stochastic quantities and time such thatdyi = ai (y1, y2 , . . . , yn, t) dt + bi (y1, y2, . . . , yn, t) dXi , where the n Wiener processes dXi have correlations ρij then We can understand this (if not entirely legitimately derive it) via Taylor series by using the rules of thumb Another

Risk Management in Trading

by Davis Edwards · 10 Jul 2014

sigma) to time n (indicated above the sigma) A common type of random process is called Brownian motion. Brownian motion may also be called the Wiener process. In this process, the percent change in price (ΔP) is a random number described by a normal distribution. The end result of this process is

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movements get smaller and will never fall below zero. (See Figure 3.9, Dispersion in a Random Series.) There are two major factors that make Wiener processes important to financial mathematics. First, the dispersion of expected results accumulates in a manner that is easy to calculate mathematically. In this type of process

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assets, this type of dispersion process is a reasonably good model for how prices actually change over time. (See Equation 3.7, Dispersion of a Wiener Process.) For a Weiner process that follows N(0, σ): Variance at time T = σ2T Standard Deviation at time T = σ T where N(0,σ

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.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

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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

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volatility are represented as annualized numbers σ Volatility. The annualized volatility that is used to scale the change in the Wiener Process (ΔWt) to the asset being modeled ΔWt 75 Change in the Wiener process. A draw from a normal distribution scaled to the appropriate time step ΔWt = N(0, 1) Δt N(0

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, drift and volatility are represented as annualized numbers Volatility. The annualized volatility that is used to scale the change in the Wiener Process (ΔWt) to the asset being modeled Change in the Wiener process. A draw from a normal distribution scaled to the appropriate time step ΔWt = N(0, 1) Δt Normal Distribution. A

Mathematical Finance: Theory, Modeling, Implementation

by Christian Fries · 9 Sep 2007

(t − s)In , where In denotes the n × n identity matrix. Then W is called (n-dimensional) P-Brownian motion or a (n-dimensional) P-Wiener process. y We have not yet discussed the question whether a process with such properties exists (it does). The question for its existence is non-trivial

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– implied volatility . . . . . . . . . . . . . . . . 97 volatility bootstrapping . . . . . . . . . . . . 268 volatility surface . . . . . . . . . . . . . . . . . . . 97 – definition . . . . . . . . . . . . . . . . . . . . . . . 97 W weak convergence . . . . . . . . . . . . . . . . . 39 weather derivative . . . . . . . . . . . . . . . . . 61 weighted Monte-Carlo . . . . . . . . . . . . 185 Wiener measure . . . . . . . . . . . . . . . . . . . 40 Wiener process . . . . . . . . . . . . . . . . . . . . 38 Y Yield . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 Z Zerlegung . . . . . . . . . . . . . . . . . . . . . . . . – Feinheit . . . . . . . . . . . . . . . . . . . . . . . Zero Coupon Bond – Cashflow Diagram . . . . . . . . . . . . . . Zero Structure . . . . . . . . . . . . . . . . . . . . zero structure – definition . . . . . . . . . . . . . . . . . . . . . . Zero Swap – Cashflow Diagram . . . . . . . . . . . . . . Zinsstrukturen . . . . . . . . . . . . . . . . . . . . 401 401 162

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

in the ﬁnancial literature over the past two decades. Most of the proposed models are particular cases of a stochastic volatility component driven by a Wiener process superposed with a pure-jump component accounting for the Handbook of Modeling High-Frequency Data in Finance, First Edition. Edited by Frederi G. Viens, Maria

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represent a general geometric Brownian motion in the form St = S0 eσ Wt +μt , where (Wt )t≥0 is the Wiener process. In the context of the above Black–Scholes model, a Wiener process can be deﬁned as the log return process of a price process satisfying the Black–Scholes conditions (1)–(3) with

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) and BarndorffNielsen (1998), respectively, to describe the log return process Xt := log St /S0 of a ﬁnancial asset. Both models can be seen as a Wiener process with drift that is time-deformed by an independent random clock. That is, (Xt ) has the representation Xt = σ W (τ (t)) + θτ (t) + bt

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, (1.1) where σ > 0, θ, b ∈ R are given constants, W is Wiener process, and τ is a suitable independent subordinator (nondecreasing Lévy process) such that Eτ (t) = t, and Var(τ (t)) = κt. In the VG model

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) β2 σ42 (t) dσi2 (t) = (ωi − φi σi2 (t))dt + αi σi2 (t)dWi (t), i = 1, . . . , 4, where {Wi (t)}6i=1 are independent Wiener processes. Moreover, we assume that the logarithmic noises η1 (t), η2 (t) are i.i.d. Gaussian, possibly contemporaneously correlated and independent from p. We also

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–226 Whittle contrast function, 225 Whittle estimator, 227 Whittle maximum likelihood estimate, 225 Whittle-type criterion, 221 Whole real line, solution construction in, 399–400 Wiener process, 3, 7, 8 WMT data series, DFA and Hurst methods applied to, 151 XOM data series, DFA and Hurst methods applied to, 152 Xu, Junyue