Numerical Solution of Stochastic Di erential Equations in ...

1 Numerical Solution ofStochastic Differential Equationsin FinanceTimothy SauerDepartment of MathematicsGeorge Mason UniversityFairfax, VA chapter is an introduction and survey of Numerical solutionmethods for Stochastic differential Equations . The solutions will be continuousstochastic processes that represent diffusive dynamics, a common modelingassumption for financial systems. We include a review of fundamental con-cepts, a description of elementary Numerical methods and the concepts ofconvergence and order for Stochastic differential equation the remainder of the chapter we describe applications of SDE solvers toMonte-Carlo sampling for financial pricing of derivatives. Monte-Carlo simu-lation can be computationally inefficient in its basic form, and so we exploresome common methods for fostering efficiency by variance reduction and theuse of quasi-random numbers. In addition, we briefly discuss the extension ofSDE solvers to coupled systems driven by correlated noise, which is applicableto multiple asset Stochastic differential equationsStochastic differential Equations (SDEs) have become standard models for fi-nancial quantities such as asset prices, interest rates, and their derivatives.

2 Un-like deterministic models such as ordinary differential Equations , which havea unique Solution for each appropriate initial condition, SDEs have solutionsthat are continuous-time Stochastic processes. Methods for the computationalsolution of Stochastic differential Equations are based on similar techniques forordinary differential Equations , but generalized to provide support for stochas-tic will begin with a quick survey of the most fundamental concepts fromstochastic calculus that are needed to proceed with our description of nu-merical methods. For full details, the reader may consult Klebaner (1998);Oksendal (1998); Steele (2001).2 Timothy SauerA set of random variablesXtindexed by real numberst 0 is called acontinuous-time Stochastic process. Each instance, orrealizationof the stochas-tic process is a choice from the random variableXtfor eacht, and is thereforea function (deterministic) functionf(t) can be trivially considered as a stochasticprocess, with varianceV(f(t)) = 0.

3 An archetypal example that is ubiquitousin models from physics, chemistry, and finance is theWiener processWt, acontinuous-time Stochastic process with the following three properties:Property 1. For eacht, the random variableWtis normally distributed withmean 0 and 2. For eacht1< t2, the normal random variableWt2 Wt1is indepen-dent of the random variableWt1, and in fact independent of allWt,0 t 3. The Wiener processWtcan be represented by continuous Wiener process, named after Norbert Wiener, is a mathematical con-struct that formalizes random behavior characterized by the botanist RobertBrown in 1827, commonly called Brownian motion. It can be rigorously de-fined as the scaling limit of random walks as the step size and time intervalbetween steps both go to zero. Brownian motion is crucial in the modeling ofstochastic processes since it represents the integral of idealized noise that is in-dependent of frequency, called white noise.

4 Often, the Wiener process is calledupon to represent random, external influences on an otherwise deterministicsystem, or more generally, dynamics that for a variety of reasons cannot bedeterministically typicaldiffusion processin finance is modeled as a differential equationinvolving deterministic, ordriftterms, and Stochastic , ordiffusionterms, thelatter represented by a Wiener process, as in the equationdX=a(t,X)dt+b(t,X)dWt(1)Notice that the SDE (1) is given in differential form, unlike the derivative formof an ODE. That is because many interesting Stochastic processes, like Brow-nian motion, are continuous but not differentiable. Therefore the meaning ofthe SDE (1) is, by definition, the integral equationX(t) =X(0) + t0a(s,y)ds+ t0b(s,y)dWs,where the meaning of the last integral, called an Ito integral, will be < t1< .. < tn 1< tn=dbe a grid of points on the interval[c,d]. The Riemann integral is defined as a limit dcf(x)dx= lim t 0n i=1f(t i) ti, Numerical Solution of Stochastic Differential Equations in Finance3where ti=ti ti 1andti 1 t i ti.

5 Similarly, theIto integralis the limit dcf(t)dWt= lim t 0n i=1f(ti 1) Wiwhere Wi=Wti Wti 1, a step of Brownian motion across the a major difference: while thet iin the Riemann integral may be chosen atany point in the interval (ti 1,ti), the corresponding point for the Ito integralis required to be the left endpoint of that random variables, so is the Ito integralI= dcf(t)dWt. ThedifferentialdIis a notational convenience; thusI= dcf dWtis expressed in differential form asdI= differentialdWtof Brownian motionWtis calledwhite noise. A typicalsolution is a combination of drift and the diffusion of Brownian solve SDEs analytically, we need to introduce the chain rule for stochas-tic differentials, called theIto formula:IfY=f(t,X), thendY= f t(t,X)dt+ f x(t,X)dx+12 2f x2(t,X)dx dx(2)where thedx dxterm is interpreted by using the identitiesdt dt= 0dt dWt=dWtdt= 0dWtdWt=dt(3)The Ito formula is the Stochastic analogue to the chain rule of conventionalcalculus.

6 Although it is expressed in differential form for ease of understanding,its meaning is precisely the equality of the Ito integral of both sides of theequation. It is proved under rather weak hypotheses by referring the equationback to the definition of Ito integral (Oksendal, 1998).Some of the important features of typical Stochastic differential equationscan be illustrated using the following historically-pivotal example from fi-nance, often called the Black-Scholes diffusion equation:{dX= X dt+ X dWtX(0) =X0(4)4 Timothy Sauerwith constants and . Although the equation is comparatively simple, thefact that it can be exactly solved led to its central importance, by makinga closed-form formula available for the pricing of simple options (Black andScholes, 1973).The Solution of the Black-Scholes Stochastic differential equation is geo-metric Brownian motionX(t) =X0e( 12 2)t+ Wt.(5)To check this, writeX=f(t,Y) =X0eY, whereY= ( 12 2)t+ Wt.}

7 Bythe Ito formula,dX=X0eYdY+12eYdY dYwheredY= ( 12 2)dt+ dWt. Using the differential identities from theIto formula,dY dY= 2dt,and thereforedX=X0eY(r 12 2)dt+X0eY dWt+12 2eYdt=X0eY dt+X0eY dWt= X dt+ X dWtas 1 shows a realization of geometric Brownian motion with constant driftcoefficient and diffusion coefficient . Similar to the case of ordinary differen-tial Equations , relatively few Stochastic differential Equations have closed-formsolutions. It is often necessary to use Numerical approximation Numerical methods for simplest effective computational method for the approximation of or-dinary differential Equations is Euler s method (Sauer, 2006). The Euler-Maruyama method (Maruyama, 1955) is the analogue of the Euler methodfor ordinary differential Equations . To develop an approximate Solution on theinterval [c,d], assign a grid of pointsc=t0< t1< t2< .. < tn= < w1< w2< .. < wnwill be determined at the respectivetpoints.

8 Given the SDE initial valueproblemNumerical Solution of Stochastic Differential Equations in Finance5{dX(t) =a(t,X)dt+b(t,X)dWtX(c) =Xc(6)we compute the approximate Solution as follows:Euler-Maruyama Methodw0=X0wi+1=wi+a(ti,wi) ti+1+b(ti,wi) Wi+1(7)where ti+1=ti+1 ti Wi+1=W(ti+1) W(ti).(8)The crucial question is how to model the Brownian motion Wi. DefineN(0,1) to be the standard random variable that is normally distributed withmean 0 and standard deviation 1. Each random number Wiis computed as Wi=zi ti(9)whereziis chosen fromN(0,1). Note the departure from the deterministicordinary differential equation case. Each set of{w0,..,wn}produced bythe Euler-Maruyama method is an approximate realization of the solutionstochastic processX(t) which depends on the random numberszithat werechosen. SinceWtis a Stochastic process, each realization will be different andso will our a first example, we show how to apply the Euler-Maruyama method tothe Black Scholes SDE (4).}

9 The Euler-Maruyama Equations (7) have formw0=X0(10)wi+1=wi+ wi ti+ wi will use the drift coefficient = and diffusion coefficient = ,which are values inferred from the series of market close share prices of Google,Inc. (NYSE ticker symbol GOOG) during the 250 trading days in 2009. Tocalculate the values and 2, the mean and variance, respectively, of the dailystock price returns were converted to an annual basis, assuming independenceof the daily exact realization, generated from the Solution (5), along with the cor-responding Euler-Maruyama approximation, are shown in Fig. 1. By corre-sponding, we mean that the approximation used the same Brownian motionrealization as the true Solution . Note the close agreement between the solutionand the approximating points, plotted as small circles every time addition, the original time series of Google share prices is shown for com-parison. Both the original time series (grey curve) and the simulation from(5) (black curve) should be considered as realizations from the same diffusionprocess, with identical , and initial priceX0= Sauer01time (years)300600priceFig.

10 1. Solution to the Black Scholes Stochastic differential equation (4).The exact Solution (5) is plotted as a black curve. The Euler-Maruyama approxima-tion with time step t= is plotted as circles. The drift and diffusion parametersare set to = and = , respectively. Shown in grey is the actual stockprice series, from which and were another example, consider theLangevin equationdX(t) = X(t)dt+ dWt(11)where and are positive constants. In this case, it is not possible to ana-lytically derive the Solution to this equation in terms of simple processes. Thesolution of the Langevin equation is a Stochastic process called theOrnstein-Uhlenbeck process. Fig. 2 shows one realization of the approximate was generated from an Euler-Maruyama approximation, using the stepsw0=X0(12)wi+1=wi wi ti+ Wifori= 1,..,n. This Stochastic differential equation is used to model systemsthat tend to revert to a particular state, in this case the stateX= 0, inthe presence of a noisy background.