Stochastic Modelling Unit 3: Brownian Motion and Diffusions

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1 Stochastic Modelling Unit 3: Brownian Motion and Diffusions Russell Gerrard and Douglas Wright Cass Business School, City University, London June 2004 Contents of Unit 3 1 Introduction 2 Brownian Motion 3 Lévy Processes 4 Diffusions 5 Stochastic Calculus 1 Introduction 1.1 General The processes examined in this module all have continuous state space and operate in continuous time. The situations which are commonly modelled by such processes include those for which the quantity observed are continuously monitored and can undergo change at any time, especially in cases where a rapid response may be essential, such as security prices. In recent times greater interest has been focused on the use of processes related to Brownian motion in areas like forecasting and control of markets. The Black-Scholes equation is a case in point. 1.2 Contents We shall begin by reviewing the random walk and its properties, in preparation for deriving Brownian motion from it by a limiting procedure. 1

2 Some of the properties of Brownian motion will be examined. It is seen that Brownian motion itself is unsuitable as a financial model, but simple transformations such as the general Wiener process and the geometric Brownian motion (or lognormal Wiener process) are popular. We investigate such problems as the probability that a Wiener process hits a particular value within a specified time limit. On the practical side, the problem of finding the right Wiener model to suit your data is addressed. Even a geometric Brownian motion, being continuous, cannot model sudden jumps in market prices. We combine Wiener processes with compound Poisson processes to form a very broad class of models known as L evy processes. Neither Brownian motion nor the Wiener process is stationary; we introduce the Ornstein-Uhlenbeck process as a stationary model related to Brownian motion. This in turn serves as a first look at the class of diffusions. Some properties of diffusions may best be addressed by expressing them in the form of stochastic differential equations. The idea of a stochastic integral will be introduced, and we shall investigate the uses of SDEs. 2 Brownian motion 2.1 From random walk to Brownian Motion The basic random walk is well understood: X n+1 = X n + J n+1, where the J n are all ±1 with probability 1 2 each. If we assume that X 0 = 0, then so that X n = n J i, IEX n = niej i = 0, and Var(X n ) = nvar(j i ) = n. We change the random walk, so that it jumps twice as often (at times 0.5, 1, 1.5,...), but not by so much (variance only 1 2 ). By time 1 the RW has made 2 jumps, each of mean 0, variance 0.5, so the overall variance is

3 Repeat the trick: the chain jumps at 0.25, 0.5, 0.75,..., and the jump height is now ±0.5, with variance Mean is still 0, variance 1 at time 1. Continue to do this. At the kth stage X(t) is the sum of 2 k t variables, each with variance 2 k. In the limit the distribution is Normal (cf Central Limit Theorem). The resulting process is Brownian motion. 2.2 Properties of Brownian motion Brownian motion, denoted by B(t), inherits the properties of the SSRW: 1. B(0) = 0 2. IEB(t) = 0 3. VarB(t) = t 4. B(t + s) B(t) is independent of B(t) (and indeed of H t ) and has the same distribution as B(s). (In other words, BM has stationary, independent increments) 5. Cov(B(s), B(t)) = min(s, t) 6. B possesses the Markov property: given B(t), the values of B before time t are irrelevant when predicting the future behaviour of B. 7. B possesses the martingale property: IE[B(t + s) H t ] = B(t). 8. B is recurrent, returning infinitely often to 0 (or any other level). In addition 9. B(t) is normally distributed 10. The trajectory of B is continuous. Any Gaussian stochastic process is fully determined by its mean, variance and covariance functions. Therefore 11. B 1 (t) = 1 c B(ct) is a Brownian motion, for any c > 0. (This is the scaling property.) 12. B 2 (t) = tb( 1 ) is a Brownian motion. (This is the time inversion property.) t The trajectory of B is not smooth. It is not differentiable anywhere. Any small piece of a Brownian motion trajectory, if expanded, looks like the whole trajectory. (Rather like fractals.) 2.3 Time to hit a point The probability that B has hit a (where a > 0) some time before t is 2P(B(t) > a) = 2Φ( a/ t). This is derived from the Reflection Principle: if B hits a before t, then at time t the Brownian motion is equally likely to be above or below a. We can deduce the density function of the first time to hit a: f Ta (t) = a ( ) exp a2 (t > 0). 2πt 3 2t As for the random walk, the expectation is infinite. 3

4 2.4 Wiener process B(t) is standard Brownian motion. We can add drift (µ) and scale (σ) to get a process I shall call this a Wiener process. W t = µt + σb(t). It is more flexible than standard BM as a model. Parameters µ and σ may be estimated by considering the observed distribution of increments. The fit of the model may be tested by testing the increments for normality testing for lack of dependence on the current value testing for correlation between successive increments The density of the hitting time of W on a(> 0) is ) a f Ta (t) = ( σ 2πt exp (a µt)2 3 2σ 2 t (t > 0). We can integrate f to show that W is certain to hit a if drift µ 0, but if µ < 0 the probability is e 2aµ. 2.5 Geometric Brownian motion A model which is frequently used for modelling the behaviour of a variety of assets is geometric Brownian motion (or the lognormally distributed Wiener process): S t = S 0 e Wt = S 0 e µt+σb(t). In order to fit a GBM to a set of data, just fit a Wiener process model to log S t. Notice that ( [ IES t = S 0 e µt IEe σb(t) = S 0 exp µ + 1 ] ) 2 σ2 t. If we assume that the price of an asset behaves like a martingale, then we must have µ+ 1 2 σ2 = Simple option pricing The value of a European option depends on the price S t of the underlying asset at the time t when the contract falls due, and in particular on whether the price is higher or lower than the exercise price c. 4

5 An option to purchase a unit at price c at time t will have value S t c at time t if S t > c, no value otherwise. The value of the option at time s will be IE[(S t c) + H s ]. If S t is a GBM, then and therefore where IE[(S t c) + H s ] = S t = S s e µ(t s)+σb(t s), z 0 ( ) S s e µ(t s)+σz t s c φ(z) dz, z 0 = µ(t s) + log(c/s s) σ t s and where φ is the standard Normal density. This integral is not hard to evaluate: IE[(S t c) + H s ] = S s e (µ+σ2 /2)(t s) Φ( z 0 + σ t s) cφ( z 0 ) American options are harder to handle. It is not just a question of finding the maximum value of S over the period up to time t, as the time when this occurs is not a stopping time. 2.7 The Ornstein-Uhlenbeck process A process related to Brownian motion, but stationary. Used for modelling processes which have a tendency to drift towards 0. We introduce it as a stationary, zero-mean Gaussian process with variance τ 2 and autocovariance function Cov(U s, U t ) = τ 2 e γ t s. It has the Markov property. A stationary distribution exists as long as γ > 0. Notice that, if U is observed only at discrete times 1, 2,..., then Corr(X t, X t+k ) = α k where α = e γ. Therefore U is an AR(1) if observed at discrete times, and may be seen as a generalisation of AR(1) to continuous time. 2.8 Planar white noise Like a Gaussian version of a 2-d Poisson process. Every region in the quadrant has a random area, Normally distributed with mean 0, variance equal to its area. We can define B(t) as the random area of the rectangle joining (0, 0) to (1, t). The Ornstein-Uhlenbeck process can be thought of as the random area of the rectangle from (0,0) to (τe γt, τe γt ). This formulation is sometimes useful for visualisation and for evaluation of covariances. 5

6 3 Lévy Processes 3.1 Introduction Brownian motion and simple transformations of it are continuous. Large jumps in short intervals are very rare. We need a model with larger jumps, at random times. A Lévy process is any continuous-time process with stationary, independent increments. Brownian motion and the Poisson process are examples. 3.2 Decomposition The definition is quite restrictive. For example, the distribution of X(t) must be infinitely divisible. In fact, every Lévy process is the sum of: a deterministic part µt a continuous random part σb(t) a discontinuous random part J(t) B(t) is standard BM. J(t) is essentially a compound Poisson process: in any time interval (t, t + s) the number of jumps of height > x has a Poisson distribution with mean sν + (x) for some function ν +. Negative jumps are also permitted, with corresponding ν (x). The definition permits ν ± (x) as x 0, giving a process more complicated than compound Poisson. 3.3 Fitting a Lévy process model A LP model is usually introduced only because of occasional large deviations, so we need not worry about infinitely many jumps. This means that J really is a compound PP. But there are problems: the process is observed only occasionally; we cannot tell if a large deviation is due to a jump or an unusual Normal variate. if a time interval contains a jump, it also contains Normal variation; how much of the deviation is due to the jump? small jumps might be occurring quite frequently, unnoticed because of the noise. Usually we would guess a parametric distribution for jump sizes (eg double exponential or Normal), then try to estimate simultaneously the parameters µ and σ, the rate λ of the Poisson process and the parameters of the jump size distribution. The estimation process can be unreliable. 6

7 4 Diffusions 4.1 Introduction A Wiener process has drift and volatility, but they are constant. A diffusion is similar, but with varying drift and volatility: it is a continuous Markov process such that (for small h) X t+h X t has expectation hµ(x) and variance hσ 2 (x), given X t = x. Time-inhomogeneous diffusions permit µ(t, x) and σ 2 (t, x). Example: GBM. Given S t = s = e µt+σb(t), we have with expectation and variance S t+h s = s(e µh+σ[b(t+h) B(t)] 1), s(e µh+σ2 h/2 1) s 2 e 2µh (e 2σ2h e σ2h ). Taking limits as h 0, we have µ(s) = s (µ + 12 ) σ2 and σ 2 (s) = s 2 σ 2. Example: Vasicek model for interest rate. µ(r) = α(b r), σ(r) = σ. This is esentially an O-U process which is drawn towards b rather than towards Diffusion approximation A diffusion is often used as a continuous approximation to a discontinuous process. Example: Birth, death and immigration process. If X t = x then X t+h X t is +1 with prob. (α + xβ)h, 1 with prob. xδh, 0 otherwise. This gives µ(x) = α + x(β δ), σ 2 (x) = α + x(β + δ). The use of an approximation must always be justified. 4.3 Transition density Many of the properties of Markov jump processes carry over, with allowance for continuous states. Examples are the Chapman-Kolmogorov equations and the Kolmogorov differential equations. Both of these affect the transition density p t (x, y), defined by P(x < X s+t < x + dx X s = y) = p t (x, y)dx + o(dx). In some cases the transition density may be evaluated. 7

8 4.4 Long-term properties As for discrete state space, it is possible that the distribution of X t converges. The limit will be a continuous distribution on a range [K, K + ], say, possibly with a density function, π(y). If so, then d dy {σ2 (y)π(y)} = 2µ(y)π(y), and of course π(y) dy = 1. Example: Ornstein-Uhlenbeck process. Using the autocovariance function, we have and IE[U t+h U t U t = x] = (e γh 1)x Var(U t+h U t U t = x) = τ 2 (1 e 2γh ). Therefore µ(x) = γx, σ(x) = σ, where σ = τ 2γ. We try to solve dπ dy = 2γ σ 2 yπ, obtaining solution π(y) exp( γy 2 /σ 2 ), so that the limiting distribution is N(0, σ2 ), as 2γ expected. 5 Stochastic Calculus 5.1 The stochastic integral We are going to attempt to deduce the properties of diffusions by representing them in terms of integrals involving Brownian motion. We shall see that the range of applicability of stochastic calculus techniques is broad, though calculations are not always simple. We use O-U as an example. Recall its formulation as a diffusion. Since B t+h B t has mean 0 and variance h, we have U t+h U t = γu t dt + σ(b t+h B t ). In a deterministic context we would divide by h, take limits and get and solve the DE as du t dt = γu t + σ db t dt, U t = e γt U 0 + σ e γ(t s) db s 0 ds ds. The problem is that B is not differentiable. But that s OK: integrate by parts to get t 0 t e γ(t s) db s ds ds = B t γ which is fine. Can this be done more generally? t 0 e γ(t s) B s ds, It turns out to be possible to define integrals such as t 0 f(s, X s) db s in a consistent way (if f satisfies a square-integrability condition). This is a stochastic integral. 8

9 5.2 Properties of the stochastic integral Write Y t = t 0 f(s, X s) db s. Since f(t, X t ) is determined by H t, and B t+h B t is independent of H t and has mean 0, it follows that Y t is a martingale. Y is continuous (since B is) ( If f(s, X s ) is deterministic, ie just f(s), then Y t N 0, ) t 0 f(s)2 ds. [ ] t Even if f depends on random X, Var(Y t ) = IE f(s, X 0 s) 2 ds. But some familiar equalities from non-stochastic integration are lost. Thus t 0 B s db s = 1 2 B2 t 1 2 t. The rules of integration shown here are those of the Itô calculus. There are other forms of stochastic calculus, such as the Stratonovich calculus, which are based on different principles and have different rules. 5.3 The stochastic differential equation The equation Y t = t 0 g(s, X s) ds + t 0 f(s, X s) db s is often abbreviated as dy t = g(t, X t ) dt + f(t, X t ) db t. This stochastic differential equation is just a shorthand notation for the integral equation. 5.4 Itô s Lemma If X t is a process satisfying and if Y t = f(t, X t ), then dx t = U t dt + V t db t, dy t = f x dx t + f t dt V t 2 2 f x dt. 2 Example: a model often used for stock prices is ds t = αs t dt + σs t db t. If this were non-stochastic, we would divide by S t and identify dst S t with d(log S t ). Using the Itô calculus, we have d(log S t ) = 1 ds t + 1 ( 1 ) (σs S t 2 St 2 t ) 2 dt, which is σdb t + (α 1 2 σ2 )dt, and we conclude that log S t = log S 0 + σb t + (α 12 ) σ2 t, so that S t is a GBM with µ = α 1 2 σ2. 9

10 5.5 Moments of Itô processes Sometimes the moments of a process can be deduced from the SDE which it satisfies, without needing to solve the equation. Example: a model of customer volume in which both mean and variance of increments are proportional to current customer base. dn t = αn t dt + β N t db t. If µ 1,t denotes IEN t and µ 2,t denotes IEN 2 t, then dµ 1,t = IE(dN t ) = αie(n t ) dt = αµ 1,t dt, solved by µ 1,t = N 0 e αt, and, if we let Y t = N 2 t, then dy t = 2N t dn t (2)β2 N t dt, so that which in turn gives dµ 2,t = IE(2αNt 2 dt + 2αβN 3/2 t db t + β 2 N t dt), dµ 2 dt = 2αµ 2 + β 2 µ 1. The autocovariance function can be found similarly, using IE(X t H s ). Take the Ornstein- Uhlenbeck process as an example. ( t ) IE[X t H s ] = IE e γ(t s) X s + e γ(t u) db u H s ; therefore IE[X t X s H s ] = e γ(t s) X 2 s, from which we deduce that the correlation of X t with X s is e γ(t s) when the process is stationary. s 10