Simulating Stochastic Differential Equations
نویسنده
چکیده
Let S t be the time t price of a particular stock. We know that if S t ∼ GBM (µ, σ 2), then S t = S 0 e (µ−σ 2 /2)t+σBt (1) where B t is the Brownian motion driving the stock price. An alternative possibility is to use a stochastic differential equation (SDE) to describe the evolution of S t. In this case we would write S t = S 0 + t 0 µS u du + t 0 σS u dB u (2) or in shorthand , dS t = µS t dt + σS t dB t. A number of observations are in order: (1) The SDE defined by (2) can be shown to be well-defined. In particular, while the first integral on the right-hand-side of (2) is a regular Riemann integral, the second integral is a stochastic integral. Without going into any technical details, it is convenient to interpret this integral as t 0 σS u dB u = lim h→0 σS t i−1 (B t i − B t i−1) (4) where h = max i |t i − t i−1 | is the width of the partition. The important feature of (4) is that the S t terms are evaluated at the left-hand point of the intervals. This feature is extremely important in finance as it may be interpreted as modelling the inability of people to see into the future. In general, we can similarly interpret the stochastic integral, X(u, B u) dB u , so that t 0 X(u, B u) dB u = lim h→0 X(t i−1 , B ti−1)(B ti − B ti−1). (2) At this point, it is not clear that (1) and (2) define the same process but we will soon see that this is indeed the case. (3) It is important to note that on its own, equation (3) has no meaning. It is only shorthand for equation (2). In general, it is often convenient to model stock prices and interest rates as SDE's. Another example is given by the assumption that X t = log(S t) is an Ornstein-Uhlenbeck (OU) process. In particular, this means that dX t = −γ(X t − α) dt + σdB t (5) where γ, α and σ are non-negative constants. Exercise 1 In what circumstances might an Ornstein-Uhlenbeck model prove useful?
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تاریخ انتشار 2004