Pn(t) = P[t/h] = P[nt/T] , t [0, T], where [x] denotes the greatest integer less than or equal to x. Pn(t) is a left continuous step function.
We need to adjust , such that Pn(t) will converge when n goes to infinity. Consider
the m= n(2-1) Var (Pn(T)) = 4n(-1) 2
We wish to obtain a continuous time version of the random walk, we should expect the mean and variance of the limiting process P(T) to be linear in T. Therefore, we must have
(althrough dB(t) is).
Lecture #9: Black-Scholes option pricing formula
• Brownian Motion
The first formal mathematical model of financial asset prices, developed by Bachelier (1900), was the continuous-time random walk, or Brownian motion. This continuous-time process is closely related to the discrete-time versions of the random walk.
the derivative of Brownian motion, B’(t) does not exist in the ordinary sense, they are nowhere differentiable.
• Stochastic differential
equations
Despite the fact, the infinitesimal increment of Brownian motion, the limit of B(t+h) = B(t) as h approaches to an infinitesimal of time (dt) has earned the notation dB(t) and it has become a fundamental building block for constructing other continuous time process. It is called white noise. For P(t) define earlier we have dP(t) = dt + dB(t). This is called stochastic differential equation. The natural transformation dP(t)/dt = + dB(t)/dt doesn’t male sense because dB(t)/dt is a not well defined
Consider the following moments:
E[P(t) | P(t0)] = P(t0) +(t-t0) Var[P(t) | P(t0)] = 2(t-t0) Cov(P(t1),P(t2) = 2 min(t1,t2) Since Var[ (B(t+h)-B(t))/h ] = 2/h, therefore,
• The discrete-time random
walk
Pk = Pk-1 + k, k = (-) with probability (1-), P0 is fixed. Consider the following continuous time process Pn(t), t [0, T], which is constructed from the discrete time process Pk, k=1,..n as follows: Let h=T/n and define the process
n(2-1) T 4n(-1) 2 T
This can be accomplished by setting
= ½*(1+h /), =h
• The continuous time limit
It cab be shown that the process P(t) has the following three properties:
1. For any t1 and t2 such that 0 t1 < t2 T: P(t1)-P(t2) ((t2-t1), 2(t2-t1))
2. For any t1, t 2 , t3, and t4 such that 0 t1 < t2 t1 < t2 t3 < t4 T, the increment P(t2)- P(t1) is statistically independent of the increment P(t4)- P(t3).
3. The sample paths of P(t) are continuous. P(t) is called arithmetic Brownian motion or Winner process.
If we set =0, =1, we obtain standard Brownian Motion which is denoted as B(t). Accordingly, P(t) = t + B(t)