定价竞争策略英文

the mean and variance of Pn(T):
E(Pn(T)) = 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
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.
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
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)
(althrough dB(t) is).
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
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).
• 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:
Consider the following moments:
E[P(t) | P(t0)] = P(t0) +(t-t0) Var[P(t) | P(t0)] = 2(t-t0) CoSince Var[ (B(t+h)-B(t))/h ] = 2/h, therefore,