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1、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 th
2、e random walk., 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 Pn(t) = Pt/h = Pnt/T , t
3、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 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
4、 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 n(2-1) T 4n(-1) 2 T This can be accomplished by setting = *(1+h /), =h, The continuous time limit,It cab be shown that the process P(t) has the following three
5、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 arith
6、metic 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) Consider the following moments: EP(t) | P(t0) = P(t0) +(t-t0) VarP(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,
7、therefore, 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) h
8、as 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
9、doesnt male sense because dB(t)/dt is a not well defined (althrough dB(t) is).,The moments of dB(t): EdB(t) =0 VardB(t) = dt E dB dB = dt VardB dB = o(dt) EdB dt = 0 VardB dt = o(dt),If we treat terms of order of o(dt) as essentially zero, the (dB)2 and dBdt are both non-stochastic variables. | dB d
10、t dB | dt 0 dt | 0 0 Using th above rule we can calculate (dP)2 = 2dt. It is not a random variable!, Geometric Brownian motion,If the arithmetic Brownian motion P(t) is taken to be the price of some asset, the price may be negative. The price process p(t)= exp(P(t), where P(t) is the arithmetic Brow
11、nian motion, is called geometric Brownian motion or lognormal diffusion., Itos Lemma,Although the first complete mathematical theory of Brownian motion is due to Wiener(1923), it is the seminal contribution of Ito (1951) that is largely responsible for the enormous number of applications of Brownian
12、 motion to problems in mathematics, statistics, physics, chemistry, biology, engineering, and of course, financial economics. In particular, Ito constructs a broad class of continuous time stochastic process based on Brownian motion now known as Ito process or Ito stochastic differential equations w
13、hich is closed under general non-linear transformation.,Ito (1951) provides a formula Itos lemma for calculating explicitly the stochastic differential equation that governs the dynamics of f(P,t): df(P,t) = f/P dP + f/t dt + 2f/P2 (dP)2, Applications in Finance,A lognormal distribution for stock pr
14、ice returns is the standard model used in financial economics. Given some reasonable assumptions about the random behavior of stock returns, a lognormal distribution is implied. These assumptions will characterize lognomal distribution in a very intuitive manner.,Let S(t) be the stocks price at date
15、 t. We subdivided the time horizon 0 T into n equally spaced subintervals of length h. We write S(ih) as S(i), i=0,1,n. Let z(i) be the continuous compounded rate of return over (i-1)h ih, ie S(i)=S(i-1)exp(z(i), i=1,2,.,n. It is clear that S(i)=S(0)expz(1)+z(2)+z(i). The continuous compounded retur
16、n on the stock over the period 0 T is the sum of the continuously compounded returns over the n subintervals.,Assumption A1. The returns z(j) are i.i.d. Assumption A2. Ez(t)=h, where is the expected continuously compounded return per unit time. Assumption A3. varz(t)=2h. Technically, these assumptions ensure that as the time decrease proportionally, the behavior of the distribution for S(t) dose not explode nor degenerat
