面试binomial model问题

发信人: tempone (jane), 信区: Quant
标  题: binomial model again
发信站: BBS 未名空间站 (Sat Apr 17 01:58:55 2010, 美东)

u = exp(sigma *sqrt(delta_t))
d = exp(-sigma *sqrt(delta_t))
p = (expt(r * delta_t) - d) / (u - d).
S_0 = 1;

At step n, whats the variance?

I follow the suggestion of this board and use the variance definition:
a node at nth is u^kd^(n-k). the probability is C_n_k * p^k * (1-p)^(n-k)

the expected stock price is exp(r * n *delta_t)
variance is sum over 0..n (C_n_k *p^k *(1-p)^(n-k) * (u^kd^(n-k)^2 - expt(-2
*n * delta_t) and I fursther simply this to:

(p*u^2 + (1-p) * d^2)^2 - expt(2*r*n*delta_t)

but after that I dont know how to further simply. I think the final anser
should be sigma^2 * T, is that?

can anybody help to give a hint how to further simply to get the solution?