W Wt t 0 denotes a standard Brownian motion starting from

W = {Wt : t 0} denotes a standard Brownian motion starting from 0. Furthermore, the cumulative distribution function of standard normal is denoted by , that is, (x) = P(N(0, 1) x) = Z x 1 2 e z 2 2 dz. 1. Compute the following quantity (you can use in your expressions):

(c) E[(Wt a) +], where (x a) + = max(x a, 0).

Solution

W = { Wt: t0 } denotes a standard Brownian motion starting from 0.

Again the cummulative distribution function of standard normal is denoted by (x) = P(N(0,1) x), where

P(Z ) = 1/(2) exp(-z²/2 )

Now E [(Wt -a)] = (Wz - a) 1/(2) exp(-z²/2 ) dz ( limit 0 to )

= w1/(2) z exp(-z²/2 ) dz( limit 0 to ) - a

Substituting z²/2 = t and putting limit, we get

E [(Wt -a)] = w/(2) - a.