## Transcribed Text

4. Suppose the functions 4,0 : [0,T X R
R
and ยง : R
R are given and we are trying to find a function
F : [0,T]
X
R
R which satisfies the following boundary value problem:
(*)
a) Show that if F(t,x) solves the above problem, then the function G(t,x) e-r(T-t) x) solves the
problem:
(#)
b) Show that if G(t,x) solves prolblem (#), then the function F(t,x) = er(T=t)G(t,r) solves problem (*) .
Note. a) shows that the probabilistic method of solving (*) (given by Feynman-Kai Theorem) will work for the
equation (#). When (x) = |X - Kl+, the problem (#) is identical with the Black-Scholes-Merton equation for the
price of a European call option.
5. Consider the following process
Y(t)=&"W(+)
Use Ito's Lemma to show that
Y(0)=0 =
=
6. Consider the standard stock price process
dS(t) =pS(t)dt+oS(t)dW(t) = (**)
Let c(t,I be a function with continuous partial derivatives and suppose that we want to investigate the process
Z(t) = c(t,S(t)). Ito's lemma states that Z(t) is an Ito process, i.e., it has the dynamic:
Z(0) = c(0,5 S0)
( dZ(t) = a(t,S(t))dt +b(t,S(t))dW(t) =
Using stochastic calculus laws (dt * dt = dW * dt = dt * dW = 0, dW * dW = dt) and Ito's Lemma, find the coefficients
a(t,r) and b(t,x).
Note: This calculation is a step in derivation of Black-Scholes-Merton equation where o(t, S(t)) is interpreted as the
price of a call option.

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