 # 4. Suppose the functions 4,0 : [0,T X R R and § : R R are g...

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4. Suppose the functions 4,0 : [0,T X R R and § : 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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