Question: Consider an option with value V(S, t), which has payoff at time T. Reduce the Black Scholes equation, with final and boundary conditions, to the

Consider an option with value V(S, t), which has payoff at time T. Reduce the Black– Scholes equation, with final and boundary conditions, to the diffusion equation, using the following transformations:

(a)

(b)
v = e^αx+βτu(x, τ ),

for some α and β. What is the transformed payoff? What are the new initial and boundary conditions? Illustrate with a vanilla European call option.

S= Ee", t= T - o2' V(S, t) = Ev(x, t), |

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