Consider a European call option C on a non$-$dividend paying stock. The price of

the underlying stock follows the geometric Brownian motion given by the stochastic

differential equation

dSt $=\backslash$mu Stdt $+\backslash$sigma StdWt

where St is the stock price, $\backslash$mu is the drift,$\backslash$sigma is the volatility, Wt is the Wiener process

and t is the time.

a$)$ Provide the Black$-$Scholes Partial Differential Equation $($PDE$)$ for the European call

option C$.[1]$

b$)$ The Feynman$-$Kac representation theorem in the context of the Black$-$Scholes PDE,

states that the solution of V the Black$-$Scholes PDE can be written as

V $($S$,$ t$)=$ E

h

e$$r$($T$$t$)$g$($ST $)$

$$

St $=$ S

i

where T is the expiry time, g is payoff function of the option V and E

h

$$

i

is the

expectation operator.

Apply the Feynman$-$Kac formula to derive an expression for the price C$($S$,$ t$)$ of

the European call option in terms of an expectation involving the stochastic process.

$[3]$

Discuss the economic interpretation of the terms in the Feynmann$-$Kac formula and

how it connects to the pricing of financial derivatives.