Question 1 - Black-Scholes-Merton PDE Proof An investment firm is developing a new Exotic Derivative Contract....

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Question 1 - Black-Scholes-Merton PDE Proof An investment firm is developing a new Exotic Derivative Contract. (5 Marks) This contract will pay off stock price at expiry squared, S,, given stock price is less than the strike price, K. That is defined by the following function: [ST K S] Payoff (G)= = ST > K 0 Given that the underlying stock is assumed to follow Geometric Brownian Motion; dS = Sdt+oSdz (A) Use Risk-Neutral Valuation to derive the fair price of the security at time t in terms of the stock price, S, at time t. This will be referred to as G. (HINT: You will first need to derive the stochastic process that is followed by Y = S, Your derivation in (a) should show that Y, also follows a GBM) (2 Marks) (B) Determine whether or not this value satisfies the Black-Scholes-Merton Partial Differential Equation; aG G S G r f G = r rfS + + as t 2 S2 (C) Explain, in words, what the result of part (B) means for this contract's tradeability. (2 Marks) (1 Mark) Question 1 - Black-Scholes-Merton PDE Proof An investment firm is developing a new Exotic Derivative Contract. (5 Marks) This contract will pay off stock price at expiry squared, S,, given stock price is less than the strike price, K. That is defined by the following function: [ST K S] Payoff (G)= = ST > K 0 Given that the underlying stock is assumed to follow Geometric Brownian Motion; dS = Sdt+oSdz (A) Use Risk-Neutral Valuation to derive the fair price of the security at time t in terms of the stock price, S, at time t. This will be referred to as G. (HINT: You will first need to derive the stochastic process that is followed by Y = S, Your derivation in (a) should show that Y, also follows a GBM) (2 Marks) (B) Determine whether or not this value satisfies the Black-Scholes-Merton Partial Differential Equation; aG G S G r f G = r rfS + + as t 2 S2 (C) Explain, in words, what the result of part (B) means for this contract's tradeability. (2 Marks) (1 Mark)