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In our first approach to deriving the Black-Scholes formula we obtained a differential equation (PDE) for the option price. The PDE, which is the same no matter the payoff, is a second-order linear PDE. We started with these assumptions: r (interest rate) is constant. q (stock's dividend yield) = 0%. Stock price evolves under geometric Brownian motion: dS = a S dt + o SdW We applied Ito's Formula to the derivative pricing function V(S.t). Then we considered a portfolio comprised of 1 unit of the derivative and units of the stock. We asked the question: What does B have to equal so that the change in value of the portfolio over [t, t+dt] is deterministic? A clever arbitrage argument led to the PDE. To solve the PDE we invoked the Feynman-Kac formula. This formula implied this solution of the PDE: V(S,t) exp(-t(T-t)) E[Payoff formula (function of S(T))] where S(T) is the random variable obtained by letting the price start at the current stock price (S) and evolve from t to T under the process: dS = r S dt + o SdW [a] Use the pricing formula to show that the pricing p.d.f." for S(T) satisfies: E[S(T)] = Forward price of the stock = exp(r(T-t))S(t) = Hint: Apply the pricing formula to a forward contract on the stock with delivery date T and invoice price equal to the forward price for T. What do we know about the value at time t (trade date) of this derivative? [b] Write down an expression (in terms of S,r, o, t, and T) for Prob{log(S(T) > log(S)}. [c] Write down a formula for the price of the log derivative," the derivative that pays log(S(T)) at time T. Call this pricing function L(St). (Hint: Write down log(S(T)) as a random variable.) [d] Is the delta of the log derivative positive or negative? [e] Does the log derivative's delta change when we change the starting stock price S

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