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probs40_53 - Michlmas 2010 Mathematical Finance Questions 6...

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Michælmas 2010 Mathematical Finance: Questions 6 40. Find the no-arbitrage price of a ( K, T ) call option at time 0 on a stock that pays dividends dS ( T i ) at times 0 < T 1 < T 2 < T . 41. Use the put-call parity relation for options on a non-dividend paying stock to relate (i) the deltas of European ( K, T ) calls and puts; (ii) the thetas of European ( K, T ) calls and puts. 42. (i) Calculate Δ for a ( K, 1 / 2) European call on a non-dividend paying stock with σ = 0 . 25 when the interest rate is r = 0 . 1 and the current stock price S = K . (ii) What does Δ = 0 . 7 mean for a European call on a non-dividend paying stock? Suppose you have a portfolio (with value function V ) which is short 1 , 000 such calls. How many shares of the stock should you buy/sell to make the portfolio delta-neutral i.e. to make ∂V/∂S = 0? 43. Suppose that a stock price follows the standard geometric Brownian motion model with volatility σ and interest rate r . What is the risk-neutral distribution of log S ( t ) α /S α 0 , α > 0? An option with payoff function [ S α - K ] + for some α > 0 when the stock price is S at the time of exercise is called a power call. Express the price of a European power call option in terms of the Black-Scholes formula for a European call on the same stock with the same strike and expiry. 44. Consider an American put option with strike K > 0 on a stock whose price evolves according to the homogeneous, risk-neutral binomial tree model with M stages. Show
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