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Unformatted text preview: Continuous Time Finance, Spring 2004 Homework 2 Distributed 2/4/04, due 2/18/04 (1) Consider a market with one source of randomness, a scalar Brownian motion w . Suppose the price S of a stock satisfies dS = ( S, t ) S dt + ( S, t ) S dw where and are known functions of S and t . Let r be the risk-free rate, assumed constant for simplicity. We know that all tradeables must have the same market price of risk = - r . Lets check this for the special case of a European option on S with maturity T and payoff f ( S T ). Use the Black-Scholes PDE to verify that this options market price of risk is the same as that of the underlying. (2) Let S 1 and S 2 be stocks with constant drift and volatility dS 1 = 1 S dt + 1 S dw 1 , dS 2 = 2 S dt + 2 S dw 2 , and assume that w 1 and w 2 have (constant) correlation . The risk-free rate is r (also constant). The Section 3 notes discuss the pricing of an exchange option with payoff ( S 2 ( T )- S 1 ( T )) + . Find....
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