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# ex_4 - by trading in the three calls Exercise 2 Consider...

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ORIE 569 Problem set 4 March 6, 2008 Homework must be turned in by Thursday, March 27, at 10 a.m. Solutions turned in at a later time will not be graded. Place homework in the 569 homework box on the bridge between Upson and Rhodes. Exercise 1. Let us denote by C ( T, K ) the observed market price of a liquid call option ( S T K ) + with maturity T and strike K , where S T is the price of the underlying at time T (we do not assume that prices follow any particular model). a) Suppose we have C ( T, K 1 ) < C ( T, K 2 ) for strikes K 1 < K 2 . Construct an arbitrage opportunity by trading in the two calls ( S T K i ) + , i = 1 , 2. b) Suppose we observe three call prices C ( T, K i ), i = 0 , 1 , 2, that violate the condition of convexity. That is, for K 0 < K 1 < K 2 we have C ( T, K 1 ) > α C ( T, K 0 ) + (1 α ) C ( T, K 2 ) , where α is such that K 1 = α K 0 + (1 α ) K 2 . Construct an arbitrage opportunity
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Unformatted text preview: by trading in the three calls. Exercise 2: Consider the market model with constant interest rate r and risk-neutral price process given by geometric Brownian motion S with volatility σ , i.e., dS t = S t ( r dt + σ d f W t ). Let v ( S ,r,σ ) denote the corresponding Black-Scholes price of a call option ( S T − K ) + at time 0 and investigate the limit lim σ ↑∞ v ( S ,r,σ ) . Exercise 3: Let v ( T,S ,r,σ ) be as in Exercise 2 and prove the following formulas for the two Greeks Rho and Vega, % = ∂ ∂r v ( S ,r,σ ) = KT e − rT N ° d − ¢ and V = ∂ ∂σ v ( S ,r,σ ) = S √ T N ° d + ¢ . Exercise 4: Exercise 6.10 on page 293 in Shreve’s book. 1...
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