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Unformatted text preview: MAFS 5030 Quantitative Modeling of Derivatives Securities Homework Five Course Instructor: Prof. Y.K. Kwok 1. Consider a European capped call option whose terminal payoff function is given by c M ( S, 0; X,M ) = min(max( S- X, 0) ,M ) , where X is the strike price and M is the cap. Show that the value of the European capped call is given by c M ( S,τ ; X,M ) = c ( S,τ ; X )- c ( S,τ ; X + M ) , where c ( S,τ ; X + M ) is the value of a European vanilla call with strike price X + M . 2. Consider the value of a European call option written by an issuer whose only asset is α ( < 1) units of the underlying asset. At expiration, the terminal payoff of this call is then given by S T- X if αS T ≥ S T- X ≥ αS T if S T- X > αS T (1) and zero otherwise. Show that the value of this European call option is given by c L ( S,τ ; X,α ) = c ( S,τ ; X )- (1- α ) c parenleftbigg S,τ ; X 1- α parenrightbigg , α < 1 , where c parenleftbigg S,τ ; X 1- α parenrightbigg is the value of a European vanilla call with strike price X 1- α . 3. Suppose the dividends and interest incomes are taxed at the rate R but capital gains taxes are zero. Find the price formulas for the European put and call on an asset which pays a continuous dividend yield at the constant rate q , assuming that the riskless interest rate r is also constant. Hint: Explain why the riskless interest rate r and dividend yield q should be replaced by r (1- R ) and q (1- R ), respectively, in the Black-Scholes formulas. 4. Consider a futures on an underlying asset which pays N discrete dividends between t and T and let D i denote the amount of the i th dividend paid on the ex-dividend date t i . Show that the futures price is given by F ( S,t ) = Se r ( T- t )- N summationdisplay i =1 D i e r ( T- t i ) , 1 where S is the current asset price and r is the riskless interest rate. Consider a European call option on the above futures. Show that the governing differential equation for thecall option on the above futures....
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- Spring '11