5. (i) State the Black-Scholes equation assuming that the underlying risky as- set that does not...
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5. (i) State the Black-Scholes equation assuming that the underlying risky as- set that does not pay any dividends nor involves a cost of carry. Explain what this equation models, and what its terms, variables, and param- eters are. Then, show by substitution that two exact solutions of the Black-Scholes equation are (a) V(S,t) = AS, (b) V(S, t) = Ae", where A is an arbitrary constant. Why would you expect these to be solutions? (ii) A European power option with expiry time T is similar to a European call option except that its payoff is V(S,T) = S", where n is a positive constant, and there is no strike price X. (a) By seeking solutions to the Black-Scholes equation stated in (i) of the form V(S, t) = g(Tt)Sm, where m is a constant, find the ordinary differential equation satisfied by g(T-t). (b) Find the value of a power option at time t and its premium. (c) Which type of power option should an investor choose if they wish to minimise the risk? 5. (i) State the Black-Scholes equation assuming that the underlying risky as- set that does not pay any dividends nor involves a cost of carry. Explain what this equation models, and what its terms, variables, and param- eters are. Then, show by substitution that two exact solutions of the Black-Scholes equation are (a) V(S,t) = AS, (b) V(S, t) = Ae", where A is an arbitrary constant. Why would you expect these to be solutions? (ii) A European power option with expiry time T is similar to a European call option except that its payoff is V(S,T) = S", where n is a positive constant, and there is no strike price X. (a) By seeking solutions to the Black-Scholes equation stated in (i) of the form V(S, t) = g(Tt)Sm, where m is a constant, find the ordinary differential equation satisfied by g(T-t). (b) Find the value of a power option at time t and its premium. (c) Which type of power option should an investor choose if they wish to minimise the risk?
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i The BlackScholes equation is a partial differential equation used to model the pricing of options Assuming that the underlying risky asset does not ... View the full answer
Related Book For
Smith and Roberson Business Law
ISBN: 978-0538473637
15th Edition
Authors: Richard A. Mann, Barry S. Roberts
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