We place ourselves within the setup of the Black-Scholes market model M = (B, S) with...

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We place ourselves within the setup of the Black-Scholes market model M = (B, S) with a unique martingale measure P. Consider a European contingent claim X with maturity T and the following payoff X = max (K, ST) - LST where K = erT So and L> 0 is an arbitrary constant. We take for granted the Black-Scholes pricing formulae for the call and put options. (a) Sketch the profile of the payoff X as a function of the stock price ST at time T and show that X admits the following representation X = K+CT(K) - LST where Cr(K) denotes the payoff at time T of the European call option with strike K. (b) Find an explicit expression for the arbitrage price (X) at time 0 < t < T in terms of Ft = ert So, St and So. Then compute the price To(X) in terms of So and use the equality N(x) - N(-x) = 2N (x) - 1 to simplify your result. := (c) Find the limit lim - To(X). (d) Find the limit lim → To(X). (e) Explain why the price of To(X) is positive when L = 1 by analysing the payoff X when L = 1. We place ourselves within the setup of the Black-Scholes market model M = (B, S) with a unique martingale measure P. Consider a European contingent claim X with maturity T and the following payoff X = max (K, ST) - LST where K = erT So and L> 0 is an arbitrary constant. We take for granted the Black-Scholes pricing formulae for the call and put options. (a) Sketch the profile of the payoff X as a function of the stock price ST at time T and show that X admits the following representation X = K+CT(K) - LST where Cr(K) denotes the payoff at time T of the European call option with strike K. (b) Find an explicit expression for the arbitrage price (X) at time 0 < t < T in terms of Ft = ert So, St and So. Then compute the price To(X) in terms of So and use the equality N(x) - N(-x) = 2N (x) - 1 to simplify your result. := (c) Find the limit lim - To(X). (d) Find the limit lim → To(X). (e) Explain why the price of To(X) is positive when L = 1 by analysing the payoff X when L = 1.

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a To sketch the profile of the payoff X as a function of the stock price ST at time T we need to consider the different cases based on the relationshi... View the full answer