We assume that the stock price S is governed under the martingale measure by the Black-Scholes...

 

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We assume that the stock price S is governed under the martingale measure by the Black-Scholes stochastic differential equation dS = St (r dt + dWt), So > 0, where > 0 is the stock price volatility and r is the instantaneous interest rate. Consider a European claim X with maturity T and payoff X where L > 0 is fixed and a = R is an arbitrary real number. (a) Show that the payoff X satisfies X assume that the strike L satisfies L = = = AST - |L - ST aST CT(L) PT(L). We henceforth - Soe. Using the Black-Scholes pric- ing formulae for call and put options, find an explicit expression for the arbitrage price To(X) of X at time 0. (b) Compute the hedge ratio at time 0 for the claim X and examine the sign of the hedge ratio as a function of a R. (c) Find a unique value of a such that T0 (X) = 0 and show that it satisfies 2 < a < 0. Compute the limits lim 0 T0 (X) and lim To(X). (d) Assume that r = 0. Show that the process S is a submartingale under P, that is, the inequality E(S | Ft) S2 holds for all dates 0 t s. We assume that the stock price S is governed under the martingale measure by the Black-Scholes stochastic differential equation dS = St (r dt + dWt), So > 0, where > 0 is the stock price volatility and r is the instantaneous interest rate. Consider a European claim X with maturity T and payoff X where L > 0 is fixed and a = R is an arbitrary real number. (a) Show that the payoff X satisfies X assume that the strike L satisfies L = = = AST - |L - ST aST CT(L) PT(L). We henceforth - Soe. Using the Black-Scholes pric- ing formulae for call and put options, find an explicit expression for the arbitrage price To(X) of X at time 0. (b) Compute the hedge ratio at time 0 for the claim X and examine the sign of the hedge ratio as a function of a R. (c) Find a unique value of a such that T0 (X) = 0 and show that it satisfies 2 < a < 0. Compute the limits lim 0 T0 (X) and lim To(X). (d) Assume that r = 0. Show that the process S is a submartingale under P, that is, the inequality E(S | Ft) S2 holds for all dates 0 t s.

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a To show that the payoff X satisfies X ST CTL PTL we can express the payoff in terms of call and put options as follows X ST L ST ST maxL ST 0 maxST ... View the full answer