Which of the following is assumed by the Black–Scholes–Merton model? A. The return from the stock in a short period of time is lognormal. B. The stock price at a future time is lognormal. C. The stock price at a future time is normal. D. None of the above

Black Scholes Model – their interpretations

e place ourselves withinthe setup of the Black-Scholes market model M = (B, S) with a unique martingalemeasure ˜P. Consider a European contingent claim Y with maturity T and thefollowing payoffY = γST − max (ST , L)where L = erT S0 and γ > 0 is a real number. We take for granted the Black-Scholespricing formulae for the call and put options.(a) Sketch the profile of the payoff Y as a function of the stock price ST at time Tand show that Y admits the representation Y = γST − CT (L) − L where CT (L)denotes the payoff at time T of the European call option with strike L.

e place ourselves withinthe setup of the Black-Scholes market model M = (B, S) with a unique martingalemeasure ˜P. Consider a European contingent claim Y with maturity T and thefollowing payoffY = γST − max (ST , L)where L = erT S0 and γ > 0 is a real number. We take for granted the Black-Scholespricing formulae for the call and put options.(a) Sketch the profile of the payoff Y as a function of the stock price ST at time Tand show that Y admits the representation Y = γST − CT (L) − L where CT (L)denotes the payoff at time T of the European call option with strike L.(b) Find an explicit expression for the arbitrage price πt(Y ) at time 0 ≤ t < T interms of Ft := ertS0, St and S0. Then compute the price π0(Y ) in terms of S0and use the equality N (x) − N (−x) = 2N (x) − 1 to simplify your result.(c) Compute and describe the hedging strategy at time 0 for the claim Y .(d) Find the limits lim σ→0 π0(Y ) and lim σ→∞ π0(Y ).(e) Explain why the price π0(Y ) is negative when γ = 1 by analysing the payoff Y .

We place ourselves withinthe setup of the Black-Scholes market model M = (B, S) with a unique martingalemeasure ˜P. Consider a European contingent claim Y with maturity T and thefollowing payoffY = γST − max (ST , L)where L = erT S0 and γ > 0 is a real number. We take for granted the Black-Scholespricing formulae for the call and put options.(a) Sketch the profile of the payoff Y as a function of the stock price ST at time Tand show that Y admits the representation Y = γST − CT (L) − L where CT (L)denotes the payoff at time T of the European call option with strike L.(b) Find an explicit expression for the arbitrage price πt(Y ) at time 0 ≤ t < T interms of Ft := ertS0, St and S0. Then compute the price π0(Y ) in terms of S0and use the equality N (x) − N (−x) = 2N (x) − 1 to simplify your result.

We place ourselves withinthe setup of the Black-Scholes market model M = (B, S) with a unique martingalemeasure ˜P. Consider a European contingent claim Y with maturity T and thefollowing payoffY = γST − max (ST , L)where L = erT S0 and γ > 0 is a real number. We take for granted the Black-Scholespricing formulae for the call and put options.(a) Sketch the profile of the payoff Y as a function of the stock price ST at time Tand show that Y admits the representation Y = γST − CT (L) − L where CT (L)denotes the payoff at time T of the European call option with strike L.(b) Find an explicit expression for the arbitrage price πt(Y ) at time 0 ≤ t < T interms of Ft := ertS0, St and S0. Then compute the price π0(Y ) in terms of S0and use the equality N (x) − N (−x) = 2N (x) − 1 to simplify your result.(c) Compute and describe the hedging strategy at time 0 for the claim Y .(d) Find the limits lim σ→0 π0(Y ) and lim σ→∞ π0(Y ).(e) Explain why the price π0(Y ) is negative when γ = 1 by analysing the payoff Y