Derive the call option price based on BSM model.Suppose you have already proved that𝐸[max(𝑉 − 𝐾, 0)] = 𝐸(𝑉)𝑁(𝑑1) − 𝐾𝑁(𝑑2)for a lognormally distributed 𝑉, where𝑑1 = ln[𝐸(𝑉)/𝐾] + 𝑤2/2𝑤𝑑2 = ln[𝐸(𝑉)/𝐾] − 𝑤2/2𝑤and 𝑤 is the standard deviation of 𝑙𝑛𝑉.

You plan to value a call option on a non-dividend-paying stock. The current stock value is𝑆0 = 50, the strike price of the option is 𝐾 = 53, the time to maturity is 𝑇 = 1. Supposethe risk-free interest rate is 5%.1) (10 points) Suppose the implied volatility for 𝐾/𝑆0 = 1 is 3%, implied volatilityfor 𝐾/𝑆0 = 1.05 is 3.5%, and implied volatility for 𝐾/𝑆0 = 1.1 is 4%.Calculate the option value based on BSM model

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

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 .