Consider the CRR model withT = 2 and S0 = 100, Su1 = 120, Sd1 = 90. Assume that the interest rate r = 0.Consider an American put option with reward process g(St, t) = (Lt − St)+ andvariable strike price L0 = 105, L1 = 116, L2 = 111.(a) Find parameters u, d, the stock price at time T = 2, and a martingale measure˜P on (Ω, F2)

Consider the CRR model withT = 2 and S0 = 100, Su1 = 120, Sd1 = 90. Assume that the interest rate r = 0.Consider an American put option with reward process g(St, t) = (Lt − St)+ andvariable strike price L0 = 105, L1 = 116, L2 = 111.(a) Find parameters u, d, the stock price at time T = 2, and a martingale measure˜P on (Ω, F2).(b) Compute the price process P a of this option using the recursive relationshipP at = max{(Lt − St)+, (1 + r)−1(˜pP aut+1 + (1 − ˜p)P adt+1)}with the terminal condition P a2 = (L2 − S2)+.(c) Find the holder’s rational exercise time τ ∗0 .(d) Find the replicating strategy for the option up to the exercise time τ ∗0 .(e) Check whether the arbitrage price process (P at ; t = 0, 1, 2) is a martingale or asupermartingale under ˜P with respect to the filtration F

A stock is currently trading at $50; its annual volatility is 0.40, the risk-free interest rate is15% per annum with continuous compounding, and ∆t is equal to three months. Use thebinomial model to answer the following questions:i. Calculate the price of a 6-month European put option with an exercise price of $105written on this stock.(5 marks)ii. Calculate the price of a 6-month American put option with an exercise price of $105written on this stock.

Consider the following information for an individual stock: Current share price is S10Risk-free rate is 5% pa compounded continuously Volatility of the stock returns (o) is 30% pa Strike price is S12 Time to maturity of the option is 9 mthsThe firm is expected to pay dividend estimated at S1 per share in 2 mths, S1.40 pershare in 8 mths and S1.80 per share in 14 mths. Use the closed-form Black-Scholes model to price the European put option with the abovecharacteristics 3.96a5.13b)1.25cdNone of the above

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 .