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 .

Static hedging with options. Consider a parametrised family of contingent claims with the payoff Y (α) at time T given by the following expression Y (α) = min 􏰀α, β + 2|β − ST | − ST 􏰁 where a real number β > 0 is fixed and the parameter α is an arbitrary real number such that α ≥ 0. (a) Foranyfixedα≥0,sketchtheprofileofthepayoffY(α)asafunctionofST ≥0 and find a decomposition of Y (α) in terms of the payoffs of standard call and put options with maturity date T (do not use a constant payoff). Notice that a decomposition of Y (α) may depend on the value of the parameter α. (b) Assume that call and put options with all strikes are traded at time 0 at some finite prices. For each value of α ≥ 0, compute the arbitrage price π0(Y (α)) at time t = 0 for the claim Y (α) using the prices at time 0 of call and put options and a suitable decomposition obtained in part (a). (c) For any α > 0, examine the sign of an arbitrage price of the claim Y (α) in any (not necessarily complete) arbitrage-free market model M = (B,S) with a finite state space Ω. Justify your answer. (d) Consider a complete arbitrage-free market model M = (B, S) defined on some finite sample space Ω. Show that the arbitrage price of Y (α) at time t = 0 is a monotone function of the variable α ≥ 0 and find the limits limα→0 π0(Y (α)), limα→∞ π0(Y (α)) and limα→3β π0(Y (α)).

Consider a world in which some stock, S, can either go up by 25% or down by 20% in one year and noother outcomes are possible. The continuously compounded risk-free interest, r, is 5.5% and the current priceof the stock, S0, is $100.1. What are the possible stock values in one year’s time, ST ?2. What are the possible payoffs of a European call option written on stock S with a strike price, X, of$100 and time-to-expiration of 1 year, T = 1 ?3. Suppose you want to form a portfolio, P , consisting of short on one call option and long on somenumber, ∆, of the stock, such that the portfolio value in one year’s time, PT , does not depend on thevalue of the stock, ST . What would be the appropriate value of ∆, also called the hedge ratio or delta?4. What would be the (certain) portfolio value in one year’s time, PT ?5. What is the arbitrage-free value of the portfolio today, P0 ?6. What is the premium of the call option today, c0, if there is no arbitrage opportunity?7. Define p = (erT − d) /(u − d), and call this the risk-neutral probability that the stock price increases.What is the value of p ?8. What is the expected value of the stock in one year’s time, E (ST ), under the risk-neutral probabilities?9. At what continuous rate would the stock price have to grow to end up at the expected value?10. What would be the expected value of the call option in one year’s time, E (cT ), under the risk-neutralprobabilities?11. At what continuous rate would the call price have to grow to end up at the expected value?

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).

The Black-Scholes option pricing model states:a.The value of an at-the-money call and put is the same.b.The risk-free interest rate has no bearing on the option price.c.The option price is independent of the underlying stock's anticipated return on investment.d.The risk-free interest rate is the most challenging metric to measure.

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)