Section A

A1   (a) Give the mathematical expression of the gamma and vega of a portfolio of derivatives.

Explain briefly the meaning of these two quantities.                                                   [3]

(b) A portfolio consists of stocks in the Asian markets. It is known that the portfolio has a    negative delta. Suppose there is a small increase in the value of the stocks. Explain         briefly the effect of this increase in the total value of the portfolio.         [2]

(c) Find the value of delta of a 9-month European call option on a stock with a strike price   equal to the current stock price (t = 0). The interest rate is 8%. The volatility is              σ = 0.10.           [4]

A2 A 9-months forward contract on a non-dividend paying stock is entered into when the stock price is e40 and the risk-free interest rate is 5%.

(a) What are the forward price and the value of the forward contract at the initial time?    [4]

(b)  Suppose that at the initial time the forward contract is being bought and sold in the         market for e5, instead of being traded at the fair price. Outline a simple trading strategy to take advantage of the arbitrage opportunity, stating how much risk-free profit will be   made per contract.                                                                                                     [5]

A3   (a) Write down the Black-Scholes-Merton partial differential equation. State the boundary

condition for the case of a derivative with payoff ST(2) + 2ST + 3.                                [2]

(b) Consider a European put option and a European call option on the same underlying       asset, with the same strike price K = $5, and same expiry date T = 1. The put is traded at $3, the call at $10, and the underlying asset at $6. The interest rate is r = 0.05.

i. Verify that put-call parity does not hold in this case.                                            [3]

ii. Describe a portfolio that exploits the arbitrage opportunity. State the risk-free profit that the portfolio will provide.          [5]

A4 Consider the Geometric Brownian Motion (GBM) process

dSt = µStdt + σStdBt,    S0 = 1.

A stock price follows the above GBM, so that for the first two years, µ = 4 and σ = 2, and for the next two years, µ = 0 and σ = 2. Express the probability P [ S4  < s ], for any s > 0, as a  function of the cumulative distribution function, N(·), of the standard normal distribution.      Hint: You can make use of the equivalent expression St = S0 exp((µ . )t + σBt).    [6]

A5 Consider a European call option on a stock. The call option will expire in 6 months. The         current stock price is $30, and the strike price of the call option is $20. At the expiration date, the stock price can either be $35 or it can be $25. The risk free interest rate is 4%. What is the value of the European call option today?                                                                            [6]

Section B

B1 Consider a given portfolio with delta equal to 2, 000 and vega equal to 60, 000. We plan to create a new portfolio that is both delta and vega neutral by adding to the given portfolio:

i) units of the underlying stock, and

ii) units of a traded option with delta equal to 0.5 and vega equal to 10.

How many units of the underlying stock and the traded option will we need?                      [5]

B2 A stock price St follows the usual model dSt = µStdt + σStdBt with expected return

µ = 0.16 and volatility σ = 0.35. The current price is e38.

(a) What is the probability that a European call option on the stock with an exercise price of e40and a maturity date in 6 months will be exercised?                                              [5]

(b) Using the Black- Scholes formula, find the price of the call option if the risk-free interest rate is µ .

Hint: You might have already calculated quantity d2 (required for finding the price of the call) in your answer in part (a). This can save you a lot of calculations.       [4]

(c) What is the probability that a European put option on the stock with the same exercise    price and maturity date will be exercised? Find the price of the put option, using put-call parity.                              [4]

B3 Consider a binomial model with T = 2 periods, S = 100, u = 1.6 and d = 0.6. The interest rate is r = 0.1. What is the price at time t = 0 for an American put option that has exercise price K = 97?

Hint: To avoid too many calculations, you are given that the risk-neutral probability of one-step up movement is pˆ = 0.505, and that Su = 160, Sd = 60, Su2 = 256, Sd2 = 36,

Sud = 96.                                                                                                                        [9]

B4    (a) Assume in this question that all the derivatives have the same maturity date and the same underlying asset. Write explicitly the payoff function for a portfolio consisting of a short position in two European put options with exercise price e20 and a long position in

three European put options with exercise price e25.                                                   [4]

(b) Consider the following payoff function at maturity T and strike price K : Payoff = min(ST . K, 4).

i. Draw the payoff diagram, that is the plot of payoff at date of maturity versus price   of underlying asset.                                                                                            [4]

ii. By using 1 unit of forward contract (in a long or short position) and 1 unit of call     option (long or short position), create a portfolio with the payoff function given       above. The strike prices of these two derivatives need not be equal to K.             [4]