Answer ALL questions.

Section A carries 40% of the total marks and Section B carries 60%. The relative weights attached to each question are as follows: Al (10), A2 (10), A3 (10), A4 (10), Bl (20), B2 (20), B3 (20). The numbers in square brackets indicate the relative weight attached to each part question.

Marks will not only be given for the final (numerical) answer but also for the accuracy and clar­ ity ofthe answer. So make sure to write down workings, e.g. formulas, calculations, reasoning.

Show your full working for all questions. Do not write formulas alone without any comment about what you are calculating. Except where otherwise stated, interest is compounded continuously and there are no transaction charges or buy-sell spreads. Assume that a positive risk-free interest rate always exists, and is the samefor all maturities and is constant over time unless otherwise stated. All risk-free rates are expressed on an annualised basis. Assume also that all assets are tradable, that limitless short selling is always allowed, and that there is no counter-party credit default risk for all transactions.

Time allowed: 2½ hours.

All data in this exam arefictional.

Formula sheet

1. Ito's formula for a function G(x, t) can be written as:

for an Ito process following dx = a(x, t)dt + b(x, t)dB, with the usual notation.

2. The Black-Scholes formula for the price of a European call option under the standard assumptions, with strike price Kand time to expiry T, is

where N( · ) denotes the cumulative distribution function of a standard Normal, and

The Black-Scholes formula for the price of a European put option with strike price K and time to expiry T, is

where the notation is the same as above.

3. The Black-Scholes-Merton partial differential equation is

with the usual notation.

Section A

Al    (a) Carefully define coupon bearing bonds.                                                                  [2]

(b) Carefully define financial instruments called derivatives and provide two examples of

such instruments.                                                                                                  [3]

(c) What is the value of a bond that pays £200 annually forever, when the interest rate is 5%

and is applied using annual discrete compounding?                                                  [2]

(d) Suppose there are two personal loans on offer. Either borrow funds at 10% with discrete compounded interest monthly or borrow the funds at 11% with discrete compounded

interest annually. Which one of these deals is better? Justify your answer.                  [3]

A2   (a) Using forward and/or plain put and call options create a portfolio with the following

payoffs at maturity T.

(ii) Payoff= -ISr - XI.                                                                                     [2]

Note: X is the strike price.

(b) Consider a four-month long forward contract written on a non-dividend paying asset.

The current price is £60. The risk-free interest rate is 3.5% per annum (with continuous compounding).

(ii) A client wants to buy a four-month forward contract wit delivery price at £55.

How much money does the client have to pay?                                                 [2]

A3  (a) A stock price is currently £30. It is known that at the end of three months it will be

either £25 or £35. The risk-free rate is 15% per annum with continuous

compounding.

Suppose that St is the stock price at the end of three months. what is the value of a

derivative that pays si + 5 at the end of three months?                                     [5]

(b) Consider a variable S that follows the process

dS = µdt + CYdB

i.e. follows generalised Brownian motion. For the first three years, µ = 3 and CY = 4 per year; for the next two years, µ = 4 and CY = 5 per year. If the initial value of the variable is 6, i.e. So  = 6 what is the probability distribution of St at the end of year

five? Explain your working.                                                                            [5]

A4 A financial institution sold for £300,000 a European call option on 100, 000 shares of a

non-dividend paying stock. We assume that the stock price is So  = 49, the strike price is K = 50 , the risk free interest rate is r = 5% per annum, the stock price volatility is

CY = 20% per annum and the time to maturity is 20 weeks.

(a) State the assumptions for the Black Scholes model.                                          [2]

(b) Assume that the assumptions of the Black Scholes model hold.

(i) Find the value of a European call option on the 100, 000 shares.

(ii) Find the value of a European put option with the same strike price and

expiration date on the 100, 000 shares.

(iii) Verify that the put-call parity holds in this case.