Maths

You must use ONE WHITE answer sheet per question as per the number supplied. To provide an answer that exceeds the space on the answer sheet, please raise your hand to request a YELLOW answer sheet.

Answer ONE question from Section A and ONE question from Section B

Section A

1. Consider the discrete-time binomial tree model with three periods of length 1, i.e. T = 3 and t = 0;1;2;3. In each period the price can move up or down, St+1 is either uSt or dSt. Assume that the factor for moving up is u = 4=3, the factor for moving down is d = 3=4, and that the interest rate is r = 0:0. The initial stock price is S0 = 1.
• Compute the price process (i.e. prices at all times and states) for a European Put option on the stock with strike price K = 1 and maturity T = 3.
• Compute the price at time t = 0 of the Australian option with

K = 1. Note: As this option is path dependent, you will not be able to use the recursive method, nor will you be able to use the CRR formula.

[50%]

1. Consider the stock price under the Black-Scholes assumption, i.e.

where r denotes the interest rate. Consider an option with payo⁄

where T is the time of maturity and K is a constant. Compute the Greeks of this option. Decide whether

is the Black-Scholes price of the option at time t assuming that St = x: Present your arguments.

[50%]

CONTINUED OVERLEAF

3

Section B

1. Critically discuss similarities and di⁄erences of American and a European call and put

[50%]

1. State and explain the Ito formula. You do not need to prove it, but you have to illustratethe application of the formula by applying it to the geometric Brownian motion. Comment on the role of geometric Brownian motion in the modelling of stocks.

[50%]