# 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**

- 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,*S*_{t}_{+1 }is either*uS*or_{t }*dS*. Assume that the factor for moving up is_{t}*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*S*_{0 }= 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

- Compute the price process (i.e. prices at all times and states) for a European

*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**%]

- 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 *S _{t}* =

*x:*Present your arguments.

[**50**%]

# CONTINUED OVERLEAF

3

**Section B**

- Critically discuss similarities and di⁄erences of American and a European
*call*and*put*

[**50**%]

- 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**%]

Before your exam answers are collected:

** **Pleaseexam paper), ensure** **yourthat** **studentyou have ID, written date of** **thebirth course and the code number (on the of** **thefront question of this that you have attempted on each answer sheet.

** **Putyellow your answer exam sheets answers follow in the the order appropriate of the question white** **answernumber, sheet. ensuring that

** **Do not place any other exam materials, including the exam paper, beside the A3 answer sheets.

END OF QUESTION PAPER

4