Question 1 (4 marks)

Consider an option on a non-dividend-paying stock when the stock price is $30, the exercise price is $29, the risk-free interest rate is 5% per annum, the volatility is 25% per annum, and the time to maturity is four months.

1. What is the price of the option if it is a European call?
2. What is the price of the option if it is an American call?
3. What is the price of the option if it is a European put?
4. Verify that put-call parity holds.

Question 2 (4 marks)

The current price of a non-dividend-paying biotech stock is $140 with a volatility of 25%. The risk-free rate is 4%. For a three-month time step:

1. What is the percentage up movement?
2. What is the percentage down movement?
3. What is the probability of an up movement in a risk-neutral world?
4. What is the probability of a down movement in a risk-neutral world?

Use a two-step tree to value a six-month European call option and a six-month European put option. In both cases the strike price is $150.

Question 3 (3 marks)

A one year European put option and a one year European call option with a strike price of $59 are both priced at $5 in the market a one year futures price is currently traded at $58. The risk free rate is 7% per annum. Is there an arbitrage opportunity, if so, show the gain on the arbitrage.

Question 4 (4 marks)

A financial institution has the following portfolio of over-the-counter options on sterling:

Type	Position	Delta of option	Gamma of option	Vega of option
Call	-1,000	0.50	2.2	1.8
Call	-500	0.80	0.6	0.2
Put	-2,000	-0.40	1.3	0.7
Call	-500	0.70	1.8	1.4

A traded option is available with a delta of 0.6, a gamma of 1.5, and a vega of 0.8.

1. What position in the traded option and in sterling would make the portfolio both gamma neutral and delta neutral?
2. What position in the traded option and in sterling would make the portfolio both vega neutral and delta neutral?