QUESTION 1 (2 + 1 + 2 + 1 + 2 + 4 + 2 = 14 marks)

Consider a 3-asset portfolio worth $100M with the following 10-day return variance-covariance matrix

Stock 1 Stock 2 Stock 3

Stock 1   0.00183 -0.00050

-0.00084

Stock 2   -0.00050 0.00112

0.00103

Stock 3

-0.00084

0.00103

0.00311

and its

corresponding correlation matrix

Stock 1

Stock 1 Stock 2 Stock 3

1

-0.35

-0.35

-0.35

1

0.55

-0.35

0.55

1

Assuming return normality

a)    What is the 100-day 99% relative value at risk (VaR) assuming the following weights:

Stock 1 -40%, Stock 2 25%, Stock 3 115%  i.e w’=[-0.4, 0.25, 1.15]’ .    [2 marks]

b)   Interpret your VaR estimate in part a).     [1 mark]

c)    What is the VaR diversification benefit for the portfolio in part a)?    [2 marks]

d)   Recalculate the 100-day VaR and the VaR diversification benefit if the weight vector is changed to w’=[0.5, 0.25, 0.25]’ .             [1 mark]

e)    Explain how and why the change in weights in part d) has affected the VaR and the diversification benefit. [2 marks]

Now assume normality is unreasonable due to the presence of fat tails in the return distributions

f)    For a given weight vector, outline how monte-carlo simulation can be used to calculate the 100- day VaR at 99% in the presence of fat tails.                         [4 marks]

g)    Is the simulated VaR in part f) likely to be larger or smaller than the estimates assuming return normality? Why/why not?           [2 marks]

QUESTION 2 (2 + 6 + 2 + 3 = 13 marks)

Consider a speculator seeking to trade volatility using American put options written against a dividend paying stock. To do so she seeks an estimate of implied volatility using the following information

Stock price: $40

Strike: $36

Time to maturity: 6 months

Dividend: paid at the end of 3 months, expected to be a constant 10% proportion of the share price at that time

Risk free rate: 5% p.a. continuously compounded

Option price: $2.01

a)   A colleague suggested she use the Black Scholes Merton (BSM) model to calculate the implied volatility. What would be the effect on the IV estimate of using the BSM model without dividends? Why?                    [2 marks]

b)   Show how you could use a two-period binomial model to calculate the implied volatility on the above American put. You are only required to show calculations for the first TWO iterations. Comment on how you would proceed to solve for IV from this point.                      [6 marks]

Now assume the trader has an econometric model that forecasts volatility to be 15% higher (annualised) than the IV estimate over the same period (the next 6 months).

c)    The same colleague suggests she should now take a long position in 6-month American puts. What are the risks associated with this strategy?                                                          [2 marks]

d)   Outline another strategy she could implement given her volatility forecast, and why the strategy you propose may be better than the strategy in part c).                                              [3 marks]

QUESTION 3 (4 + 2 + 5 + 2 + 2 = 15 marks)

Consider an institution that has just sold to a client an at-the-money American put option with a 6-      month maturity. The underlying is a non-dividend paying stock priced at $30. Assuming a continuously compounded risk-free rate of 6% p.a. and stock price movements of +/- 10% each period

a)    Price the option using a three-period binomial option pricing model.                      [4 marks]

b)   Given the pricing model you used in part a), when should the client exercise the option?Why? [2 marks]

c)    Show how the institution would hedge their exposure to the stock if the stock price fell every period. [5 marks]

d)    Explain the intuition behind the evolution of the hedging strategy over the life of the hedge. [2 marks]

e)    Is the price for an otherwise equivalent European put likely to be higher or lower than the price of the American put? Why?           [2 marks]

QUESTION 4 (3 + 3 + 3 + 2 = 11 marks)

a)    In your own words outline the concepts underlying put-call parity.                          [3 marks]

b)   Consider a 9-month European call option with a strike of $80 that is written against a non-dividend paying stock. At t=0, the call is priced at $14, and the stock is valued at $85. An otherwise equivalent put is priced at $2. Assuming a continuously compounded risk-free rate of 5% p.a., demonstrate (show) an appropriate arbitrage where the mispricing is realised immediately (i.e., at time t=0). [3 marks]

Now consider otherwise equivalent American calls and puts priced at $5 and $2 respectively.

c)    Is an arbitrage possible? If so, show how it may be established at time t=0. [3 marks]

d)    Now assume the stock price falls to $70 the next day and the arbitrageur is forced to close out. Show the associated cashflows. [2 marks]

QUESTION 5 (1+ 6 + 3 = 10 marks)

Consider a stock currently priced at $100. The market expects the volatility of the stock over the next year to be 25%. The risk-free rate is 6.5% p.a. continuously compounded and a dividend of $5.50 is expected in 4 months. There are no exchange traded options available.

Assume you work for a bank and have a client with a short position in 1000 units of the stock. The client wants to buy 1000 over-the-counter (OTC) calls with a strike of $105 and maturity of 12 months. Each    call is on one unit of the underlying.

a)   Why do you think the client wants to buy the calls?                                                     [1 mark]

b)   Using an appropriate continuous time option pricing model, quantify the extra cost to the client if you give them the right to exercise the options early.                                              [6 marks]

c)    Comment on any risks associated with the client’s strategy and outline any other advice on how they could manage their exposure.                                                                                  [3 marks]

QUESTION 6 (2 + 3 + 1 + 3 + 1 + 2 = 12 marks)

Consider an institution with three option positions against the same underlying stock. The maturities of  all the options exceed one year. The institution seeks to hedge their option exposure over the next three weeks by taking a position in the stock. You are provided with the following information

a)   What does the evolution of the deltas imply about the trajectory of the underlying stock price over the next two weeks? Why?                   [2 marks]

b)   Calculate the dynamic position in the stock (to 2 decimal places) that is required to delta hedge the overall exposure now (week 0), and at the end of weeks 1 and 2.                      [3 marks]

c)    Given that units of stock cannot be purchased or sold to 2 decimal places, what are the likely hedging implications of rounding the stock position to the nearest integer? [1 mark]

d)   Show how the hedging strategy in b) changes now and at the end of weeks 1 and 2, if each option position is hedged independently of the other positions, i.e. consider independently hedging option 1, independently hedge option 2 etc. [3 marks]

e)    Is the hedging strategy in part d) likely to be optimal? Why/why not?   [1 mark]

f)    Briefly explain two strategies that could improve the hedge in part b).    [2 marks]