N1559 FINANCIAL DERIVATIVES

SECTION A

There are 2 questions in this section. Attempt BOTH questions. This section is worth 50 marks.

1. For each statement, state whether you agree or disagree, and briefly explain why in either case (eg, 1-2 sentences). (Note: Some statements may be completely true, some completely false, and some may be partially true. Read carefully.)

(a) The holder of a long futures position will receive an early cashflow from a margin call if the stock price moves upwards. [ 4 marks ]

(b) Using a binomial tree model for temperature changes is insufficient to price a weather derivative (e.g. a call option on temperature). [ 4 marks ]

(c) The convenience yield describes cashflows which are received by holders of physical commodities, just like the dividend yield for equity holders. [ 4 marks ]

(d) Accurate pricing of Asian options in a binomial tree model is difficult because they are strongly path dependent. [ 4 marks ]

(e) A risk-averse copper mine might choose to either sell copper forwards or buy copper call options to hedge its profit from future production. [ 4 marks ]

(f) In the Black-Scholes model, the Delta of a European put option is always positive and typically highest at the money. [ 4 marks ]

2. A call bull spread can be constructed by simutaneously buying a call option at a strike price of K1 and selling a call option at a strike price of K2, where K1 < K2, and the options have the same maturity date. Assume that the bank lends and borrows at some rate continuously compounded rate r > 0, and the stock doesn’t provide dividends, or any other intermediate cashflows before the options’ maturity.

(a) Explain why this strategy must always involve some initial investment (i.e. cost some money now). [ 4 marks ]

(b) Suppose the current price of the underlying stock is S0 = 50, the strikes for the call options are K1 = 48, and K2 = 52 respectively. Plot the payoff function with axes clearly labelled (x-axis does not have to start at zero). [ 4 marks ]

(c) In addition to part (b), suppose the call options are quoted as c(K1) = 4.64, and c(K2) = 2.33. Plot the profit and loss diagram with axes clearly labelled. [ 4 marks ]

(d) Explain how you can construct a strategy with the same payoff with positions in only European puts and the bank account. [ 4 marks ]

(e) In addition to information in parts (b) and (c), if a put option with strike K1 is quoted as p(K1) = 0.87, calculate both the bank’s interest rate and p(K2). Assume that both calls and both puts mature in one year. [ 6 marks ]

(f) Identify the upfront payment (if any) for the put bull spread. Is this the same as the call bull spread? If different, why? [ 4 marks ]

SECTION B

Attempt only TWO of the three questions in this section. Each question is worth 25 marks.

3. A stock is currently trading at S0 = 100, a bank provides borrowing and lending at a continuously compounded risk-free interest rate of r = 5% annually. The volatility of the stock is 20% annually. Use a two-step binomial tree model to answer the following:

(a) Identify the up and down parameters of the binomial model for an option that matures in 6 months. Draw the stock price tree. [ 5 marks ]

(b) Price a European call option with strike K = 105 that matures at T = 0.5 [ 8 marks ]

(c) Construct an arbitrage-free synthetic call at t = 0 using other available financial products. Does this strategy replicate the option’s payoff at T? Why or why not? [ 7 marks ]

(d) If the call option in part (b) is American, price this option. [ 5 marks ]

4. Throughout this question, consider a non-dividend paying stock, and let ct(K, T) denote the price of a European call option on the stock, with strike K and maturity T. Similarly, pt(K, T) for a put. The bank borrows and lends at a constant annual continuously compounded rate of r.

(a) Suppose that ct(Ki , T) and pt(Ki , T) are quoted for a list of strike prices Ki on a particular maturity. Discuss how implied volatility (IV) can be found, explain the typical shape of the IV curve for stock options and why this shape exists. [ 6 marks ]

(b) Suppose three options are avaliable for strikes K1 < K2 < K3, and that K3 − K2 = K2 − K1. Show or explain why the following must hold: [ 8 marks ]

(c) Suppose S0 = K2. Propose a cheap options strategy using calls and/or puts at any strike that allows you to profit from large stock price movements either up or down. Discuss why this strategy could be referred to as ‘long vega’. [ 5 marks ]

(d) Using no-arbitrage arguments, derive the upper and lower bounds of an American put option price. [ 6 marks ]

5. A company’s stock price S0 is $60 today and the continuously compounded interest rate is r = 3% per year. Suppose the company pays a dividend of $1 to its shareholders every quarter (four per year). The next dividend is due in one month’s time.

(a) Find the price of an eight-month forward contract on the stock. [ 6 marks ]

(b) Suppose the forward contract is trading at exactly $60, just like the stock. Is an arbitrage opportunity available? If so, how? If not, why not? [ 5 marks ]

(c) Assume instead a continuous dividend yield δ, estimated from next year’s planned dividends. By how much does your answer to (a) change? [ 5 marks ]

(d) Now suppose that S0 = 60 is instead the spot price of crude oil, r = 3% per year, and storage is paid at a continuous annualized rate of c = 5% of the spot price. What are the possible values for the eight-month forward price of oil? [ 5 marks ]

(e) Give an example of a supply or demand effect which could produce an oil forward price of $50. [ 4 marks ]