It is currently 1 January. Rugby plc stock is expected to pay its annual dividend of 0.58

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It is currently 1 January. Rugby plc stock is expected to pay its annual dividend of £0.58 per share with ex-dividend date = payment date in 3 months’ time on 1 April. Rugby plc stock is currently trading at £14.53 and has an annualised volatility of returns of 34.05% per annum. The continuously compounded risk-free interest rate is 7% per annum. Consider options on Rugby plc stock with strike price £14.00 and maturity 1 June (in 5 months’ time).

a) Calculate the current Black-Scholes value of a European call option on Rugby plc stock with maturity five months and strike price £14.00 and deduce the value of the corresponding European put option on Rugby plc with the same strike price and maturity. Give one assumption that is required for both (i) the put-call parity relationship for European options and (ii) the relationship between the Black-Scholes value of a European call option and the spot price, and one assumption that is required for (ii) but not (i). Discuss the extent to which each assumption holds in practice.

b) Discuss the general conditions (including an indication of potential timing of any early exercise and stock price ranges) under which it may be optimal to exercise before maturity an American-style call option on a stock paying a dividend D with ex-dividend date = payment date between now and the option’s maturity 0 < TD < T . Does the size of the dividend make a difference? Hence deduce whether it may be worthwhile exercising before maturity an American call option on Rugby plc stock with strike price £14.00 and maturity 5 months. Assuming early exercise may be worthwhile, sketch a graph of the American call option value vs stock price on the ex-dividend date showing:

1. the value if the option is exercised,

2. the value if the option is not exercised,

3. the American option value.

Assuming early exercise may be worthwhile, give upper and lower arbitrage based bounds on the current value of an American call option relative to the corresponding European call option with the same strike price and maturity. If the upper bound was violated, explain how you could trade to obtain an arbitrage profit, check all future cash flows to the strategy are non-negative and calculate the arbitrage profit. Calculate upper and lower bounds on the value of a five-month American call option on Rugby plc stock with strike price £14.00.