1. (i) List the factors involved in the contract specification of a futures contract for a

commodity. [3]

(ii) Describe contango and backwardation, giving formulae and stating any

conditions necessary for these market situations to exist. [4]

2

(i) Define a complete market. [1] The price process of a traded security satisfies the following stochastic differential equation dS, = us,di + oS,dW. where W, is a Brownian motion under the real-world probability measure P. Let r > 0 be the continuously compounded risk-free rate of interest, with r u. (ii) Show that the discounted stock price e "S, is not a martingale under the real- world probability measure P. [3] (iii) Demonstrate how the discounted asset price e "'S, can be a martingale under an equivalent martingale measure O. [3]The current price of a non-dividend paying stock is $65 and its volatility is 25% per annum. The continuously compounded risk-free interest rate is 2% per annum. Consider a European call option on this share with strike price $55 and expiry date in six months' time. Assume that the Black-Scholes model applies. (i) Calculate the price of the call option. (ii) Define algebraically the delta of the call option. (iii) Calculate the value of the delta of the call option.For three different risks, an actuary is modelling the monthly claim numbers with three different Poisson distributions. Risk Exposure Number of claims Risk 1 36 months 20 Risk 2 30 months 18 Risk 3 24 months 16 (i) Derive the maximum likelihood estimates of the parameter for each of the three individual Poisson distributions fitted. (ii) Test the hypothesis that the three risks have the same monthly claim rate