Economics;

N (i) Define an Arrow-Debreu security. Define any terms used. [1] (ii) Consider an arbitrage-free one-period model with three states. There are (only) three assets in the market and their current prices and payoffs in each of the three states at time 1 are as follows: Asset Price at time 0 Payoff at time I State I 3 100 200 110 50 UNE 90 100 100 60 75 75 75 75 The (real world) probability of state 2 occurring is 50% and the state price deflator for state 1 is 0.5. Calculate the (real world) probability of state 3 occurring.(i) Define three Greeks other than Delta and explain briefly what they measure about an option. Define any symbols used. [3] (ii) A non-dividend paying stock has volatility o = 25% p.a. and the continuously compounded risk-free rate of interest is / = 12% p.a. An option on the stock has a theoretical price of $327+3g-1 , when the current price is S and the time to expiry is t. (a) Use the Cox-Ross-Rubinstein binomial lattice approximation with q = (ersi- d)/(u - d), u = ed and d = e-GM (with At = 1 day) to estimate I' when S = 1 and f = 1 year. Assume that there are 365 days in a year. (b) Compare the estimate in (a) to the value obtained by differentiating (as appropriate) the formula for the option price and comment. [5]A non-dividend paying stock has a current price of 1000p. In any unit of time the price of the stock either increases by 25% or decreases by 20%. The risk-free rate of interest is 5% per unit of time. (i) Find the risk-neutral probability measure for the model. [3] (ii) Denoting the stock price after / time units by S; (a) Find the price of a path-dependent option on the stock with expiry date 1 = 2 which pays $2 - M2, where My = mingsis2 S, (b) Find the hedging portfolio for the option at f = 1 if the stock price S, is 800p. [71The model for the price of a non-dividend paying asset at time t, S, is given by dS, = S, (udt + odZ,) where Z is a standard Brownian Motion and u and o are fixed parameters. (i) Describe the relationship between u and the risk free rate of interest if the model is an equivalent martingale measure. [1] (ii) Use Ito's formula to prove that the solution is S, = So exp((H -62/2) t+ oz,) [3] (iii) Discuss the properties of the solution and its plausibility as a model of a non- dividend paying share price. [3] (iv) Describe, with justification, how to simulate monthly returns from this model