7 The table below shows the closing prices (represented by letters) on a particular day for a series yla of European call options with different strike prices and expiry dates on a particular non-dividend-paying security. Strike price 125 150 3 months W Y 6 months X Z Write down, with reasons, the strictest inequalities that can be deduced for the relative values of W, X,Y, Z , assuming that the market is arbitrage-free. (Your inequalities should not involve any other quantities.) [4] (ii) Calculate numerical values for a lower and an upper bound for X , given that the current share price is 120 and the continuously compounded risk-free interest rate is 6% pa. [2] [Total 6] 8 Let pr be the price of a European put option exercisable at time 7 with a strike price K on an yle underlying non-dividend-paying share with price S, at time t . By considering a suitably chosen notional portfolio or portfolios (which should be specified carefully), show that the price p, satisfies the inequality: Pt 2 Ke "(T-1) - 5, [4] (ii) Explain how you would modify your inequality if you knew that holders of the share on the day before the option expires are entitled to receive a cash dividend of 0.025, payable at time T . (2] [Total 6] 9 Consider an asset with price S, at time t , paying a dividend at a constant dividend yield, D . yle Dividends are paid at the end of each year and are immediately reinvested in the asset. The continuously compounded risk-free rate of interest is r pa. Derive the forward price, K, of a contract issued at time t , with maturity at time 7 , to trade one unit of the asset, where 7-t is an integer number of years. State any assumptions you make. [6] 10 By constructing two portfolios with identical payoffs at the exercise date of the options, derive an expression for the put-call parity of European options on a share that has a dividend of known amount d payable prior to the exercise date. [6]1 Explain what is meant by the continuous-time lognormal model of security prices. 2 When valuing derivatives it is often assumed that the price of the underlying security follows a geometric Brownian motion with stochastic differential equation: dS, = S,(udt + adz,) where Z, represents a standard Brownian motion. List the advantages and disadvantages of this assumption. 3 An investor has decided to model PPB plc shares using the continuous-time lognormal model. Using historical data, the investor has estimated the annual drift and volatility parameters to be 6% and 25% respectively. PPB's current share price is $2. (0 Calculate the mean and variance of PPB's share price in one year's time. [3] Calculate the probability that: (a] PPB's shares fall in value over the next year. (b) PPB's shares yield a return of greater than 30% over the next year. Assume that no dividends are to be paid over the next year. [4] [Total 7]