Question1

(i) Sketch rough graphs on the same diagram showing the price of a call option on a non-interest bearing stock as a function of the volatility parameter o, in the cases where the option is: (a) deep in-the-money (b) at-the-money (c) deep out-of-the-money. [4] (ii) The following table relates to the prices for 6-month call options on Grade A Copper on a particular day: Strike price Option Price Implied Volatility 1,350 0.255 1,400 102.5 0.260 1,450 85.2 ? The current spot price is 1,370 and the risk-free interest rate is 0.05 (expressed as a continuously-compounded annual rate). Assuming that the Black-Scholes approach is valid here, complete the table by calculating the option price for a contract with a strike price of 1,350 and the implied volatility for a contract with a strike price of 1,450. [8] [Total 12]Consider the single index model of investment returns in which for any security i : R; = 0 + BR, + 6, where E(E;) =0, E(86; ) =0 for i# j, E(R 2;)=0 and R, is the return on the market. (i) Assuming that this model applies, derive expressions for the mean investment return on security i, and the mean investment return on a portfolio P. containing a securities, with a proportion x, invested in security i . [3] (ii) Show that Cip = Ex, Cy, where Cip and Cy are the covariance of investment j=1 returns between security / and portfolio P and securities / and / respectively. [2] (iii) State a general expression for the variance of of portfolio P in terms of the covariances Cy. [1] (iv) Use your results from (ii) and (iii) to show that: Bip = dop 1 dx; Op where Pip "iP and comment briefly on this result. [7] [Total 13]