The question.

Consider two securities, F and G, that are both positively dependent on the same underlying source of uncertainty. Security F has price f that follows the process of = ufdt + ofdw , where w is a standard Brownian motion. Security G has price g that follows the process dg = vgdt + Ogdw, with the same underlying Brownian motion. It has been found that the volatility of G is a damped version of the volatility of F, as defined by the formula 0=o e for some constant B. The risk-free rate r is 4% per annum. (i) Explain the significance of the "market price of risk" 2 = = [2] (ii) If F has a constant expected return of 7% per annum, i.e. u = 7%, and if B= 0.05, calculate the expected return of G at time t = 10. [3] A squared payoff call on a security with strike price K is an option where the payoff at exercise is the excess, if any, of the square of the stock price over K. (iii) (a) By creating a suitable risk-free portfolio, or otherwise, derive the differential equation for the price of an American style squared payoff call based on security F. (b) State the boundary conditions required to solve the equation. (A solution to the equation is not required.) [6] (iv) Explain how you would adapt the differential equation derived in (iii)(a) to price an American style squared payoff call based on G. [3] [Total 141