A stock price is currently $10. It is known that at the end of three months it will be either $11 or $8.5. Th risk-free interest rate is 5% per annum with continuous compounding. Suppose Sr is the stock price at the end of three months. (a) What is the value of a derivative that pays off In(ST) at this time? Use both the no arbitrage and risk neutral valuation approach. No Arbitrage Approach: Risk Neutral Valuation Approach: At the end of three months the value of the At the end of three months the value of the derivative will be either 121 (if the stock price is derivative will be either $72.25 (if the stock 11) or 72.25 (if the stock price is 8.5). Consider a price is $8.5) or $121 (if the stock price is $11). portfolio consisting of: Consider a portfolio consisting of: The value of the portfolio is either 1 1 shares - u = 11 = 1.1 and d = 8.5 121 or 8.5 shares - 72.25 in three months. If 10 10 = 0.85 1 1 shares - 121 = 8.5 shares - 72.25 so that, 2.5 shares = 48.75 p =. erf x (t - d ) u -d Shares = 19.5 0.05X12 - 0.85 P= the value of the portfolio is certain to be 1.1 - 0.85 (19.5 x 11) - 121 = 93.5 For this value of P = 1.012578 - 0.85 0.25 share the portfolio is therefore risk less. The current value of the portfolio is: 0.162578 P = 0.25 19.5 x 25 - f p = 0.6503 where f is the value of the derivatives. Since the and portfolio must earn the risk-free rate of interest f = e-rfxt [p x higher value + (19.5 x 10 - f)e 0.05X12 = 93.5 (1 - p) lower value] (195 - f) 1.012578 = 93.5 f = e-0.05x12( 0.6503 * 121 + (1 - 0.6503) x 72.25 197.45271 - 1.012578 f = 93.5 f = 0.987577 X (78.6863 + 25.2658) 103.95271 = 1.012575 f f = 102.66 f = 102.66 The value of the option is therefore $102.66 The value of the option is therefore $102.66