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The price of a non-dividend paying share, St , follows geometric Brownian motion, such that:

1. The price of a non-dividend paying share, St, follows geometric Brownian motion, such that under the risk-neutral probability measure Q Let the current price of the share be So-1, the annual volatility of the share price process ? : 10% and the continuously compounded constant annual risk-free interest rate is 5%. (a) Determine the distribution of SFt, where Fi is the filtration up to time t. 5 marks] (b) Show that the current price of the derivative contract, which promises to pay S2 at the end of 2 years, is given by: W= exp (0, 12) [5 marks] (c) Calculate the current price of the derivative contract, which promises to pay (S2 1)2 at the end of 2 years (d) [5 marks] contract with calculate the price of a European call option Using Black-Scholes formula, strike price El and 2 years to maturity. (e) Using put-call parity, or otherwise, caleulate the price of a European put option contract with strike price 1 and 2 years to maturity (f) Using parts (d) and (e), calculate the current price of the derivative contract, which promises to pay S 1 at the end of 2 years. (Note: 1?l is defined as: x ifz > 0 and-?if z-0.) (g) Explain the differences between the prices of the derivatives in parts (c) and (f). [5 marks] 3 marks] [2 marks] 4 marks] [2 marks] (h) Without doing any further calculations, discuss the significance of the volatility assump- tion in terms of the derivative prices calculated in parts (c) andf) [Total: 31 marks] 1. The price of a non-dividend paying share, St, follows geometric Brownian motion, such that under the risk-neutral probability measure Q Let the current price of the share be So-1, the annual volatility of the share price process ? : 10% and the continuously compounded constant annual risk-free interest rate is 5%. (a) Determine the distribution of SFt, where Fi is the filtration up to time t. 5 marks] (b) Show that the current price of the derivative contract, which promises to pay S2 at the end of 2 years, is given by: W= exp (0, 12) [5 marks] (c) Calculate the current price of the derivative contract, which promises to pay (S2 1)2 at the end of 2 years (d) [5 marks] contract with calculate the price of a European call option Using Black-Scholes formula, strike price El and 2 years to maturity. (e) Using put-call parity, or otherwise, caleulate the price of a European put option contract with strike price 1 and 2 years to maturity (f) Using parts (d) and (e), calculate the current price of the derivative contract, which promises to pay S 1 at the end of 2 years. (Note: 1?l is defined as: x ifz > 0 and-?if z-0.) (g) Explain the differences between the prices of the derivatives in parts (c) and (f). [5 marks] 3 marks] [2 marks] 4 marks] [2 marks] (h) Without doing any further calculations, discuss the significance of the volatility assump- tion in terms of the derivative prices calculated in parts (c) andf) [Total: 31 marks]

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