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

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2. The price of a non-dividend paying share, St, follows geometric Brownian motion, such that: ST = S₂ exp [σ (WT — W₁) + (r − ¹/20²) (T − t)], where t <T, under the risk-neutral probability measure Q, where r is the continuously compounded risk- free rate of interest per annum and o is the volatility per annum. (a) Consider the following derivative contract, B, with maturity payoff at time T as follows: BT = max [log ST - log St, 0], Show that the price, at time t, of the derivative contract B is given by: e-r(T-t) [(r — ¹1/20²) (T − t) Þ(d) + o √T − t ø(d)], where (.) and (.) are cumulative distribution function and density function of standard Normal distribution respectively and d is a constant. [8 marks] The company is negotiating a senior executive's pay structure which will include a bonus payable at the end of 5 years. The bonus will be based on her performance over the next 5 years. The following two alternative bonus approaches are under consideration: Bonus A: At-the-money 5-year European call option on the share. Bonus B: Derivative contract B defined in part (a) with T = 5 and t = 0. Once one of these two bonus approaches is agreed, a certain number of the chosen contract will be issued to the senior executive. Let the current share price be So = £0.50. (b) In a single figure, plot the maturity payoffs of Bonus A and Bonus B, as functions of ST. [4 marks] (c) Discuss the rationale behind the specific formulations of the two bonus approaches. [3 marks] Suppose r = 5% per annum. (d) Determine the numerical values of Bonus A and Bonus B separately for each of the following cases: o = 30% per annum and o= 50% per annum. [4 marks] (e) Discuss which of the bonus structures will be preferred by the senior executive. 2. The price of a non-dividend paying share, St, follows geometric Brownian motion, such that: ST = S₂ exp [σ (WT — W₁) + (r − ¹/20²) (T − t)], where t <T, under the risk-neutral probability measure Q, where r is the continuously compounded risk- free rate of interest per annum and o is the volatility per annum. (a) Consider the following derivative contract, B, with maturity payoff at time T as follows: BT = max [log ST - log St, 0], Show that the price, at time t, of the derivative contract B is given by: e-r(T-t) [(r — ¹1/20²) (T − t) Þ(d) + o √T − t ø(d)], where (.) and (.) are cumulative distribution function and density function of standard Normal distribution respectively and d is a constant. [8 marks] The company is negotiating a senior executive's pay structure which will include a bonus payable at the end of 5 years. The bonus will be based on her performance over the next 5 years. The following two alternative bonus approaches are under consideration: Bonus A: At-the-money 5-year European call option on the share. Bonus B: Derivative contract B defined in part (a) with T = 5 and t = 0. Once one of these two bonus approaches is agreed, a certain number of the chosen contract will be issued to the senior executive. Let the current share price be So = £0.50. (b) In a single figure, plot the maturity payoffs of Bonus A and Bonus B, as functions of ST. [4 marks] (c) Discuss the rationale behind the specific formulations of the two bonus approaches. [3 marks] Suppose r = 5% per annum. (d) Determine the numerical values of Bonus A and Bonus B separately for each of the following cases: o = 30% per annum and o= 50% per annum. [4 marks] (e) Discuss which of the bonus structures will be preferred by the senior executive.

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Answer a To find the price of the derivative contract B at time t we need to calculate its expected payoff at maturity discounted to time t using the ... View the full answer