MATH4143 Page 1 of 17 Winter 2007 MATH4143: Scientific Computations for Finance Applications Final exam Time: 9:00 am - 12:00 noon, April 18, 2007 Student Name (print): Student Signature: Student ID: Question Full Mark Mark 1 15 2 15 3 10 4 15 5 15 6 10 7 20 Total (100): Notes: - Closed book exam - Calculator and one-page single-sided cheat sheet of size 8x11 are allowed; no other aids are permitted
MATH4143 Page 2 of 17 Winter 2007 1. Questions related to the basics of options a) (3 marks) Which of the following are always positively related to the price of a European put option on a stock i The stock price ii The strike price iii The time to expiration iv The volatility v The risk-free rate vi The magnitude of dividends anticipated during the life of the option b) (3 marks) Which of the following are true i It is always optimal to exercise American call options early if the underlying asset price is sufficiently high ii It is always optimal to exercise American put options early if the underlying asset price is sufficiently low iii A call option will always be exercised at maturity if the underlying asset price is greater than the strike price iv A put option will always be exercised at maturity if the strike price is v greater than the underlying asset price No matter what happens, the call option price can never be worth more than the strike price vi No matter what happens, the put option price can never be worth more than the strike price c) (6 marks) Explain carefully the difference between hedging, speculation and arbitrage
MATH4143 Page 3 of 17 Winter 2007 d) (3 marks) A trader buys 100 European call options with a strike price of $20 and a time to maturity of one year. The cost of each option is $2. The price of the underlying asset proves to be $25 in one year. What is the trader s gain or loss? (show how you derive the answer)
MATH4143 Page 4 of 17 Winter 2007 2. a) (10 marks) Consider a European call option on a non-dividend-paying stock where the stock price is $40, the strike price is $40, the risk-free rate is 4% per annum, the volatility is 30% per annum, and the time to maturity is 6 months. Use a two-step tree to calculate the value of this option. b) (1 mark) What is meant by the delta of a stock option? c) (3 marks) Based on the results in part (a), estimate the delta s corresponding to the stock movements over the first and second time steps. d) (1 mark) If the derivative in part (a) is American style, should it be exercise early?
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MATH4143 Page 6 of 17 Winter 2007 3. (10 marks) A 3-month American call option on a stock has a strike price of $20. The stock price is $20, the risk-free rate is 3% per annum, and the volatility is 25% per annum. A dividend of $1 per share is expected at the end of the second month. Use a three-step binomial tree to calculate the option price.
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MATH4143 Page 8 of 17 Winter 2007 4. a) (6 marks) Consider an Asian average rate call option with discrete arithmetic N 1 averaging. The option payoff is max S( t i ) K, 0 where t i = i t, N k = 1 T t =, K is the strike price and T is the life of the option. Assuming that the N risk-free interest rate as well as volatility is constant, describe in detail how to use Monte Carlo simulation to value the option. b) (3 marks) The accuracy of the result given by Monte Carlo simulations depends on the number of trials. How do you determine the number of trials in the Monte Carlo simulations? To increase the accuracy by a factor of N, by what factor the number of trials must increase? c) (6 marks) Explain the control variate and antithetic variable techniques and how they could be used to improve numerical efficiency of the Monte Carlo simulation?
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MATH4143 Page 10 of 17 Winter 2007 5. a) (3 marks) Suppose that we know 10-day 99% VaR for our portfolio in Microsoft is 1 million. What does it mean? What are the advantages and disadvantages of VaR? b) (12 marks) Stock A has a daily volatility of 1.2% and stock B has a daily volatility of 1.8%. The correlation between the two stock price returns is 0.2. (Note that N(-1.96) = 0.025, where N(x) is the cumulative standard normal distribution function.) i What is the 97.5% 5-day VaR for a 1 million dollar investment in stock A? ii What is the 97.5% 5-day VaR for a 1 million dollar investment in stock B? iii What is the 97.5% 5-day VaR for a 1 million dollar investment in stock A and a 1 million dollar investment in stock B? iv What is the benefit of diversification to the 97.5% 5-day VaR?
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MATH4143 Page 12 of 17 Winter 2007 6. a) (4 marks) A variable, x, starts at 10 and follows a generalized Wiener process ds = adt + bdz where the draft rate a = 2 per annum and the variance rate b 2 = 3 2 per annum and dz is a Wiener process. i. What is the mean value of the variable after three years? ii. What is the standard deviation of the value of the variable after three years? b) (6 marks) The process followed by a stock price is ds = µ Sdt + σ Sdz. What 3 kind of process does Q= S follow?
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MATH4143 Page 14 of 17 Winter 2007 7. Finite difference methods for option pricing. a) (4 marks) Briefly describe the finite difference approach in valuing an option. Comment on their advantages and disadvantages. b) (4 marks) It is computationally more efficient to use finite difference methods with ln S rather than S as the underlying variable. Define Z = ln S. Show that the Black-Schole equation becomes 2 2 2 f σ f σ f + r + = rf. (7.1) 2 t 2 Z 2 Z c) (4 marks) Consider a European vanilla put option. Set up proper final and boundary conditions for Z = ln S in Equation (7.1). The finite different mesh grid then evaluates the derivative for equally spaced values of Z rather than for equally spaced values of S. d) (5 marks) Use proper finite differences for time derivative and Z derivatives in (7.1) to derive the explicit finite difference scheme for a European put option. e) (3 marks) Make comments on their truncation/discretization errors and stability issues in part (d).
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