MATH6911 Page 1 of 16 Winter 2007 MATH6911: Numerical Methods in Finance Final exam Time: 2:00pm - 5:00pm, April 11, 2007 Student Name (print): Student Signature: Student ID: Question Full Mark Mark 1 12 2 15 3 10 4 10 5 10 6 6 7 12 8 25 Total (100): Notes: - Closed book exam - Calculator and one-page single-sided cheat sheet of size 8x11 are allowed; no other aids are permitted
MATH6911 Page 2 of 16 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) (3 marks) Explain carefully the difference between hedging, speculation and arbitrage
MATH6911 Page 3 of 16 Winter 2007 d) (3 marks) A one-year call option on a stock with a strike price of $30 costs $3; a one-year put option on the stock with a strike price of $30 costs $4. Suppose that a trader buys two call options and one put option. i What is the breakeven stock price, above which the trader makes a profit? (show how you derive the answer) ii What is the breakeven stock price below which the trader makes a profit? (show how you derive the answer)
MATH6911 Page 4 of 16 Winter 2007 2. a) (10 marks) A stock price is currently $30. During each 2-month period for the next 4 months it will increase by 8% or reduce by 10%. The risk-free interest rate is 5%. Use a two-step tree to calculate the value of a European power option that pays off max[(30-s T ), 0] 2, where S T is the stock price in 4 months. 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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MATH6911 Page 6 of 16 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.
MATH6911 Page 7 of 16 Winter 2007 4. a) (4 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 ti = 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 are constant, describe in detail how to use Monte Carlo simulation to value the option. b) (2 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) (4 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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MATH6911 Page 9 of 16 Winter 2007 5. a) (3 marks) Suppose that we know 10-day 99% VaR for our portfolio in Microsoft is $1,000,000. What does it mean? What are the advantages and disadvantages of VaR? b) (7 marks) Consider a position of a $300,000 investment in gold and a $500,000 investment in silver. Suppose that the daily volatilities of these two assets are 1.8% and 1.2%, respectively, and that the coefficient of correlation between their returns is 0.6. What is the 10-day 97.5% VaR for the portfolio? By how much does diversification reduce the VaR? (Note: N(-1.96) = 0.025, where N(x) is the cumulative standard normal distribution function.)
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MATH6911 Page 11 of 16 Winter 2007 6. a) (3 marks) A variable, x, starts at 10 and follows a generalized Wiener process ds = adt + bdz. During the first two years a=4 and b=3. During the following three years a=6 and b=4. i. What is the mean value of the variable at the end of the five years? ii. What is the standard deviation of the variable at the end of the five years? b) (3 marks) The process followed by a stock price is ds = µ Sdt + σ Sdz 3 i. What kind of process does Q= S follow?
MATH6911 Page 12 of 16 Winter 2007 7. a) (4 marks) What are the direct and iterative methods for solving Ax = b? Discuss about their advantages and disadvantages of both methods. b) (8 marks) Consider the following 3-by-3 linear system Ax=b, where 2 1 0 1 A= 1 3 1 and b= 1. 0 1 2 1 Compute an approximate solution for two iterations using Jacobi and Gauss- 0 ( 0) Seidel methods, respectively. Please use the initial guess is x = 0. 0
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MATH6911 Page 15 of 16 Winter 2007 8. Finite difference methods for option pricing. a) (3 marks) Briefly describe the finite difference approach in valuing an option. Comment on their advantages and disadvantages. b) (3 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. (8.1) 2 t 2 Z 2 Z c) (3 marks) Consider a European vanilla put option. Set up proper final and boundary conditions for Z = ln S in Equation (8.1). The finite different mesh grid then evaluates the derivative for equally spaced values of Z rather than for equally spaced values of S. d) (3 marks) Use proper finite differences for time derivative and Z derivatives in (8.1) to derive the explicit finite difference scheme for a European put option. e) (3 marks) Use proper finite differences for time derivative and Z derivatives in (8.1) to derive the implicit finite difference scheme for a European put option. f) (3 marks) Use proper finite differences for time derivative and Z derivatives in (8.1) to derive the Crank-Nicolson finite difference scheme for a European put option. g) (3 marks) Make comments on their truncation/discretization errors and stability issues in (c), (d) and (e). h) (4 marks) Describe the projected SOR method to value the American version of the above European put option. Use the Crank-Nicolson finite-difference approximations of the Black-Scholes equation.
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