.5 M339W/389W Financial Mathematics for Actuarial Applications University of Texas at Austin Sample In-Term Exam 2.5 Instructor: Milica Čudina

Transcription

1 .5 M339W/389W Financial Mathematics for Actuarial Applications University of Texas at Austin Sample In-Term Exam 2.5 Instructor: Milica Čudina Notes: This is a closed book and closed notes exam. Time: 50 minutes TRUE/FALSE 1 (2) TRUE FALSE 2 (2) TRUE FALSE 3 (2) TRUE FALSE 4 (2) TRUE FALSE 5 (2) TRUE FALSE MULTIPLE CHOICE 1 (5) a b c d e 2 (5) a b c d e 3 (5) a b c d e 4 (5) a b c d e 5 (5) a b c d e 6 (5) a b c d e FOR GRADER S USE ONLY: T/F M.C. Σ

2 TRUE/FALSE QUESTIONS. Please note your answers on the front page. Problem 2.1. A market-maker writes a put option on a stock. To delta-hedge, (s)he needs to buy shares of the underlying stock. True or false? Problem 2.2. A market-maker writes a call option on a stock. To decrease the delta of his position, (s)he can write a call on the underlying stock. True or false? Problem 2.3. Call gamma tends to increase for at-the-money options as the time to expiry increases. True or false? Problem 2.4. Call delta is increasing as a function of the underlying asset price. True or false? Problem 2.5. Consider a European call and an otherwise identical put. Then, the call vega is strictly greater than the put vega. True or false? Problem 2.6. Call elasticity is at least 1 in the Black-Scholes model. True or false? Problem 2.7. Gamma of a call bull spread is always positive.true or false? 2.2. FREE-RESPONSE PROBLEMS. Please, explain carefully all your statements and assumptions. Numerical results or single-word answers without an explanation (even if they re correct) are worth 0 points. Problem 2.8. (13 points) Assume that the market-maker is trading options on an underlying asset with the price process S. Let S(0) = 40, σ = 0.3, r = 0.08 and δ = 0. The market-maker buys a bull spread on S with 91 days to expiration. To construct this particular bull spread the market maker uses only calls: He buys a 40 strike call and sells a 45 strike call on S. i. (5 pts) What is the Black-Scholes cost of this purchase? Then the market-maker decides to delta-hedge the above position by trading the shares of stock S. ii. (5 pts) What investment is required to do so? (Describe the type of the investment and find its cost.) The market-maker finances his trading by investing in the money market at the risk-free rate r. iii. (3 pts) What is the net-effect of the above two trades on the market-maker s wealth, i.e., how much does (s)he need to borrow from/invest in the money market?

3 Problem 2.9. (10 points) The current price of a non-dividend-paying stock is $25 per share. A market-maker writes a three-month European put option on this stock and proceeds to delta-hedge it. The put premium is $2.50, its delta is 0.30, its gamma is 0.04, and its theta is 0.01 per day. The continuously compounded risk-free interest rate is Assuming that the stock price does not change, what is the approximate overnight profit for the market-maker? 3 Problem (10 points) Assume the Black-Scholes framework. The current stock price is $50 per share. Its dividend yield is 0.01 and its volatility is The continuously compounded risk-free interest rate is Consider a one-year, $55-strike European put option on the above stock. What is the volatility of the put option? Problem (15 points) Assume the Black-Scholes framework for the evolution of a stock price. The stock pays no dividends. Consider a one-year European call on this stock. You are given the following: the call s delta is , under the risk-neutral probability the probability that the option is in the money at expiration is What is the volatility of this call option? Problem (15 points) An investor buys a time T European call option on this stock at time 0 and creates a delta-neutral, fully-leveraged portfolio by trading in the shares of the underlying stock and borrowing/lending at the continuously compounded risk-free interest rate r. The current price of the underlying stock is $50 and its dividend yield is equal to the continuously compounded risk-free interest rate. The time 0 premium of the above call option is $7.50 and its delta is The premium for the otherwise identical put option is $5.60. At time t (prior to the call s exercise date T ), the investor decides to liquidate her portfolio. She sees that the current stock price is unchanged from time 0, the above call premium is $4.50 and the above put premium is $2.40. What is the investor s profit after liquidation?

4 MULTIPLE CHOICE QUESTIONS. Please note your answers on the front page. Problem Assume the Black-Scholes framework. For an at-the-money, T year European call option on a non-dividend-paying stock you are given that its delta equals What is the delta of an otherwise identical option with exercise date at time 2T? (a) 0.62 (b) 0.66 (c) 0.70 (d) 0.74 Problem Which of the following greeks is usually negative? (a) Call delta. (b) Call gamma. (c) Call theta. (d) Call vega. Problem A market-maker sells option I for $10. This option s delta is and its gamma is The market maker proceeds to delta-gamma hedge this commitment by trading in the underlying and also in option II on the same stock. The latter option s price is $4.70, its delta is and its gamma is What is the market-maker s resulting position in option II? (a) Buy 0.5 of option II. (b) Write 0.5 of option II. (c) Buy 2 of option II. (d) Write 2 of option II. Problem Consider the following portfolio: 5 long options of type I, 4 long options of type II, 1 written option of type III. The prices of the three options are 0.75, 1.00, and 1.50, respectively, while the option elasticities are 10, 7, and 2, respectively. What is the elasticity of the above portoflio? (a) 5 (b) 7 (c) 10 (d) 12

5 Problem Assume the Black-Scholes framework. The currrent price of a non-dividendpaying stock is $40 per share. Consider a $42-strike, quarter-year put option on this stock whose current delta is The continuously compounded risk-free interest rate is What is the stock s volatility? (a) 0.60 (b) 0.64 (c) 0.68 (d)