Derivative Markets Midterm Exam, Exams of Credit and Risk Management

Download Derivative Markets Midterm Exam and more Exams Credit and Risk Management in PDF only on Docsity! UCLA Anderson School of Management Daniel Andrei, Derivative Markets MGMTMFE 406, Winter 2018 MFE – Midterm February 2018 Date: Your Name: Your Signature:1 • This exam is open book, open notes. You can use a calculator or a computer, but be sure to show or explain your work. • You are not allowed to communicate with anyone (verbally, in writing, or elec- tronically), except for me, during the exam period. • You may present calculations in non-reduced form (e.g., as “e0.095 − 1”). • If you are stuck on something, make an assumption, tell me what it is, and do the best you can. I give partial credit if you provide enough correct information. TIME LIMIT: 2 hours TOTAL POINTS: 100 1As a member of the UCLA Anderson academic community, the highest standards of academic behavior are expected of you. It is your responsibility to make yourself aware of these standards (specifically regarding plagiarism, individual work, and team work) and adhere to them. By signing the exam: (i) you certify your presence, and (ii) you state that you neither gave nor received help during the examination. 1 (24 points) Answer the following questions. a. (4 points) The effective annual interest rate is 8%. What is the equivalent continuously compounded interest rate? Interest rate b. (4 points) A stock has an annual volatility of 24%. What is the 1-month volatil- ity? 1-month volatility 2 1 Solution a. ln(1.08) = 0.077. b. The 1-month volatility is 0.24 × √ 1/12 = 0.0693. c. The number of nodes is 20 + 21 + 22 + ...2N = 2N+1 − 1 (1) d. You want to be short a call if you expect the call to fall in price (e.g., the underlying is expected to fall or the volatility is expected to fall). e. The required rate of return on the stock is the riskless interest rate. Thus, the option finishes out-of-the money: 100 × e0.05 = 105.127 > 104, (2) and is therefore worthless. f. The number of nodes is 1 + 2 + ...+N + 1 = (N + 1)(N + 2) 2 (3) 5 2 Risk management (16 points) Goldfield is a gold-mining firm planning to sell 10,000 ounces of gold precisely 1 year from today. Goldfield hopes that the gold price will rise over the next year. Gold insurance (i.e., a put option) provides a way to have higher profit at high gold prices while still being protected against low prices. That is, the put option provides a floor on the price. a. (4 points) Suppose that the market price for a 400-strike put option is $6.775/oz. Goldfield decides to buy a put option for every ounce of gold it plans to sell. Draw the payoff diagram for this hedged position (i.e., the payoff resulting from selling one ounce of gold and exercising one 400-strike put option). 250 300 350 400 450 500 550 Gold Price in 1 year ($)250 300 350 400 450 500 550 Payoff ($) 6 b. (4 points) A disadvantage to buying a put option is that Goldfield pays the premium even when the gold price is high and insurance was, after the fact, unnecessary. One way to avoid this problem is a paylater strategy, where the premium is paid only when the insurance is needed. Consider the following strategy for Goldfield : Buy two 400-strike puts and sell a 450-strike put. The market price for a 450-strike put option is $13.55/oz. What is the net option premium of this strategy? Net option premium c. (4 points) Draw the payoff diagram for this option position (i.e., buy two 400-strike puts and sell a 450-strike put). 300 350 400 450 500 550 Gold Price in 1 year ($) -100 -50 0 50 100 150 200 Payoff ($) 7 250 300 350 400 450 500 550 Gold Price in 1 year ($)250 300 350 400 450 500 550 Payoff ($) When the price of gold is greater than $450, neither put is exercised, and Gold- field ’s payoff is the same as if it were unhedged. When the price of gold is between $400 and $450, because of the written 450-strike put, the firm loses $2 of payoff of every $1 decline in the price of gold. Below $400, the purchased 400-strike puts are exercised, and the payoff becomes constant. The net result is an insurance policy that is not paid for unless it is needed (in contrast with point a, where the put is paid even when insurance is unnecessary). However, when the gold price is below $450, the paylater hedging strategy does worse because it offers less insurance. 10 3 Asymmetric Butterfly Spread (16 points) Below is a payoff diagram for a position. All options have 1 year to maturity and the stock price today is $40. The underlying asset (the stock) is not paying any dividends. 30 40 50 60 Stock Price ($) -10 0 10 20 30 Payoff ($) a. (4 points) Can the portfolio corresponding to the above payoff have zero or negative initial premium? Justify your answer. b. (4 points) The “kinks” of the payoff diagram are located at $35, $39, and $49. Consider 3 options: A call option with strike $35, a call option with strike $39, and a call option with strike $49. Using these options, find the quantities that construct the diagram. Asset Call(35) Call(39) Call(49) Position 11 c. (4 points) Consider 3 options: A call option with strike $35, a put option with strike $39, and a call option with strike $49. Using these options and the stock, find the quantities that construct the diagram. (Hint: Start from the more familiar situation of point b; then use the put-call parity.) Asset Stock Call(35) Put(39) Call(49) Position d. (4 points) What should be the position in the risk-free asset in order to get exactly the payoff above? (assume an interest rate of 0%) Position in the risk-free asset 12 For the rest of this exercise, assume that the annual continuously compounded interest rate is r = 15%, the annual dividend yield on the stock is δ = 10%, and the annual volatility of returns is 50%. b. (4 points) The price of an European log call option today is $0.348987. Draw the profit diagram of the log call option. What is the break-even point (i.e., the point where the profit of the position is zero)? Clearly indicate this point on the diagram. 20 40 60 80 Stock Price -0.4 -0.2 0.2 0.4 0.6 0.8 1.0 Profit ($) 15 The price of the stock today is S0 = $30. Consider a two-period binomial tree (i.e., there are 6 months between binomial nodes). c. (4 points) Find u and d. u and d: d. (4 points) What is the risk-neutral probability of an up move? Risk-neutral probability: 16 e. (8 points) At each node in the tree, fill in the prices for the American and European versions of this option (when you exercise the American version at time t, you receive max [ln(St/20), 0], as at expiration). Put an asterisk at each intermediary node where the American option is exercised. S0 = 30 Eur = Amer = Su 1 = Eur = Amer = Sd 1 = Eur = Amer = Suu 2 = Eur = Amer = Sud 2 = Eur = Amer = Sdd 2 = Eur = Amer = f. (4 points) Comparing the price of the European log call option that you found on the tree above with the market price of the option from point b, is there an arbitrage opportunity? If yes, does it involve buying or selling the market option? 17 S0 = 30 Eur = 0.3603 Amer = 0.4055∗ Su 1 = 43.8051 Eur = 0.6932 Amer = 0.7840∗ Sd 1 = 21.5989 Eur = 0.1743 Amer = 0.1743 Suu 2 = 63.9630 Eur = 1.1626 Amer = 1.1626 Sud 2 = 31.5381 Eur = 0.4555 Amer = 0.4555 Sdd 2 = 15.5505 Eur = 0 Amer = 0 f. Yes, there is an arbitrage opportunity. The option is undervalued by the market ($0.348987< $0.3603), so an arbitrage strategy would involve buying the option. 20 5 Arbitrage (8 points) The premium difference between otherwise identical Euro- pean calls with different strike prices cannot be greater than the difference in prices. That is, if K1 < K2, then C(K1) − C(K2) ≤ K2 −K1 (14) Consider two European call options with the same maturity. The first option has a strike of $60 and a premium of $17. The second option has a strike of $65 and a premium of $11. Is there an arbitrage opportunity? If yes, describe the arbitrage strategy. 5 Solution Yes, the inequality (14) is violated: the change in the option premium ($6) exceeds the change in the strike price ($5). The arbitrage strategy involves selling the low strike call and buying the high-strike call (a call bear spread), coupled with lending the proceeds at the risk-free rate: Transaction Time 0 ST < 60 60 ≤ ST ≤ 65 65 < ST Sell 60-strike call 17 0 60 − ST 60 − ST Buy 65-strick call -11 0 0 ST − 65 Lend $6 -6 FV(6) FV(6) FV(6) TOTAL 0 FV (6) > 0 60 + FV (6) − ST > 0 FV (6) − 5 > 0 where FV (6) means “the future value of $6.” 21 6 Physical vs. risk-neutral probabilities (8 points) Consider a non-dividend paying stock whose price today is S0 and in h years from now it can go up to Su or down to Sd. The physical probability of the stock going up is denoted by p. The risk-free rate is r and the expected return of the stock is α > r (both are continuously compounded). The risk-neutral probability of the stock going up is denoted by p∗. Find A in the following relationship between p∗ and p: p∗ = p− A (15) Interpret this relationship. 6 Solution We know that S0 = e−αh[pSu + (1 − p)Sd] (16) from which we obtain p = eαhS0 − Sd Su − Sd (17) Similarly p∗ = erhS0 − Sd Su − Sd (18) Therefore, we can write p∗ = p− S0 ( eαh − erh ) Su − Sd (19) This relationship shows that the actual probability, p, and the risk-neutral prob- ability, p∗, differ by a term that is proportional to the dollar risk premium on the stock, S0 ( eαh − erh ) . The actual probability, p, is reduced by the dollar risk premium per dollar of risk, Su − Sd. Thus, the real and risk-neutral probabilities differ by an amount that is determined by investors’ attitude toward risk. 22