Notes: This is a closed book and closed notes exam. The maximal score on this exam is 100 points. Time: 75 minutes

Transcription

1 M375T/M396C Introduction to Financial Mathematics for Actuarial Applications Spring 2013 University of Texas at Austin Sample In-Term Exam II Post-test Instructor: Milica Čudina Notes: This is a closed book and closed notes exam. The maximal score on this exam is 100 points. Time: 75 minutes Part I. TRUE/FALSE QUESTIONS Please note your answers on the front page. 1. (2 points) A chooser option with the choice date coinciding with the exercise date is equivalent to a straddle with the same strike. Solution: TRUE 2. (2 points) Prices of otherwise identical call options on non-dividend-paying stocks are decreasing as a function of time to exercise. Solution: FALSE Part II. MULTIPLE CHOICE QUESTIONS Please note your answers on the front page. 1. Let K 1 = 50, K 2 = 60 and K 3 = 65 be the strikes of three European call options on the same underlying asset and with the same exipration date. Let V C (K i ) denote the price at time 0 of the option with strike K i for i = 1, 2, 3. We are given that V C (K 1 ) = 12 and V C (K 3 ) = 5. What is the maximum possible value of V C (K 2 ) which still does not violate the convexity property of call option prices? (a) About $16/3 (b) About $7 (c) About $22/3 (d) About $8 Solution: (c) From the given parameters, we see that K 2 = λk 1 + (1 λ)k 3 for λ = 1/3. So, we must have λv C (K 1 ) + (1 λ)v C (K 3 ) V C (K 2 ). The extreme case, i.e., the equality is obtained above for V C (K 2 ) = (1/3) 12 + (2/3) 5 = 22/3. 2. (5 pts) Which of the following statements is incorrect? (a) The payoff of a call bull spread is always nonnegative. (b) The payoff of a butterfly spread is never negative.

2 2 (c) The payoff of a strangle is never negative. (d) The payoff of a call bear spread is always nonnegative. Solution: (d) We drew the payoff of the call bull spread in class repeatedly. It is always nonnegative. The call bear spread is the short call bull spread. So, its payoff is always nonpositive and is negative for the final stock price below the higher of the two strike prices. 3. (5 pts) Which of the following models always satisfies the no-arbitrage condition for the construction of the binomial stock-price tree, regardless of the choice of parameters r, δ, h, σ, T, n, S(0)? (a) The forward binomial tree. (b) The Cox-Ross-Rubinstein model. (c) The lognormal tree. (d) Any of the three models. (e) None of the three models. Solution: (a) It was discussed in class that the forward binomial tree always satisfies the no-arbitrage condition. So, the only two acceptable choices above are (a) and (d). To discard the Cox-Ross-Rubinstein (CRR) model as always being arbitrage free, consider the situation in which σ = 0.1, r = 0.3, and δ = 0. In the graph below, the red line in the is the value of the u in the CRR model while the blue line stands for the e (r δ)h both as functions of the length of a single period h As you can see, if we choose too small a value for h, the up factor u falls below e rh so that the no-arbitrage condition is violated. This was discussed (theoretically) in class as well as in the textbook. Now you have an actual choice of parameters which can backfire in a manner of speaking. This means that the offered choice (d) is no longer acceptable. 4. (5 points) Consider the one-period binomial option pricing model. Let V C (0) > 0 denote the price of a European call on a stock which pays continuous dividends. What is the impact on the value of European call option prices if the company decides to increase the dividend yield paid to the shareholders?

3 (a) The call option price will drop. (b) The call option price will increase. (c) The call option price will always remain constant. (d) The impact on the price of the call cannot be determined using the binomial option pricing model. (e) There is not enough information provided. Solution: (a) Let δ < δ be the two dividend yields. Then, the risk-neutral price of the European call on the stock with the dividend yield δ equals V C (0) = e rt [p (S u K) + + (1 p )(S d K) + ] with p = (e (r δ)h d)/(u d). On the other hand, the risk-neutral price of the European call on the stock with the dividend yield δ equals Ṽ C (0) = e rt [ p (S u K) + + (1 p )(S d K) + ] with p = (e (r δ)h d)/(u d). We have δ < δ e (r δ)h > e (r δ)h p > p V C (0) > ṼC(0). 5. (5 points) An investor acquires a call bull spread consisting of the call with strike K 1 = 100 and K 2 = 110 and with expiration in one year. The initial price of the 100 strike call option equals $11.34, while the price of the 110 strike option equals $7.74. At expiration, it turns out that the stock price equals $105. Given a continuously compouned annual interest rate of 5.0%, what is the profit to the investor? (a) $3.78 loss (b) $1.22 loss (c) $1.22 gain (d) $5 gain Solution: (c) The total initial cost of establishing the investor s position is = The future value of this amount at expiration is 3.60e 0.05 = The payoff at expiration is (S(T ) 100) + (S(T ) 110) + = ( ) + ( ) + = 5. So, the profit is = (5 points) Assume that the continuously compounded interest rate equals Stock S has the current price of S(0) = 100 and pays continuous dividends at the rate δ S. Stock Q has the current price of Q(0) = 100 and it pays continuous dividends at the rate of An exchange option gives its holder the right to give up one share of stock Q for a share of stock S in exactly one year. The price of this option is $ Another exchange option gives its holder the right to give up one share of stock S for a share of stock Q in exactly one year. The price of this option is $

4 4 Find δ S. (a) 0 (b) 0.02 (c) 0.05 (d) 0.07 Solution: (c) By the generalized put-call parity, So, V EC (Q(0), S(0), 0) + F P 0,T (S) = V EC (S(0), Q(0), 0) + F P 0,T (Q) e δ S = e We get ( ) e 0.02 δ S = ln = (5 points) Which of the following American-type options will never be exercised early to get strictly higher profit? (a) Put on a dividend-paying stock (b) Call on a dividend-paying stock (c) Call on a non-dividend-paying stock (d) Put on a non-dividend-paying stock (e) All of the above should be exercised early in a certain scenario Solution: (b) 8. (5 pts) For a two-period binomial model, you are given that: (1) each period is one year; (2) the current price of a non-dividend paying stock S is S 0 = $20; (3) u = 1.3, with u as in the standard notation for the binomial model; (4) d = 0.9, with d as in the standard notation for the binomial model; (5) the continuously compounded risk-free interest rate is r = Find the price of an American call option on the stock S with T = 2 and the strike price K = $22. (a) $1 (b) $2 (c) $3 (d) $4 (e) None of the above Solution: (b) Since the stock does not pay dividends, we can price the option as if it were European, i.e., without taking into account the possibility of early exercise.

5 5 The risk-neutral probability is p = When one constructs the two-period binomial tree, one gets S u = 26, S d = 17, S uu = 33, S ud = S dd = 22, S dd = 15. Hence, the payoffs at the end of the second period are V uu = 11, V ud = V dd = 0. So, taking the expected value at time 0 of the payoff with respect to the risk-neutral probability, we get that the price of the call should be e 0.05 [11 (p ) 2 ] 2. Part II. 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. 1. (8 points) Assume that one of the no-arbitrage conditions in the binomial model for pricing options on a non-dividend paying stock S is violated. Namely, let e r h d < u. Illustrate that the above inequalities indeed violate the no-arbitrage requirement. In other words, construct an arbitrage portfolio and show that your proposed arbitrage portfolio is, indeed, and arbitrage portfolio. Solution: There are multiple ways to illustrate arbitrage opportunities in the above set-up. We provide just one simple example. Let today s stock-price be denoted by S(0). We simply borrow S(0) from the money market and buy one share of stock. After one period, according to the binomial model, the stock-price either rises to S u = us(0) or drops to S d = ds(0). Let us denote the value of our portfolio on the second day by X u in the case the stock price went up and by X d if the stock price went down. The values of our portfolio in those two cases are X u = e rh S(0) + us(0) > 0 X d = e rh S(0) + ds(0) 0 We have non-negative payoffs in both cases and a strictly positive payoff in one of the cases. Hence, the above strategy constitutes arbitrage. 2. (13 points) (i) (5 pts) Calculate the price of a long butterfly spread using the following call options: (1) a 3, 925 strike call on the FTSE100 index which is being sold for ; (2) a 4, 325 strike call on the FTSE100 index which is being sold for ; (3) a 4, 725 strike call on the FTSE100 index which is being sold for

6 6 (ii) (8 pts) Assume that the index pays no dividends. Use the put-call parity to derive the price of the corresponding butterfly spread in terms of the prices of put options analogous to the call options listed above. Solution: (i) The only thing one needs to remember here is how to construct a butterfly spread using calls. The butterfly spread above is symmetric. Its total cost can be evaluated as = (ii) First, we recall the general form of the put-call parity: V C (t, K) V P (t, K) = F P t,t (S) P V t,t (K), where V C (t, K) denotes the time t value of a European call with strike K and V P (t, K) denotes the time t value of a European put with strike K. The price of the butterfly spread can be rewritten as V C (t, 3925) 2 V C (t, 4325) + V C (t, 4725) = V P (t, 3925) + F P t,t (S) P V t,t [3925] 2 (V P (t, 4325) + F P t,t (S) P V t,t [4325]) + V P (t, 4725) + F P t,t (S) P V t,t [4725] = V P (t, 3925) 2 V P (t, 4325) + V P (t, 4725) P V t,t [ ] = V P (t, 3925) 2 V P (t, 4325) + V P (t, 4725) P V t,t [0]. As one would expect, the answer must be the same as in part (i), i.e., Part IV. PROBLEMS WITH TABLES 1. (18 points) Let S(0) = $100, K = $95, r = 8%, T = 1 and δ = 0. Assuming that u = 1.3 and d = 0.8, construct a two-period binomial tree for a call option. Provide all the entries in the following table: 1 UP 1 DOWN 0 Solution: Note: See Problem from the textbook!

7 7 1 UP DOWN (18 points) Let S(0) = $80, K = $95, r = 8%, T = 1 and δ = 0. Assuming that u = 1.3 and d = 0.8, construct a two-period binomial tree for a call option. Provide all the entries in the following table: 1 UP 1 DOWN 0 Solution: Note: See Problem from the textbook! 1 UP DOWN