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 - Solutions This problem set is aimed at making up the lost 75 minute class! 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 1. (2 pts) All else being equal, American put options are at least as valuable as European put options. TRUE 2. (2 points) In our usual notation, let S(0) = 40, r = 0.08, σ = 0.3, δ = 0. You need to construct a 2 period forward binomial tree for the above stock with every period in the tree of length h = 0.5. Then, u > FALSE u = exp{(0.08 0) } (2 points) In the usual notation for the binomial asset pricing model, we always have FALSE 1 < d < u. 4. (2 pts) In the setting of the Binomial Model, with i denoting the effective interest rate per period and assuming that the underlying asset pays no dividends: If then there is no possibility for arbitrage. FALSE d < u 1 + i 5. (2 pts) An American call option on a non-dividend paying stock should never be exercised early. More precisely, it is never more profitable to early exercise an American call option on a stock which pays no dividends prior to the expiry date of the option. TRUE 6. (2 pts) Suppose that the European options with the same maturity and the same underlying assets have the following prices: (1) a 50 strike call costs $9; (2) a 55 strike call costs $10;

2 2 Then, some of the monotonicity conditions for no-arbitrage are violated by the above premiums. TRUE We know that for strikes K 1 < K 2, the price of a call with strike K 1 should be greater than or equal to the price of a call with strike K 2 (see equation (9.13) in the book). This condition is violated. 7. (2 pts) Let V A (0, T ) denote the price at time 0 of an American option with expiration date T. Then, we always have TRUE V A (0, T ) V A (0, 2T ). 8. (2 pts) In the Cox-Ross-Rubinstein tree, we always have u = 1/d. TRUE Part II. FREE-RESPONSE PROBLEMS Please, explain carefully all your statements and assumptions. Numerical results or singleword answers without an explanation (even if they re correct) are worth 0 points. 1. (20 points) Consider a two-period forward binomial tree, where the length of each period is 6 months. Assume the stock price is $50.00, σ = 0.20, r = 0.06 and the dividend yield δ = What is the lowest strike price for which early exercise could occur after the first time step and with an American put option? With the given data, we get that approximately e σ h = 1.15 and e (r δ)h = Hence, u = 1.16 and d = S uu = ; S ud = S du = ; S dd = Let us denote the strike by K. The payoffs, if there is no early exercise, as a function of K are V uu = (K 62.28) +, V ud = (K 51.04) +, and V dd = (K 38.72) +. The risk-neutral probability is p = = So, the continuation values of the put after taking a single step back in the binomial tree are Vu cont = e 0.03 [0.46 (K 62.28) (K 51.04) + ] ; Vd cont = e 0.03 [0.46 (K 51.04) (K 38.72) + ]. On the other hand, the stock prices at these two nodes are S u = and S d =

3 3 So, the values of immediate/early exercise are V imm u = (K 58) + and V imm d = (K 44) +. At any of these two nodes, early exercise will happen if V imm node > V node cont. Hence, we must have that K > 44 otherwise, no early exercise should happen at either of the two nodes. Then, for K > 44, we have that V cont u = 0 and V cont d = (K 38.72). So, we need to figure out for which K the inequality K 44 > (K 38.72) K( ) > holds. We get K > (30 points) Consider a two-period binomial model with S(0) = $50, u = 2 and d = 0.5. (i) (5 points) Draw the binomial tree modeling the future evolution of this stock price with the given u and d. Your goal is to price an at-the-money European call option with two periods to maturity under the following assumptions: the underlying stock does not pay any dividends, the effective risk-free interest rate per period equals i = 25%. (ii) (5 pts) Find the risk-neutral probability. (iii) (8 pts) Find the fair price of the call using the risk-neutral pricing formula. (iv) (12 pts) Find the which should be used at every node in the tree in order to form a replicating portfolio. More precisely, in the notation used in class, calculate u, d and 0.

4 4 The binomial tree looks like this: The risk-neutral probability of the stock price going up in a single period is simply: p (1 + i) d = = 0.5. u d We proceed backwards through the inner nodes of the tree. At the up node, we have that the value of the call is V u = 1 1 [ ] = On the other hand, the remaining two final payoffs are both zero, which yields that the value of the call at the down node equals V d = 0. We have now reduced the pricing problem to a one-step binomial tree. The usual calculation gives us that the fair price of the above call is V C (0) = 1 1 [60 + 0] = With the usual notation, we have u = = 1; d = 0; 0 = = (20 points) Consider a one-period Binomial model with S(0) = $105, S u = $130 and S d = $80. Your goal is to determine if there is an arbitrage opportunity in a market in which a European call option on S with strike of K = $120 and exercise date T = 1

5 year is sold for $5. Assume that the continuously compounded risk-free interest rate equals r = 10%. If you believe that there is an arbitrage opportunity, describe the arbitrage portfolio and show that it is, indeed, an arbitrage portfolio. If you believe that there is no arbitrage opportunity, explain your reasoning. First, we need to figure out the no-arbitrage price of the given call option. Since no dividends are mentioned, we set δ = 0. The risk-neutral probability of the stock price going up is p = e(r δ)h d u d = e0.1 (80/105) (130 80)/105 = So, using the risk-neutral pricing formula, we get V C (0) = e ( ) = The observed price of the call is given to be C = 5. We conclude that there is an arbitrage opportunity since the no-arbitrage price (i.e., the value at time 0 of the replicating portfolio for the call) is different from the observed call premium. Moreover, since the observed premium is lower than the no-arbitrage price, we conclude that he observed call is relatively cheap when comapred to its replicating portfolio. To take advantage of this situation, we construct the following arbitrage portfolio: (1) long call, (2) short = Vu V d S u S d = 0.2 shares of stock, (3) invest B = V C (0) S(0) = = in the money market at the risk-free interest rate r. Note: Since B is a negative number, the interpretation is that we lend B = B to the money market; this can be interpreted as purchasing zero-coupon bonds in the required amout to be redeemed at time T. The initial cost of this portfolio is C S(0) + ( (V C (0) S(0))) = C V C (0) < 0. So, there is an initial inflow of money. At the expiration date, the call and the short replicating portfolio have payoffs which exactly cancel out (by construction!). Since there was an initial inflow of money, we indeed constructed an arbitrage portfolio. The arbitrage portfolio above is just an example of an arbitrage portfolio it is by no means unique! 4. (10 points) Let S(0) = 40, r = 0.08, σ = 0.3, δ = 0. You need to find the up and down factors in a 2 period forward binomial tree modeling the price of this stock during the following year. (a) (4 pts) What are u and d? (b) (6 pts) What is the risk-neutral probability of the stock price going up in a single period? 5

6 6 (a) (b) u = e (r δ)h+σ h = e = 1.29; d = e (r δ)h σ h = e = p = e σ h = Part III. MULTIPLE CHOICE QUESTIONS 1. (5 pts) Let S be a non-dividend paying stock with a current price equal to S 0. You know that in the one-period binomial tree for this stock, S u = 150 and S d = 120. An actuary calculates the volatility of S based on the provided values and gets σ Which of the models we covered in class did this actuary use to obtain the volatility? (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. (d) The ratio S u /S d = u/d is always e 2σ for all of the models we covered. In this case, u = 1.25 = d e2σ σ = 1 ln(1.25)