ACT4000, MIDTERM #2 ADVANCED ACTUARIAL TOPICS MARCH 16, 2009 HAL W. PEDERSEN You have 70 minutes to complete this exam. When the invigilator instructs you to stop writing you must do so immediately. If you do not abide by this instruction you will be penalised. All invigilators have full authority to disqualify your paper if, in their judgement, you are found to have violated the code of academic honesty. Each question is worth 10 points. Provide sufficient reasoning to back up your answer but do not write more than necessary. This exam consists of 8 questions. Answer each question on a separate page of the exam book. Write your name and student number on each exam book that you use to answer the questions. Good luck! Question 1. You have written a European call option on a share of stock which expires in six months and has a strike price of $55.00. You are given the following information. • The current price of the stock is S0 = $50.00. • The stock pays no dividends. • The continuous time interest rate r =0.08. • The volatility on the stock is σ =0.25. • The price of the 55-call is $2.37535. • The delta of the 55-strike call you have written is ∆ = 0.41119. • The gamma of the 55-strike call you have written is Γ = 0.04401. You have decided to delta-gamma hedge your position using a European put option on a share of stock which expires in six months and has a strike price of $60.00. You are also given the following information. • The price of the 60-put is $8.77541. • The delta of the 60-strike put is ∆ = −0.76322. • The gamma of the 60-strike put is Γ = 0.03491. • The price of a 55-call with remaining maturity of six months less a day when the underlying stock is trading at $51.00 is $2.79428. • The price of a 60-put with remaining maturity of six months less a day when the underlying stock is trading at $51.00 is $8.03167. (a) [4 points] Compute the position you must take in 60-strike puts to effect the delta-gamma hedge. (b) [6 points] One day after you effect the delta-gamma hedge the stock price has increased by $1.00. What is your profit or less on the hedged portfolio. 1 2 ACT4000 – MIDTERM #2 Question 2. Assume that the Black-Scholes framework holds. The price of a stock that pays no dividends is $30.00. The price of a put option on this stock is $4.00. You are given: • ∆=−0.28, and • Γ=0.20. Using the delta-gamma approximation, determine the price of the put option if the stock price changes to $31.50. Question 3. Consider a model with two stocks. Each stock pays dividends continu- ously at a rate proportional to its price. Sj(t) denotes the price of one share of stock j at time t. Consider a claim maturing at time 3. The payoff of the claim is: Maximum S1(3),S2(3) . You are given the following information • S1(0) = 100 • S2(0) = 200 • Stock 1 pays dividends of amount (0.05)S1(t) dt between time t and time t+dt. • Stock 2 pays dividends of amount (0.10)S2(t) dt between time t and time t+dt. • The price of a European option to exchange Stock 2 for Stock 1 at time 3 is $10.00. Calculate the price of the claim. Question 4. Consider a “forward start option” which, 1 year from today, will give its owner a 1-year European call option with a strike price equal to the stock price at that time. You are given: • The European call option is on a stock that pays no dividends. • The stock’s volatility is 30%. • The forward price for delivery of 1 share of the stock 1 year from today is $100. • The continuously compounded risk-free interest rate is 8%. Under the Black-Scholes framework, determine the price today of the forward start option. ACT4000 – MIDTERM #2 3 Question 5. Let S(t) denote the price at time t of a stock that pays dividends continuously at a rate proportional to its price. Consider a European gap option with expiration date T for T>0. If the stock price at time T is greater than $100, the payoff is S(T ) − 90 otherwise the payoff is 0. You are given: (i) S(0) = $80 (ii) The price of a European call option with expiration date T and strike price $100 is $4. (iii) The delta of the call option in (ii) is 0.2. Calculate the price of the gap option. Question 6. Explain the notion of a compound call option. Be sure to clearly identify what the underlying is and to indicate what the payoff of the compound option is at expiration. Question 7. The stochastic process X follows the SDE dXt =0.14 dt + σdWt Xt where W is a standard Brownian motion. −t 2 Consider the new process Y defined by Yt = f(t, Xt)wheref(t, x)=e x . Apply Itˆo’s lemma to write an expression for dYt. Question 8. An asset price follows the diffusion process defined by St =80· exp 0.08 t +0.25 Wt where W is a standard Brownian motion. The risk-free interest rate is r =0.05. A second asset, denoted X, is available for trade with asset price dynamics dXt = µdt+0.20 dWt Xt and X0 = 20. Assuming that the market is arbitrage-free, compute µ. We- Jl.,L~- jdf"~lI dt.)t.d d"J )d .....r--4 0+ to -t~.... J d-/;" "•..J ~II> ~Ll'" /;Jr,l{llA. 6, ;:. #- ~lJrc...J ~t S.foc-/z. e1...~.If bO-~&.f.) e 1- ~ ~ = 00• Lf/l '1 e7, (I) + ()'l- (- .1~32.1-) , otf'l.e.0 J e'L (.o~LfqJ) [" . At •. 1: J 1 ) (p) P"1 () ~ ss - c~II 1 . 3'7 S 3.5 - I 2-. 1Cf Lf L 8 S-l <> c, Jt- .so S I '.31$,-/ ~ - f~ 8.1'1$'11 t 8'.o~ib7 1. 2b07 ~l\" It A t: c.f -17.31, Ot\~ Vl\i f~L - I (l,"7 9 'f 2 8 - 2.. S 75' .s 5") + J. 37.$1(51->0) + I. If,07(f.o'}/k7-8.?75'11) - 7'7. ), ( e. .,,813'S _ I) - . ()OOI2 . Question 2 The delta-gamma approximation is merely the Taylor series approximation with up to the quadratic term. In terms of the Greek symbols, the first derivative is Δ, and the second derivative is Γ. The approximation formula is 1 2 P(S + ε) ≈ P(S) + ε Δ + ε Γ. (13.2 & 13.5) 2 With P(30) = 4, Δ = −0.28, Γ = 0.10, and ε = 1.50, we have P(31.5) ≈ 4 + (1.5)(−0.28) + 1 (1.5)2(0.1) 2 = 3.6925 ≈ 3.70. Question 3 Because of the identity Maximum( S1(3), S2(3) ) = Maximum( S1(3) – S2(3), 0) + S2(3), the payoff of the claim can be decomposed as the sum of the payoff of the exchange option in statement (v) of the problem and the price of stock 2 at time 3. In a no- arbitrage model, the price of the claim must be equal to the sum of the exchange option price (which is 10) and the prepaid forward price for delivery of stock 2 at −δ2 ×3 time 3 (which is e ×S2(0)). So, the answer is 10 + e−0.1×3×200 = 158.16. Remark: If one buys e−δ2 ×3 share of stock 2 at time 0 and re-invests all dividends, one will have exactly one share of stock 2 at time 3. Question 4 This problem is based on Exercise 14.21 on page 465 of McDonald (2006). Let S1 denote the stock price at the end of one year. Apply the Black-Scholes formula to calculate the price of the at-the-money call one year from today, conditioning on S1. 2 2 d1 = [ln (S1/S1) + (r + σ /2)T]/(σ T ) = (r + σ /2)/σ = 0.417, which turns out to be independent of S1. d2 = d1 − σ T = d1 − σ = 0.117 The value of the forward start option at time 1 is −r C(S1) = S1N(d1) − S1e N(d2) −0.08 = S1[N(0.417) − e N(0.117)] −0.08 ≈ S1[N(0.42) − e N(0.12)] -0.08 = S1[0.6628 − e ×0.5438] = 0.157S1. (Note that, when viewed from time 0, S1 is a random variable.) Thus, the time-0 price of the forward start option must be 0.157 multiplied by the time-0 price of a security that gives S1 as payoff at time 1, i.e., multiplied by the prepaid forward P price F 1,0 (S) . Hence, the time-0 price of the forward start option is P −0.08 −0.08 0.157× F 1,0 (S) = 0.157×e × F 1,0 (S) = 0.157×e ×100 = 14.5 Remark: A key to pricing the forward start option is that d1 and d2 turn out to be independent of the stock price. This is the case if the strike price of the call option will be set as a fixed percentage of the stock price at the issue date of the call option. Question 5 In terms of the notation in Section 14.15, K1 = 90 and K2 = 100.