ACT4000, MIDTERM #1 ADVANCED ACTUARIAL TOPICS FEBRUARY 9, 2009 HAL W. PEDERSEN

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1 ACT4000, MIDTERM #1 ADVANCED ACTUARIAL TOPICS FEBRUARY 9, 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 are considering investments in a single-period binomial market. (As you know, this means that we are currently at time 0 and at time 1 the world will be in one of two states which we will call the upstate and the downstate.) There are two assets available for trade. The first asset currently sells for 10 and at time 1 will be worth 12 in the upstate and 8 in the downstate. The second asset currently sells for 10 and at time 1 will be worth 15 in the upstate and 2 in the downstate. (i) (3 points) What is the current price of an asset that at time 1 pays 1 in the upstate and 0 in the downstate? (ii) (2 points) What is the current price of an asset that at time 1 pays 0 in the upstate and 1 in the downstate? (iii) (5 points) If you do not want any risk, is it possible for you to deposit 100 at time 0 and receive a certain payoff at time 1? Explain how this can be done or why it cannot be done. If this can be done, what is the effective interest-rate you will earn over the period? Question 2. An equity securities market model follows a multi-period binomial model. At each node of the binomial tree the current stock price S will branch to us in the upstate and ds in the downstate. You are given that the initial stock price is 10, u =1.35, d =0.85 and the interest-rate is 5% effective per period. The stock does not pay dividends. (i) (4 points) Compute the price of a European put option on the stock which expires in 4 periods and has a strike price of 9.0. (ii) (6 points) Compute the price of an American put option on the stock which expires in 4 periods and has a strike price of 9.0 and describe the optimal exercise policy for this American put option. 1

2 2 ACT4000 MIDTERM #1 Question 3. The insurance company you work for has recently began issuing a stock index GIC. The essence of the contract is that the investor places an amount of principal in an account for two years and the investor is guaranteed some minimum effective return over the two-year period. The investor s returns are based on the returns on the TSE 35 index. As pricing actuary, you are told that the product is to guarantee a 0% return and you are to set the maximum total return the investor will receive over the two years so that the insurance company will break even. The continuously compounded risk-free interest rate is 4% and the stock index is currently at 50 and will go to either 25 or 75 at the end of the two years. (i) (5 points) Compute the maximum total return the investor will receive over the two years so that the insurance company will break even. If this break even return does not exist then explain why. (ii) (2 points) What are the embedded options in this contract? (iii) (3 points) Write a general algebraic expression for the cost to the insurance company per $1 invested of providing a maximum return of R, foreachr>0? Question 4. Assume the Black-Scholes option pricing model. Consider a standard European call option on a stock. The strike price of the option is equal to the current price of the stock (i.e. the option is at-the-money). The option has one year to maturity and the stock does not pay dividends. Is the option s delta greater than 0.5, less than 0.5, or equal to 0.5. Justify your answer. Question 5 (Text Question 12.7, page 407). You are given the following data. S = $100 K = $95 σ = 30% r =0.08 δ =0.03 T =0.75 Compute the Black-Scholes price of a call.

3 ACT4000 MIDTERM #1 3 Question 6 (Text Question 13.2, page 439). You are given the following data assuming a Black-Scholes model. S = $40 σ = 30% r =0.08 δ =0 Suppose you sell a 40-strike put with 91 days to expiration. (i) (5 points) What is delta? (ii) (5 points) If the option is on 100 shares, what investment is required for a deltahedged portfolio? Question 7 (Text Question 10.1, page 338). You are given the following data. S = $100 K = 105 r = 8% (continuously compounded) T =0.5 δ =0 You are given u =1.3 andd =0.8. For a single-period binomial model compute the premium, and B for a European call option. Question 8. For a two-period binomial model, you are given the following data. Each period is one year. The current price for a non-dividend paying stock is $20. u =1.2840, where u is one plus the rate of capital gain on the stock per period if the stock price goes up. d =0.8607, where d is one plus the rate of capital loss on the stock per period if the stock price goes down. The continuously compounded risk-free interest rate is 5%. Calculate the price of an American call option on the stock with a strike price of $22.

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9 u p* American European d Put Price Put Price cont. int_rate n 4 delta K 9.00 American European h Call Price Call Price Stock Prices 10 r=ln(1.05) (i) Answer = (ii) Answer = Exercise at period 3 if stock has fallen three times (Exercise when intrinsic value is greater than value of future cash flows if not exercised.) S_ Put Intrinsic Value European Put Payoff American Put Prices European Put Prices Price Price Call Intrinsic Value European Call Payoff American Call Prices European Call Prices Price Price

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17 Ddt6;; f'j(rl,) J,. t-( s/1<-)!'-j L A,) = A 7'". \ I 7 z: Without dividends, the standard Black and Scholes (1973) pricing formula for the European call opt.ion is given by c( t) S(t)N(dd - e-r(t-t) J( N(d2), where In ( ) + (r + 4a2)(T - t) a.jy=t T - t d1 - avt - t. ' and The option's "delta" is given by g~~~~ = N(d1). With the option struck at-the-money, S(t) = K, and thus, In e~») = 0 [remember that In(l) = 0]. All other terms in d1 are positive. Therefore, d1 > 0, and N(dd > 0.5 (remember that N(O) = 0.5 and N( ) is an increasing function of its argument). Thus, an at-the-money option on a non-dividend-paying stock always has a delta slightly greater than one-half.

18 Black-Scholes Option Pricing Model S_0 Stock Price Summary K Strike Price Call Price sigma Volatility Call Delta r Interest Rate Call Gamma T Time to Expiration (Years) Put Price delta Dividend Yield Put Delta Put Gamma d_ d_ C(S,K,sigma,r,T,delta) = S_0*exp(-delta*T)*N(d_1)-K*exp(-r*T)*N(d_2) N(d_1) N(d_2) P(S,K,sigma,r,T,delta) = K*exp(-r*T)*N(-d_2)-S_0*exp(-delta*T)*N(-d_1) Answer: Call Price Call Price = Put Price

19 Black-Scholes Option Pricing Model S_0 Stock Price Summary K Strike Price Call Price sigma Volatility Call Delta r Interest Rate Call Gamma T Time to Expiration (Years) Put Price delta Dividend Yield Put Delta Put Gamma d_ d_ C(S,K,sigma,r,T,delta) = S_0*exp(-delta*T)*N(d_1)-K*exp(-r*T)*N(d_2) N(d_1) N(d_2) P(S,K,sigma,r,T,delta) = K*exp(-r*T)*N(-d_2)-S_0*exp(-delta*T)*N(-d_1) Answer: Call Price (i) Put Delta = Put Price (ii) Short 100*Delta = shares & deposit short sale proceeds plus put option premiums in the bank.

20 One Period Interest Rate Cash Flow Matrix Traded Asset -- Bank Account Cash Flow Matrix Inverse Time 0 1 Replicating Portfolio (Bank Account & Stock) Traded Asset -- Stock Price of Cash Flows Time 0 1 Answer: Cash Flows to Price Delta = 0.5, B= , Price= Time 0 1 Note: One Period Interest Rate = Exp(0.08*0.5)-1

21 u p* American European d Put Price Put Price int_rate n 2 delta K American European h Call Price Call Price Stock Prices Put Intrinsic Value 10 Answer: American Call Price = S_ European Put Payoff American Put Prices European Put Prices Price Price Call Intrinsic Value European Call Payoff American Call Prices European Call Prices Price Price

22 Solution to (4) Answer: (C) First, we construct the two-period binomial tree for the stock price. Year 0 Year 1 Year The calculations for the stock prices at various nodes are as follows: S u = = S d = = S uu = = S ud = S du = = S dd = = The risk-neutral probability for the stock price to go up is rh 0.05 e d e p* = = = u d Thus, the risk-neutral probability for the stock price to go down is If the option is exercised at time 2, the value of the call would be C uu = ( ) + = C ud = ( ) + = C dd = ( ) + = 0 If the option is European, then C u = e 0.05 [0.4502C uu C ud ] = and C d = e 0.05 [0.4502C ud C dd ] = But since the option is American, we should compare C u and C d with the value of the option if it is exercised at time 1, which is 3.68 and 0, respectively. Since 3.68 < and 0 < , it is not optimal to exercise the option at time 1 whether the stock is in the up or down state. Thus the value of the option at time 1 is either or Finally, the value of the call is C = e 0.05 [0.4502(4.7530) (0.0440)] =

23 Remark: Since the stock pays no dividends, the price of an American call is the same as that of a European call. See pages of McDonald (2006). The European option price can be calculated using the binomial probability formula. See formula (11.17) on page 358 and formula (19.1) on page 618 of McDonald (2006). The option price is e r(2h) 2 [ p * Cuu + p *(1 p*) Cud + (1 p*) 2 Cdd ] = e 0.1 [(0.4502) ] =