ActuarialBrew.com. Exam MFE / 3F. Actuarial Models Financial Economics Segment. Solutions 2014, 1 st edition

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1 ActuarialBrew.com Exam MFE / 3F Actuarial Models Financial Economics Segment Solutions 04, st edition Brewing Better Actuarial Exam Preparation Materials ActuarialBrew.com 04

2 Please visit our website, for a description of our exam preparation materials. We also offer: Live Seminars Online Seminar Study Manual Questions Set of 4 Practice Exams Flashcards Our products can be ordered from Actex or the Actuarial Bookstore. Links to order our products can be found on our website, Please exammfesupport@actuarialbrew.com with any content-related questions. For more general questions about any of our products, please info@actuarialbrew.com.

3 Exam MFE/3F Solutions Overview Solutions to the Exam MFE/3F Questions Solutions to Questions Overview The Solutions to ActuarialBrew.com s MFE/3F Questions provide full solutions to the approximately 650 exam-style practice questions contained in the Questions. If you have access to the online normal distribution calculator or a spreadsheet when working the Questions, the SOA advises students to use 5 decimal places for both the inputs and the outputs of the normal distribution calculator. If you don t have access to either one, we provide the old printed Normal Distribution Table at the end of the Questions for your convenience. Please note that your results from using the printed table may be different due to rounding. Practice Questions Each solution has the following key to indicate the question s degree of difficulty. The more boxes that are filled in, the more difficult the question: Easy: Very Difficult: Table of Contents The MFE/3F Questions cover the entire required course of reading for Exam MFE/3F: Chapter 0: Put-Call Parity and Replication Chapter 0: Comparing Options Chapter 03: Binomial Trees: Part I Chapter 04: Binomial Trees: Part II Chapter 05: Lognormally Distributed Prices Chapter 06: Histograms and Normal Probability Plots Chapter 07: The Black-Scholes Formula Chapter 08: The Greeks and Other Measures Chapter 09: Delta-Hedging Chapter 0: Exotic Options: Part I Chapter : Exotic Options: Part II Chapter : Monte Carlo Simulation Chapter 3: Volatility Chapter 4: Brownian Motion Chapter 5: The Sharpe Ratio and Itô s Lemma Chapter 6: The Black-Scholes Equation Chapter 7: The Black Model for Options on Bonds Chapter 8: Binomial Short Rate Models Chapter 9: Continuous-Time Models of Interest Rates ActuarialBrew.com 04 Page O-

4 Exam MFE/3F Solutions Overview Errata The errata for the MFE/3F Study Manual and the Solutions to the MFE/3F Questions can be found on our website at Please let us know about any errata you find by ing us at Other Products Please visit our website for a description of our other exam preparation products: Live and Online Seminars Study Manual Questions Set of 4 Practice Exams Flashcards Good Luck! ActuarialBrew.com 04 Page O-

5 Chapter Put-Call Parity and Replication Chapter Solutions Solution.0 C Put-Call Parity Since the underlying stock does not pay dividends, the value of a European call is equal to the value of the American call: CEur( K, T) = CAmer( K, T) CEur(50,0.5) = 3.48 We can use put-call parity to solve for the value of the European put option: -rt CEur( K, T) + Ke = S0 + PEur( K, T) -0.08(0.5) e = 50 + PEur(50,0.5) PEur(50,0.5) =.49 Solution.0 B Put-Call Parity The European call premium can be found using put-call parity: -rt Eur + = 0-0, T + Eur C ( K, T) Ke S PV ( Div) P ( K, T) -0.08(0.5) -0.08(0.5) C (48,0.5) + 48e = 50-5e C Eur Eur (48,0.5) = 3.36 Solution.03 B Put-Call Parity We ignore the dividend occurring in 7 months, because it occurs after the call option expires. The European call premium can be found using put-call parity: -rt Eur + = 0-0, T + Eur -0.05(0.5) -0.05(4 /) Eur C ( K, T) Ke S PV ( Div) P ( K, T) C (80,0.5) + 80e = 75-4e C (80,0.5) = 3.4 Eur Solution.04 C Put-Call Parity This question may appear tricky since the options are euro-denominated. There is no other currency involved though, so we can use the standard put-call parity formula. ActuarialBrew.com 04 Page.0

6 Chapter Put-Call Parity and Replication We can use put-call parity to solve for the risk-free rate of return: -rt -d T CEur( K, T) + Ke = S0e + PEur( K, T) -r() -0.04() e = 75e r e = r = Solution.05 D Put-Call Parity The dividend occurs after the expiration of the options, so we can ignore it. Furthermore, when there are no dividends prior to the expiration of an American call option, the option has the same value as a European call option. We can use put-call parity to solve for the value of the call option: -rt CEur( K, T) + Ke = S0 + PEur( K, T) -0.07(0.5) CEur( K, T) + 70e = CEur( K, T) = 0.48 Solution.06 A Put-Call Parity Since the options are at-the-money, the strike price is equal to the stock price. We can use put-call parity to find the price of the stock: -rt CEur( K, T) + Ke = S0 - PV0, T ( Div) + PEur ( K, T) -rt S0 = CEur( K, T) - PEur( K, T) + Ke + PV0, T ( Div) -0.07(0.75) -0.07(0.5) S0 = Ke + 3e -0.07(0.75) -0.07(0.5) S0 = S0e + 3e -0.07(0.5) e S0 = = (0.75) - e Solution.07 D Synthetic Stock We can rearrange put-call parity so that it is a guide for replicating the stock: -rt S0 = CEur( KT, ) + Ke + PV0, T ( Div) -PEur( KT, ) ActuarialBrew.com 04 Page.0

7 Chapter Put-Call Parity and Replication To replicate the stock, we must purchase the call, sell the put and lend the present value of the strike plus the present value of the dividends. The present value of the strike plus the present value of the dividends is: -rt -0.09(0.75) -0.09(0.5) Ke + PV0, T ( Div) = 8e + 3e = 9.04 Solution.08 D Synthetic T-bills As an alternative to the method below, we could solve for r and then use it to find the present value of $,000. We can rearrange put-call parity so that it is a guide for replicating a T-bill: -rt -d T Ke = S0 e - CEur( K, T) + PEur( K, T ) To create an asset that matures for the strike price of $50, we must purchase of the stock, sell a call option, and buy a put option. The cost of doing this is: -rt -d T Ke = S0e - CEur( K, T ) + PEur( K, T ) -0.07(0.75) = 5e = d T e shares Since it costs $ to create an asset that is guaranteed to mature for $50, it must cost 0 times as much to create an asset that is guaranteed to mature for $,000: = 97.8 Solution.09 A Synthetic Stock To answer this question, we don t need to know the risk-free rate of return or the time until the options expire. We can rearrange put-call parity so that it is a guide for replicating the stock: -rt S0 = CEur( KT, ) + Ke + PV0, T ( Div) -PEur( KT, ) To replicate the stock, we must purchase the call, sell the put and lend the present value of the strike plus the present value of the dividends. We can use the equation above to find the present value of the strike plus the present value of the dividends: -rt Ke PV0, T Div -rt Ke PV0, T Div 5 = ( ) ( ) = = Solution.0 E Currency options The current exchange rate is in the form of euros per dollar, which is the inverse of our usual form of dollars per euro. ActuarialBrew.com 04 Page.03

8 Chapter Put-Call Parity and Replication The exchange rate in dollars per euro is: x 0 = = Put-call parity for currency options can be used to find the value of the call option: -rt -rt f CEur( K, T) + Ke = x0e + PEur ( K, T) -0.06() -0.04() CEur(0.94,) e =.0467e CEur(0.94,) = Solution. A Currency options We treat the Swiss franc as the base currency for the first part of this question. But then at the end we must convert its value into dollars. With the Swiss franc as the base currency, we have the following exchange rate in terms of francs per dollar: x 0 = = Put-call parity for currency options can be used to find the value of the put option: -rt -rt f CEur( K, T) + Ke = x0e + PEur( K, T) -0.06() -0.04() e =.5 e + PEur(.5,) P Eur (.5,) = Therefore the cost of the put option is Swiss francs. But the possible choices are all expressed in dollars, so we must convert the value to dollars at the current exchange rate. One Swiss franc costs $0.80, so Swiss francs must cost (in dollars): = Solution. C Currency options We treat the Iraqi dinar as the base currency. Put-call parity for currency options can be used to find interest rate on the dinar: -0.5r -rt -rt f 0 Eur -0.5r -0.08(0.5) C ( K, T) + Ke = x e + P ( K, T) Eur e = 00e e = r = 0.9 ActuarialBrew.com 04 Page.04

9 Chapter Put-Call Parity and Replication Solution.3 A Options on Bonds Here s a quick refresher on compound interest. The annual effective interest rate is denoted by i: r + i = e v = + i n - v a = ni i The price of the bond is: B = 0a +,000e (5) -0.0(5) - e = 0 +,000e 0.0 e - = 0(7.3867) =, (5) Only one coupon occurs before the expiration of the options, and it occurs one year from now. Using put-call parity, we have: -rt Eur + = 0-0, T + Eur -0.0(.5) -0.0() C ( K, T) Ke B PV ( Coupons) P ( K, T) 50 +,000e =, e + PEur(,000,.5) P (,000,.5) = 3.54 Eur Solution.4 E Options on Bonds The 6-month effective interest rate is: 0.08(0.5) e - = 4.08% ActuarialBrew.com 04 Page.05

10 Chapter Put-Call Parity and Replication The price of the bond one month after it is issued is: -0.08() 0.08(/) B0 = È45a +,000e e Î % È 4 Í - (.048) -0.08() 0.08(/) Í45 +,000e e Í Î È -0.04(4) - e -0.08() 0.08(/) = Í45 +,000e e 0.04 ÍÎ e (/) = [ 45(5.) ] e 0.08(/) =, e =, Two coupons occur prior to the expiration of the option, the first of which occurs in 5 months and the second of which occurs in months. Using put-call parity, we have: -rt CEur( K, T) + Ke = B0 - PV0, T ( Coupons) + PEur ( K, T) -0.08() -0.08(5 /) -0.08(/) CEur(950,) + 950e =, e - 45e + 5 CEur(950,) = 33.6 Solution.5 B Options on Bonds The price of the bond is equal to $,000. Since the bond price is equal to its par value, its yield must be equal to its coupon rate. Therefore the yield is 7%. Since 7% is the only interest rate provided in the problem, we use 7% as the risk-free interest rate. The 7% interest rate is compounded twice per year since coupons are paid semi-annually. Therefore, the semiannual effective interest rate is 3.5%. Using put-call parity for bonds, we have: -rt Ke = B0 - PV0, T( Coupons) + PEur( K, T )-CEur( K, T ) K (.035) =, K = Solution.6 D Exchange Options The first step is to pick one of the assets to be the underlying asset. You can choose either one. We chose Stock X below, so a call option costs $.70. If we chose Stock Y to be the underlying asset, then the same option would be a put option. ActuarialBrew.com 04 Page.06

11 Chapter Put-Call Parity and Replication Let s assume that Stock X is the underlying asset and Stock Y is the strike asset. In that case, the option to give up Stock Y for Stock X is a call option: CEur( Xt, Y t,0.5) =.70 We can now use put-call parity for exchange options: P P Eur t t Eur t t t, T t t, T t C ( X, Y,0.5)- P ( X, Y,0.5) = F ( X )- F ( Y ).70 - P = È - Eur( Xt, Yt,0.5) 50 3e -5e Î P ( X, Y,0.5) = 5.9 Eur t t -0.06( /) -0.03(0.5) Solution.7 A Exchange Options We didn t need to know the risk-free rate of return. That was included in the question as a red herring. The put option that allows its owner to give up Stock B in exchange for Stock A has Stock B as its underlying asset: PEur( Bt, A t,) =.49 Using put-call parity, we can find the value of the call option having Stock B as its underlying asset: P P CEur( Bt, At,)- PEur ( Bt, At,) = Ft, T ( Bt )- Ft, T ( At ) -0.05() CEur( Bt, At,) -.49 = 67e -70 CEur( Bt, At,) = 5. We can describe the call option as a put option by switching the underlying asset and the strike asset: CEur( Bt, At,) = PEur( At, Bt,) PEur( At, Bt,) = 5. Therefore, the value of a put option giving its owner the right to give up a share of Stock A in exchange for a share of Stock B is $5.. Solution.8 D Exchange Options Let s choose Stock X to be the underlying asset. In that case, Option A is a put option and Option B is a call option. Using put call parity, we have: P P CEur( St, Yt,5/) - PEur( St, Yt,5/) = Ft, T ( Xt ) - Ft, T ( Yt ) P P Option B - Option A = FtT, ( Xt) - FtT, ( Yt) -0.04(5 /) -0.04(5 /) = 40e -Y0e Y0 = ActuarialBrew.com 04 Page.07

12 Chapter Put-Call Parity and Replication Solution.9 E Exchange Options Let s establish two portfolios. Portfolio X consists of share of Stock A and shares of Stock B. Portfolio Y consists of share of Stock C and share of Stock D. The first call option described in the question has Portfolio X as its underlying asset: CEur ( Xt, Y t,) = 0 We can find the prepaid forward prices for both portfolios: P -0.04() FtT, ( Xt) = 40e + (50) = P -0.0() FtT, ( Yt) = e = We can now use put-call parity to find the value of the corresponding put option: P P CEur( Xt, Yt,)- PEur( Xt, Yt,) = Ft, T ( Xt )- Ft, T ( Yt ) 0 - PEur( Xt, Yt,) = PEur( Xt, Yt,) = We can describe a put option as a call option by switching the underlying asset with the strike asset: PEur( Xt, Yt,) = CEur( Yt, Xt,) CEur( Yt, Xt,) = Therefore, the call option giving its owner the right to acquire Portfolio Y in exchange for Portfolio X has a value of $ Solution.0 B Exchange Options This question is easier if we assume that Stock X is the underlying asset. Let s assume that Stock X is the underlying asset. The American option is then an American call option. Since Stock X does not pay dividends, the American call option is not exercised prior to maturity. Therefore the American call option has the same value as a European call option: CEur( Xt, Y t,7 /) = 0. We are asked to find the value of the corresponding European put option. We can make use of put-call parity: P P CEur( Xt, Yt,7/) - PEur ( Xt, Yt,7/) = Ft, T ( Xt ) - Ft, T ( Yt ) -0.04(7 /) 0. - PEur( Xt, Yt,7 /) = 00-00e PEur( Xt, Yt,7 /) = 7.9 ActuarialBrew.com 04 Page.08

13 Chapter Put-Call Parity and Replication Solution. C Options on Currencies We can use put-call parity to determine the spot exchange rate: -rt -rt f CEur( K, T) + Ke = x0e + PEur ( K, T) -0.05(0.5) (0.5) e = x0e x0 = 0.90 Solution. B Exchange Options Neither the time to maturity nor the risk-free rate of return is needed to answer the question. They were provided as red herrings. The prepaid forward price of Stock A is $. The prepaid forward price of Stock B is $6. Stock A is the underlying asset, giving rise to the following generalized form of put-call parity: P P CEur( A0, B0, T) - PEur ( A0, B0, T) = F0, T ( A) - F0, T ( B) CEur( A0, B0, T) - PEur ( A0, B0, T) = -6 CEur( A0, B0, T) - PEur ( A0, B0, T) = -4 Therefore the put price exceeds the call price by $4. By inspection, only Choice B has a put price that is $4 greater than the call price. Solution.3 C Exchange Options, Options on Currencies To answer this question, let s use the yen as the base currency. In that case, the U.S. Dollar and the Canadian dollars are assets with prices denominated in the base currency. The U.S. dollar is the underlying asset for the options, and the Canadian dollar is the strike asset: P P CEur( St, Qt, T -t)-peur( St, Qt, T - t) = FtT, ( S) - FtT, ( Q) -0.08(0.5) -0.07(0.5) CEur( St, Qt,0.5) - PEur ( St, Qt,0.5) = 0e -05e CEur( St, Qt,0.5) - PEur ( St, Qt,0.5) = 3.9 Solution.4 A Put-Call Parity Let each dividend amount be D. The first dividend occurs at the end of months, and the second dividend occurs at the end of 5 months. ActuarialBrew.com 04 Page.09

14 Chapter Put-Call Parity and Replication We can use put-call parity to find D: rt CEur( K, T) Ke S0 PV0, T ( Div) PEur ( K, T) 0.05(0.5) 0.05( /) 0.05(5 /).55 4e 40 De De ( /) 0.05(5 /) 0.05(0.5) De [ e ] e.9708D D Solution.5 D Early Exercise For each put option, the choice is between having the exercise value now or having a - year European put option. Therefore, the decision depends on whether the exercise value is greater than the value of the European put option. The value of each European put option is found using put-call parity: -rt -d T CEur( K, T) + Ke = S0e + PEur( K, T) -rt -d T PEur( K, T) = CEur( K, T) + Ke -S0e The values of each of the -year European put options are: -0.05() -0.09() P (5,) = e - 50e = 0.0 Eur -0.05() -0.09() P (50,) = e - 50e = 5.6 Eur -0.05() -0.09() P (75,) = e - 50e = 5.86 Eur -0.05() -0.09() P (00,) = e - 50e = Eur In the third and fourth columns of the table below, we compare the exercise value with the value of the European put options. The exercise value is Max( K - S 0,0). Exercise European K C Value Put $5.00 $ $50.00 $ $75.00 $ $00.00 $ The exercise value is less than the value of the European put option when the strike price is $75 or less. When the strike price is $00, the exercise value is greater than the value of the European put option. Therefore, it is optimal to exercise the special put option with an exercise price of $00. ActuarialBrew.com 04 Page.0

15 Chapter Put-Call Parity and Replication Solution.6 B Exchange Options The first page of the study note, Some Remarks on Derivatives Markets, tells us that for each share of the stock the amount of dividends paid between time t and time t + dt is assumed to be S(t ) d dt. Therefore, the continuously compounded dividend rate for Stock is 7%, and the continuously compounded dividend rate for Stock is 3%. The claim has the following payoff at time 4: [ (4), (4)] Max S S A portfolio consisting of a share of Stock and the option to exchange Stock for Stock effectively gives its owner the maximum value of the two stocks. If the value of Stock is greater than the value of Stock at time 4, then the owner keeps Stock and allows the exchange option to expire unexercised. If the value of Stock is greater than the value of Stock, then the owner exercises the option, giving up Stock for Stock. Since Stock has a continuously compounded dividend rate of 3%, the cost now of a share of Stock at time 4 is: P -d T F0, T ( S) = e S0 P -0.03(4) F0,4 ( S ) = e 85 = The cost of an exchange option allowing its owner to exchange Stock for Stock at time 4 is $43. Adding the costs together, we obtain the cost of the claim: = 8.39 Solution.7 E Reverse Conversion -0.75r -0.50r A T-bill with a current value of ( 55e Xe ) + can be replicated with a conversion by purchasing a share of stock, selling a call option, and purchasing a put option: -rt Ke + PV ( Div) = S - C ( K, T ) + P ( K, T ) 0, T 0 Eur Eur -0.75r -0.50r 55e + Xe = = Doing the opposite creates a reverse conversion. Selling a share of stock, buying a call option, and selling a put option is a reverse conversion and provides us with $56.60 now 0.5r 55 + Xe in 9 months. and an obligation to pay ( ) ActuarialBrew.com 04 Page.

16 Chapter Put-Call Parity and Replication To borrow $,000 now, we must sell:,000 = 7.67 shares of stock Solution.8 E Minimum of Assets The European option has a payoff of: Max 7 Min S(), S(),0 We recognize the European option as a put option on the minimum of the two stocks. Let s call the underlying asset X: [ ] X() Min S (), S () = The price of the option is $0.67: PEur ( X,7,) = 0.67 We are told that a claim that pays the minimum of X () and 7 has a current value of $5.50: P ( [ ]) ( [ ]) P 0, 0, F Min S (), S (), 7 = F Min X (), 7 = 5.50 We can express the value of the claim as the value of the strike asset minus the value of the put option: P ( ) tt - ( ) P FtT, Min[ XT, QT ] = F, ( Q) -PEur( Xt, Qt, T -t) P F0, Min[ X(),7] = 7e r r 5.50 = 7e r 6.7 = 7e r = 0.05 Solution.9 A Maximum of Assets The European option has a payoff of: Max Max S(), S(),0 We recognize the European option as a call option on the maximum of the two stocks. Let s call the underlying asset X: [ ] X() = Max S(), S () ActuarialBrew.com 04 Page.

17 Chapter Put-Call Parity and Replication The price of the option is $.83: CEur ( X,,) =.83 We are told that a claim that pays the maximum of X () and has a current value of $3.34: P ( [ ]) ( [ ]) P F0, Max S (), S(), = F0, Max X (), = 3.34 We can express the value of the claim as the value of the call option plus the value of the strike asset: ( ) ( [ -,]) =.83 + P P FtT, Max[ XT, QT ] = CEur( Xt, Qt, T - t) + FtT, ( Q) P r F0, Max X e -r 3.34 =.83 + e -r 0.5 = e r = Solution.30 D Maximum of Assets The European option has a payoff of: Max Max S(3), S(3) 5,0 We recognize the European option as a call option on the maximum of the two stocks. Let s call the underlying asset X: [ ] X(3) = Max S(3), S (3) We are told that a claim that pays the maximum of X (3) and 5 has a current value of $5.50: P ( [ ]) ( [ ]) P F0,3 Max S (3), S(3), 5 = F0,3 Max X (3), 5 = 5.50 We can express the value of the claim as the value of the call option on X plus the value of the strike asset: ( ) P P tt, [ T, T ] = Eur( t, t, - ) + tt, ( ) P ,3 ( [ 3,5] ) = Eur ( 3,5,3) = CEur X3 e Eur ( 3,5,3) = F Max X Q C X Q T t F Q F Max X C X e e 5.50 (,5,3) C X e ActuarialBrew.com 04 Page.3

18 Chapter Put-Call Parity and Replication Solution.3 B Maximum of Assets The European option has a payoff of: Max 5 Min X(5),3 Y (5),0 We recognize the European option as a put option on the minimum of the two stocks. Let s call the underlying asset Z: ( ) Z(5) = Min X(5),3 Y (5) The rainbow option pays the minimum of Z (5) and 5, and it has a current value of $.67: P ( [ ]) ( [ ]) P 0,5 0,5 F Min X(5), 3Y(5), 5 = F Min Z (5), 5 =.67 We can express the value of the claim as the value of the prepaid forward price of the strike asset minus the value of the European put option: P ( [ T, T ]) = tt( )- Eur( t, t, - ) - - ( ) = - P tt,, P ,5 Eur t = -PEur Zt e Eur ( t,5,5) = 6.80 F Min Z Q F Q P Z Q T t F Min[ Z(5),5] 5 e P ( Z,5 e,5) (,5,5) P Z e Solution.3 B Put-Call Parity We can rewrite the payoff of the first option with parentheses in the first Max funtion: [ 0, () -( -)] - [ 0, () - ] Max S K Max S K We can now see that the special option consists of a long position in a call with a strike price of ( K -) and a short position in a call with a strike price of K: CK ˆ( ) = CK ( - ) - CK ( ) Likewise, the second option consists of a long position in a put with a strike price of K and a short position in a put with a strike price of ( K -): PK ˆ ( ) = PK ( ) - PK ( - ) ActuarialBrew.com 04 Page.4

19 Chapter Put-Call Parity and Replication We can add the value of the two special options together and use put-call parity to show that the value of the sum does not depend on K: CK ˆ( ) + PK ˆ ( ) = CK ( -)- CK ( ) + PK ( )-PK ( -) = We can now find P ˆ (09) : [ CK ( ) PK ( ) ] [ PK ( ) CK ( )] -r -r = ÈS(0) -( K - ) e + ÈKe -S(0) Î Î -r -0. = = = e e CK ˆ( ) + PK ˆ ( ) = Cˆ (09) + Pˆ (09) = Pˆ (09) = Pˆ (09) = = Solution.33 A Put-Call Parity To answer this question, we use put-call parity: P -r CK ( )- PK ( ) = F0,( S) - Ke Let s subtract the value of Set from the value of Set and use put-call parity to simplify: [ C P ] [ P C ] [ C P ] [ C P ] (48) - (54 - (48) - (54) =.9 -(-0.53) (48) - (48) + (54) - (54) =.44 È P -r È P -r F = Î 0, S e F Î 0, S e P -r F0,( S) - 0e =.44 ( ) 48 ( ) Subtracting Set 4 from Set 3 and again using put-call parity to simplify, we obtain: [ C(55) -P(47] -[ P(55) - C(47) ] = [ P(55) -C(47)] [ C(55) - P(55) ] + [ C(47) - P(47) ] = [ P(55) -C(47)] È P -r È P -r F ( S) - 55 e + F ( S) - 47e = [ P(55) -C(47)] Î 0, Î 0, P -r F0,( S) - 0e = [ P(55) -C(47)].44 = [ P(55) -C(47)] [ P(55) - C(47) ] = -.4 ActuarialBrew.com 04 Page.5

20 Chapter Put-Call Parity and Replication Solution.34 A Currency Options We didn t need to know the dollar-denominated or the euro-denominated rates of return. They were included in the question as red herrings. The first option allows its owner to give up $.5 in order to get.00. The second option allows its owner to give up $.00 in order to get Therefore, the value of the second option is 0.80 times the value of the first option: 0.80 $0.083 = $ The value of the second option in euros is: $ = $.0 Solution.35 E Currency Options The first option allows its owner to give up 0.80 in order to get $.00. The second option allows its owner to give up.00 in order to get $.5. Therefore, the value of the second option is.5 times the value of the first option: = 0.5 The value of the second option in dollars is: $ 0.5 = $ Solution.36 B Currency Options The first option allows its owner to give up.00 in order to get.60 francs. The second option allows its owner to give up 0.65 in order to get.00 francs. Therefore, the value of the second option is 0.65 times the value of the first option: francs = francs The value of the second option in euros is: francs = francs ActuarialBrew.com 04 Page.6

21 Chapter Put-Call Parity and Replication Solution.37 D Currency Options The pound-denominated put allows its owner to give up $.00 in order to get The dollar-denominated put allows its owner to give up.00 in order to get $.50. We can find the value of a dollar-denominated call that allows its owner to give up $.50 in order to get.00. The value of this call is.5 times the value of the pounddenominated put. Below, the value of this call is expressed in dollars: Ê ˆ $ C $ Á,.50, = = $ Ë We can use put-call parity to find the value of the dollar-denominated put: Ê ˆ Ê ˆ C$ Á,.50, +.50 e = e + P$ Ë Á,.50, Ë Ê ˆ e = e + P$ Á,.50, 0.4 Ë0.4 Ê ˆ P$ Á,.50, = 0.3 Ë0.4 Solution.38 B Currency Options The euro-denominated call allows its owner to give up 0.65 in order to get $.00. The dollar-denominated call allows its owner to give up $.60 in order to get.00. We can find the value of a dollar-denominated put that allows its owner to give up.00 in order to get $.60. The value of this put is.6 times the value of the euro-denominated call. Below, the value of this put is expressed in dollars: Ê ˆ $ P $ Á,.60, = = $ Ë We can use put-call parity to find the value of the dollar-denominated call: Ê ˆ Ê ˆ C$ Á,.60, e = e + P$ Ë Á,.60, Ë0.7 Ê ˆ = C$ Á,.60,0.5.60e e Ë Ê ˆ C$ Á,.60,0.5 = Ë0.7 The value of the Canadian dollar-denominated call is Canadian dollars. The answer choices are expressed in euros though, so we must convert this amount into euros: = 0.7 $ $ ActuarialBrew.com 04 Page.7

22 Chapter Put-Call Parity and Replication The value of the Canadian dollar-denominated call is euros. Solution.39 B Currency Options The put option gives its owner the right to give up in order to get 00. The call option gives its owner the right to give up 0.0 in order to get. Therefore, the value of the call option is 0.0 times the value of the put option: Ê ˆ C Á,0.0, T = = Ë0 We can convert this value from yen into euros: Ê ˆ C Á,0.0, T = = Ë0 0 Solution.40 C Currency Options The exchange rate can be expressed in two equivalent ways: $ or $ The dollar-denominated call allows its owner to give up $0.008 in order to get.00. The yen-denominated call allows its owner to give up 5.00 in order to get $.00. We can find the value of a yen-denominated put that allows is owner to give up $.00 in order to get The value of this put is 5 time the value of the dollardenominated call. Below, the value of this put is expressed in yen: 5 P ( 5,5,) = 5 $ = $ We can use put-call parity to find the value of the yen-denominated call: ( 5,5,) + 5 = 5 + ( 5,5,) ( 5,5,) + 5 = ( 5,5,) =.408 C e e P C e e C ActuarialBrew.com 04 Page.8

23 Chapter Comparing Options Chapter Solutions Solution.0 E Bounds on Option Prices This question was written in terms of euros to make it look more confusing, but since there is no other currency involved, we don t have to worry about exchange rates. Unless arbitrage is available, it must be the case that: È P -r( T-t) Max 0, F - - Î tt, ( St) Ke C (,, ) Eur St K T t In fact, the inequality above is not satisfied since: P tt, t -r( T-t) Eur t -0.9(0.5) F ( S )- Ke > C ( S, K, T -t) 00-98e > > 8 The call option has too low of a price relative to the forward prices of the underlying asset and the strike asset. Therefore, we can earn arbitrage profits by buying the right side of the inequality and selling the left side of the inequality. Time 0.5 Transaction Time 0 S S > Buy call 8 0 S Sell stock 00 -S 0.5 -S 0.5 Lend the present value of the strike -0.9(0.5) -98e Total S The ending price of the stock is 99, so the cash flow at time 0 is.94, while the cash flow at time 0.5 is 0. The accumulated value of the arbitrage profits is: 0.9(0.5).944e =.4 Solution.0 C Comparing Options With Different Strikes and Maturities The solution below is lengthy, making this problem look harder than it really is. The solution is thorough to assist those that might have trouble with this question. The strike price grows at the interest rate: 0.7(0.5) rt ( -t) T t = 98 e fi K K e ActuarialBrew.com 04 Page.0

24 Chapter Comparing Options Therefore, to preclude arbitrage it should be the case that: P(0.75) P (0.5) But the longer option costs less than the shorter option, indicating that arbitrage is possible: 3.35 < 4.03 The arbitrageur buys the longer option for $3.35 and sells the shorter option for $4.03. The difference of $0.68 is lent at the risk-free rate of return. The 6-month option After 6 months, the stock price is $96. Therefore, the shorter option is exercised against the arbitrageur. The arbitrageur borrows $98 and uses it to buy the stock. As a result, at the end of 9 months the arbitrageur owns the stock and must repay the borrowed funds. This position results in the following cash flow at the end of 9 months: 0.7(0.5) S - 98e = S = = -.60 The 9-month option The stock price at the end of 9 months is less than $06.60, so the 9-month put option has the following cash flow at the end of 9 months: Payoff table S = = We don t necessarily recommend constructing the entire payoff table during the exam, but we ve included it below to assist you in understanding this problem. Below is the payoff table for the strategy. Time 0.75 S 0.75 < S Transaction Time 0 S 0.5 < 98 S S 0.5 < 98 S Sell P (0.5) 4.03 S S Buy P (0.75) S S Total S 0.75 S The cash flow of the two options nets to zero after 9 months, since the stock price path makes the first column applicable. The accumulated value of the arbitrage strategy is the difference between the cost of the two options, accumulated for 9 months: 0.7(0.75) ( ) e = ActuarialBrew.com 04 Page.0

25 Chapter Comparing Options Solution.03 E Comparing Options With Different Strikes and Maturities The solution below is lengthy, making this problem look harder than it really is. The solution is thorough to assist those that might have trouble with this question. The strike price grows at the interest rate: 0.9(0.5) rt ( -t) = 98e fi KT Kte Therefore, to preclude arbitrage it should be the case that: C(0.75) C (0.5) But the longer option costs less than the shorter option, indicating that arbitrage is possible:.00 < 3.50 The arbitrageur buys the longer option for $.00 and sells the shorter option for $3.50. The difference of $.50 is lent at the risk-free rate of return. The 6-month option After 6 months, the stock price is $78. Therefore, the shorter option is exercised against the arbitrageur. The arbitrageur borrows a share of stock and sells it for $98. As a result, at the end of 9 months the arbitrageur owes the stock and has the accumulated value of the $98. This position results in the following cash flow at the end of 9 months: 0.9(0.5) - S e = - S = =.63 The 9-month option The stock price at the end of 9 months is less than $07.63, so the 9-month call option expires worthless. Payoff table We don t necessarily recommend constructing the entire payoff table during the exam, but we ve included it below to assist you in understanding this problem. Below is the payoff table for the strategy. ActuarialBrew.com 04 Page.03

26 Chapter Comparing Options Time 0.75 S 0.75 < S Transaction Time 0 S 0.5 < 98 S S 0.5 < 98 S Sell C (0.5) S S Buy C (0.75) S S Total S S After 9 months, the net cash flow from the two options is: - S = =.63 Adding this to the accumulated value of the $.50 obtained at the outset, we have the accumulated arbitrage profits at the end of 9 months: 0.9(0.75).50e +.63 = 4.36 Solution.04 C Comparing Options With Different Strikes and Maturities The option has the total return portfolio as its underlying asset, so it should be the case that: PPort ( t, Kt,.5) PPort ( t, K t,.0) But the longer option costs less than the shorter option, indicating that arbitrage is possible: 7.00 < 7.85 The arbitrageur buys the longer option for $7.00 and sells the shorter option for $7.85. The difference of $0.85 is lent at the risk-free rate of return. The -year option After year, the stock price is $60. This means that the total return portfolio has a value of: 0.03() Port.0 = 60e = 6.87 The strike price is: 0.06() K = 6.47e = Therefore, the -year put option is exercised against the arbitrageur. The arbitrageur borrows $65 and uses it to buy the total return portfolio. ActuarialBrew.com 04 Page.04

27 Chapter Comparing Options As a result, at the end of.5 years the arbitrageur owns the total return portfolio and must repay the borrowed funds. This position results in the following cash flow at the end of.5 years: 0.06(0.5) 0.03(.5).5.5 Port - 65e = Port = 66e =.058 The.5-year option The stock price at the end of.5 years is $66. This means that the total return portfolio has a value of: 0.03(.5) 66e = The strike price is: 0.06(.5) K.5 = 6.47e = Therefore, the.5-year put option expires unexercised. This position results in zero cash flow at the end of.5 years. Net accumulated cash flows Buying the longer put option and selling the shorter put option results in a cash flow of 0.85 at time 0. The -year option resulted in a cash flow of.058 at time.5 years. The.5 year option resulted in a cash flow of 0 at time.5 years. The accumulated net cash flows are: 0.06(.5) 0.85e =.99 Solution.05 A Proposition The prices of the options violate Proposition : because: P ( K )- P ( K ) K - K Eur Eur > (55-50) e 4.75 > () Arbitrage is available using a put bull spread: Buy P (50) Sell P (55) ActuarialBrew.com 04 Page.05

28 Chapter Comparing Options Here s a tip for remembering which option to buy and which option to sell. Notice that Proposition fails because the 55-strike put premium is too high relative to the 50-strike put premium. To earn arbitrage profits, we sell the asset that is priced too high and buy the asset that is priced too low. This is a recurring theme in arbitrage strategies. This strategy produces the following payoff table: Time Transaction Time 0 S < S < S Buy P (50) S 0 0 Sell P (55) (55 -S ) -(55 -S ) 0 Total (55 -S ) 0 If the final stock price is $48, then the accumulated arbitrage profits are: 0.09 X = 4.75e = If the final stock price is $5, then the accumulated arbitrage profits are: 0.09 Y = 4.75 e -(55-5) =.9733 The ratio of X to Y is: X Y = = Solution.06 C Proposition 3 The prices of the options violate Proposition 3: PK ( ) - PK ( ) P( K3) - P( K) K - K K3 - K because: > > 5 6 ActuarialBrew.com 04 Page.06

29 Chapter Comparing Options Arbitrage is available using an asymmetric butterfly spread: Buy l of the 50-strike options Sell of the 55-strike options Buy ( - l) of the 6-strike options where: K3 - K l = = = K - K In the payoff table below, we have scaled the strategy up by multiplying by : Time Transaction Time 0 S < S S 6 6 < S Buy 6 of P (50) 6(3.00) 6(50 - S ) Sell of P (55) (7.00) -(55 -S ) -(55 -S ) Buy 5 of P (6) 5(.00) 5(6 - S ) 5(6 - S ) 5(6 - S ) 0.00 Total S S 0.00 If the final stock price is $5, then the accumulated arbitrage profits are: X = 4.00e + 6S = 4.00e + 6(5) = If the final stock price is $60, then the accumulated arbitrage profits are: Y = 4.00e S = 4.00e (60) = The ratio of X to Y is: X Y = = Solution.07 E Proposition The prices of Option A and Option B violate Proposition : -rt CEur ( K) - CEur ( K) ( K - K) e because: > (55-50) e 4.75 > () ActuarialBrew.com 04 Page.07

30 Chapter Comparing Options Arbitrage is available using a call bear spread: Buy C (55) Sell C (50) This strategy produces the following payoff table: Time Transaction Time 0 S < S < S Buy C (55) S - 55 Sell C (50) ( S -50) -( S -50) Total ( S -50) 5.00 If the final stock price is $5, then the accumulated arbitrage profits are: X = 4.75 e -( S - 50) = 4.75 e -(5-50) = 4.75e - = If the final stock price is $60, then the accumulated arbitrage profits are: 0.07 Y = 4.75e - 5 = The ratio of X to Y is: X Y = = Solution.08 D Proposition The prices of Option A and Option B violate Proposition : P( K ) P( K ) This is because: P(7) < P(0) 0 < Therefore, arbitrage profits can be earned by buying the 7-strike put and selling the 0-strike put. This is a put bear spread. ActuarialBrew.com 04 Page.08

31 Chapter Comparing Options Solution.09 C Proposition 3 The prices of the options violate Proposition 3: CK ( ) - CK ( ) CK ( ) - CK ( 3) K - K K - K This is because: 3 C(45) -C(50) C(50) -C(53) < < < 5 3 Arbitrage is available using an asymmetric butterfly spread: Buy l of the 45-strike options Sell of the 50-strike options Buy ( - l) of the 53-strike options where: K3 - K l = = = K - K In the payoff table below, we have scaled the strategy up by multiplying by 8: Time Transaction Time 0 S < S S < S Buy 3 of C (45) 3(7.00) ( S - 45) 3( S - 45) 3( S - 45) Sell 8 of C (50) 8(5.00) ( S -50) -8( S -50) Buy 5 of C (53) 5(3.00) ( S - 53) Total ( S - 45) 65-5S 0.00 ActuarialBrew.com 04 Page.09

32 Chapter Comparing Options The cash flows at time for each of the possible stock prices are in the table below: Answer Choice Stock Price Time Cash Flow A B 46 3( S - 45) = 3(46-45) = 3.00 C 50 3( S - 45) = 3(50-45) = 5.00 D S = 65-5(5) = 5.00 E The highest cash flow at time, $5.00, occurs if the final stock price is $50. This results in the highest arbitrage profits since the time 0 cash flow is the same for each future stock price. Solution.0 A Bid-Ask Prices This question might seem a bit unfair since bid and ask prices are not discussed in the assigned reading in the Derivatives Markets textbook. On the other hand, Problem 9.6 at the end of the second edition of Derivatives Markets Chapter 9 uses bid and ask prices. Someone writing an exam question just might think that it is a clever idea for an exam question. The arbitrageur must pay the ask price to purchase assets. The arbitargeur receives the bid price when selling assets. Let s write put-call parity as follows: - S0 = CEur( K, T) + Ke rt -PEur( K, T ) This shows that the purchase of a share of stock can be replicated by: buying a call option lending the present value of the strike price selling a put option Let s see if arbitrage profits can be earned by selling the stock and then replicating the purchase of the stock. The cash flows at time 0 are broken out below: Sell Stock Buy Call Lend 90e Sell Put 5.00 Net Cash Flow Selling the stock and replicating the purchase of the stock results in a time 0 cash flow of The cash flow at the end of the year will be zero regardless of the stock price. Therefore, the time 0 cash flow of is an arbitrage profit. ActuarialBrew.com 04 Page.0

33 Chapter Comparing Options At this point, we have answered the question, but for thoroughness, we also consider whether arbitrage profits can be earned by buying the stock and replicating the sale of the stock. The sale of a share of stock can be replicated by doing the opposite of the actions needed to replicate the purchase of a share of stock. The sale of a share of stock is replicated by: selling a call option borrowing the present value of the strike price buying a put option Let s see if arbitrage profits can be earned by buying the stock and then replicating the sale of the stock. The cash flows at time 0 are broken out below: Buy Stock Sell Call Borrow 90 e Buy Put Net Cash Flow Buying the stock and replicating the sale of the stock results in a time 0 cash flow that is negative, so it does not produce arbitrage profits. Solution. C Application of Option Pricing Concepts We are told that the price of a European call option with a strike price of $54.08 has a value of $0.6. The payoff of the call option at time is: Max 0, S() Once we express the payoff of the single premium deferred annuity in terms of the expression above, we will be able to obtain the price of the annuity. The payoff at time is: S() Time Payoff P( y%) Max, P( y%) MaxS(), P( y%) MaxS(), P( y%) MaxS() 54.08, ActuarialBrew.com 04 Page.

34 Chapter Comparing Options The current value of a payoff of Max 0, S() at time is $0.6. The current value of $54.08 at time is: e = Therefore, the current value of the payoff is: Current value of payoff = P( - y %) { } 50 For the company to break even on the contract, the current value of the payoff must be equal to the single premium of P: P( y%) P 50 ( y%).649 y% 3.978% Solution. C Early Exercise Early exercise should not occur if the interest on the strike price exceeds the value of the dividends obtained through early exercise: -rt ( -t) No early exercise if: K - Ke > PVtT, ( div ) The present value of the dividends is: -0.0(0.75) PVtT, ( div) = 3 + e = 4.86 The interest cost of paying the strike price early is shown in the rightmost column below: Option Strike Price T -rt ( -0) K - Ke A B C D E Only Option C and Option E have interest on the strike price that is less than the present value of the dividends of $4.86. Option E is not in the money though, because its strike price exceeds the stock price of $58, so it is not optimal to exercise Option E. Therefore, Option C is the only option for which early exercise might be optimal. ActuarialBrew.com 04 Page.

35 Chapter Comparing Options Solution.3 B Comparing Options With Different Strikes and Maturities The underlying asset is the total return portfolio. The strike price grows at the interest rate: 0.08(0.5) rt ( -t) 5,0 = 5,000e fi KT Kte Therefore, to preclude arbitrage it should be the case that: C(0.75) C (0.5) But the longer option costs less than the shorter option, indicating that arbitrage is possible: 470 < 473 The arbitrageur buys the longer option for $470 and sells the shorter option for $473. The difference of $3 is lent at the risk-free rate of return. The 6-month option After 6 months, the stock price is $49. Therefore, the value of the total return portfolio is: 0.5(0.06) 00 e 49 = 5,049.3 Since the strike price of the shorter option is $5,000, the shorter option is exercised against the arbitrageur. The arbitrageur sells the total return portfolio short for $5,000 and lends the $5,000 at the risk-free rate. After 9 months, the stock price is $5. Therefore, the value of the total return portfolio is: 0.06(0.75) 0.06(0.75) e S = 00 e 5 = 5, At the end of 9 months the arbitrageur owes the total return porfolio and owns the accumulated value of the $5,000. The position results in the following cash flow at the end of 9 months: 0.08(0.5) - 5, ,000e = The 9-month option The strike price of the 9-month option is $5,0, so the 9-month call option has the following cash flow at the end of 9 months: 0.06(0.75) 00 e S0.75 5,0 5, , The net cash flow Since the 6-month option and the 9-month options have offsetting cash flows at the end of 9 months, the accumulated value of the arbitrage strategy is the difference between the cost of the two options, accumulated for 9 months: 0.08(0.75) ( ) e = 3.9 ActuarialBrew.com 04 Page.3

36 Chapter Comparing Options Solution.4 E Bounds on Option Prices Let s begin by noting the prepaid forward price of the stock and the present value of the strike price: P -d ( T -t) (0.5) FtT, ( S) = e S = e S = Se -rt ( -t) -0.08(0.5) Ke = 00e = For the European call option, the bounds are: È P -r( T-t) - P Max 0, F - Î tt, ( S) Ke C Eur( St, K, T t) FtT, ( S) È Max 0, Se C (,, - ) Î Eur St K T t Se This describes Graph I, so the European call option corresponds to Graph I. For the American call option, the bounds are: È P -r( T -t) Max 0, F Î tt, ( S) Ke, S K C Amer ( St, K, T t) St È Max 0, Se , S - 00 C (,, - ) Î Amer St K T t S Max 0, S - 00 C ( S, K, T - t) S [ ] Amer t This describes Graph II, so the American call option corresponds to Graph II. For the European put option, the bounds are: È -rt ( -t) P -rt ( -t) Max 0, Ke - F - Î tt, ( S) P (,, ) Eur St K T t Ke È Max 0, Se P (,, - ) Î Eur St K T t This describes Graph IV, so the European put option corresponds to Graph IV. For the American put option, the bounds are: È -rt ( -t) P Max 0, Ke -F - - Î tt, ( S), K S P Amer ( St, K, T t) K È Max 0, Se, 00 - S P (,, - ) 00 Î Amer St K T t Max 0, 00 - S P ( S, K, T - t) 00 [ ] Amer t This describes Graph III, so the American put option corresponds to Graph III. ActuarialBrew.com 04 Page.4

37 Chapter Comparing Options Solution.5 C Propositions and 3 Choice A is true. Proposition can be used to determine the following relationship for the first derivative of a call price: CK ( ) - CK ( ) K - K CK ( ) - CK ( ) K - K CK ( - - ) CK ( ) K - K dc fi- dk dc 0 + dk Choice B is true, because Proposition implies that the first derivative of a put price is positive, and the first derivative of a call option is negative. Choice C is false because Proposition 3 implies that the second derivative of the call price is positive: CK ( ) - CK ( ) CK ( - ) CK ( 3) K - K K3 - K CK ( ) - CK ( ) CK ( - 3) CK ( ) K - K K3 - K CK ( 3) - CK ( ) CK ( - - ) CK ( ) 0 K3 - K K - K dc fi 0 dk Choice D and Choice E are true because the second derivative of a European put price is equal to the second derivative of a European call price. We can show this by taking the derivative of both sides of the put-call parity expression: -rt -d T C + Ke = Ste + P dc -rt dp + e = dk dk dc dp = dk dk ActuarialBrew.com 04 Page.5

38 Chapter Comparing Options Solution.6 A Option Payoffs A butterfly spread involves options with 3 different strike prices. Part one of the investor s position is a symmetric butterfly spread with strike prices $, $3 and $4. Part two of the investor s position is a symmetric butterfly spread with strike prices $4, $5 and $6. The payoff when buying a butterfly spread is never less than zero. In this case, each butterfly spread was sold, which means the payoffs are less than zero. The payoff of the butterfly spread looks the same whether the position is composed of calls or puts. Part one of the position s payoff is shown as the bold line below at the left, and part two of the position s payoff is shown as the bold line below at the right. Stock Price Stock Price Payoff Payoff When combined, the investor s position looks like Choice A. Solution.7 D Comparing Options With Different Strikes and Maturities The prices of the options decrease as time to maturity increases. Therefore, if the strike price increases at a rate that is less than the risk-free rate, then arbitrage is available. Option B expires 0.5 years after Option A, so let s accumulate Option A s strike price for 0.5 years at the risk-free rate: e 5.57 = Since the strike price of Option B is $5, which is greater than $5.57, arbitrage is not indicated by the prices of Option A and Option B. Option C expires 0.5 years after Option B, so let s accumulate Option B s strike price for 0.5 years at the risk-free rate: e = Since the strike price of Option C is $53, the strike price grows from time to time.5 at a rate that is less than the risk-free rate of return. Consequently, arbitrage can be earned by purchasing Option C and selling Option B (i.e., buy low and sell high). The arbitrageur buys the -year option for $7.50 and sells the.5-year option for $7.70 The difference of $0.0 is lent at the risk-free rate of return. ActuarialBrew.com 04 Page.6

39 Chapter Comparing Options The.5-year option After.5 years, the stock price is $5.50. Therefore, the.5-year option is exercised against the arbitrageur. The arbitrageur borrows a share of stock and sells it for the strike price of $5. As a result, at the end of years the arbitrageur owes the share of stock and has the accumulated value of the $5. This position results in the following cash flow at the end of years: e =.0836 The -year option The stock price of $5.50 at the end of years is less than the strike price of the -year option, which is $53. Therefore, the -year call option expires worthless, and the resulting cash flow is zero. The net cash flow The net cash flow at the end of years is the sum of the accumulated value of the $0.0 that was obtained by establishing the position, the $.0836 resulting from the.5-year option, and the $0.00 resulting from the -year option: e = Solution.8 D Arbitrage Let X be the number of puts with a strike price of $60 that are sold for Jill s portfolio. The fact that the net cost of establishing the portfolio is zero allows us to solve for X: - P(45) + 3 P(55) + - P(60) X = X = 0 X = An arbitrage strategy does not allow the cash flow at expiration to be negative. But suppose that only the $60-strike put option is in-the-money at expiration. Since Jill is short the $60-strike put option, this results in a negative cash flow. In particular, if the stock price at expiration is between $55 and $59, then the payoff from Jill s strategy will be negative. For example, if the stock price at expiration is $57, the payoff from Jill s strategy will be: [45-strike payoff ] 3[55-strike payoff ] [60-strike payoff ] [Proceeds from loan] 0 0 (60 57) 5 Since Jill s strategy can result in a negative payoff at expiration, Jill s strategy is not arbitrage. ActuarialBrew.com 04 Page.7