Course MFE/3F ractice Exam 3 Solutions The chapter references below refer to the chapters of the ActuarialBrew.com Study Manual. Solution 1 C Chapter 15, repaid Forward rice of S a The payoff consists of Q () shares of stock. The quantity of shares is therefore: Q() [ S ()] The price of each share at time is S (). The value of the payoff at the end of years is equal to the quantity of shares times the price of each share: ayoff Quantity rice Q() S() [ S()] S() [ S ()] 3 The value of the derivative is therefore the prepaid forward price of ST ( ) a, where a 3 and T. The prepaid forward price is: rt ( r) 0.5 ( 1) T F a 0, T S( T) a a a a e S(0) e ra( r) 0.5 a( a1) T e a S(0) [ 0.043(0.040.07) 0.5(3)(31)0.3 ] 3 e (4) 0.8 3 e (4) 84.6803 Solution E Chapter 16, Black-Scholes Equation The partial derivatives are: 3 0.17t V S e 0.17t VS 3S e 0.17t VSS 6Se 3 0.17t Vt 0.17S e 0.17V ActuarialBrew.com 016 age 1
The Black-Scholes equation can be used to find the risk-free rate: 0.5 St ( ) V ( r) StV ( ) V Dt ( ) rvt ( ) SS S t 0.17t 0.17t 3 0.17t 3 0.17t 0.5(0.0) S 6 Se ( r 0.03) S 3S e 0.17V 0 rv 0.1 S e (3r 0.09) S e 0.17V rv 0.1 (3r 0.09) 0.17 r r 0.14 r 0.07 Solution 3 B Chapter 4, Options on Futures Contracts The values of u F and d F are: h 0.18 0.5 uf e e 1.13573 h 0.18 0.5 df e e 0.88049 The risk-neutral probability of an upward movement is: 1 df 1 0.88049 p* 0.468 uf df 1.13573 0.88049 The futures price tree and the put option tree are below: F 0, TF F 0.5, TF F 1, TF American ut Option 16.7686 0.0000 14.7645 0.593 13.0000 13.0000 1.1835 0.5000 11.4463.0537 10.0784 3.416 If the futures price reaches $11.4463 at the end of 6 months, then early exercise of the American put option is optimal. The price of the American put option is: e 0.05(0.5) (0.468)(0.593) (1 0.468)(.0537) 1.1835 ActuarialBrew.com 016 age
Solution 4 B Chapter 7, Black-Scholes ut rice The first step is to calculate d 1 and d : We have: 1 ln( S/ K) ( r 0.5 ) T ln(10.67 / 9.75) (0.10 0.00 0.5 0.40 ) 0.5 d1 T 0.40 0.5 0.67584 d d T 0.67584 0.40 0.5 0.47584 Nd ( ) N (0.67584) 0.75043 1 Nd ( ) N (0.47584) 0.6891 The value of the European put option is: rt T Eur Ke N( d) Se N( d1) 0.10(0.5) 0.00(0.5) 9.75 e (1 0.6891) 10.67 e (1 0.75043) 0.354 Solution 5 C Chapter 11, Asset-or-Nothing Options The investor can replicate the payoff by purchasing a -year asset call and selling 100 - year cash puts. Therefore, the current value of the payoff is: ( Tt) r( Tt) AssetCall(100) 100 Cashut(100) Ste N( d1) 100 e N( d) r 9 e N( d1) 100 e N( d) We can use the -year forward price to find ( r )(0) F0, S0e ( r ) 100 9e r 100e 9e r Substituting 100e in for 9e, we have: AssetCall(100) 100 Cashut(100) r 9 e N( d1) 100 e N( d) r r 100 e N( d1) 100 e N( d) r 100 e N( d1) N( d) 9e : ActuarialBrew.com 016 age 3
We can make use of the substitution and d : r 9e 100e in determining the values of d 1 ( T t) Se 9e ln 0.5 ( T t) ln ( ) 0.5 rt t r Ke 100e 1 T t d Therefore: r 100e ln 0.5 r 100e 00.5 0.5 d d T 0.5 0.5 d 1 d 1 The current value of the payoff is: r 100 e r N( d1) N( d) 100 e N( d1) N( d 1) 0 Solution 6 C Chapter 15, Itô s Lemma The drift is the expected change in the asset price per unit of time. For the first equation, the partial derivatives are: UZ 10 UZZ 0 Ut 0 This results in: 1 du( Z, t) UZdZ UZZ( dz) Utdt 10dZ 00 10dZ Since there is no dt term, the drift is zero for du. For the second equation, the partial derivatives are: VZ 10 Z VZZ 10 Vt 5 This results in: 1 1 dv ( Z, t) VZdZ VZZ( dz) Vtdt 10 Z( t) dz 10( dz) 5dt 10 Z( t) dz 5dt 5dt 10 Z( t) dz (since ( dz) dt) Since there is no dt term, the drift is zero for dv. For the third equation, the partial derivatives are: W t W 0 W Z Z ZZ t ActuarialBrew.com 016 age 4
This results in: 1 1 Ztdt () tdzt () dw( Z, t) WZdZ WZZ( dz) Wtdt tdz 0 ( dz) Z( t) dt Since the dt term is nonzero, the drift is nonzero for dw. Solution 7 D Chapter 10, Gap Options For the gap option, we have: K K 1 Strike rice 100 Trigger rice 110 The delta of the regular put option is: T T ut e N( d1) 0.888 e N( d1) 0.888 For the regular European put option, we have: rt T Eur( K, T) Ke N( d) Se N( d1) rt 3.43 110 e N( d ) 85(0.888) rt e N( d ) 0.853436 Since the regular put option and the gap put option have the same values for d 1 and d, we can substitute the final line above into the equation for the value of the gap put option: rt T Gap put price Ke 1 N( d) Se N( d1) 100(0.853436) 85(0.888) 14.90 Solution 8 E Chapter 9, Market-Maker rofit The market-maker s profit is zero if the stock price movement is one standard deviation: One-standard-deviation move St h ActuarialBrew.com 016 age 5
To answer the question, we must determine the value of. We first determine the value of Nd ( 1 ): T ut e N( d1 ) 0.0(1.00) 0.3446 e N( d1 ) 0.3446 [1 Nd ( 1)] 0.3446 1 Nd ( 1) Nd ( 1) 1 0.34458 Nd ( 1) 0.6554 From the inverse cumulative normal distribution table, we have: d1 0.40000 We can use the formula for d 1 to solve for the value of : ln( S/ K) ( r 0.5 ) T d1 T ln(65 / 65) (0.08 0 0.5 )1.00 0.40000 1.00 0.4 0.08 0.5 0.5 0.4 0.08 0 0.8 0.16 0 ( 0.4)( 0.4) 0 0.4 The price of the stock moves by: 1 St h 0.4 65 1.36 365 The stock price moves either up or down by 1.36. Solution 9 C Chapter 5, robability that Stock rice is > K The current time is t 0.5, and we are interested in the price at the end of 9 months, so T 0.75. ActuarialBrew.com 016 age 6
The value of ˆd is: S ln t ( 0.5 )( T t) ˆ K T t 75 ln (0.1 0.05 0.5 0.3 )(0.75 0.5) 77 0.3 0.75 0.5 0.07476 d The probability that S (0.75) is greater than $77 is: rob S ˆ T K N( d ) N ( 0.07476) 0.4700 Solution 10 D Chapter 19, Vasicek Model The process can be written as: dr 0.15(0.08 r) dt 0.04dZ The expected change in the interest rate is: Edr [ ] E[0.15(0.08 rdt ) 0.04 dz] (0.01 0.15 rdt ) 0.04 0 (0.01 0.15 rdt ) Since r 0.07, we have: E[ dr] (0.01 0.15 r) dt 0.01 0.15(0.07) dt 0.0015dt To convert the expected change in the interest rate into an annual rate, we must divide by the increment of time: 0.0015dt 0.0015 dt Solution 11 B Chapter 3, Multiple-eriod Binomial Tree If we work through the entire binomial tree, this is a very time-consuming problem. Even using the direct method takes a lot of time. But if we notice that the value of the corresponding put option can be calculated fairly quickly, then we can use put-call parity to find the value of the call option. The values of u and d are: ( r) h h u e (0.080.00)(1 /1) 0.30 1 /1 e 1.09776 ( r) h h d e (0.080.00)(1 /1) 0.30 1 /1 e 0.9318 ActuarialBrew.com 016 age 7
The risk-neutral probability of an upward movement is: ( r ) h (0.080.00)(1/1) e d e 0.9318 p* 0.47836 ud 1.09776 0.9318 If the stock price decreases each month, then at the end of 7 months, the price is: 7 7 130 130(0.9318) 74.905 d If the stock price decreases for 6 of the months and increases for 1 of the months, then the price is: 6 6 130 130(1.09776)(0.9318) 88.3396 ud Consider the corresponding European put option with a strike price of $88. Since the price of $88.3396 is out-of-the-money, all higher prices are also out-of-the-money. This means that the only price at which the put option expires in-the-money is the lowest possible price, which is $74.905. Calculating the price of the European call option is daunting, but the corresponding European put option price can be found fairly easily. The value of the put option is: n rhn ( ) n j nj j nj V( S0, K,0) e ( p*) (1 p*) V( S0u d, K, hn) j j0 0.08(7 /1) 7 e [(1 0.47836) (88 74.905) 0000000] 0.1375 Now we can use put-call parity to find the value of the corresponding call option: rt T 0 Eur 0.08(7 /1) 0.00(7 /1) C ( K, T) Ke S e ( K, T) Eur CEur( K, T) 88e 130e 0.1375 C ( K, T) 46.1498 Eur The value of the European call option is $46.15. Since the stock does not pay dividends, the value of the American call option is equal to the value of the European call option. Solution 1 C Chapter, Different Strike rices The prices of the options violate roposition 3: K ( ) K ( 1) ( K3) ( K) K K K K 1 3 ActuarialBrew.com 016 age 8
This is because: (75) (70) (95) (75) 75 70 95 75 14 10 5 14 75 70 95 75 4 11 5 0 0.8 0.55 Arbitrage is available using an asymmetric butterfly spread: Buy of the 70-strike options Sell 1 of the 75-strike options Buy (1 ) of the 95-strike options The value of is: K3 K 95 75 0 0.8 K3 K1 95 70 5 To provide the convenience of dealing in integers, let's multiply by 5 to scale the strategy up. The strategy is therefore: Buy 4 of the 70-strike options Sell 5 of the 75-strike options Buy 1 of the 95-strike options The payoffs for each of the possible answer choices are: Year-end Stock rice 69 74 77 93 96 Buy 4 of the 70-strike options 4 0 0 0 0 Sell 5 of the 75-strike options 30 5 0 0 0 Buy 1 of the 95-strike options 6 1 18 0 Strategy ayoff 0 16 18 0 The highest cash flow at time 1, $18.00, occurs if the final stock price is $77. This results in the highest arbitrage profits since the time 0 cash flow is the same for each future stock price. Shortcut The highest payoff for an aymmetric butterfly spread occurs at the middle strike price, and we can use this fact to narrow down the answer choices. ActuarialBrew.com 016 age 9
The graph of the asymmetric butterfly spead is shown below: ayoff Asymmetric Butterfly Spread ayoff 4 Slope = 4 Slope = 1 0 70 75 95 S T It isn't necessary to sketch the graph above to answer this question. It is provided to illustrate that the highest payoff occurs at a final stock price of $75. Although $75 is not one of the answer choices, we can narrow the solution down to the possible answer choices on either side: $74 and $77. Instead of filling out the entire table above, it is sufficient to obtain the portion below: Year-end Stock rice 74 77 Buy 4 of the 70-strike options 0 0 Sell 5 of the 75-strike options 5 0 Buy 1 of the 95-strike options 1 18 Strategy ayoff 16 18 Since the highest payoff of $18 occurs when the final stock price is $77, the answer to the question is $77. Solution 13 C Chapter 11, Forward Start Option In one year, the value of the call option will be: rt 1 1 1 0.10 SNd 1 1 Se 1 Nd C ( S ) S N( d ) Ke N( d ) Eur ( ) ( ) ActuarialBrew.com 016 age 10
In one year, the values of d 1 and d will be: F ( S) t 1, T ln 0.5 ( T t1 ) F ( K) t1, T d1 T t1 S 1 ln 0.101 0.5(0.3) (1) Se 1 0.3 1 0.48333 d d T t 0.48333 0.30 1 0.18333 1 1 From the normal distribution table: Nd ( 1) N(0.48333) 0.68557 Nd ( ) N(0.18333) 0.5773 In one year, the value of the call option can be expressed in terms of the stock price at that time: 0.10 1 1 1 1 0.10 S1 S1e C ( S ) S N( d ) S e N( d ) Eur 0.68557 0.5773 0.16734 S 1 In one year, the call option will be worth 0.1674 shares of stock. The prepaid forward price of one share of stock is: 0, t1 0,1 F ( S) S V ( Div) 0 0, t 0.10(0.5) F ( S) 50 5e 45.439 1 The value today of 0.1674 shares of stock in one year is: 0.16734 45.439 7.571 Solution 14 E Chapter 13, Estimating Volatility Since we have 7 months of data, we can calculate 6 monthly returns. Each monthly return is calculated as a continuously compounded rate: r i Si ln Si 1 ActuarialBrew.com 016 age 11
The next step is to calculate the average of the returns: k ri i 1 r k The returns and their average are shown in the third column below: Si h Date rice ri ln S i ( ri r ) 1 85 81 0.0480 0.005668 3 87 0.071459 0.001969 4 93 0.066691 0.001569 5 10 0.09373 0.0046 6 104 0.019418 0.000059 7 100 0.0391 0.004397 r 0.07086 6 ( ri r) 0.01794 i 1 The fourth column shows the squared deviations and the sum of squares. The estimate for the standard deviation of the monthly returns is: ˆ h i 1 ( r r) k 1 0.01794 ˆ 0.059873 5 1 1 k i We adjust the monthly volatility to obtain the annual volatility: 1 ˆ ˆh 0.059873 1 0.074 h This problem isn t very difficult if you are familiar with the statistical function of your calculator. On the TI-30X IIS, the steps are: [ nd ][STAT] (Select 1-VAR) [ENTER] [DATA] 81 X1= ln 85 [ENTER] (Hit the down arrow twice) X= 87 ln [ENTER] 81 93 X3= ln [ENTER] 87 ActuarialBrew.com 016 age 1
X4= X5= X6= 10 ln [ENTER] 93 104 ln [ENTER] 10 100 ln [ENTER] 104 [STATVAR] (Arrow over to Sx) (1) [ENTER] The result is: 0.0740485 To exit the statistics mode: [ nd ] [EXITSTAT] [ENTER] On the BA II lus calculator, the steps are: [ nd ][DATA] [ nd ][CLR WORK] X01 81/85 = LN [ENTER] (Hit the down arrow twice) X0 87/81 = LN [ENTER] X03 93/87 = LN [ENTER] X04 10/93 = LN [ENTER] X05 104/10 = LN [ENTER] X06 100/104 = LN [ENTER] [ nd ][STAT] (1) = The result is: 0.0740485 To exit the statistics mode: [nd][quit] On the TI-30XS MultiView, the steps are: [data] [data] 4 (to clear the data table) (enter the data below) L1 L L3 85 81 ----------- 81 87 87 93 93 10 10 104 104 100 (place cursor in the L3 column) [data] (to highlight FORMULA) 1 [ln] [data] / [data] 1 ) [enter] [ nd ] [quit] [ nd ] [stat] 1 ActuarialBrew.com 016 age 13
DATA: (highlight L3) FRQ: (highlight one) (select CALC) [enter] 3 (to obtain Sx) Sx 1 [enter] The result is 0.0740485 Solution 15 D Chapter 7, Currency Options and Black-Scholes The currency option is a put option with a euro as its underlying asset. The domestic currency is dollars, and the current value of the underlying asset is: 1 x 0 1.5 dollars 0.80 Since the option is at-the-money, the strike price is equal to the value of one euro: 1 K 1.5 dollars 0.80 The domestic interest rate is 3%, and the foreign interest rate is 7%: r 3% rf 7% The volatility of the euro per dollar exchange rate is: 0.08 The values of d 1 and d are: ln( x 0 / K) ( r rf 0.5 ) T ln(1) (0.03 0.07 0.5 0.08 )1 d1 0.46000 T 0.08 1 d d1 T 0.46 0.08 1 0.54000 In this case, we don t need to round d 1 and d when using the normal distribution table: N( d1 ) N(0.46000) 0.6774 N( d ) N(0.54000) 0.70540 The value of the put option is: rt rt 1 0.03(1) 0.07(1) f ( S, K,, r, T, ) Ke N( d ) Se N( d ) Eur 1.5 e (0.70540) 1.5 e (0.6774) 0.06637361 Since the option is for 100,000,000, Company A purchases 100,000,000 put options, and the value of the put options in dollars is: 0.06637361 100,000,000 6,637,36 ActuarialBrew.com 016 age 14
The solution is $6,637,36. Solution 16 C Chapter 1, ut-call arity The options expire in 6 months, so we do not need to consider dividends paid after 6 months. Therefore, only the first dividend is included in the calculations below. We can use put-call parity to understand this problem: rt Eur (, ) 0 0, T ( ) Eur (, ) C K T Ke S V Div K T ut-call parity tells us that the following strategies produce the same cash flow at the end of 6 months: urchase 1 call option and lend the present value of the $90 strike price. The cost of this strategy is: rt 0.06(0.5) Eur (, ) 7. 90 94.5601 C K T Ke e urchase 1 share of stock, borrow the present value of the first dividend, and purchase 1 put option. The cost of this strategy is: 0.06( /1) 0 0, T( ) Eur(, ) 9 5 5.50 9.5498 S V div K T e Since the first strategy costs more than the second strategy, arbitrage profits can be earned by shorting the first strategy and going long the second strategy. The net payoff in 6 months is zero, and the arbitrage profit now is: 94.5601 9.5498.0103 Solution 17 A Chapter 9, Delta-Hedging For writing the 100 options at the end of Day 1, the market-maker receives: 100 3.65 365.00 The market-maker wrote 100 of the call options, so the delta at the end of Day 1 is: 100 0.4830 48.30 The market-maker hedges this position by purchasing 48.30 shares of the stock for: 48.30 7.00 3,477.60 Since the cost of the stock exceeds the amount received for writing the options, the market-maker borrows: 3,477.60 365.00 3,11.60 ActuarialBrew.com 016 age 15
At the end of Day, the market maker's profit is the value of the position, which consists of the sum of the following: the value of the options, the value of the stock, and the value of the borrowed funds. Value of options: 100 3.16 316.00 Value of stock: 48.30 71.00 3,49.30 0.10 / 365 Value of borrowed funds: 3,11.60 e 3,113.45 0.15 The second day s profit is $0.15. Solution 18 C Chapter 19, Cox-Ingersoll-Ross Model We begin with the Sharpe ratio and parameterize it for the CIR model: (,, rtt) r ( rt, ) qrtt (,, ) r ( r, t, T) r B(, tt) ( r) r ( r, t, T) r B(, tt) r ( rtt,, ) r r BtT (, ) We use the value of (0.08,1,5) provided in the question: 0.0818538 0.08 0.08 B(1,5) 0.0818538 0.08 B(1,5) (0.08) B(1,5) 0.03175 Making use of the fact that Shortcut B(1,5) B (,6), we have: (0.07,,6) 0.07 0.07 B(,6) (0.07,,6) 0.07 B (,6) (0.07) 0.07 0.03175(0.07) 0.07161 In the CIR Model, when two bonds have the same time until maturity, the ratio of the expected return to the short rate is the same for the two bonds. ActuarialBrew.com 016 age 16
The first bond matures at time 5 and is being valued at time 1, so it is a 4-year bond. The second bond matures at time 6 and is being valued at time, so it is also a 4-year bond. Therefore, we can use the following shortcut: (0.08,1,5) (0.07,,6) 0.08 0.07 0.0818538 (0.07,,6) 0.08 0.07 (0.07,,6) 0.07161 Solution 19 D Chapter 7, Black-Scholes Formula w/ Discrete Dividends The prepaid forward price can be written in terms of the forward price: r( Tt) r( Tt) tt, tt, tt, tt, F ( S) e F F e F ( S) The variance of the natural log of the prepaid forward price is equal to the variance of the natural log of the forward price:,1 r(1 t) t t,1 r(1 t) Var ln e ln Ft,1( S) 0 ln t,1( ) 0.04 Var ln F ( S) Var ln e F ( S) Var F S t The prepaid forward volatility is: Var ln Ft,1( S) 0.04t F t t 0.0 The prepaid forward prices of the stock and the strike price are: 0.10(3 /1) F0, T ( S) 80 5.00e 75.135 0.101 F0, T ( K) 85e 76.911 ActuarialBrew.com 016 age 17
We use the prepaid forward volatility in the Black-Scholes Formula: d1 d We have: F 0, T ( S) ln 0.5 75.135 FT F ln 0.5 0.0 1 0, T ( K) 76.911 0.01759 T 0.0 1 F F 0, T ( S) ln 0.5 75.135 FT F ln 0.5 0.0 1 0, T ( K) 76.911 0.1759 T 0.0 1 F N( d1 ) N(0.01759) 0.5070 N( d ) N(0.1759) 0.58613 The value of the European put option is: rt 0, 0, T T 0 0, T 1 Eur F ( S ), F ( K ),, T Ke N ( d ) S V ( Div ) N ( d ) 76.911 0.58613 75.135 0.5070 6.9909 Solution 0 C Chapter 19, Risk-Neutral CIR Model The standard form of the CIR model is: dr a ( b r) dt rdz From (i), we observe that: 10a b The risk-neutral version of the CIR model is: a( ) ( ) (, ) ( ) dr r r r t dt r dz r a( br) r dt rdz ( br) rdt rdz a Subtracting the drift of the realistic process in (i) from the drift of the risk-neutral process in (ii) allows us to solve for : aba( br) r ar r a ActuarialBrew.com 016 age 18
We can substitute this value into the formula for : ( a ) ( aa ) In the CIR model, as the maturity of a zero-coupon bond approaches infinity, its yield approaches: r ln[ r (,0, T)] ab ab ab Lim 0.1414 T T ( a ) ( aa ) 10ab 10 Solution 1 A Chapter 5, Comparing Stock with a Risk-free Bond Elizabeth s $1,000 investment in the stock purchases the following quantity of stock at time 0: 1,000 S 0 Since Elizabeth reinvests the dividends, each share purchased will result in owning e T shares at time T. Therefore, the time T payoff (including dividends) for Elizabeth is: 1,000 T e S T S 0 Sadie s $1,000 investment at the risk-free rate produces a time T payoff of: 1,000 rt e The probability that Sadie s investment outperforms Elizabeth s investment is equal to the probability that the risk-free investment outperforms the stock: 1,000 T rt ( r) T rob e S ˆ T 1,000e rob S T S0e N( d ) S0 ( r ) T where: K S0e The value of ˆd is: S S 0 0 ln ( 0.5 ) T ln ( 0.5 ) T ( r ) T ˆ K S0e d T T [( r) 0.5 ] T The probability that Sadie s investment outperforms Elizabeth s investment is: T rt ˆ 0.5 ( r) rob e S T S0e N( d ) N T ActuarialBrew.com 016 age 19
The question tells us that this probability is equal to 50%: 0.5 ( r) N T 50% This implies that the value in parentheses above is 0: 0.5 ( r) 0.5 ( r ) 0 T 0.00000 0.5 (0.1 0.08) 0 0.08 The Sharpe ratio is: r 0.1 0.08 0.1414 0.08 Solution A Chapter 1, Simulating Lognormal Stock rices Dolores uses the following steps to obtain a standard normal random variable based on the inverse cumulative normal distribution: 1. Obtain an observation û from U (0,1). This observation is 0.6.. Find the value of ẑ that is the û quantile of the normal distribution. In this case, we obtain the 6% quantile: ˆ 1 1 z N uˆ N 0.6 0.64335 Thus 0.64335 is the observation from the standard normal distribution. The Black-Scholes framework implies that the stock price follows a lognormal distribution, so Dolores estimate is: ST ( 0.5 )( Tt) z Tt Ste (0.10.040.50.3 ) (30) 0.3 ( 0.64335) 30 75e 59.631 Solution 3 C Chapter 15, Itô's Lemma The usual formula for the forward price is: ( r )( T t) FtT, S() t e ActuarialBrew.com 016 age 0
In this case, the underlying asset is one South African rand, and the South African interest rate is analogous to the dividend yield: ( r )( T t) ( rr*)( T t) (0.06 0.10)( T t) Gt () FtT, Ste () Ste () Ste () 0.04( T t) Ste () The partial derivatives are: GS 0.04( T t) e GSS 0 Gt 0.04( T t) 0.04 S( t) e 0.04 G( t) From Itô's Lemma, we have: 0.04( T t) 0.04( T t) Gt ( )0.11 dt 0.35 dz() t dg() t GSdS() t 0.5 GSS ds() t Gtdt 0.04( T t) e ds() t 0.50 ds() t 0.04 G() t dt e 0.07 Stdt ( ) 0.35 StdZt ( ) ( ) 0 0.04 Gtdt ( ) e S( t) 0.07dt 0.35 dz( t) 0.04 G( t) dt G( t) 0.07dt 0.35 dz( t) 0.04 G( t) dt Solution 4 B Chapter 5, Conditional and artial Expectations The risk-neutral partial expectation of the time 1 stock price, conditional on the time 1 stock price being above 98 is: * * * E ST ST K STg ( ST; St) dst S1g ( S1; S0.5) ds 1 64.10 K 98 The risk-neutral probability that the final stock price is above 98 is found using the relationship below: * E S * T ST K E ST ST K * rob ST K * E S 1 S1 98 * 64.10 rob S1 98 0.546 * E S1 S1 98 118.14 The risk-neutral expected value of the call at time 1 is therefore: Call ayoff rob * * * E E ST ST K K ST K 118.14 98 0.546 10.975 ActuarialBrew.com 016 age 1
Since we used the risk-neutral probability to obtain this expected value, we can use the risk-free rate of return to discount it back to time 0.5: 0.090.5 10.975 e 10.45 Solution 5 A Chapter 9, Re-hedging Frequency If R is the profit from delta-hedging a short position in 1 call, then 100R is the profit from delta-hedging a short position in 100 calls. The variance in the position is therefore: Var 100R 100 Var R When 100 calls are written and delta-hedged, the variance of the return earned over a period of length nh years, when re-hedging occurs every h years is: 1 100 Var R nh n re-hedgings 100 n S h Let's convert the formula for variance into a formula for standard deviation by taking the square root: n 100 StdDev Rnh n re-hedgings 100 S h For Doug, we have nh 1/ 365 and h 1/ 365 : 1 1 X 100 StdDev R 1 1 re-hedgings 100 S 365 365 For Bruce, we have nh 1/ 365 and h 1/(365 4) : 4 1 Y 100 StdDev R 1 4 re-hedgings 100 S 365 365 4 The ratio of X to Y is: X Y 1 1 100 S 365 1 4 4.8990 4 1 1 100 S 365 4 4 Solution 6 D Chapter 14, Volatility of repaid Forward The prepaid forward price at time t is equal to the stock price minus the present value of the dividend: 0.100.5,1 ( ) ( ) 1 t Ft S S t e ActuarialBrew.com 016 age
We can take the differential of both sides. For 0 t 0.5, we have: 0.100.5t df t,1( S) ds( t) 1.e dt 0.100.5t F t,1( S)0.15 dt dz() t S() t () t dt () t dz() t 1.e dt 0.100.5t 0.15 F t,1 ( S) dt Ft,1 ( S) dz( t) ( t) S( t) 1. e dt ( t) S( t) dz( t) The coefficients to the dz( t ) terms above are equal for 0 t 0.5: t,1 F ( S) ( t) S( t) () tst () F ( S) t,1 Since we know the time-0 values on the right side of the equation, we can find : (0) S(0) 0.30 90 0.3436 F ( S) 0.100.50 90 1e 0,1 Solution 7 E Chapter 11, Cash-Or-Nothing Call Option The cash-or-nothing put option pays: 1,000 if S T 80 The question provides information about a gap call option, but consider a gap put option with a strike price of 70 and a trigger price of 80. This gap put pays: 70 ST if S T 80 A regular put option with a strike price of 80 pays: 80 ST if S T 80 The cash-or-nothing put option can be replicated by purchasing 100 of the put options and selling 100 of the gap put options: 100 ut Gaput 100 (80 ST ) (70 ST ) 100 10 1,000 if S T 80 We use put-call parity for gap calls and gap puts to find the theta of the gap put: rt ( t) ( Tt) GapCall K1e Se Gaput 0.15(1 t) 0.04(1 t) GapCall 70e 70e Gaput 0.15(1 t) 0.04(1 t) GapCall (0.15)70 e (0.04)70e Gaput 0.15(1 t) 0.04(1 t) 7.0 (0.15)70 e (0.04)70e Gaput ActuarialBrew.com 016 age 3
Evaluating at time t 0, we have: 0.15(1 0) 0.04(1 0) 7.0 (0.15)70 e (0.04)70e 0.658 Gaput Gaput The theta of the cash-or-nothing put is the theta of a position consisting of 100 long puts and 100 short gap puts: 100 ut Gaput 100.0 ( 0.658) 100.658 65.8 Solution 8 B Chapter 19, Interest Rate Derivative The realistic process for the short rate follows: dr a( r) dt ( r) dz where: a ( r) 0.5(0.16 r) & ( r) 0. r The risk-neutral process follows: dr a ( r) ( r) ( r, t) dt ( r) dz We can use the coefficient of the first term of the risk-neutral process to solve for the Sharpe ratio, ( rt, ): 0.04 0.1 r a( r) ( r) ( r, t) 0.04 0.1r 0.5(0.16 r) 0. r( r, t) 0.04 0.1r 0.04 0.5r 0. r( r, t) 0.15r 0. r( r, t) ( rt, ) 0.75 r The derivative, like all interest-rate dependent assets, must have a Sharpe ratio of 0.75 r. Let s rearrange the differential equation for g, so that we can more easily observe its Sharpe ratio: dg r 0.1 r r 0.1 r r 0.1 r dt dz ( r, t) g Since the Sharpe ratio is 0.75 r : 0.75 r 0.16 0.1 r ActuarialBrew.com 016 age 4
Solution 9 D Chapter 7, Arbitrage with Options on Futures The current value of the futures price is: ( r ) T (0.030.08)4 F0, F T S0e 100e 81.8731 F Gail uses put-call parity to see if arbitrage is available: rt rt Eur 0, TF 0, TF Eur 0, TF 0.031 0.031 5 95e 81.8731e 0 97.193 99.4533 C ( F, K,, rtr,, ) Ke F e ( F, K,, rtr,, ) As shown above, the left side is less than the right side. Therefore, Gail buys the left side and sells the right side (buy low, sell high). In order to buy the left side, at time 0 Gail does the following: Buy the call. Lend 0.03 95e dollars. In order to sell the right side at time 0, Gail does the following: 0.03 Sell stock with a value of 81.8731e 79.45. Since the current stock price is $100, this means selling 0.7945 shares of stock Sell the put. These steps are described in Choice (D). Additional Explanation To see that shares of stock (and not futures contracts) are sold at time 0, consider the putcall parity expression at time 1: Max 0, F1,4 95 95 F1,4 Max 0,95 F 1,4 In order to buy the left side, at time 0 Gail does the following: Buy the call Lend 0.03 95e In order to sell the right side at time 0, Gail does the following: Sell the put Sell something that will have a payoff of F 1,4 at time 1. ActuarialBrew.com 016 age 5
At time 1, the futures price will be: ( r )( T F T) FTT, S F Te (0.030.08)(41) 0.15 F1,4 S1 e S1 e This suggests that having e 0.15 shares of stock at time 1 is equivalent to having F 1,4 0.15 dollars at time 1. The prepaid forward price of e shares of stock is: 0.15 1 0.15 0.08 0.15 F0,1 ( S) e e S0e e S0e 0.7945S 0 Selling something now that will have a value of F 1,4 at time 1 is equivalent to selling 0.7945 shares of stock now. Solution 30 D Chapter 1, Early Exercise For each put option, the choice is between having the exercise value now or having a 1- 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 T CEur( K, T) Ke S0e Eur( K, T) rt T Eur( K, T) CEur( K, T) Ke S0e The values of each of the 1-year European put options are: 0.05(1) 0.06(1) Eur (5,1) 3.3 5e 50e 0.01 0.05(1) 0.06(1) Eur (50,1) 4.47 50e 50e 4.94 0.05(1) 0.06(1) Eur (70,1) 0.58 70e 50e 0.08 0.05(1) 0.06(1) Eur (100,1) 0.01 100e 50e 48.04 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 ut $5.00 $3.3 0 0.01 $50.00 $4.47 0 4.94 $70.00 $0.58 0 0.08 $100.00 $0.01 50 48.04 ActuarialBrew.com 016 age 6
The exercise value is less than the value of the European put option when the strike price is $70 or less. When the strike price is $100, 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 $100. ActuarialBrew.com 016 age 7