Course MFE/3F Practice Exam 2 Solutions

Size: px

Start display at page:

Download "Course MFE/3F Practice Exam 2 Solutions"

Geoffrey Alexander
5 years ago
Views:

1 Course MFE/3F Practice Exam Solutions The chapter references below refer to the chapters of the ActuarialBrew.com Study Manual. Solution 1 A Chapter 16, Black-Scholes Equation The expressions for the value of the derivative and the partial derivatives are: 1 V S S VS 1 S 3 VSS S Vt 0 From Statement (ii), we observe that 0.4. From the Black-Scholes equation, we have: 0.5 SVSS ( r) SVS Vt rv (0.4) S S ( r 0) S 1 S 0 r S S S rs rs rs rs S rs rs 0.16 r r 0.08 Solution E Chapter 10, Compound Options Using the formula for an American call option, we can find the value of the call on a put. Then we can use compound option parity to find the value of the ordinary put option. We know that the value of an American call option is: C ( S, K, T ) S Ke 1 CallOnPut Amer 0 0 rt The call on a put expires at time t 1 and has a strike price of: rt ( t1 ) x D K 1 e In this case, we have: rt ( t1 ) 0.085(1 0.5) x D K 1 e e 3.63 ActuarialBrew.com 016 Page 1

2 Therefore the value of the compound call with a strike price of $3.63 can be found as follows: rt C 1 Amer ( S0, K, T ) S0 Ke CallOnPut 0.085(0.5) e CallOnPut CallOnPut 6.37 We can now use the parity relationship to find the value of the European put option: rt CallOnPut PutOnPut Put xe 0.085(0.5) Put 3.63e Put Solution 3 B Chapter 3, Two-Period Binomial Model Since no dividend is mentioned, we assume that the dividend rate is zero. The values of u and d are: ( r) h h ( )(1) u e e ( r) h h ( )(1) d e e The risk-neutral probability of an upward movement is: ( r ) h ( )(1) e d e p* ud The stock price tree and its corresponding tree of option prices are: Stock European Put The put option can be exercised only if the stock price falls to $ At that point, the put option has a payoff of $ Although the entire table of put prices is filled in above, we can directly find the value of the put option as follows: n rhn ( ) n j nj j nj 0 0 j j (1)() V( S, K,0) e ( p*) (1 p*) V( S u d, K, hn) e 0 0 ( ) (13.678) ActuarialBrew.com 016 Page

3 Solution 4 A Chapter 7, Black-Scholes Call Price The Black-Scholes Formula uses the risk-free interest rate, not the expected annual return. The first step is to calculate d 1 and d : We have: ln( S/ K) ( r 0.5 ) T ln(48.96 / 50) ( ) 0.5 d1 T d d T Nd ( ) N ( ) Nd ( ) N ( ) The value of the European call option is: T rt CEur Se N( d1) Ke N( d) 0.06(0.5) 0.07(0.5) 48.96e e Solution 5 E Chapter 5, Effect of Parameters on Call Option Statement B is true because options on nondividend-paying stocks increase in value as their time to maturity increases when the strike price grows at the risk-free interest rate. The probability that the call option expires in the money is: ˆ rt Prob S K N( d ) where: K S e T The value of ˆd is: 0 S S 0 0 ln ( 0.5 ) T ln ( ) T rt ˆ K S0e d T T [( r) 0.5 ] T The probability that the call option expires in the money is: rt ˆ ( r) 0.5 Prob S T S0e N( d ) N T ActuarialBrew.com 016 Page 3

4 The question tells us that the risk premium is greater than 0.5 : r 0.5 ( r ) Therefore ˆd is positive: [( r) 0.5 ] T 0 We have established that the numerator in the fraction below is positive: rt ( r) 0.5 Prob S T S0e N T Therefore, the value in the parentheses becomes more positive as T increases, meaning that an increase in T increases the probability that the call option expires in the money. Therefore Statement A is true. The value in the parenthesis becomes more positive as increases, so an increase in increases the probability that the call option expires in the money. Therefore, Statement D is true. The risk-neutral probability is obtained by substituting r for : ( r r) N T N T The value in the parentheses is now negative, meaning that an increase in T decreases the risk-neutral probability that the call option expires in the money. Therefore, Statement C is true. The value in the parentheses is not affected by changes in r, so Statement E is false. Solution 6 B Chapter 9, Delta-Hedging The delta of a put option covering 1 share is: T Put e N( d1 ) e N( ) e ( ) The market-maker sold 100 put options, each of which covers 100 shares. Therefore, from the perspective of the market-maker, the delta of the position to be hedged is: ( ) 3,4.74 To delta-hedge the position, the market-maker sells 3,4.74 shares of stock. ActuarialBrew.com 016 Page 4

5 When the stock price increases, the market-maker has a loss on the stock and a gain on the put options: Gain on Stock: 3,4.74 (5 51) 3,4.74 Gain on Puts: ,13.00 Net Profit: The market-maker has a net profit of Solution 7 A Chapter 9, Delta-Gamma Approximation The delta-gamma approximation for the new price is: Vt ( h) Vt ( ) t t The change in the stock price is: St h St The delta-gamma approximation is: Vt ( h) Vt ( ) t t (1.50)( 0.418) (0.05) Solution 8 B Chapter 15, Sharpe Ratio When the price follows geometric Brownian motion, the natural log of the price follows arithmetic Brownian motion: dst () Stdt () StdZt () () dln St () 0.5 dt dzt () Therefore: dy () t Adt BdZ() t Yt () d ln Y() t A 0.5 B dt BdZ() t The arithmetic Brownian motion provided in the question for dln Y( t ) allows us to find an expression for C and to solve for B. ActuarialBrew.com 016 Page 5

6 This implies that B is 0.0: d ln Y () t Cdt 0.0 dz() t and d ln Y () t A 0.5 B dt BdZ() t A 0.5B C B 0.0 Since X and Y are perfectly positively correlated, they must have the same Sharpe ratio. This allows us to solve for A: A A 0.11 We can now determine C: C A 0.5B Solution 9 C Chapter 4, Binomial Model & Currency Options The values of u and d are: ( rrf ) h h ( ) u e e ( rrf ) h h ( ) d e e The risk-neutral probability of an upward movement is: 1 1 p* h e 1e The tree of prices for one British pound is below: The tree of prices for an American put option on one British pound is below: The bold entry indicates that early exercise is optimal at that node. ActuarialBrew.com 016 Page 6

7 If the pound goes down in value in the first month, then the value of holding the put option is: e ( ) The value of exercising the option then is: The value of the option is the maximum of the value of holding it and the value of exercising it, so its value after 1 downward movement is $ Continuing with the recursive calculations, the value of the put option at time 0 is: e ( ) The price of a put option on 100 British pounds is: Solution 10 A Chapters 3 and 4, Greeks in the Jarrow-Rudd Binomial Model How do we know that gamma in the question refers to and not? Because we would need to know the realistic probability of an upward movement in order to determine the expected return on the option,, but there is no way of knowing the realistic probability of an upward movement. The values of u and d are: ( r0.5 ) h h ( )(1) u e e ( r0.5 ) h h ( )(1) d e e The stock price tree and its corresponding tree of option prices are: Stock Call There is no need to calculate the current value or the time 1 values of the call to answer this question, but they are provided in the tree above for completeness. We need to calculate the two possible values of delta at time 1: h Vuu Vud ( Su, h) e e Su Sud h Vud Vdd ( Sd, h) e e Sud Sd ActuarialBrew.com 016 Page 7

8 We can now calculate gamma: ( Su, h) ( Sd, h) ( S,0) ( Sh, h) Su Sd Solution 11 D Chapter 19, Vasicek Model The Vasicek model of short-term interest rates is: dr a ( b r) dt dz Therefore, we can determine the value of a: dr 0.4( b r) dt dz a 0.4 In the Vasicek model, the Sharpe ratio is constant: ( rt, ) Therefore, for any r, t, and T, we have: ( rtt,, ) r qrtt (,, ) Since the Sharpe ratio is constant: (0.08,0,3) 0.08 (0.09,,6) 0.09 q(0.08,0,3) q(0.09,,6) We now make use of the following formula for qrtt (,, ) : qrtt (,, ) BtT (, ) ( r) BtT (, ) ActuarialBrew.com 016 Page 8

9 Substituting this expression for qrtt (,, ) into the preceding equation allows us to solve for (0.09,,6) : (0.08,0,3) 0.08 (0.09,,6) 0.09 q(0.08,0,3) q(0.09,,6) (0.08,0,3) 0.08 (0.09,,6) 0.09 B(0,3) B(,6) (0.09,,6) (30) 0.4(6) 1e 1e (0.09,,6) (3) 0.4(4) 1 e 1 e (0.09,,6) (0.09,,6) Solution 1 A Chapter 10, Collect-on-Delivery Call The COD call is a gap-call that has a strike price of K1 75 P and a trigger price of K 75. The first step to pricing the gap call is to calculate d 1 and d : We have: ln( S/ K ) ( r 0.5 )( T t) ln(80 / 75) ( ) 1 d1 T t d d T Nd ( ) N ( ) Nd ( ) N ( ) The initial price of the gap call is: T rt GapCall Se Nd ( 1) Ke 1 Nd ( ) e (75 P) e ActuarialBrew.com 016 Page 9

10 Since this gap call is a COD call, its initial price is zero: e (75 P) e e P 0.04 e P If the final stock price is $111, then the payoff is: ST ( ) KP Solution 13 B Chapter 1, Exchange Options Let Stock A be the strike asset and let 8 shares of Stock B be the underlying asset. The value of 8 shares of Stock B in dollars is: dollars We can use put-call parity to find the value of an option that allows its owner to exchange 8 shares of Stock B for 1 share of Stock A: Eur P 10 Eur ,30, P P , 0, /1 CEur 40,30, P 40,30, F (8 B) F ( A) ,30, 40 30e PEur 1 The put option gives its owner the right to give up 8 shares of Stock B and receive 1 share of Stock A. The right to give up 4 shares of Stock B and receive 3 shares of Stock A is the same as owning 3 of these put options: The value of a an exchange option that gives its owner the right to exchange 4 shares of Stock B for 3 shares of Stock A at the end of 10 months is $ Solution 14 E Chapter 18, Lognormal Prediction Intervals For a 90% lognormal prediction interval, we set p 10% in the expression below: U U ( 0.5 )( T t) z T t U p ST St e where: P z z ActuarialBrew.com 016 Page 10

11 First, we use the standard normal calculator to determine U P z z U P z z U 1P z z 0.05 U P z z 0.95 U z The upper bound is: U z : U ST ( (0.30) )(10) 0.3( ) 10 50e Solution 15 E Chapter 4, Utility Values and State Prices The payoffs of the put option are: V V u d Max(0,60 80) 0 Max(0,60 35) 5 The price of the put option is: V Q V Q V pu 0 (1 pu ) (0.86) (0.97) 5 u u d d u d The expected return on the put option can now be calculated: 0.5 pvu (1 p) Vd (1 Put ) V (1 Put ) Put Solution 16 A Chapter 8, Volatility of an Option The first step is to calculate d 1 and d : ln( S/ K) ( r 0.5 ) T d1 T ln(50 / 55) ( ) d d1 T ActuarialBrew.com 016 Page 11

12 We have: N( d1 ) N ( ) N( d ) N ( ) The value of the put option is: rt T PEur( S, K,, r, T, ) Ke N( d) Se N( d1) 0.03(1) 0.0(1) 55e e The value of delta is: T 0.0(1) Put e N( d1 ) e ( ) The elasticity of the option is: S 50 ( ).8064 V The volatility of the put option is: Option Stock Option Solution 17 E Chapter 19, Cox-Ingersoll-Ross Model In the Cox-Ingersoll-Ross Model, we have: B(, tt) r PrtT AtTe (,, ) (, ) We use the following two facts about the CIR model: A(, tt ) and B( tt, ) do not depend on r. AtT (, ) A(0, Tt ) and B(, tt) B(0, T t ). This implies: A(0,3) A(,5) A(3,6) B(0,3) B(,5) B(3,6) We have two equations and two unknowns: B(0,3)(0.08) A(0,3) e B(0,3)(0.10) A(0,3) e ActuarialBrew.com 016 Page 1

13 Dividing the second equation into the first equation allows us to find B (0,3) : B(0,3)(0.08) B(0,3)(0.10) e B(0,3)(0.0) ln B(0,3).1809 We can now solve for the value of A (0,3) : B(0,3)(0.08) A(0,3) e B(0,3)(0.08) A(0,3) e.1809(0.08) A(0,3) e A(0,3) We can now solve for r * : B(0,3)( r*) A(0,3) e r* e r* ln r* Solution 18 B Chapter 1, Synthetic T-bills The T-bill is replicated by purchasing stock, selling a call option, and buying a put: rt 0 T Eur r(0.75) 0.04(0.75) Ke S e C ( K, T) P ( K, T) 40e 36e e r(0.75) r ln( ) r Eur Solution 19 B Chapter 7, Black-Scholes Formula The volatility parameter is: Var[ln St ] 0.3t t t 0.3 We can use the version of the Black-Scholes formula that is based on prepaid forward prices to find the value of the put option: ActuarialBrew.com 016 Page 13

14 d 1 P F 0, ( ) ln T S 0.5 T 75 P ln F0, T ( K) T d d T N( d ) N( ) N( d ) N(0.6137) The price of the put option is: P P P P Eur 0, T 0, T 0, T 0, T 1 P F ( S), F ( K),, T F ( K) N( d ) F ( S) N( d ) Solution 0 B Chapter 15, Forward Price of S a The drift of the risk-neutral Itô process for the stock is equal to the difference between the risk-free rate and the dividend yield: 0.06 dt ( r ) dt 0.06 r The expected value of the claim under the risk-neutral valuation measure is equal to the forward price of the claim: ( ) ( ) (0) * E a S T F a 0, T S T a S e 5 1 1(0.06) 0.5( 1)( 11)0.5 4 e e 0.00 a( r) 0.5 a( a1) T Solution 1 B Chapter 5, Expected Value and Median of Stock Price From the distribution of the natural log of the stock price, we observe that: The Sharpe ratio of the call option is equal to the Sharpe ratio of the stock: r ActuarialBrew.com 016 Page 14

15 The expected value of S(3) is: ( ) t ESt [ ( )] S(0) e ( )3 ES [ (3)] 65e The median value of S(3) is: ( 0.5 ) t ( )3 S(0) e 65e 65 The expected value exceeds the median by: Solution B Chapter 19, Theta in the Cox-Ingersoll-Ross Model The process describes the Cox-Ingersoll-Ross-Model with: a( r) a( br) 0.13(0.07 r) ( r) r 0.18 r The price of a 15-year bond is: BtT (, ) r PrtT (,, ) AtTe (, ) e The formula for the price of a bond must satisfy the following partial differential equation: 1 rp [ ( r )] Prr a ( r ) ( r ) ( r, t ) Pr Pt Delta and gamma for the bond are: BtT (, ) r PrtT (,, ) AtTe (, ) Delta: P BtT (, ) PrtT (,, ) r Gamma: P [ BtT (, )] PrtT (,, ) (4.7193) rr We can now solve for theta: 1 rp [ ( r )] Prr a ( r ) ( r ) ( r, t ) Pr Pt 1 [ ( )] Pt rp r Prr a ( r ) ( r ) ( r, t ) Pr 1 Pt ( ) (0) (.0007) ActuarialBrew.com 016 Page 15

16 Solution 3 C Chapter 1, Variance of Control Variate Estimate The formula from the ActuarialBrew.com Study Manual that has X as the control variate and Y * as the control variate estimate is: * 1 X, Y Var Y Var Y In this question, however, the control variate is denoted by A and the control variate estimate is denoted by B *, so we have: Var B* Var B AB, Solution 4 D Chapter 14, Geometric Brownian Motion and Mutual Funds Since b is greater than 1, the mutual fund takes a leveraged position in the stock by borrowing at the risk-free rate. The instantaneous percentage increase of the mutual fund is the weighted average of the return on the stock (including its dividend yield) and the return on the risk-free asset: dw() t ds() t b dt(1 b) rdt Wt () St () dt 0.5 dz( t) 0.05dt 0.3(0.06) dt dt 0.5 dz( t) 0.3(0.06) dt 0.156dt 0.35 dz( t) 0.018dt 0.138dt 0.35 dz( t) As written above, we see that Wt () is a geometric Brownian motion. Therefore, the expected value can be expressed as: EWt () 0.138t W(0) e EW(5) e EW(5) Solution 5 D Chapter 17, Caplets and the Black Model This question s method of presenting forward price volatilities is the same as that used in Problem 5. at the end of the Derivatives Markets textbook Chapter 5. The price of the caplet is: Caplet price (1.11) (Put Option Price) ActuarialBrew.com 016 Page 16

17 The put option: has a zero-coupon bond that expires at time 4 as its underlying asset, so T s 4 expires at time 3, so T 3 has a strike price of: 1 1 K KR 1.11 The forward price volatilities are: ln t(1,) ln t(,3) ln t(3,4) Var P Var P Var P t t t The appropriate volatility for an option that expires in 3 years on a bond that matures in 4 years is: 0.15 The bond forward price is: P(0, T s) P(0,4) 0.68 F P0 ( T, T s) P(0, T) P(0,3) 0.77 The values of d 1 and d are: We have: F ln 0.5 T ln 0.5(0.15) (3) K d T d d1 T N( d1 ) N ( ) N( d ) N (0.0665) The Black formula for the put price is: P P(0, T) K N( d) F N( d1) 0.77[ ] The price of the caplet for $1 of borrowing is: Caplet price (1.11) (Put Option Price) The price of the caplet for $1,000 of borrowing is: 1, ActuarialBrew.com 016 Page 17

18 Solution 6 C Chapter 4, American Call Option & State Prices The stock price tree is: 50 The strike price K, for which an investor will exercise the call option at the beginning of the period must be less than $50, since otherwise the payoff to immediate exercise would be zero. Since we are seeking the largest strike price that results in immediate exercise, let s begin by determining whether there is a strike price that is less than $50 but more than $35 that results in immediate exercise. If 35 K 50, then the rightmost portion of the tree for the call option is: K If there is strike price that is greater than $35 that results in immediate exercise, then the value of exercising now must exceed the value of holding the option: 50 K (66 K) Q H 50 K (66 K) K K K K The inequality above suggests that $ is the largest strike price for which the investor will exercise the call option immediately. The largest integer that satisfies this inequality is: INT( ) 38 The difference is: K INT( K) ActuarialBrew.com 016 Page 18

19 Solution 7 B Chapter 15, Sharpe Ratio and Option Volatility The stock and the option must have the same market price of risk at all times, including time : V When an option is purchased, it is delta-hedged by purchasing shares of stock. The cost of the shares required to delta-hedge the option is the number of shares required,, times the cost of each share, S. Therefore, based on statement (vi) in the question, we have: S 7 S 7 The fact that is negative is not surprising in light of the fact that the option s expected return is negative. We can find the volatility parameter of the option in terms of the volatility of the underlying stock: S 7 V V V 5.30 We can now solve for : The general form for a dividend-paying stock that follows geometric Brownian motion is: ds() t ( ) dt dz( t) St () We can use the differential equation provided in the question to find the dividend yield: ds() t 0.15 dt dz( t) St () Solution 8 B Chapter 11, Cash-or-Nothing Call Options The option can be replicated by purchasing 00 cash calls with a strike price of $50 and selling 100 cash calls with a strike price of $100. Therefore, the price of the option is: 00 CashCall(50) 100 CashCall (100) ActuarialBrew.com 016 Page 19

20 First, we find the value of the cash call with a strike price of $50: T Se 100e 0.4 ln T ln 1 rt Ke 50e d T Nd ( ) N(1.6070) rt CashCall(50) e N( d ) e ( ) Next, we find the value of the cash call with a strike price of $100: T Se 100e 0.4 ln T ln 1 rt Ke 100e d T Nd ( ) N( ) rt CashCall(100) e N( d ) e (0.487) The value of the option described in the question is: 00 CashCall(50) 100 CashCall(100) e ( ) 100 e (0.487) Solution 9 E Chapter 1, Currency Options Greg s call option gives him the right to give up $1.5 to receive Marcia s put option gives her the right to give up $1.00 to receive Note that the payoff of Greg s options is 1.5 times the payoff of Marcia s option. Therefore, Greg s option is worth 1.5 times Marcia s option. The value of Greg s option in dollars is Z, and the value of Greg s option in pounds is: Z 1.41 The value of Greg s option in pounds is 1.5 times as much as the value of Marcia s option: Z Y We can now solve for the ratio of Z to Y: Z Z 1.5 Y Y ActuarialBrew.com 016 Page 0

21 Solution 30 A Chapter 7, Options on Futures We use put-call parity to find the current futures price: rt rt Eur 0, TF 0, TF Eur 0, TF e F0,3e F0, C ( F, K,, rtr,, ) Ke F e P ( F, K,, rtr,, ) We can use the futures price to solve for the stock price: ( r ) T F F 0, T S0e F ( ) S0e S ActuarialBrew.com 016 Page 1

Course MFE/3F Practice Exam 1 Solutions

Course MFE/3F Practice Exam 1 Solutions he chapter references below refer to the chapters of the ActuraialBrew.com Study Manual. Solution 1 C Chapter 16, Sharpe Ratio If we (incorrectly) assume that the