Deeper Understanding, Faster Calc: SOA MFE and CAS Exam 3F. Yufeng Guo

Deeper Understanding, Faster Calc: SOA MFE and CAS Exam 3F Yufeng Guo

Contents Introduction ix 9 Parity and other option relationships 1 9.1 Put-callparity... 1 9.1.1 Optiononstocks... 1 9.1. Optionsoncurrencies... 11 9.1.3 Optionsonbonds... 13 9.1.4 Generalizedparityandexchangeoptions... 13 9.1.5 Comparing options with respect to style, maturity, and strike... 19 10 Binomial option pricing: I 35 10.1One-periodbinomialmodel:simpleexamples... 35 10.Generalone-periodbinomialmodel... 36 10..1 Twoormorebinomialtrees... 49 10.. Optionsonstockindex... 64 10..3 Optionsoncurrency... 67 10..4 Optionsonfuturescontracts... 71 11 Binomial option pricing: II 79 11.1Understandingearlyexercise... 79 11.Understandingrisk-neutralprobability... 80 11..1 Pricing an option using real probabilities... 81 11.. Binomialtreeandlognormality... 88 11..3 Estimate stock volatility... 91 11.3Stockspayingdiscretedividends... 95 11.3.1 Problemswithdiscretedividendtree... 97 11.3. Binomialtreeusingprepaidforward... 98 1 Black-Scholes 105 1.1IntroductiontotheBlack-Scholesformula...105 1.1.1 Callandputoptionprice...105 1.1. WhenistheBlack-Scholesformulavalid?...107 1.DerivetheBlack-Scholesformula...107 iii

iv CONTENTS 1.3Applyingtheformulatootherassets...11 1.3.1 Black-Scholes formula in terms of prepaid forward price. 11 1.3. Optionsonstockswithdiscretedividends...11 1.3.3 Optionsoncurrencies...1 1.3.4 Optionsonfutures...13 1.4OptiontheGreeks...14 1.4.1 Delta...14 1.4. Gamma...15 1.4.3 Vega...15 1.4.4 Theta...16 1.4.5 Rho...16 1.4.6 Psi...16 1.4.7 Greekmeasuresforaportfolio...16 1.4.8 Option elasticity and volatility...17 1.4.9 OptionriskpremiumandSharperatio...18 1.4.10Elasticityandriskpremiumofaportfolio...19 1.5 Profit diagramsbeforematurity...19 1.5.1 Holding period profit...19 1.5. Calendarspread...13 1.6Impliedvolatility...133 1.6.1 Calculatetheimpliedvolatility...133 1.6. Volatility skew...134 1.6.3 Usingimpliedvolatility...135 1.7PerpetualAmericanoptions...135 1.7.1 Perpetualcallsandputs...135 1.7. Barrierpresentvalues...140 13 Market-making and delta-hedging 143 13.1Deltahedging...143 13.ExamplesofDeltahedging...143 13.3TextbookTable13....15 13.4TextbookTable13.3...154 13.5MathematicsofDeltahedging...155 13.5.1 Delta-Gamma-Thetaapproximation...155 13.5. Understanding the market maker s profit...156 14 Exotic options: I 159 14.1Asianoption(i.e.averageoptions)...159 14.1.1 Characteristics...159 14.1. Examples...160 14.1.3 Geometricaverage...160 14.1.4 Payoff at maturity...160 14.Barrieroption...161 14..1 Knock-inoption...161 14.. Knock-outoption...161

CONTENTS v 14..3 Rebateoption...161 14..4 Barrierparity...16 14..5 Examples...16 14.3Compoundoption...163 14.4Gapoption...165 14.4.1 Definition...165 14.4. Pricingformula...165 14.4.3 Howtomemorizethepricingformula...165 14.5Exchangeoption...166 18 Lognormal distribution 169 18.1Normaldistribution...169 18.Lognormaldistribution...170 18.3Lognormalmodelofstockprices...170 18.4Lognormalprobabilitycalculation...171 18.4.1 Lognormal confidenceinterval...17 18.4. Conditionalexpectedprices...176 18.4.3 Black-Scholesformula...177 18.5Estimatingtheparametersofalognormaldistribution...177 18.6Howareassetpricesdistributed...179 18.6.1 Histogram...179 18.6. Normalprobabilityplots...180 18.7Sampleproblems...18 19 Monte Carlo valuation 187 19.1 Example 1 Estimate ( )...187 19. Example Estimate...191 19.3 Example 3 Estimate the price of European call or put options.. 194 19.4Example4Arithmeticandgeometricoptions...198 19.5 EfficientMonteCarlovaluation...07 19.5.1 Controlvariancemethod...07 19.6Antitheticvariatemethod...11 19.7 Stratifiedsampling...13 19.7.1 Importancesampling...13 19.8Sampleproblems...13 0 Brownian motion and Ito s Lemma 19 0.1Introduction...19 0.1.1 Bigpicture...0 0.Brownianmotion...1 0..1 Deterministicprocessvs.stochasticprocess...1 0.. DefinitionofBrownianmotion... 0..3 Martingale...4 0..4 PropertiesofBrownianmotion...31

vi CONTENTS 0..5 Arithmetic Brownian motion and Geometric Brownian motion...36 0..6 Ornstein-Uhlenbeckprocess...37 0.3 Definitionofthestochasticcalculus...38 0.3.1 Whystochasticcalculus...38 0.4Propertiesofthestochasticcalculus...48 0.5Ito slemma...51 0.5.1 Multiplicationrules...51 0.5. Ito slemma...5 0.6GeometricBrownianmotionrevisited...53 0.6.1 Relativeimportanceofdriftandnoiseterm...54 0.6. CorrelatedItoprocesses...54 0.7 Three worlds:,, and...61 0.8Sharperatio...305 0.9Girsanov stheorem...307 0.10Riskneutralprocess...317 0.11Valuing a claim on...39 0.11.1 Process followed by...39 0.11. Formula for () and E ()...330 0.11.3 Expected return of a claim on ()...331 0.11.4 Specific examples...331 1 Black-Scholes equation 341 1.1 Differential equations and valuation under certainty...341 1.1.1 Valuationequation...341 1.1. Bonds...34 1.1.3 Dividendpayingstock...34 1.Black-Scholesequation...34 1..1 HowtoderiveBlack-Scholesequation...34 1.. Verifyingtheformulaforaderivative...343 1..3 Black-Scholesequationandequilibriumreturns...346 1.3Risk-neutralpricing...348 Exotic options: II 349.1 All-or-nothingoptions...349 3 Volatility 351 4 Interest rate models 353 4.1Market-makingandbondpricing...353 4.1.1 Reviewofdurationandconvexity...353 4.1. Interestrateisnotsosimple...360 4.1.3 Impossiblebondpricingmodel...361 4.1.4 Equilibrium equation for bonds...365 4.1.5 Delta-Gammaapproximationforbonds...370 4.Equilibriumshort-ratebondpricemodels...371

CONTENTS vii 4..1 ArithmeticBrownianmotion(i.e.Mertonmodel)...371 4.. Rendleman-Barttermodel...37 4..3 Vasicekmodel...373 4..4 CIRmodel...395 4.3Bondoptions,caps,andtheBlackmodel...400 4.3.1 Blackformula...400 4.3. Interestratecaplet...403 4.4Binomialinterestratemodel...404 4.5Black-Derman-Toymodel...408

Introduction This study guide is for SOA MFE and CAS Exam 3F. Before you start, make sure you have the following items: 1. Derivatives Markets, the nd edition.. Errata of Derivatives Markets. You can download the errata at http:// www.kellogg.northwestern.edu/faculty/mcdonald/htm/typose_01. html. Don t miss the errata about the textbook pages 780 through 788. 3. Download the syllabus from the SOA or CAS website. 4. Download the sample MFE problems and solutions from the SOA website. 5. Download the recent SOA MFE and CAS Exam 3 problems. Please report any errors to yufeng.guo.actuary@gmail.com. ix

Chapter 1 Black-Scholes You probably have memorized the famous Black-Scholes call and put price formulas and can readily calculate the price of a plain vanilla European call or put option. But what if SOA throws a tricky derivative at you? Here are a few examples of ad hoc contracts: An option allows you to pay and receive ln at. What s its price? An option allows you to pay and receive at.what sitsprice? An option pays ( ) at only if. What s its price? To tackle non-standard derivatives, you need to do more than memorize the Black-Scholes formula. In this chapter, you ll learn how to derive the Black- Scholes formula from the ground up and how to price an ad hoc contract. The math behind the Black-Scholes formula is simple. All you need to know is (1) some Exam P level calculus, and () the risk neutral pricing (an option is worth its expected payoff discounted at the risk free rate). First, though, let s review the basics of the Black-Scholes formula. 1.1 Introduction to the Black-Scholes formula 1.1.1 Call and put option price The price of a European call option is: ( ) = ( 1 ) ( ) (1.1) The price of a European put option is: ( ) = ( 1 )+ ( ) (1.) 105

106 CHAPTER 1. BLACK-SCHOLES 1 = ln + µ + 1 (1.3) Notations used in Equation 1.1, 1.3, and 1.4: = 1 (1.4), the current stock price (i.e. the stock price when the option is written), thestrikeprice, the continuously compounded risk-free interest rate per year, the continuously compounded dividend rate per year, the annualized standard deviation of the continuously compounded stock return (i.e. stock volatility), option expiration time () = ( ) where is a standard normal random variable ( ), the price of a European call option with parameters ( ) ( ), the price of a European put option with parameters ( ) Tip 1.1.1. To help memorize Equation 1., we can rewrite Equation 1. similar to Equation 1.1 as ( ) =( ) ( 1 ) ( ) ( ). In other words, change,, 1,and in Equation 1.1 and you ll get Equation 1.. Example 1.1.1. Reproduce the textbook example 1.1. This is the recap of the information. =41, =40, =008, =03, =05 (i.e. 3 months), and =0. Calculate the price of the price of a European call option. ln µ + + 1 1 = ln 41 µ008 40 + 0+ 1 03 05 = 03 =03730 05 = 1 =03730 03 05 = 030 ( 1 )=0645 4 ( )=0588 =41 0(05) 0645 4 40 008(05) 0588 = 3 399

1.. DERIVE THE BLACK-SCHOLES FORMULA 107 Example 1.1.. Reproduce the textbook example 1.. This is the recap of the information. =41, =40, =008, =03, =05 (i.e. 3 months), and =0. Calculate the price of the price of a European put option. ( 1 )=1 ( 1 )=1 0645 4 = 0354 6 ( )=1 ( )=1 0588 = 0411 8 = 41 0(05) 0354 6 + 40 008(05) 0411 8 = 1 607 1.1. When is the Black-Scholes formula valid? Assumptions under the Black-Scholes formula: Assumptions about the distribution of stock price: 1. Continuously compounded returns on the stock are normally distributed (i.e. stock price is lognormally distributed) and independent over time. The volatility of the continuously compounded returns is known and constant 3. Future dividends are known, either as a dollar amount (i.e. and are known in advance) or as a fixed dividend yield (i.e. is a known constant) Assumptions about the economic environment 1. The risk-free rate is known and fixed (i.e. is a known constant). There are no transaction costs or taxes 3. It s possible to short-sell costlessly and to borrow at the risk-free rate 1. Derive the Black-Scholes formula By learning how to derive the Black-Scholes formula, we ll be able to remove the black-box behind the formula and recreate the formula instantly. First, some basics. What s the density function of a standard normal random variable? If (0 1), then () = 1 05, Φ () = ( ) = R () How can we convert a normal variable to a standard normal variable? If,thenset = ; = + What s the stock price under the Black-Scholes assumption in the real world? DM 0.13: = 0 exp 05 +, (0 1) What s under the Black-Scholes assumption in the risk neutral world? Set =. Then = 0 exp 05 +, (0 1)

108 CHAPTER 1. BLACK-SCHOLES Next, let s derive some normal random variable related integral shortcuts. The first shortcut. For a standard normal random variable (0 1) and a constant, Z () = ( )=Φ ( ) (1.5) Proof. Clearly, R () = ( ). We just need to prove that ( )= Φ ( ). ( )=1 ( )=1 Φ (). Notice that ( = ) =0(the probability for a continuous random variable to take on a single value is zero). Hence ( )=1 ( ) ( = ) =1 ( ). Then using the formula Φ ()+Φ( ) =1, we get 1.5. The second integral shortcut: Z () = 05 Z 1 05( ) (1.6) Proof. (0 1) and () = 1 05 R () = R 1 05 = R 1 05 + 05 + = 05 + +05 = 05( ) +05 R 1 05 + = R 1 05( ) +05 = 05 R 1 05( ) The third shortcut is: Z Proof. R R () = 05 Set =. R () = 05 Φ ( ) (1.7) 1 05( ) 1 05( ) = R 1 05 1 05 is the density of (0 1) R 1 05 = Φ ( ( )) = Φ ( ). R () = 05 Φ ( ).

1.. DERIVE THE BLACK-SCHOLES FORMULA 109 Problem 1.1. For a normal random variable,calculate. = +, (0 1). = R + () = R () Use Equation R () = 05 Φ ( ), set = : R () = 05 Φ ( )= 05 Φ ( ) = 05 1= 05 = = 05 = +05 = ()+05() ³ ) ( = ()+05() = +05 (1.8) Problem 1.. For a normal random variable,calculate R () where (0 1) and () = 1 05. R () () = R + () = R () = 05 Φ ( ) = +05 Φ ( ) = Φ ( ) Z () = Φ (std dev of ) = Φ ( ) (1.9) Tip 1..1. Whenever you need to calculate R () where, first calculate = +05.Next,multiply by Φ ( ). Problem 1.3. For a normal random variable, calculate R () where (0 1) and () = 1 05.

110 CHAPTER 1. BLACK-SCHOLES R () = R () R () = Φ ( ) = [1 Φ ( )] = Φ [ ( )] Z () = Φ [ (std dev of )] = Φ [ ( )] (1.10) Tip 1... Whenever you need to calculate R () where, first calculate = +05. Next, multiply by Φ [ ( )]. Problem 1.4. Derive the Black-Scholes call option formula 1.1. Consider two contracts: Contract #1 pays at if. The payoff at is 1 = ½ If 0 If. ½ If Contract # pays at if.thepayoffat is = 0 If Let 1 and represent the price of the Contract #1 and # respectively. Under the risk neutral pricing, the price of a contract is just the expected payoff discounted at the risk free rate. Hence 1 = 1 and =,where ½ is the risk-neutral world. Since 1 = If is the payoff of a call option, the 0 If call option price is equal to = 1. = 0, = 05 = h Solve () = 0 exp 05 + i ln 05 0 = ln 0 Notice = + 05 in the Black-Scholes formula. Then () isthesameas. Hence we can write 1 and as follows:

1.. DERIVE THE BLACK-SCHOLES FORMULA 111 ½ ½ 1 = If 0 If = If 0 If ½ ½ If = If 0 If = 0 If R = 0 () +R () = R () = R () = Φ ( ) = = ( )= Φ ( ) Notice Φ ( )= R () = ( ) is the risk neutral probability of. R 1 = () () + R 0 () = R () () = R R R () () = 0 () () = 0 ( 05 ) + () Use the formula: R () = Φ (std dev of ) = 0 R () = 0 Φ (std dev of + ) = ()+05() = ( 05 ) +05 = ( ) Φ (std dev of + )=Φ ³ + = Φ ( 1 ) = 1 = 0 ( ) Φ ( 1 ) = = 1 = 0 Φ ( 1 ) Φ ( ) Problem 1.5. Derive the Black-Scholes put option formula 1.. Consider two contracts: ½ Contract #1 pays at if. The payoff at is 1 = If 0 If. ½ If Contract # pays at if. The payoff at is = 0 If Let 1 and represent the price of the Contract #1 and # respectively. 1 = 1 and = ½,where is the risk-neutral world. Since 1 = If is the payoff of a put option, the put 0 If option price is equal to = 1.

11 CHAPTER 1. BLACK-SCHOLES Solve (). ln 05 0 = Then () isthesameas. Hence we write 1 and as follows: ½ ½ 1 = If 0 If = If 0 If ½ ½ If = If 0 If = 0 If R = () + R 0 () = R () = Φ ( ) = = ( )= Φ ( ) Φ ( )= R () = ( ) is the risk neutral probability of. R 1 = () () + R 0 () = R () () = R R 0 () = 0 () where = 05 = R () = Φ [ (std dev of ( ))] = h Φ ³ + i = Φ ( 1 ) = ()+05() = ( 05 ) +05 = ( ) = 1 = ( ) Φ ( 1 ) = 1 = Φ ( 1 ) = = 1 = Φ ( ) 0 Φ ( 1 ) Problem 1.6. What s the meaning of Φ ( ) and Φ ( ) in the Black-Scholes formula? Φ ( )= ( ) is the risk neutral probability of. Φ ( )= ( ) is the risk neutral probability of. Problem 1.7.

1.. DERIVE THE BLACK-SCHOLES FORMULA 113 Since Φ ( ) is the risk neutral probability of, I thought the call price should be 0 Φ ( ) Φ ( ), but the Black-Scholes formula is 0 Φ ( 1 ) Φ ( ).Why? The wrong formula = 0 Φ ( ) Φ ( )= 0 Φ ( ) can easily produce a negative or zero call price when 0. For example, set = = 0 and 0 =. The wrong formula is = ( 0 ) Φ ( )=0, but a call price is always positive. Here s another example. To simplify calculation, set =0, =006, 0 = 50, = 100, =1, =1. = ln + µ 1 133 15 1 = + = 1 133 15 + 1 1= 0133 15 ( )=NormalDist( 1 133 15) = 018 58 ( 1 )=NormalDist( 0133 15) = 0447 04 = ln 50 100 + 006 0 05 1 1 1 1 = 1 ³ Notice that 1 = + ( ) because the cumulative density function () is an increasing function of. Thecorrectcallpriceis = 0 Φ ( 1 ) Φ ( )=50 0447 04 100 006 018 58 = 10 4 The wrong call price is = 0 Φ ( ) Φ ( )= 50 100 006 018 58 = 5 68 Third example. Set =0, =006, 0 =1, = 100, =1, =1 ln µ + 1 ln 1 = = 100 + 006 0 05 1 1 1 1 045 17 1 = + = 5 045 17 + 1 1= 4045 17 ( )=NormalDist( 5 045 17) = 65 59 10 7 ( 1 )=NormalDist( 4 045 17) = 614 6 10 5 ( 1 ) ( ) 614 6 10 5 = = 115 389 8 65 59 10 7 Thecorrectcallpriceis = 5

114 CHAPTER 1. BLACK-SCHOLES µ = 0 Φ ( 1 ) Φ ( 1 ) Φ ( )=Φ( ) 0 Φ ( ) =65 59 10 7 1 115 389 8 100 006 =48 10 6 Φ ( 1 ) In this example, as becomes much greater than 0, gets big as µ Φ ( ) Φ ( 1 ) well, making the call price Φ ( ) 0 Φ ( ) positive. In contrast, the wrong formula = 0 Φ ( )= 1 100 006 65 59 10 7 = 11 10 5 produces a negative call price. Similarly, though Φ ( )= ( ) is the risk neutral probability of, the put price is NOT = Φ ( ) 0 ( ) but = Φ ( ) 0 ( 1 ) Since = 1,wehave Φ ( ) ( 1 ). This makes the put price Φ ( ) 0 ( 1 ) positive. Problem 1.8. Verify that the first term of the Black-Scholes call price formula 0 ( 1 ) is equal to ( ) ( ),notequalto ( ) ( ) the nd term of the Black-Scholes put price formula 0 ( 1 ) is equal to ( ) ( ),notequalto ( ) ( ) isthesameas Notice that Z () () = {z } ( ) where 1 and 1 Z () () {z } ( 1 )= ( ) ( ) are the payoffs of the following contracts: + Z () () {z } ( 1 )= ( ) ( ) Contract #1 pays at if. The payoff at is 1 = ½ If 0 If. ½ Contract #1 pays at if. The payoff at is 1 = If 0 If

1.. DERIVE THE BLACK-SCHOLES FORMULA 115 1 = ( ) ( )= 0 ( 1 )= 0 ( ) ( 1 ) = 1 = ( ) ( ) = 0 ( 1 ) 1 = ( ) ( )= 0 ( 1 )= 0 ( ) ( ) = 1 = ( ) ( ) = 0 ( 1 ) Problem 1.9. What s the meaning of ( 1 ) and ( 1 )? Laterinthechapteraboutthe world (where is used as the numeraire) we ll learn that ( 1 )= ( ) (the V world probability of ) and ( 1 )= ( ) (the V world probability of ). Now let s interpret ( 1 ) and ( 1 ) in a different way. Consider two contracts: Contract #1 pays at if. The payoff at is 1 = ½ If 0 If. ½ Contract # pays at if. The payoff at is 1 = If 0 If Notice that ( )= R () () = Z () () {z } ( 1 )= ( )Φ( 1) Z + () () {z } ( 1 )= ( )Φ( 1 ) ( ) consists of two parts, one to pay for 1 = ( ) Φ ( 1 ) and the other to pay for 1 = ( ) Φ ( 1 ).Weseethat Φ ( 1 ) is the fraction of ( ) to pay for 1 and Φ ( 1 ) = 1 Φ ( 1 ) is the remaining fraction of ( ) to pay for 1 ½ Alternatively, notice that 1 + 1 = If If = Hence 1 + 1 = ( ). So ( ) consists of two parts, one to pay for 1 = ( ) Φ ( 1 ) and the other to pay for 1 = ( ) Φ ( 1 ),whereφ( 1 ) and Φ ( 1 ) represent the fraction of ( ) used to pay for 1 and 1 respectively.

116 CHAPTER 1. BLACK-SCHOLES Problem 1.10. A silly option gives its owner the right to receive ln at by paying. The assumptions under the Black-Scholes formula hold. Calculate the option price. ½ ln If ln The option payoff at is =. The option 0 If ln priceattimezerois = ( ). h = 0 exp 05 + i h Solve ln ln =ln 0 + 05 + i ln 0 05 = where = ln 0 + 05 ( )= R (ln ) () = R h ln 0 + 05 + i () To calculate R (), notice that for (0 1) and () = 1 05 () = 1 05 = µ 1 05 = [ ()] Hence R () = R [ ()] = R [ ()] = () = () ( ) = () 0= () R Z () = () = 1 05 (1.11) h ln 0 + 05 + i () ln 0 + 05 () + R () = R = ln 0 + 05 Φ ( )+ ( ) The option price is = ( )= ln 0 + 05 Φ ( )+ ( ) Problem 1.11.

1.. DERIVE THE BLACK-SCHOLES FORMULA 117 A silly option gives its owner the right to receive at by paying. The assumptions under the Black-Scholes formula hold. Calculate the option price. We ll calculate a generic option where the option owner has the right to get cash equal to ( ) = (where 6= 0)at by paying. ½ The option payoff at is = If 0 If. The option price at time zero is = ( ). h = 0 exp 05 + i h = 0 exp 05 + i Solve : ln ln ln ln 0 05 = ln ln 0 05 = = ln 0 + 05 ( )= R ( ) () = R () R () R () = Φ ( ) R () = R = 0 ( 05 ) 05 Φ 0 h exp 05 + i ³ + () Set + = 1 = ( )= 0 ( 05 ) 05 Φ ( 1 ) Φ ( ) If =05, then = 0 05( 05 ) 053 Φ ( 1) Φ ( ) where = ln 0 +05 05 05 1 =05 + Problem 1.1.

118 CHAPTER 1. BLACK-SCHOLES A special contract pays ( ) at if. The assumptions under the Black-Scholes formula hold. Calculate the contract price. Method 1 Borrow as much as you can from the Black-Scholes formula ½ ( ) = The contract payoff is = + If 0 If Consider three contracts: ½ Contract #1 pays if. The payoff is 1 = If 0 If. ½ Contract # pays if. The payoff is = If 0 If ½ If Contract #3 pays at if.thepayoffis 3 = 0 If Let 1 and 3 represent the price of the above contracts respectively. Let represent the price of the contract with payoff. We replicate by buying 1 unit of Contract #1, selling units of Contract #, and buying units of Contract #3: = 1 + 3 = = 1 + 3 is equal to the first component of the Black-Scholes call price: = 0 Φ ( 1 ) where = + 3 is equal to the second component of the Black-Scholes call price: ln 0 µ 3 = + 1 Φ ( ) where = The remaining work is to calculate 1 = R 1 = R () () = R h 0 ( 05 ) + i () = 0 ) +05( ) ( 05 Φ ³ + ³ + = 0 ( ) + Φ = 0 ( ) Φ ³ + 0 Φ ( 1 )+ Φ ( ) () (). By the way, from the previous problem, we know the price of an option that allows you to pay and receive is 0 ( 05 ) 05 Φ ( 1)

1.. DERIVE THE BLACK-SCHOLES FORMULA 119 Φ ( ). You may be attempted to use this formula to calculate 1 by setting =, but that won t work. The contract in the previous problem pays if. In contrast, Contract #1 in this problem pays if (so vs. ). Method Calculate from scratch is the same as. The contract payoff is as () = ½ () ()+ If 0 If The contract price is = ( ) = ( ) ( )= R () () = R () () R () () + R () Evaluating this integral, you should get = 0 ( ) Φ ³ + 0 Φ ( 1 )+ Φ ( ) Problem 1.13. A special contract pays, at,thegreaterof and. Calculate its price. The payoff is ½ ½ () If () =max( )= If = () If If Method 1 ( )= R () () + R () = 0 ( ) ( 1 )+ ( ) = ( )= 0 ( 1 )+ ( ) Method () =max( )=max( 0) + max ( 0) is the payoff of a call option. It s price is 0 ( 1 ) ( ). The price of is. Hence the contract price is = 0 ( 1 ) ( )+ = 0 ( 1 )+ [1 ( )] = 0 ( 1 )+ ( ) Problem 1.14.

10 CHAPTER 1. BLACK-SCHOLES A special contract pays at. Calculate its price. The payoff is ½ () = = If If Method 1 Notice that If is the payoff of a call option; If is the payoff of a put option. Hence the contract price is = 0 ( 1 ) ( )+ ( ) 0 ( 1 ) = 0 [ ( 1 ) ( 1 )] [ ( ) ( )] = 0 [ ( 1 ) 1] [ ( ) 1] Method =max( 0) ( ) max( 0) is twice the payoff of a call option. Its price is 0 ( 1 ) ( ) The price of ( ) is 0 The contract price is = 0 ( 1 ) ( ) 0 Problem 1.15. Derive the gap call price formula DM 14.15. 1 is the ½payment amount; is the payment ½ trigger. The payoff is: () () = 1 If () = 1 If 0 If 0 If ln 0 + µ 1 where = is calculated by solving: h = 0 exp 05 + i ln µ 1 0 ln 0 + µ 1 = = ( )= R [ () 1 ] () = R () () R 1 () = 0 ( 1 ) 1 ( ) where 1 = +