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www.downloadslide.com www.downloadslide.com www.downloadslide.com OPTIONS, FUTURES, AND OTHER DERIVATIVES TENTH EDITION www.downloadslide.com This page intentionally left blank A01_GORD2302_01_SE_FM.indd 28/05/15 7:33 pm www.downloadslide.com OPTIONS, FUTURES, AND OTHER DERIVATIVES John C Hull Maple Financial Group Professor of Derivatives and Risk Management Joseph L Rotman School of Management University of Toronto TENTH EDITION New York, NY www.downloadslide.com Vice President, Business Publishing: Donna Battista Director of Portfolio Management: Adrienne D’Ambrosio Director, Courseware Portfolio Management: Ashley Dodge Senior Sponsoring Editor: Neeraj Bhalla Editorial Assistant: Kathryn Brightney Vice President, Product Marketing: Roxanne McCarley Director of Strategic Marketing: Brad Parkins Strategic Marketing Manager: Deborah Strickland Field Marketing Manager: Ramona Elmer Product Marketing Assistant: Jessica Quazza Vice President, Production and Digital Studio, Arts and 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Education, Inc., or its affiliates, authors, licensees, or distributors Library of Congress Cataloging-in-Publication Data Hull, John, 1946–, author Options, futures, and other derivatives / John C Hull, University of Toronto Tenth edition New York: Pearson Education, [2018] Revised edition of the author’s Options, futures, and other derivatives, [2015] Includes index 2016051230 | 013447208X Futures Stock options Derivative securities HG6024.A3 H85 2017 | 332.64/5–dc23 LC record available at https://lccn.loc.gov/2016051230 10 ISBN-10: 013447208X ISBN-13: 9780134472089 www.downloadslide.com To Michelle www.downloadslide.com CONTENTS IN BRIEF 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 vi List of Business Snapshots xviii List of Technical Notes xix Preface xx Introduction Futures markets and central counterparties 24 Hedging strategies using futures 49 Interest rates 77 Determination of forward and futures prices 107 Interest rate futures 135 Swaps 155 Securitization and the credit crisis of 2007 184 XVAs 199 Mechanics of options markets 209 Properties of stock options 231 Trading strategies involving options 252 Binomial trees 272 Wiener processes and Itoˆ’s lemma 300 The Black–Scholes–Merton model 319 Employee stock options 352 Options on stock indices and currencies 365 Futures options and Black’s model 381 The Greek letters 397 Volatility smiles 430 Basic numerical procedures 449 Value at risk and expected shortfall 493 Estimating volatilities and correlations 520 Credit risk 543 Credit derivatives 569 Exotic options 596 More on models and numerical procedures 622 Martingales and measures 652 Interest rate derivatives: The standard market models 670 Convexity, timing, and quanto adjustments 689 Equilibrium models of the short rate 702 No-arbitrage models of the short rate 715 HJM, LMM, and multiple zero curves 738 Swaps Revisited 757 Energy and commodity derivatives 772 Real options 789 Derivatives mishaps and what we can learn from them 803 Glossary of terms 815 DerivaGem software 838 Major exchanges trading futures and options 843 Tables for NðxÞ 844 Credits 846 Author index 847 Subject index 851 www.downloadslide.com Contents List of Business Snapshots xviii List of Technical Notes xix Preface xx Chapter Introduction 1.1 Exchange-traded markets 1.2 Over-the-counter markets 1.3 Forward contracts 1.4 Futures contracts 1.5 Options 1.6 Types of traders 11 1.7 Hedgers 11 1.8 Speculators 14 1.9 Arbitrageurs 16 1.10 Dangers 17 Summary 18 Further reading 19 Practice questions 19 Further questions 21 Chapter Futures markets and central counterparties 2.1 Background 2.2 Specification of a futures contract 2.3 Convergence of futures price to spot price 2.4 The operation of margin accounts 2.5 OTC markets 2.6 Market quotes 2.7 Delivery 2.8 Types of traders and types of orders 2.9 Regulation 2.10 Accounting and tax 2.11 Forward vs futures contracts Summary Further reading Practice questions Further questions 24 24 26 28 29 32 36 38 39 40 41 43 44 45 45 47 Chapter Hedging strategies using futures 3.1 Basic principles 3.2 Arguments for and against hedging 3.3 Basis risk 3.4 Cross hedging 49 49 51 54 58 vii www.downloadslide.com viii Contents 3.5 3.6 Stock index futures 62 Stack and roll 68 Summary 70 Further reading 70 Practice questions 71 Further questions 73 Appendix: Capital asset pricing model 75 Chapter Interest rates 77 4.1 Types of rates 77 4.2 Swap rates 79 4.3 The risk-free rate 80 4.4 Measuring interest rates 81 4.5 Zero rates 84 4.6 Bond pricing 84 4.7 Determining zero rates 85 4.8 Forward rates 89 4.9 Forward rate agreements 92 4.10 Duration 94 4.11 Convexity 98 4.12 Theories of the term structure of interest rates 99 Summary 101 Further reading 102 Practice questions 102 Further questions 105 Chapter Determination of forward and futures prices 5.1 Investment assets vs consumption assets 5.2 Short selling 5.3 Assumptions and notation 5.4 Forward price for an investment asset 5.5 Known income 5.6 Known yield 5.7 Valuing forward contracts 5.8 Are forward prices and futures prices equal? 5.9 Futures prices of stock indices 5.10 Forward and futures contracts on currencies 5.11 Futures on commodities 5.12 The cost of carry 5.13 Delivery options 5.14 Futures prices and expected future spot prices Summary Further reading Practice questions Further questions 107 107 108 109 110 113 115 115 117 118 120 124 126 127 127 130 131 131 133 Chapter Interest rate futures 6.1 Day count and quotation conventions 6.2 Treasury bond futures 6.3 Eurodollar futures 6.4 Duration-based hedging strategies using futures 6.5 Hedging portfolios of assets and liabilities Summary Further reading 135 135 138 143 148 150 150 151 www.downloadslide.com 714 CHAPTER 31 31.13 Suppose that in a risk-neutral world the CIR parameters are a ¼ 0:15, b ¼ 0:025, and ¼ 0:075 What is the price of a 5-year zero-coupon bond with a principal of $1 when the short rate is 2.5%? pffiffi pffiffi 31.14 Suppose that the market price of risk of the short rate is 1 = r ỵ 2 r Show that if the real world process for the short rate is the one assumed by CIR, the risk-neutral process has the same functional form Derive the relationship between (a) the real-world reversion rate and the risk-neutral reversion rate and (b) the real-world reversion level and the risk-neutral reversion level 31.15 In the two-factor extension of Vasicek given in Section 31.5, derive the differential equations which must be satified by a bond price, Pðt; T Þ Use this to derive differential equations that must be satisfied by Aðt; T Þ, Bðt; T Þ, and Cðt; T ị in Pt; T ị ẳ At; T ịeBt;T ÞrÀCðt;T Þu Show that the expressions given for Bðt; T Þ in equation (31.7) and Cðt; T Þ in equation (31.14) satisfy these equations [Hint: Use equation (14A.10) to obtain the drift of Pðt; T Þ and set this drift equal to rPðt; T Þ.] 31.16 Use the result in Problem 31.7 to determine the best fit parameters for the Vasicek model using the same data as in Section 31.4 (see www-2.rotman.utoronto.ca/$hull/ VasicekCIR) Verify that the regression approach in Section 31.4 and the maximumlikelihood approach give the same answer 31.17 What is the result corresponding to that given in Problem 31.7 for the CIR model Use maximum likelhood methods to estimate the a, b, and parameters for the CIR model using the same data as that used for the Vasicek model in Section 31.4 (see www-2.rotman.utoronto.ca/$hull/VasicekCIR) Setting the market price of risk equal pffiffi to r use the market data in Table 31.1 to estimate the best fit www.downloadslide.com 32 C H A P T E R No-Arbitrage Models of the Short Rate The disadvantage of the equilibrium models presented in Chapter 31 is that they not automatically fit today’s term structure of interest rates By choosing the parameters judiciously, they can be made to provide an approximate fit to many of the term structures that are encountered in practice But the fit is not an exact one Most traders, when they are valuing derivatives, find this unsatisfactory Not unreasonably, they argue that they can have very little confidence in the price of a bond option when the model used does not price the underlying bond correctly A 1% error in the price of the underlying bond may lead to a 25% error in an option price A no-arbitrage model is a model designed to be exactly consistent with today’s term structure of interest rates The essential difference between an equilibrium and a noarbitrage model is therefore as follows In an equilibrium model, today’s term structure of interest rates is an output In a no-arbitrage model, today’s term structure of interest rates is an input In an equilibrium model, the drift of the short rate is not usually a function of time In a no-arbitrage model, the drift is, in general, dependent on time This is because the shape of the initial zero curve governs the average path taken by the short rate in the future in a no-arbitrage model If the zero curve is steeply upward-sloping for maturities between t1 and t2 , then r has a positive drift between these times; if it is steeply downward-sloping for these maturities, then r has a negative drift between these times 32.1 EXTENSIONS OF EQUILIBRIUM MODELS It turns out that some equilibrium models can be converted to no-arbitrage models by including a function of time in the drift of the short rate Here, we consider the Ho–Lee, Hull–White (one- and two-factor), Black–Derman–Toy, and Black–Karasinski models The Ho–Lee Model Ho and Lee proposed the first no-arbitrage model of the term structure in a paper in 1986.1 They presented the model in the form of a binomial tree of bond prices with two See T S Y Ho and S.-B Lee, ‘‘Term Structure Movements and Pricing Interest Rate Contingent Claims,’’ Journal of Finance, 41 (December 1986): 1011–29 715 www.downloadslide.com 716 CHAPTER 32 Figure 32.1 The Ho–Lee model Short rate r Initial forward curve r r r Time parameters: the short-rate standard deviation and the market price of risk of the short rate It has since been shown that the continuous-time limit of the model in the traditional risk-neutral world is dr ẳ tị dt ỵ dz ð32:1Þ where , the instantaneous standard deviation of the short rate, is constant and ðtÞ is a function of time chosen to ensure that the model fits the initial term structure The variable ðtÞ defines the average direction that r moves at time t This is independent of the level of r Ho and Lee’s parameter that concerns the market price of risk is irrelevant when the model is used to price interest rate derivatives Technical Note 31 at www-2.rotman.utoronto.ca/$hull/TechnicalNotes shows that tị ẳ Ft 0; tị ỵ t ð32:2Þ where Fð0; tÞ is the instantaneous forward rate for a maturity t as seen at time zero and the subscript t denotes a partial derivative with respect to t As an approximation, ðtÞ equals Ft ð0; tÞ This means that the average direction that the short rate will be moving in the future is approximately equal to the slope of the instantaneous forward curve The Ho–Lee model is illustrated in Figure 32.1 Superimposed on the average movement in the short rate is the normally distributed random outcome Technical Note 31 also shows that Pt; T ị ẳ At; T ịertịT tị where ln At; T ị ẳ ln 32:3ị P0; T ị ỵ T tịF0; tị À 12 tðT À tÞ2 Pð0; tÞ From Section 4.8, F0; tị ẳ @ ln P0; tị=@t The zero-coupon bond prices, Pð0; tÞ, are www.downloadslide.com 717 No-Arbitrage Models of the Short Rate known for all t from today’s term structure of interest rates Equation (32.3) therefore gives the price of a zero-coupon bond at a future time t in terms of the short rate at time t and the prices of bonds today The Hull–White One-Factor Model In a paper published in 1990, Hull and White explored extensions of the Vasicek model that provide an exact fit to the initial term structure.2 One version of the extended Vasicek model that they consider is dr ẳ ẵtị ar dt ỵ dz or dr ẳ a 32:4ị ! tị r dt ỵ dz a where a and are constants This is known as the Hull–White model It can be characterized as the Ho–Lee model with mean reversion at rate a Alternatively, it can be characterized as the Vasicek model with a time-dependent reversion level At time t, the short rate reverts to ðtÞ=a at rate a The Ho–Lee model is a particular case of the Hull–White model with a ¼ The model has the same amount of analytic tractability as Ho–Lee Technical Note 31 shows that 2 32:5ị tị ẳ Ft 0; tị ỵ aF0; tị ỵ e2at ị 2a Figure 32.2 The Hull–White model r Short rate Initial forward curve r r r Time See J C Hull and A White, ‘‘Pricing Interest Rate Derivative Securities,’’ Review of Financial Studies, 3, (1990): 573–92 www.downloadslide.com 718 CHAPTER 32 The last term in this equation is usually fairly small If we ignore it, the equation implies that the drift of the process for r at time t is Ft ð0; tị ỵ aẵF0; tị r This shows that, on average, r follows the slope of the initial instantaneous forward rate curve When it deviates from that curve, it reverts back to it at rate a The model is illustrated in Figure 32.2 Technical Note 31 shows that bond prices at time t in the Hull–White model are given by Pt; T ị ẳ At; T ịeBt;T ịrtị 32:6ị where eaT tị 32:7ị Bt; T ị ẳ a and P0; T ị ln At; T ị ẳ ln ỵ Bt; T ịF0; tị ðeÀaT À eÀat Þ2 ðe2at À 1Þ ð32:8Þ Pð0; tÞ 4a As we show in the next section, European bond options can be valued analytically using the Ho–Lee and Hull–White models A method for representing the models in the form of a trinomial tree is given later in this chapter This is useful when American options and other derivatives that cannot be valued analytically are considered The Black–Derman–Toy Model In 1990, Black, Derman, and Toy proposed a binomial-tree model for a lognormal short-rate process.3 Their procedure for building the binomial tree is explained in Technical Note 23 at www-2.rotman.utoronto.ca/$hull/TechnicalNotes It can be shown that the stochastic process corresponding to the model is d ln r ẳ ẵtị atị ln r dt ỵ tị dz with atị ẳ ðtÞ ðtÞ where ðtÞ is the derivative of with respect to t This model has the advantage over Ho–Lee and Hull–White that the interest rate cannot become negative The Wiener process dz can cause lnðrÞ to be negative, but r itself is always positive One disadvantage of the model is that there are no analytic properties A more serious disadvantage is that the way the tree is constructed imposes a relationship between the volatility parameter ðtÞ and the reversion rate parameter aðtÞ The reversion rate is positive only if the volatility of the short rate is a decreasing function of time In practice, the most useful version of the model is when ðtÞ is constant The parameter a is then zero, so that there is no mean reversion and the model reduces to d ln r ẳ tị dt ỵ dz This can be characterized as a lognormal version of the Ho–Lee model See F Black, E Derman, and W Toy, ‘‘A One-Factor Model of Interest Rates and Its Application to Treasury Bond Prices,’’ Financial Analysts Journal, January/February (1990): 33–39 www.downloadslide.com 719 No-Arbitrage Models of the Short Rate The Black–Karasinski Model In 1991, Black and Karasinski developed an extension of the Black–Derman–Toy model where the reversion rate and volatility are determined independently of each other.4 The most general version of the model is d ln r ẳ ẵtị atị ln r dt þ ðtÞ dz The model is the same as Black–Derman–Toy model except that there is no relation between aðtÞ and ðtÞ In practice, aðtÞ and ðtÞ are often assumed to be constant, so that the model becomes d ln r ẳ ẵtị a ln r dt ỵ dz ð32:9Þ As in the case of all the models we are considering, the ðtÞ function is determined to provide an exact fit to the initial term structure of interest rates The model has no analytic tractability, but later in this chapter we will describe a convenient way of simultaneously determining ðtÞ and representing the process for r in the form of a trinomial tree The Hull–White Two-Factor Model In Section 31.4, we presented a two-factor equilibrium model which is an extension of Vasicek’s model Hull and White show how this model can be converted into a noarbitrage model by adding ðtÞ in the drift of r.5 This model provides a richer pattern of term structure movements and a richer pattern of volatilities than one-factor models of r For more information on the analytical properties of the model and the way a tree can be constructed for it, see Technical Note 14 at www-2.rotman.utoronto.ca/$hull/TechnicalNotes 32.2 OPTIONS ON BONDS Some of the models just presented allow options on zero-coupon bonds to be valued analytically For the Vasicek, Ho–Lee, and Hull–White one-factor models, the price at time zero of a call option that matures at time T on a zero-coupon bond maturing at time s is ð32:10Þ LPð0; sÞNðhÞ À KPð0; T ÞNðh À P Þ where L is the principal of the bond, K is its strike price, and h¼ LPð0; sị P ỵ ln P P0; T ịK The price of a put option on the bond is KPð0; T ịNh ỵ P ị LP0; sịNhị See F Black and P Karasinski, ‘‘Bond and Option Pricing When Short Rates are Lognormal,’’ Financial Analysts Journal, July/August (1991): 52–59 See J C Hull and A White, ‘‘Numerical Procedures for Implementing Term Structure Models II: TwoFactor Models,’’ Journal of Derivatives, 2, (Winter 1994): 37–48 www.downloadslide.com 720 CHAPTER 32 Technical Note 31 shows that, in the case of the Vasicek and Hull–White models, sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi À eÀ2aT P ẳ ẵ1 easT ị 2a a and, in the case of the Ho–Lee model, pffiffiffiffi P ¼ ðs À T Þ T Equation (32.10) is essentially the same as Black’s model for pricing pffiffiffiffi bond options in Section 29.1 with the forward bond price volatility equaling P = T As explained in Section 29.2, an interest rate cap or floor can be expressed as a portfolio of options on zero-coupon bonds It can, therefore, be valued analytically using the equations just presented There are also formulas for valuing options on zero-coupon bonds in the Cox, Ingersoll, and Ross model, which we presented in Section 31.2 These involve integrals of the noncentral chi-square distribution Options on Coupon-Bearing Bonds In a one-factor model of r, all zero-coupon bonds move up in price when r decreases and all zero-coupon bonds move down in price when r increases As a result, a onefactor model allows a European option on a coupon-bearing bond to be expressed as the sum of European options on zero-coupon bonds The procedure is as follows: Calculate rà , the critical value of r for which the price of the coupon-bearing bond equals the strike price of the option on the bond at the option maturity T Calculate prices of European options with maturity T on the zero-coupon bonds that comprise the coupon-bearing bond The strike prices of the options equal the values the zero-coupon bonds will have at time T when r ¼ rà Set the price of the European option on the coupon-bearing bond equal to the sum of the prices on the options on zero-coupon bonds calculated in Step This allows options on coupon-bearing bonds to be valued for the Vasicek, Cox, Ingersoll, and Ross, Ho–Lee, and Hull–White models As explained in Business Snapshot 29.2, a European swap option can be viewed as an option on a coupon-bearing bond It can, therefore, be valued using this procedure For more details on the procedure and a numerical example, see Technical Note 15 at www-2.rotman.utoronto.ca/$hull/ TechnicalNotes 32.3 VOLATILITY STRUCTURES The models we have looked at give rise to different volatility environments Figure 32.3 shows the volatility of the 3-month forward rate as a function of maturity for Ho–Lee, Hull–White one-factor and Hull–White two-factor models The term structure of interest rates is assumed to be flat For Ho–Lee the volatility of the 3-month forward rate is the same for all maturities In the one-factor Hull–White model the effect of mean reversion is to cause the www.downloadslide.com 721 No-Arbitrage Models of the Short Rate Figure 32.3 Volatility of 3-month forward rate as a function of maturity for (a) the Ho–Lee model, (b) the Hull–White one-factor model, and (c) the Hull–White twofactor model (when parameters are chosen appropriately) Volatility Volatility Volatility Maturity (a) Maturity (b) Maturity (c) volatility of the 3-month forward rate to be a declining function of maturity In the Hull–White two-factor model when parameters are chosen appropriately, the volatility of the 3-month forward rate has a ‘‘humped’’ look The latter is consistent with empirical evidence and implied cap volatilities discussed in Section 29.2 32.4 INTEREST RATE TREES An interest rate tree is a discrete-time representation of the stochastic process for the short rate in much the same way as a stock price tree is a discrete-time representation of the process followed by a stock price If the time step on the tree is Át, the rates on the tree are the continuously compounded Át-period rates The usual assumption when a tree is constructed is that the Át-period rate, R, follows the same stochastic process as the instantaneous rate, r, in the corresponding continuous-time model The main difference between interest rate trees and stock price trees is in the way that discounting is done In a stock price tree, the discount rate is usually assumed to be the same at each node or a function of time In an interest rate tree, the discount rate varies from node to node It often proves to be convenient to use a trinomial rather than a binomial tree for interest rates The main advantage of a trinomial tree is that it provides an extra degree of freedom, making it easier for the tree to represent features of the interest rate process such as mean reversion As mentioned in Section 21.8, using a trinomial tree is equivalent to using the explicit finite difference method Illustration of Use of Trinomial Trees To illustrate how trinomial interest rate trees are used to value derivatives, consider the simple example shown in Figure 32.4 This is a two-step tree with each time step equal to year in length so that Át ¼ year Assume that the up, middle, and down probabilities are 0.25, 0.50, and 0.25, respectively, at each node The assumed Át-period rate is shown as the upper number at each node.6 We explain later how the probabilities and rates on an interest rate tree are determined www.downloadslide.com 722 CHAPTER 32 Figure 32.4 Example of the use of trinomial interest rate trees Upper number at each node is rate; lower number is value of instrument The tree is used to value a derivative that provides a payoff at the end of the second time step of maxẵ100R 0:11ị; where R is the t-period rate The calculated value of this derivative is the lower number at each node At the final nodes, the value of the derivative equals the payoff For example, at node E, the value is 100 0:14 0:11ị ẳ At earlier nodes, the value of the derivative is calculated using the rollback procedure explained in Chapters 13 and 21 At node B, the 1-year interest rate is 12% This is used for discounting to obtain the value of the derivative at node B from its values at nodes E, F, and G as ẵ0:25 ỵ 0:5 ỵ 0:25 0e0:121 ẳ 1:11 At node C, the 1-year interest rate is 10% This is used for discounting to obtain the value of the derivative at node C as 0:25 ỵ 0:5 ỵ 0:25 0ịe0:11 ẳ 0:23 At the initial node, A, the interest rate is also 10% and the value of the derivative is 0:25 1:11 ỵ 0:5 0:23 ỵ 0:25 0ịe0:11 ẳ 0:35 Nonstandard Branching It sometimes proves convenient to modify the standard trinomial branching pattern that is used at all nodes in Figure 32.4 Three alternative branching possibilities are shown in Figure 32.5 The usual branching is shown in Figure 32.5a It is ‘‘up one/straight along/ down one’’ One alternative to this is ‘‘up two/up one/straight along’’, as shown in www.downloadslide.com No-Arbitrage Models of the Short Rate Figure 32.5 723 Alternative branching methods in a trinomial tree Figure 32.5b This proves useful for incorporating mean reversion when interest rates are very low A third branching pattern shown in Figure 32.5c is ‘‘straight along/down one/ down two’’ This is useful for incorporating mean reversion when interest rates are very high The use of different branching patterns is illustrated in the following section 32.5 A GENERAL TREE-BUILDING PROCEDURE Hull and White have proposed a robust two-stage procedure for constructing trinomial trees to represent a wide range of one-factor models.7 This section first explains how the procedure can be used for the Hull–White model in equation (32.4) and then shows how it can be extended to represent other models, such as Black–Karasinski First Stage The Hull–White model for the instantaneous short rate r is dr ẳ ẵtị ar dt ỵ dz We suppose that the time step on the tree is constant and equal to Át.8 Assume that the Át rate, R, follows the same process as r dR ẳ ẵtị aR dt ỵ dz Clearly, this is reasonable in the limit as Át tends to zero The first stage in building a tree for this model is to construct a tree for a variable Rà that is initially zero and follows the process dR ẳ aR dt ỵ dz This process is symmetrical about R ẳ The variable R t ỵ ÁtÞ À Rà ðtÞ is normally distributed If terms of higher order than Át are ignored, the expected value of R t ỵ tị R tị is aR tịt and the variance of R t ỵ tị R ðtÞ is Át See J C Hull and A White, ‘‘Numerical Procedures for Implementing Term Structure Models I: SingleFactor Models,’’Journal of Derivatives, 2, (1994): 7–16; and J C Hull and A White, ‘‘Using Hull–White Interest Rate Trees,’’ Journal of Derivatives, (Spring 1996): 26–36 See Technical Note 16 at www-2.rotman.utoronto.ca/$hull/TechnicalNotes for a discussion of how nonconstant time steps can be used www.downloadslide.com 724 CHAPTER 32 Figure 32.6 Tree for Rà in Hull–White model (first stage) E B F C G D H A I Node: à R ð%Þ pu pm pd A B 0.000 0.1667 0.6666 0.1667 1.732 0.1217 0.6566 0.2217 C D 0.000 À1.732 0.1667 0.2217 0.6666 0.6566 0.1667 0.1217 E F 3.464 0.8867 0.0266 0.0867 1.732 0.1217 0.6566 0.2217 G H I 0.000 À1.732 À3.464 0.1667 0.2217 0.0867 0.6666 0.6566 0.0266 0.1667 0.1217 0.8867 The spacing between interest rates on the tree, ÁR, is set as pffiffiffiffiffiffiffiffi ÁR ¼ 3Át This proves to be a good choice of ÁR from the viewpoint of error minimization The objective of the first stage of the procedure is to build a tree similar to that shown in Figure 32.6 for Rà To this, it is first necessary to resolve which of the three branching methods shown in Figure 32.5 will apply at each node This will determine the overall geometry of the tree Once this is done, the branching probabilities must also be calculated Define ði; j Þ as the node where t ¼ i Át and Rà ¼ j ÁR (The variable i is a positive integer and j is a positive or negative integer.) The branching method used at a node must lead to the probabilities on all three branches being positive Most of the time, the branching shown in Figure 32.5a is appropriate When a > 0, it is necessary to switch from the branching in Figure 32.5a to the branching in Figure 32.5c for a sufficiently large j Similarly, it is necessary to switch from the branching in Figure 32.5a to the branching in Figure 32.5b when j is sufficiently negative Define jmax as the value of j where we switch from the Figure 32.5a branching to the Figure 32.5c branching and jmin as the value of j where we switch from the Figure 32.5a branching to the Figure 32.5b branching Hull and White show that probabilities are always positive if jmax is set equal to the smallest integer greater than 0:184=ða ÁtÞ and jmin is set equal to Àjmax Define The probabilities are positive for any value of jmax between 0:184=ða ÁtÞ and 0:816=ða ÁtÞ and for any value of jmin between À0:184=ða ÁtÞ and À0:816=ða ÁtÞ Changing the branching at the first possible node proves to be computationally most efficient www.downloadslide.com No-Arbitrage Models of the Short Rate 725 pu , pm , and pd as the probabilities of the highest, middle, and lowest branches emanating from the node The probabilities are chosen to match the expected change and variance of the change in Rà over the next time interval Át The probabilities must also sum to unity This leads to three equations in the three probabilities As already mentioned, the mean change in Rà in time Át is ÀaRà Át and the variance of the change is Át At node ði; j Þ, Rà ¼ j ÁR If the branching has the form shown in Figure 32.5a, the pu , pm , and pd at node ði; j Þ must satisfy the following three equations to match the mean and standard deviation: pu ÁR À pd ÁR ¼ Àaj ÁR Át pu ÁR2 þ pd ÁR2 ¼ Át þ a2 j2 R2 t2 p u ỵ pm ỵ pd ẳ pffiffiffiffiffiffiffiffi Using ÁR ¼ 3Át, the solution to these equations is pu ẳ 16 ỵ 12 a2 j2 t2 aj tị pm ẳ 23 a2 j2 t2 pd ẳ 16 ỵ 12 a2 j2 t2 ỵ aj ÁtÞ Similarly, if the branching has the form shown in Figure 32.5b, the probabilities are pu ẳ 16 ỵ 12 a2 j2 t2 ỵ aj tị pm ẳ 13 À a2 j2 Át2 À 2aj Át pd ¼ 76 þ 12 ða2 j2 Át2 þ 3aj ÁtÞ Finally, if the branching has the form shown in Figure 32.5c, the probabilities are pu ẳ 76 ỵ 12 a2 j2 t2 3aj tị pm ẳ 13 a2 j2 t2 ỵ 2aj t pd ẳ 16 ỵ 12 a2 j2 Át2 À aj ÁtÞ To illustrate the first stage of the tree construction, suppose that ¼ 0:01, a ¼ 0:1, pffiffiffi and Át ¼ year In this case, ÁR ¼ 0:01 ¼ 0:0173, jmax is set equal to the smallest integer greater than 0.184/0.1, and jmin ¼ Àjmax This means that jmax ¼ and jmin ¼ À2 and the tree is as shown in Figure 32.6 The probabilities on the branches emanating from each node are shown below the tree and are calculated using the equations above for pu , pm , and pd Note that the probabilities at each node in Figure 32.6 depend only on j For example, the probabilities at node B are the same as the probabilities at node F Furthermore, the tree is symmetrical The probabilities at node D are the mirror image of the probabilities at node B Second Stage The second stage in the tree construction is to convert the tree for Rà into a tree for R This is accomplished by displacing the nodes on the Rà -tree so that the initial term www.downloadslide.com 726 CHAPTER 32 structure of interest rates is exactly matched Define tị ẳ Rtị R tị The tịs that apply as the time step Át on the tree becomes infinitesimally small can be calculated analytically from equation (32.5).10 However, we want a tree with a finite Át to match the term structure exactly, so we use an iterative procedure to determine the ’s Define i as ði ÁtÞ, the value of R at time i Át on the R-tree minus the corresponding value of Rà at time i Át on the Rà -tree Define Qi;j as the present value of a security that pays off $1 if node ði; jÞ is reached and zero otherwise The i and Qi;j can be calculated using forward induction in such a way that the initial term structure is matched exactly Illustration of Second Stage Suppose that the continuously compounded zero rates in the example in Figure 32.6 are as shown in Table 32.1 The value of Q0;0 is 1.0 The value of 0 is chosen to give the right price for a zero-coupon bond maturing at time Át That is, 0 is set equal to the initial Át-period interest rate Because Át ¼ in this example, 0 ¼ 0:03824 This defines the position of the initial node on the R-tree in Figure 32.7 The next step is to calculate the values of Q1;1 , Q1;0 , and Q1;À1 There is a probability of 0.1667 that the ð1; 1Þ node is reached and the discount rate for the first time step is 3.82% The value of Q1;1 is therefore 0:1667eÀ0:0382 ¼ 0:1604 Similarly, Q1;0 ¼ 0:6417 and Q1;À1 ¼ 0:1604 Once Q1;1 , Q1;0 , and Q1;À1 have been calculated, 1 can be determined It is chosen to give the right price for a zero-coupon bond maturing at time 2Át Because ÁR ¼ 0:01732 and Át ¼ 1, the price of this bond as seen at node B is e 1 ỵ0:01732ị Similarly, the price as seen at node C is eÀ 1 and the price as seen at node D is eÀð 1 À0:01732Þ The price as seen at the initial node A is therefore Q1;1 e 1 ỵ0:01732ị ỵ Q1;0 e 1 ỵ Q1;À1 eÀð 1 À0:01732Þ ð32:11Þ From the initial term structure, this bond price should be eÀ0:04512Â2 ¼ 0:9137 Table 32.1 Zero rates for example in Figures 32.6 and 32.7 10 Maturity Rate (%) 0.5 1.0 1.5 2.0 2.5 3.0 3.430 3.824 4.183 4.512 4.812 5.086 To estimate the instantaneous ðtÞ analytically, we note that dR ẳ ẵtị aR dt ỵ dz and dR ẳ aR dt ỵ dz so that d ẳ ẵtị a tị dt Using equation (32.7), it can be seen that the solution to this is tị ẳ F0; tị ỵ 2 eat Þ2 : 2a2 www.downloadslide.com 727 No-Arbitrage Models of the Short Rate Figure 32.7 Tree for R in Hull–White model (the second stage) Node: A B C D E F G H I R ð%Þ pu pm pd 3.824 0.1667 0.6666 0.1667 6.937 0.1217 0.6566 0.2217 5.205 0.1667 0.6666 0.1667 3.473 0.2217 0.6566 0.1217 9.716 0.8867 0.0266 0.0867 7.984 0.1217 0.6566 0.2217 6.252 0.1667 0.6666 0.1667 4.520 0.2217 0.6566 0.1217 2.788 0.0867 0.0266 0.8867 Substituting for the Q’s in equation (32.11), 0:1604e 1 ỵ0:01732ị ỵ 0:6417e 1 ỵ 0:1604e 1 0:01732ị ẳ 0:9137 or or e 1 0:1604e0:01732 ỵ 0:6417 ỵ 0:1604e0:01732 ị ẳ 0:9137 ! 0:1604e0:01732 ỵ 0:6417 ỵ 0:1604e0:01732 1 ẳ ln ẳ 0:05205 0:9137 This means that the central node at time Át in the tree for R corresponds to an interest rate of 5.205% (see Figure 32.7) The next step is to calculate Q2;2 , Q2;1 , Q2;0 , Q2;À1 , and Q2;À2 The calculations can be shortened by using previously determined Q values Consider Q2;1 as an example This is the value of a security that pays off $1 if node F is reached and zero otherwise Node F can be reached only from nodes B and C The interest rates at these nodes are 6.937% and 5.205%, respectively The probabilities associated with the B–F and C–F branches are 0.6566 and 0.1667 The value at node B of a security that pays $1 at node F is therefore 0:6566eÀ0:06937 The value at node C is 0:1667eÀ0:05205 The variable Q2;1 is 0:6566eÀ0:06937 times the present value of $1 received at node B plus 0:1667eÀ0:05205 times the present value of $1 received at node C; that is, Q2;1 ¼ 0:6566e0:06937 0:1604 ỵ 0:1667e0:05205 0:6417 ẳ 0:1998 Similarly, Q2;2 ¼ 0:0182, Q2;0 ¼ 0:4736, Q2;À1 ¼ 0:2033, and Q2;À2 ¼ 0:0189 www.downloadslide.com 728 CHAPTER 32 The next step in producing the R-tree in Figure 32.7 is to calculate 2 After that, the Q3;j ’s can then be computed The variable 3 can then be calculated, and so on Formulas for ’s and Q’s To express the approach more formally, suppose that the Qi;j have been determined for i m (m > 0) The next step is to determine m so that the tree correctly prices a zerocoupon bond maturing at m ỵ 1ị t The interest rate at node m; jị is m ỵ j ÁR, so that the price of a zero-coupon bond maturing at time m ỵ 1ịt is given by nm X Pmỵ1 ẳ Qm;j expẵ m ỵ j Rịt 32:12ị jẳnm where nm is the number of nodes on each side of the central node at time m Át The solution to this equation is Pm ln nj¼Àn Qm;j eÀjÁRÁt À ln Pmỵ1 m m ẳ t Once m has been determined, the Qi;j for i ẳ m ỵ can be calculated using X Qm;k qk; jị expẵ m ỵ k Rị t Qmỵ1;j ẳ k where qk; jị is the probability of moving from node m; kị to node m ỵ 1; jÞ and the summation is taken over all values of k for which this is nonzero Extension to Other Models The procedure that has just been outlined can be extended to more general models of the form df ðrÞ ẳ ẵtị af rị dt ỵ dz 32:13ị where f is a montonic function of r This family of models has the property that they can fit any term structure.11 As before, we assume that the Át period rate, R, follows the same process as r: df ðRÞ ẳ ẵtị af Rị dt ỵ dz We start by setting x ẳ f Rị, so that dx ẳ ẵtị ax dt ỵ dz The rst stage is to build a tree for a variable xà that follows the same process as x except that ðtÞ ¼ and the initial value is zero The procedure here is identical to the procedure already outlined for building a tree such as that in Figure 32.6 11 Not all no-arbitrage models have this property For example, the extended-CIR model, considered by Cox, Ingersoll, and Ross (1985) and Hull and White (1990), which has the form pffiffi dr ¼ ẵtị ar dt ỵ r dz cannot t yield curves where the forward rate declines sharply This is because the process is not well defined when ðtÞ is negative ... Futures options and Black’s model 381 18.1 Nature of futures options 381 18.2 Reasons for the popularity of futures options 384 18.3 European spot and futures options. .. Education, [2018] Revised edition of the author’s Options, futures, and other derivatives, [2015] Includes index 2016051230 | 013447208X Futures Stock options Derivative securities HG6024.A3 H85 2017... www.downloadslide.com OPTIONS, FUTURES, AND OTHER DERIVATIVES TENTH EDITION www.downloadslide.com This page intentionally left blank A01_GORD2302_01_SE_FM.indd 28/05/15 7:33 pm www.downloadslide.com OPTIONS, FUTURES,