Pricing of Stochastic Interest Bonds using Affine Term Structure Models: A Comparative Analysis

Transcription

1 Dottorato di Ricerca in Matematica er l Analisi dei Mercati Finanziari - Ciclo XXII - Pricing of Stochastic Interest Bonds using Affine Term Structure Models: A Comarative Analysis Dott.ssa Erica MASTALLI Relatore: Chiar.mo Prof. Marcello MINENNA Relatore: Chiar.mo Prof. Fabio BELLINI Anno Accademico

2 Pricing of Stochastic Interest Bonds using A ne Term Structure Models: A Comarative Analysis Erica MASTALLI June, 010

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4 Contents 1 Term structure models Introduction An historical ersective Term structure of interest rates Risk neutral valuation and no-arbitrage condition Equilibrium models The Vasicek model The volatility of the short rate in the Vasicek model Valuation of euroean otions on zero-couon bonds in the Vasicek model No arbitrage models The Ho and Lee model The volatility of the short rate in the Ho and Lee model Valuation of euroean otions on zero-couon bonds in the Ho-Lee model The Hull and White model The volatility of the short rate in the Hull and White model Valuation of euroean otions on zero-couon bonds in the Hull-White model Aendix A.1 The volatility of the rice of a zero couon bond underlying an euroean otion in the Vasicek model Aendix A. The volatility of the rice of a zero couon bond underlying an euroean otion in the Ho and Lee model Aendix A.3 The volatility of the rice of a zero couon bond underlying an euroean otion in the Hull and White model.. 43 Ca and oor ricing using a ne term structure models 45.1 Interest rate cas Cas as ortfolios of interest rate calls Ca as ortfolios of zero couon bond uts Pricing of an interest rate ca in the Vasicek model

5 4 CONTENTS Pricing of an interest rate ca in the Vasicek model: an examle Pricing of an interest rate ca in the Ho and Lee model Pricing of an interest rate ca in the Ho and Lee model: an examle Pricing of an interest rate ca in the Hull and White model Pricing of an interest rate ca in the Hull and White model: an examle Interest rate oors Floors as ortfolios of interest rate uts Floors as ortfolios of zero couon bond calls Pricing of an interest rate oor in the Vasicek model Pricing of an interest rate oor in the Vasicek model: an examle Pricing of an interest rate oor in the Ho and Lee model Pricing of an interest rate oor in the Ho and Lee model: an examle Pricing of an interest rate oor in the Hull and White model Pricing of an interest rate oor in the Hull and White model: an examle Aendix B.1 Black Formula Credit risk and defaultable bonds valuation Introduction Credit default swas Credit default swas ricing Bootstraing default robabilities from CDS sreads Pricing of a defaultable couon bond The collared oaters Introduction General features and risk ro le Unbundling of a generic collared oater Pricing of some stochastic interest bonds Introduction Concrete examles Descrition and unbundling of the bond BNL_ Descrition and unbundling of the bond BNL_ Descrition and unbundling of the bond BNL_ Descrition and unbundling of the bond Poolare_ Descrition and unbundling of the bond Poolare_ Descrition and unbundling of the bond Unicredit_ Descrition and unbundling of the bond Unicredit_ Descrition and unbundling of the bond Unicredit_

6 CONTENTS Descrition and unbundling of the bond Unicredit_ Descrition and unbundling of the bond Intesa_ Calibration of the Vasicek model Pricing with the Vasicek model Calibration of the Hull and White Model Pricing with the Hull and White Model Comarison with the rices ublished in the rosectus Conclusions 15

7 6 CONTENTS Preface The aim of this work is to use one-factor stochastic term structure models to evaluate stochastic interest bonds, that are bonds bundled together some interest rate derivative, and to comare them with the theoretical value that the issuer indicates in the rosectus for the ublic o ering. Stochastic interest bonds are a sub-set of the big family of structured bonds, the latter being bonds that resent seci c algorithms driving couons comutation and ayment at maturity, mainly due to the resence of one or more derivative comonents embedded in their nancial structure. Structured bonds are mainly issued by banks. Over the last two decades the o ering of structured bonds to retail investors has consistently increased, with a contextual rise in the variety of the ayo structures. Chater 1, after a brief exosure of the evolution of term structure models and their classi cation, is devoted to analyze several one-factor a ne term structure models: the Vasicek model, the Ho-Lee model and the Hull-White model. Chater shows how to use the above models to rice some tyical interest rate derivatives (namely cas and oors) that are often embedded in the structure of stochastic interest bonds like those that will be considered in Chater 5, which in fact, will include either a ca or a oor or both these two tyes of interest rate derivatives. Chater 3 is devoted to analyze some key concets about credit risk in order to take into account the imact of this risk factor on the bond value. To this aim, we will illustrate some key results regarding credit derivatives, and, seci cally, credit default swas whose market quotes allow to infer reliable estimates of the cumulative and intertemoral default robabilities of an issuer at various maturities by using the so-called bootstraing technique. Once these default robabilities are estimated they can be used to derive a general ricing formula for defaultable bonds which will be used to erform the fair evaluation of the ten stochastic interest bonds analyzed in Chater 5. Chater 4 is devoted to study in detail the nancial engineering of a seci c kind of stochastic interest bonds, namely the so-called collared oaters, which are oating-rate couon bonds whose couons are subject to both an uer and a lower bound, hence embedding two interest rate derivatives, either a long ca and a short ca or a long oor and a short ca deending on the seci c unbundling choice we make. In articular, the unbundling of a generic collared oater into its various elementary comonents is examined, as it will be useful to the ricing of many bonds included in the set of securities analyzed in Chater 5. Chater 5 is focused on the ricing of ten stochastic interest bonds recently issued by four of the major Italian banks: six of them are ure collared oaters, two of them are mixed xed- oating couon bonds, whose oating couons have the tyical structure of collared oaters, one bond is a oating-rate couon bond embedding a oor, and one bond is a oating-rate couon bond embedding a oor for the rst half of its life and a ca for the second half of its life.

8 CONTENTS 7 After the illustration of their unbundling, these bonds are riced by means of two alternative ricing methodologies. The rst methodology is based on the unbundling of their nancial structure which reveals how these bonds can be seen as the comosition of one or more ure bond comonents and of one or more interest rate derivatives, namely cas and/or oors, whose closed formulas - in the framework of the one-factor a ne term structure models of Chater 1 develoed under the risk neutral robability measure - have been resented in Chater. The second methodology relies instead on Monte Carlo simulations, erformed again under the risk neutral robability measure; in this case the fair value of a bond is determined by discounting back at the evaluation date the nal value of the security over each simulated trajectory and, then, by averaging these discounted values. The two ricing methodologies are imlemented both in the framework of the Vasicek model and in that of the Hull and White model. Their results turn out to be consistent and, comared with the theoretical value indicated in the nal terms of the rosectus ublished by the issuers, are a useful instrument to exlore the reliability and the accuracy of the informative set included in this document that investors use to take their nancial decisions.

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10 Chater 1 Term structure models 1.1 Introduction The term structure models, or yield curve models, describe the time evolution of the yield curve, that is the curve that, for each maturity T, evaluated at date t, exresses the yield to maturity (sot rate), r(t; T ): The sot rate is the rate de ned at time t for a nancial oeration that starts at time t and terminates at time T, with t T. From a ractical oint of view, most of the term structures of the interest rates are comuted from the Treasury bond rices, these kind of bonds, in general and in normal market conditions, for the develoed countries, being considered risk-less. Government bonds, like Treasury securities, are nancial instruments that rovide xed and certain cash ows, couons and rincial, on a sequence of re-seci ed dates. The return rate corresonding to various maturities of these bonds can be deduced by a seci c method called bootstraing from the market rice of the most frequently traded couon-bearing-bonds, sometimes called benchmark issues. This set of yields to maturity associated with di erent maturities de nes the term structure of interest rates. The shae of the term structure changes over time. Most of times it is uward sloing, meaning that the return on long term bonds is greater than the return on short term bonds. The term structure can also be downward sloing; it can deend on macroeconomic state variables. Unlike Treasury bonds, all structured bonds, like stochastic interest bonds, have ayo s that are neither xed nor certain. In the case of structured bonds, these ayo s deend on the future levels of interest rates that are all unknown variables at the time of the evaluation date. As a consequence, the ricing of these bonds requires seci c assumtions on the future evolution of the interest rates, which usually rely on a seci c model for the dynamics of interest rates. 9

11 10 CHAPTER 1. TERM STRUCTURE MODELS 1. An historical ersective The rst research oriented to the term structure modelling has recognized the imortance of the stochastic nature of interest rates and has modelled the sot rate evolution as a random walk. On 1977 Vasicek introduced a general no arbitrage model for the ricing of zero couon bonds and he roosed a seci c model in which the instantaneous sot rate is described by a mean-reverting Ornstein-Uhlenbeck rocess. On 1985 Cox, Ingersoll and Ross have shown how to use the yield curve theory in a realistic economic world and they roosed a model only for ositive interest rates, based on a square root rocess. In articular, their model (CIR model), as Vasicek model, is a mean reverting model for the instantaneous sot rate but here the variance is not constant. Here the variance of the short rate changes over time roortional to the level of the short rate. Both of the above mentioned models found out a seci c time evolution for the interest rate and they used economic fundamentals to describe that systematic variation of the term structure. These kinds of models are known in literature as Equilibrium Models because they secify the market rice of risk and they can be suorted by an economic equilibrium model. In this category there are also the models develoed by Brennan and Schwartz (1979), Fong and Vasicek (1991) and Longsta and Schwartz (199). All these models are known as multifactor equilibrium models because they assume that the evolution of the term structure of the interest rates deends on the dynamic of more than one factor and that the yield to maturity deends on all these factors too. Equilibrium models can be calibrated by using historical data on interest rates and bond rices and, then, they can be used to evaluate the rice of both lain vanilla and structured bonds and bond otions. Often there is a misricing between the rice obtained from this family of models and the market rice of a nancial roduct. This roblem generated the requirement of term structure models to allow a bond ricing more coherent with the term structure observed on the market. At this oint some authors develoed the no arbitrage models, i.e. models that use a no arbitrage condition to recisely de ne the relationshi between the drift and the di usion coe cients of the sot rate. The rst contribution in this direction comes from Ho and Lee (1986) and, later, from the models develoed resectively by Hull and White (1990) and by the Black, Derman and Toy(1990). In this no arbitrage category there is also the Heath, Jarrow and Morton model (199). It is a term structure model that deends on the evolution of the entire forward rate curve, starting from the current interest rate curve observed on the market.

12 1.3. TERM STRUCTURE OF INTEREST RATES 11 The imlementation of these kinds of models for the interest rate derivatives has encountered several di culties. One of them is that the instantaneous forward rate term structure is not directly observable and then the Heath-Jarrow- Morton Models are di cult to aly. To solve these roblems some authors 1 develoed new term structure models known as market models that study the observable interest rates alying over nite maturities, such as the LIBOR (London Interbank O ered Rate) or the swa rates, directly within the Heath-Jarrow-Morton framework. In this work we will analyze in detail the equilibrium model develoed by Vasicek and two no arbitrage models, namely those develoed by Ho and Lee and by Hull and White. Equilibrium and no arbitrage models have similar features. The main distinction between the two categories comes from the di erent inut that are used to calibrate the model arameters. Equilibrium models exlicitly secify the market rice of risk; the model arameters, assumed constant over time, are estimated statistically from historical data. No arbitrage models are calibrated to match the observed rice on the market and the model rice. We have to oint out that some equilibrium models (for examle the Vasicek model) and some no arbitrage models (for examle the Ho-Lee model and the Hull-White model) belong to a more general class of term structure models known as A ne Term Structure Models. A ne term structure models, introduced in 1996 by Du e and Kan 3, are models in which the yield to maturity of a zero-couon bond is a linear function of the underlying variables. Du e and Kan described and analyzed a simle multifactor term structure model of the interest rates where the factors are the returns X 1 ; X ; :::; X n of n zero-couon bonds with di erent maturity, T 1 ; T ; :::; T n. The models is called A ne because, for each maturity T, there exists an a ne function Y T : R n! R such that, for each date t, the return on a zero-couon bond with maturity T is equal to Y T (X t ). 1.3 Term structure of interest rates The base for all the term structure models for the interest rates is the concet of the zero couon bond. 1 See Brace, Gatarek and Musiela (1997), Jamshidian (1997), Miltersen, Sandmnann and Sondermann (1997), and others. As Black ointed out, this classi cation may be a misuse of the term, because equilibrium models like the CIR model do not admit arbitrage in the economic environment seci ed in the model. Moreover, arbitrage models, such as the Hull and White one, are constructed by making time-varying the coe cients of some equilibrium models 3 See Du e, D. e Kan, R., (1996), A Yield-Factor Model of Interest Rates, Mathematical Finance, n. 4.

13 1 CHAPTER 1. TERM STRUCTURE MODELS A zero couon bond or ure discount bond is a bond that entitles its holders to a single certain cash ow of F V (rincial value or face value or notional amount) in a certain future date. Let P (t; T ) be the rice at time t of a zero couon bond maturing at time T with face value F V = 1$ and let r(t; T ) be the sot rate over the eriod [t; T ], then, the relationshi between the bond rice and its sot rate is: or, in terms of the sot rate: P (t; T ) = e r(t;t )(T t) (1.1) r (t; T ) = ln P (t; T ) T t (1.) meaning that the rice of a zero couon bond is the discounted value of its future cash ow. In articular, the sot rate of instantaneous maturity, i.e. the short rate r t, is simly the limit of r(t; T ) when T collases to t: r t = lim T!t r (t; T ) (1.3) The continuously comounded forward rate at time t for the future eriod between time T; T t, and time T +, > 0, is de ned by the following equation for the forward rice of the zero couon bond exiring at time T + and denoted by P (t; T; T + ) P (t; T; T + ) P (t; T + ) P (t; T ) Therefore, the exlicit formula for the forward rate is: F (t; T; T + ) = = e (F (t;t;t +)) (1.4) ln P (t; T + ) ln P (t; T ) (1.5) The limit of F (t; T; T + ) for! 0 is the instantaneous forward rate that we denote with F (t; T ). In other terms we have: and, given equation 1.5, we have: or, in integral form: F (t; T ) = lim!0 F (t; T; T + ) (1.6) F (t; T ) = P (t; T ) = ln P (t; T (1.7) R T t F (t;s)ds (1.8) From equations 1. and 1.8 we can easily deduce the following equation for the yield to maturity of a zero couon bond: r (t; T ) = R T t F (t; s) ds T t (1.9)

14 1.4. RISK NEUTRAL VALUATION AND NO-ARBITRAGE CONDITION13 This equation shows how the yield to maturity of a zero couon bond can be interreted as an average of the instantaneous forward rates on the time interval corresonding to the time to maturity of the bond. 1.4 Risk neutral valuation and no-arbitrage condition Derivative securities and structured bonds are nancial roducts whose ayo at one or more future dates corresonds to the state of nature that has occurred at that date(s), which usually deends on the rice of the underlying nancial instruments or on the evolution of these rices over a given receding time interval. In articular the ricing of interest rate derivatives and of stochastic interest bonds deends on the dynamics of the term structure of the interest rates and it is governed by the no-arbitrage condition. The no-arbitrage condition says that a strategy that has a ositive future ayo in at least one state of nature and no negative future ayo s in all the other ossible states of nature must have a current value higher than zero. This condition imlies that a contingent claim whose ayo can be relicated by a ortfolio of securities should have, under the risk-neutral measure, a rice equal to the value of the relicating ortfolio. If this equivalence is not satis ed, it will be ossible to set u an arbitrage strategy based on the di erence between the two rices. This evaluation rincile based on the construction of a relication ortfolio led to the Black and Scholes formula (1973) for the Euroean otion ricing on stocks and it is also the foundation of the ricing frameworks for interest rate derivatives. The risk neutral ricing methodology requires to comute the exected value of the discounted future ayo s of a given nancial instrument, using a seci c robability measure, P, known as Equivalent Martingale Measure or Risk Neutral Measure. Intuitively, this means that, under the risk neutral measure, the rate of return on a nancial security is equivalent to the istantaneous short rate, r t. The rice at time t of a zero couon bond with face value equal to one, i.e. F V = 1; and maturity T is equal to the exected value, under the risk neutral robability measure P; of its discounted ayo, i.e.: P (t; T ) = E P h e i t rsds 1 F t where r s is the instantaneous short rate at time s; s [t; T ]. From equations 1.1 and 1.10 we have the following equality: R T h e r(t;t )(T t) = E P e R i T t rsds 1 F t (1.10)

15 14 CHAPTER 1. TERM STRUCTURE MODELS and then: h ln ne P e R io T t rsds 1 F t r (t; T ) = (1.11) T t Equation 1.11 allows to model the whole term structure of interest rates by using r t and its risk neutral rocess. Under the robability measure P, the term e R T t rsds aearing in equation 1.10 is the discount factor that characterizes the saving account or money market account. Moreover, under P the rice of any zero-couon bond is a martingale, meaning that its conditional exected value at some time t, given all the observations u to some earlier time s, is equal to the value of that bond observed at the earlier time s. 1.5 Equilibrium models Equilibrium models start from recise assumtions about the dynamics of the state variables that describe the economic conditions and model the behavior of the term structure of interest rates into such economic context. An imortant asect of these models is that they exlicitly secify the market rice of risk, (t; r t ). Equilibrium models can be both one factor and multifactor models. One factor models are based on the assumtion that it is su cient to model only the behavior of one state variable to deduce the whole yield curve. Multifactor models as those of Brennan and Schwartz (1979), Fong and Vasicek (1991), and Longsta and Schwartz (199) assume that the evolution of the term structure of the interest rates is governed by the dynamics of more than one factor. In this work we will focus on one factor models, mainly because the emirical evidence roves that almost the 90% of the variability of the yield curve movements is determined by the movements of rst exlanatory variable, which is assumed to be the current level of the yield curve. As a consequence, each oint on the yield curve can be used as roxy of this level. Most of one factor models use as roxy the instantaneous short rate, r t, and for this reason they are also known as Short Rate Models. The main assumtion behind Short Rate Models is that, under the real-world robability measure Q, the dynamics of the instantaneous short rate follow a Markov di usive rocess as: where: dr t = (t; r t ) dt + (t; r t ) dw t (1.1) the drift coe cient, (t; r t ) ; and the di usion coe cient, (t; r t ) ; are function of two variables, the instantaneous short rate r t and the time t; W t is a standard Brownian motion under the real-world robability measure Q.

16 1.5. EQUILIBRIUM MODELS 15 Equation 1.1 shows how the short rate variation can be decomosed in a drift comonent, (t; r t ) ; on the time interval (t; t+dt); and in a random shock comonent given by the roduct of the standard Brownian motion increment dw t and an instantaneous volatility (so-called di usion coe cient) equal to (t; r t ) : The assumtion of only one risk factor is not so restrictive as it could aear. In fact, a one factor model imlies that all interest rates will move in the same direction of any small time interval but not that they will move all of the same amount, and hence, the shae of the yield curve can change over time. In these models the rice at time t of a zero couon bond that ays 1 at time T, i.e. P (t; T ) ; is a function of the instantaneous interest rate and of the time to maturity of the bond. In other words we have: P (t; T ) := P (r t ; t; T ) (1.13) It is clear that by using equation 1.13 and alying the Itô s formula, we are able to derive the stochastic di erential equation that describes the dynamics of the zero couon bond rice starting from the stochastic di erential equation that governs the dynamics of the instantaneous short rate (i.e. equation 1.1). In articular, in order to obtain the stochastic di erential equation that describes the dynamics of the zero couon bond rice, we de ne the following artial (r t ; t; T t = P rt (r t ; t; T (r t ; t; T = P t (r t ; t; T P (r t ; t; T = P rtr t (r t ; t; T ) then, by alying Itô s Lemma: dp (r t ; t; T ) = (t; r t ) P rt (r t ; t; T ) + P t (r t ; t; T ) + 1 (t; r t ) P rtr t (r t ; t; T (1.14) ) dt + (t; r t ) P rt (r t ; t; T ) dw t De ning the quantities P (r t ; t; T ) (t; r t ) and P (r t ; t; T ) (t; r t ) as follows: P (r t ; t; T ) (t; r t ) = (t; r t ) P rt (r t ; t; T ) + P t (r t ; t; T ) + 1 (t; r t ) P rtr t (r t ; t; T ) P (r t ; t; T ) (t; r t ) = (t; r t ) P rt (r t ; t; T ) equation 1.14 can be written as: dp (r t ; t; T ) = P (r t ; t; T ) (t; r t ) + P (r t ; t; T ) (t; r t ) dw t (1.15) From equation 1.15 and using a no-arbitrage argument it can be shown that there exists a stochastic rocess for the market rice of risk, (t; r t ) ; such that: (t; r t ) (t; r t ) r t = (t; r t ) (1.16)

17 16 CHAPTER 1. TERM STRUCTURE MODELS for any maturity T. Equation 1.16 shows that the stochastic rocess (t; r t ) deends on the time t and on the short rate r t, but it doesn t deend on the maturity date T. This stochastic rocess is di erent across di erent models because it deends on the hyothesis about the investors references and on the roductivity. Once the market rice of risk is determined, we can nd the risk neutral robability measure P linked with the real-world robability measure Q by the following conditions: Q and P are two equivalent robability measures 4 ; the quantity: dp dq = t e(r 0 (s;rs)dw (s) R 1 t 0 (s;r s)ds) (1.17) is the Radon-Nikodym derivative of the robability measure P with resect to the robability measure Q: Given the relationshi between P and Q exressed in equation 1.17, the Girsanov theorem allows to state that, if W t is a standard Brownian motion under Q, then the rocess: fw t = W t Z t 0 (s; r s ) ds is a standard Brownian motion under P. At this oint we can say that, under the risk neutral robability measure P, the rocess for r t evolves according to the following stochastic di erential equation: dr t = b (t; r t ) dt + (t; r t ) d f W t (1.18) where: b (t; r t ) = (t; r t ) (t; r t ) (t; r t ) (1.19) As a consequence, under the risk neutral robability measure P, the rocess P (r t ; t; T ) is described by the following stochastic di erential equation: dp (r t ; t; T ) = P (r t ; t; T ) r t dt + P (r t ; t; T ) (t; r t ) d f W t (1.0) Equation 1.0 shows how, under the risk neutral robability measure, the exected return of any zero couon bond is equal to the risk free rate r t. Using equation 1.18 and alying some stochastic calculus results, namely the Feynman-Kac formula, we can determine the zero couon bond rice as 4 Two robability measures, P and Q, are said to be equivalent if: are de ned on the same measurable sace (; F); Q (A) = 0, P (A) = 0; 8A F

18 1.5. EQUILIBRIUM MODELS 17 the exected value, under the risk neutral robability measure P; of the future ayo of the bond discounted back from time T to time t, as shown in equation 1.10, i.e.: h P (t; T ) = E P e R i T t rsds 1 F t (1.10) The Vasicek model The Vasicek model (1977) was the rst term structure model with a mean reverting dynamic for the short rate. In fact, in this model, under the real-world robability measure Q; the short rate is described by the following Ornstein-Uhlenbeck 5 rocess: where: dr t = a (b r t ) dt + dw t (1.1) a (b r t ) is the drift of the stochastic rocess of the short rate and it is mean reverting, a measures the seed of mean reversion and b is the longrun mean to which the short rate is reverting; both these arameters are ositive and constant W t is a standard Brownian motion under the robability measure Q; is the instantaneous volatility of the short rate and it is a ositive constant. The mean reverting roerty characterizes most of the one factor models based on the instantaneous sot rate dynamic. Economically the mean reverting roerty means that, when the interest rates are too high or too low, with resect to their long run level, they will move towards this level. Equation 1.1 imlies that the instantaneous short rate has a conditional Normal robability distribution with mean and variance resectively equal to: E (r t ) = b + (r b) e at V ar (r t ) = a 1 e at In this model the market rice of risk is assumed constant i.e. (t; r t ) = and then, alying equation 1.18, we can nd the stochastic rocess for the short rate under the risk neutral robability measure P: dr t = a (b 0 r t ) dt + d f W t (1.) 5 In general, an Ornstein-Uhlenbeck, X t, can be exressed by the following stochastic differential equation: dx t = qx tdt + dw t where: - q and are ositive arameters - dw t = "dt e " N(0; 1) is white noise.

19 18 CHAPTER 1. TERM STRUCTURE MODELS where: b 0 = b a (1.3) Equations 1. and 1.3 show that the instantaneous short rate rocess under P is similar to the rocess under Q, as the only di erence consists in the translation of the long run level of the short rate. Given the ricing formula exressed by equation 1.10 and the evolution of the short rate ointed out by equation 1., the rice at time t of a zero couon bond with face value equal to 1 and maturity equal to T is, for a > 0 : where 6 : P (t; T ) = A (t; T ) e B(t;T )rt (1.4) [B(t;T ) A (t; T ) = e T +t] a b 0 B(t;T ) a and: a(t t) 1 e B (t; T ) = a Substituting the RHS of equation 1.4 into equation 1. we have: r (t; T ) = ln A (t; T ) e B(t;T )rt T t and then: r (t; T ) = = ln A (t; T ) + ln e B(t;T )r t ln [A (t; T )] T t T t + B (t; T ) r t T t 4a (1.5) (1.6) (1.7) (1.8) Once we have calibrated the arameters a, b 0 and, we can determine the entire term structure as a function of r t and we can use the Vasicek model to comute the rice of interest rate derivatives as well as to evaluate both lain vanilla and structured bonds, including the stochastic interest bonds that will be analyzed in Chater 5. Given equation 1.8, the yield to maturity r (t; T ) is a linear function of the ln[a(t;t )] instantaneous short rate r t with intercet equal to T t and sloe equal to B(t;T ) T t : For this reason the Vasicek model belongs to the family of A ne term structure models. 6 If a = 0; the formulas 1.5 and 1.6 become: A (t; T ) = e (T t) 3 6 and: B (t; T ) = T t

20 1.5. EQUILIBRIUM MODELS 19 The Vasicek model is consistent with term structure that can be either uward sloing, downward sloing or humed. The model can generate negative interest rates, due to the fact that the conditional distribution of the short rate is Gaussian. This is not necessarily a roblem for real interest rates, but it is a roblem when modelling nominal rates and ricing interest rate derivatives. However, it can be xed (at least in rst aroximation) by imosing some suitable conditions The volatility of the short rate in the Vasicek model For convenience, we rewrite equation 1. as follows: dr s = a (b 0 r s ) ds + d ~ W s, where a; > 0 and t < s (1.9) In order to comute the solution for equation 1.9, given its initial condition: we de ne the following Itô s rocess: r t = r; t < s Y s = (b 0 r s )e as (1.30) To obtain the stochastic di erential equation for Y s, we comute the following artial s = a (b0 and then we aly the Itô s lemma: dy s = a (b 0 or in integral Y s e as = 0 r s ) e as r s s Y ds s d ~ s s = [ a (b 0 r s ) e as + a (b 0 r s ) e as ] ds e as d ~ W s = e as d ~ W s Y s = (b 0 r s )e as = (b 0 r t )e at Z s t e au d ~ W u from which we nd out the following exression for the instantaneous short rate: r s = b 0 (b 0 r)e a(t s) + Z s t e a(u s) d ~ W u (1.31)

21 0 CHAPTER 1. TERM STRUCTURE MODELS The term e a(u s) inside the integrating function in the third term of the RHS of equation 1.31 is a deterministic function and then we can exloit one of the roerties of the Itô s integral 7 and say that, given a xed s, r s has a conditional variance 8 equal to: Z s V ar(r s j r t ) = V ar e a(u s) d W ~ u = Z s t t e a(u = e a(s s) a s) du e a(t s) a = a (1 e a(s t) ) (1.3) Valuation of euroean otions on zero-couon bonds in the Vasicek model Let us consider an euroean ut otion, with strike rice K and maturity T, written on a zero-couon bond with face value equal to 1 and maturity s > T. The rice of this otion at time t < T < s is: where: and where 9 : zcb t = K P (t; T ) N ( d ) P (t; s) N ( d 1 ) (1.33) d = d 1 = ln P (t;s) KP (t;t ) + ln P (t;s) KP (t;t ) = d 1 r 1 e a(s T ) 1 e a(t t) = a a (1.34) In order to rove the validity of equation 1.33, we have to rove that the ut otion value at time t is equal to the conditional exected value of its ayo at maturity, under the risk neutral robability measure P, discounted at the 7 Given a stochastic integral: I t = R t 0 f(u;!)dwu(!), if f(;!) = f() - i.e. if f is a deterministic function - the following relations are true: E(I t) = 0 ; E " Z (I t) T # Z T = E f(t;!)dw t(!) = E f(t;!) dt = V ar(i t). 0 0 See. Øksendal, B., (003), Stochastic Di erential Equations, Sringer, ages I.e. conditional variance to the information set at time t and then to the initial condition r t = r. 9 See. Aendx A.1 of this Chater

22 1.5. EQUILIBRIUM MODELS 1 risk free rate. Because the rice of the zero-couon bond at time T - i.e. the underlying value at maturity - is P (T; s), the rice at time t of such otion is: zcb t = P (t; T ) E P t [max(k P (T; s); 0)] (1.35) where the value P (t; T ) in the RHS of equation 1.35 is the rice at time t of a zero-couon bond with F V = 1 and maturity equal to T. By exloiting the roerty of the maximum and minimum functions according to which: min(f(x); d(x)) max( f(x); d(x)) equation 1.35 becomes: zcb t = P (t; T ) E P t [ min(p (T; s) K; 0)] = P (t; T ) E P t [min(p (T; s) K; 0)] (1.36) Let g(p (T; s)) be the robability density function of P (T; s). We have that: Z K zcb t = P (t; T ) (P (T; s) K)g(P (T; s))dp (T; s) (1.37) 1 Being P (T; s) a lognormal random variable 10, the variable ln P (T; s) is conditionally distributed as a normal random variable with standard deviation equal to, whose value is exressed in equation Given the lognormal distribution roerties 11, the conditional exected value of ln P (T; s) is: E P t (ln P (T; s)) = ln E P t (P (T; s)) (1.38) 10 Being P (T; s) = A(T; s)e B(T;s)r T (see 1.4) and having the instantaneous short rate r t a conditional normal robability distribution (see ), we can conclude that P (T; s) has a lognormal distribution. 11 Let X be a lognormal random variable with density function: 8 < 1 f(x) = 1 x e 1 ln x for x > 0 : 0 oherwise whose exected value is E(X) = e + and whose variance is V ar(x). Then, the random variable Y = ln X has a normal robability distribution with exected value E(Y ) = V ar(y ) E(ln X) = ln E(X) and variance V ar(y ). In fact we have: hence: exliciting by : E(X) = e + ln E(X) = + = ln E(X) where = E(Y ) and = V ar(y ).

23 CHAPTER 1. TERM STRUCTURE MODELS By the martingale roerty of the zero couon bond rice, the conditional exected value of the sot rice P (T; s), evaluated at time t, with t < T < s, corresonds to the forward rice P (t; T; s), and therefore equation 1.38 becomes: E P t (ln P (T; s)) = ln P (t; T; s) (1.39) Using the de nition of the forward rice from which 1 : P (t; T; s) = P (t; s) P (t; T ) the exected value of ln P (t; T; s); shown in equation 1.39, can be exressed as: E P t (ln P (T; s)) = ln P (t; s) P (t; T ) (1.40) We now de ne a new random variable Q, obtained by standardizing the normal random variable ln P (T; s): ln P (T; s) Q = EP t (ln P (T; s)) (1.41) Then, Q has a standard normal distribution whose robability density function h(q) is: h(q) = 1 e Q (1.4) Solving equation 1.41 for P (T; s) we have: P (T; s) = e Q+EP t (ln P (T;s)) (1.43) Using equations 1.41 and 1.43 to transform the integral in P (T; s) aearing in the RHS of equation 1.37 into an integral in Q, we obtain: Z ln K zcb t = P (t; T ) 1 0 = P (t; T Z ln K K 1 Z ln K 1 EP t (ln P (T;s)) EP t (ln P (T;s)) EP t (ln P (T;s)) e Q+EP t (ln P (T;s)) K 1 h(q)dq e Q+EP t (ln P (T;s)) h(q)dq+ h(q)dqa (1.44) 1 See. equation 1.4, with T + = s.

24 1.5. EQUILIBRIUM MODELS 3 Substituting the value of h(q) given in equation 1.4, the rst term in the RHS of equation 1.44 becomes: e Q+EP t (ln P (T;s)) h(q) = = = 1 e Q+EP t (ln P (T;s)) Q 1 e Q +Q+E P t (ln P (T;s)) 1 e (Q ) +E P t (ln P (T;s))+ = e EP t (ln P (T;s))+ 1 e (Q ) (1.45) 1 From equation 1.4, we see that the quantity is the robability density function of the random variable (Q ), whose conditional distribution is a normal with arameters ( ; 1). Therefore, equation 1.45 can be written as: e Q+EP t (ln P (T;s)) h(q) = e EP t (ln P (T;s))+ h(q ) and, hence, equation 1.44 becomes: 8 < t = P (t; T ) zcb Z ln K K 1 = P (t; T ) : eep t ln K KN Z ln K (ln P (T;s))+ EP t (ln P (T;s)) 1 9 = h(q)dq ; e (Q EP t (ln P (T;s)) ) h(q e EP t (ln P (T;s))+ ln K E P N t (ln P (T; s)) E P t (ln P (T; s)) )dq+ + (1.46) where N(x) is a standard normal random variable. Substituting the value of Et P (ln P (T; s)) given from 1.40, equation! 1.46 becomes: zcb t = P (t; T ) e ln P (t;s) P (t;t ) KN ln K ln P (t;s) P (t;t ) + = P (t; T ) KN ln( KP (t;t ) P (t;s) )+!! P (t;s) P (t;t ) N ln( KP (t;t ) P (t;s) )+!! + ln K ln P (t;s) N P (t;t ) +! + +

25 4 CHAPTER 1. TERM STRUCTURE MODELS and nally: 0 ln zcb B t = P (t; KP (t;t ) P (t;s) 1 C A+KP (t; T )N 0 B KP (t;t ) P (t;s) + 1 C A Denoting by d 1 and d the quantities: d = equation 1.47 becomes 13 : d 1 = ln P (t;s) KP (t;t ) + ln P (t;s) KP (t;t ) = d 1 (1.47) zcb t = K P (t; T ) N ( d ) P (t; s) N ( d 1 ) (1.93) We can see that equation 1.93 is similar to the traditional Black-Scholes formula. In fact, under the Black-Scholes formula, the rice at time t, under the risk neutral robability measure P, of a ut otion written on a stock is: BS t = K e r(t;t )(T t) N ( d ) S t N ( d 1 ) (1.51) where: d 1 = ln St K + (r t + )(T t) T t d = St ln K + (r t T )(T t) t = d 1 T t and S t is the rice of the underlying asset at time t. In both cases the rice of the underlying asset has a lognormal distribution and has the same role of (T t), that reresents the conditional variance of the logarithm of the stock rice at maturity. The rice P (t; T ) corresonds 13 Notice that, if the underlying zero couon bond has face value di erent from one, the formula 1.93 to comute zcb t changes as follows: where: zcb t = K P (t; T ) N ( d ) F V P (t; s) N ( d 1 ) (1.48) ln d = d 1 = ln F V P (t;s) + KP (t;t ) (1.49) F V P (t;s) KP (t;t ) = d 1 (1.50)

26 1.6. NO ARBITRAGE MODELS 5 r(t;t )(T to the discount factor e t), in which r(t; T ) is the risk neutral interest rate 14. Using a rocedure similar to that one used to determine the rice of a ut otion, we can nd out the formula for the evaluation of an euroean call otion with strike rice K and maturity T, written on a zero-couon bond with face value equal to one and maturity s > T, starting from the equality: P (t; T )E P t [max(p (T; s) K; 0)] = P (t; T ) Z +1 and obtaining the call rice c zcb t exressed as 15 : K (P (T; s) K)g(P (T; s))dp (T; s) c zcb t = P (t; s) N (d 1 ) K P (t; T ) N (d ) (1.53) 1.6 No arbitrage models Equilibrium models may be derived from some equilibrium framework which would reclude the existence of arbitrage in the seci ed economy. These models usually calibrated with historical data. This aroach is not ractical for ricing interest rate derivatives because, it will not guarantee that the model term structure matches the current term structure obtained from market rices. For this reason, signi cant researches have been done to make one factor models matching the current yield curve before they are used to rice interest rate derivatives. One way to match the current term structure is to allow to the coe cient in a factor model to vary deterministically over time. This tye of models, known as no arbitrage models, takes the market rice of bonds as given and rices interest rate derivatives accordingly. We roceed to analyze two imortant models of this category, the Ho-Lee model (1986) and the Hull-White model (1991). 14 Recall that, in a risk neutral world, the exected return of any nancial asset is equal to the risk-free rate. 15 As for the ut case, notice that, if the underlying zero-couon bond has a face value di erent from one, the formula 1.94 to comute c zcb t changes as follows: c zcb t = F V P (t; s) N (d 1) K P (t; T ) N (d ) (1.5) where: F V P (t;s) ln + d 1 = KP (t;t ) (1.49) F V P (t;s) ln d = KP (t;t ) = d 1 (1.50)

27 6 CHAPTER 1. TERM STRUCTURE MODELS The Ho and Lee model In 1986 Ho and Lee ublished a one factor term structure model for the interest rates where the exlanatory factor is, as in the Vasicek model, the instantaneous short rate. Starting from the assumtion that the short rate follows a random walk, this model seci es the stochastic rocess for r t under the risk neutral robability measure P as follows: where: dr t = (t) dt + d f W t (1.54) (t) is the drift of the short rate rocess and it is a deterministic function of time; f W t is a standard Brownian motion under the risk neutral robability measure P; is the instantaneous standard deviation of the short rate and it is constant. In this model (t) is the exected direction of the short rate r t movement and it doesn t deend on the level of r t. Equation 1.54 shows that at any time t the exected variation of the interest rates in the immediately following in nitesimal time interval is always the same, no matter if interest rates are high or low. Comuting analytically the variable (t) we nd the following equality: (t) = F t (0; t) + t (1.55) where F t (0; t) is the artial derivative with resect to t of the instantaneous forward rate F (0; t) observed at time zero for the maturity t: In a rst aroximation (t) is equal to F t (0; t) ; meaning that the exected variation of the short rate is aroximately equal to the sloe of the instantaneous forward rate curve. Given the ricing formula 1.10 and the short rate dynamic exressed in equation 1.54, the rice at time t of a zero-couon bond with face value equal to 1 and maturity T is: P (t; T ) = A (t; T ) e rt(t t) (1.56) where: ln [A (t; T )] = ln P (0; T ) P (0; t) (T ln [P (0; 1 t (T t) (1.57) In equations 1.56 and 1.57, the current time is zero and the times t and T are general future times with T t > 0.

28 1.6. NO ARBITRAGE MODELS 7 Substituting equation 1.57 into equation 1.56 we can nd the exlicit formula for the rice of a bond in the Ho-Lee model, i.e.: P (t; T ) = P (t; T ) = P (0; T ) P (0; ln[p (0;t)] [(T t) + 1 t(t t) +r t(t t)] (1.58) Given equation 1.58 the zero-couon bond rice, at a given future time t, is a function of the short rate that will be observed at time t, of the istantaneous forward rate F (0; t) 16 and of the market rices, at time zero, of the zero-couon bonds with maturity t and T: We oint out that, substituting the RHS of equation 1.56 into equation 1. we have: r (t; T ) = ln A (t; T ) e B(t;T )rt (1.59) T t and then: ln A (t; T ) + ln e B(t;T )r t r (t; T ) = = ln [A (t; T )] T t T t + B (t; T ) r t T t (1.60) Given equation 1.60 the yield to maturity r (t; T ) is a linear function of ln[a(t;t )] the instantaneous short rate r t with intercet equal to: T t and sloe B(t;T ) equal to: T t. For this reason the Ho-Lee model belongs to the A ne term structure Models. Through a discretization of equation 1.57, we can comute a discrete time ricing formula for a zero-couon bond. Let t be a very short time interval, for examle one day, and let R(t) be the continuously comounded interest rate relative to this time interval. Then, from equation 1.56 we can derive the following exression 17 : where: h i ln ba (t; T ) = ln P (0; T ) P (0; t) P (t; T ) = b A (t; T ) e R(t)(T t) (1.61) (T t) ln [P (0; t + t)] P (0; t) 1 t (T t) [T t (t)] t (1.6) Equation 1.61 is more used than equation 1.56 because equations 1.61 and 1.6 require only to know the zero-couon bond rices at time zero. Moreover, it is obvious that, since the quantity t is negligible, the term 1 t (T t) [T t (t)] in the RHS of the 1.6 can be aroximated to the quantity: 1 t (T t), and then, equations 1.61 and 1.6 become resectively: P (t; T ) = b A 0 (t; T ) e R(t)(T t) (1.63) 16 See equation 1.7 of section See. Hull, J., (008), Otions, Futures, and Other Derivatives, Prentice Hall, ages

29 8 CHAPTER 1. TERM STRUCTURE MODELS and: h i ln ba P (0; T ) 0 (t; T ) = ln P (0; t) (T t) t ln [P (0; t + t)] P (0; t) 1 t (T t) (1.64) A drawback of the Ho-Lee model is that it is not a mean-reverting model, since, as shown by equation 1.54, indeendently of the interest rates level, the mean direction of the istantaneous short rate in the immediately following in- nitesimal time interval is always the same. Another inconvenience of this model is that it allows to reresent a reduced set of volatility structures. In articular: 1. the volatility at time t of a zero-couon bond with maturity T is a linear function of T ;. the instantaneous standard deviation at time t of the sot rate of return of a zero-couon bond with maturity T is constant; 3. the instantaneous standard deviation of the instantaneous forward rate with maturity T is constant. The Ho and Lee model can generate negative interest rates, due to the fact that the conditional distribution of the short rate is Gaussian. This is not necessarily a roblem for real interest rates, but it is a roblem when modelling nominal rates and ricing interest rate derivatives. However, it can be xed (at least in rst aroximation) by imosing some suitable conditions The volatility of the short rate in the Ho and Lee model We rewrite the stochastic rocess of r t ; under the risk neutral robability measure P, for the Ho and Lee model as: dr s = (s) ds + d ~ W s (1.65) with the initial condition: r t = r; t < s The solution of equation 1.65 is: r s = r t + Z s (s) ds + Z s t t d ~ W u (1.66)

30 1.6. NO ARBITRAGE MODELS 9 The rst two terms of equation 1.66 are deterministic functions and, therefore, the conditional variance of the short rate is equal to: Z s V ar (r s ) = V ar d W ~ u (1.67) t Z s = V ar d W ~ u = (s t) Valuation of euroean otions on zero-couon bonds in the Ho-Lee model The evaluation of euroean otions on zero couon bonds in the Ho and Lee model 18 is based on a formula quite similar to that derived in Black model 19. Let us consider an euroean ut otion, with strike rice K and maturity T, written on a zero-couon bond with face value equal to 1 and maturity s > T. The rice at time t of this ut otion is denoted with zcb t and it is equal to 0 : t where: zcb t = K P (t; T ) N ( d ) P (t; s) N ( d 1 ) (1.68) P (t; T ) is the rice at time t of a zero couon bond with maturity T, and it is an inut required by the model; P (t; s) is the rice at time t of a zero couon bond with maturity s > T, and it is an inut required by the model; N(x) is the value in x of the standard normal distribution function; and the quantities d 1 and d are resectively given by: d 1 = 1 ln P (t; s) K P (t; T ) + (1.69) 18 See. Jamshidian, F., 1989, An Exact Bond Otion Pricing Formula, Journal of Finance n See Aendix B.1. of this Chater 0 If the underlying zero couon bond has a face value di erent from one, the 1.68, the 1.69 and the 1.70 changes as follows: zcb t = K P (t; T ) N ( d ) F V P (t; s) N ( d 1 ) d 1 = 1 F V P (t; s) ln + K P (t; T ) d = 1 ln F V P (t; s) K P (t; T )

31 30 CHAPTER 1. TERM STRUCTURE MODELS and: where: d = 1 P (t; s) ln K P (t; T ) (1.70) = (s T ) T t (1.71) Analogously, the rice at time t of an euroean call otion with strike K and maturity T written on a zero couon bond with face value equal to 1 and maturity s; s > T, is denoted with c zcb t and it is equal to 1 : where: c zcb t = P (t; s) N (d 1 ) K P (t; T ) N (d ) (1.7) P (t; T ) is the rice at time t of a zero couon bond with maturity T, and it is an inut required by the model; P (t; s) is the rice at time t of a zero couon bond with maturity s > T, and it is an inut required by the model; N(x) is the value in x of the standard normal distribution function; and the quantities d 1 and d are given by the same equations seen for the euroean ut otion, i.e. equations 1.69 and The Hull and White model In 1990 Hull and White ublished an extension of the Vasicek model in which the short rate rocess is mean reverting as in the Vasicek model and it is consistent with the initial term structure of interest rates. In this model, under the risk neutral robability measure, P; the instantaneous short rate dynamics are governed by the following stochastic di erential equation: dr t = [ (t) ar t ] dt + d f W t (1.73) or: where: (t) dr t = a a r t dt + d f W t (1.74) 1 If the underlying zero couon bond has a face value di erent from one, the 1.7, the 1.69 and the 1.70 changes as follows: c zcb t = P (t; s) N (d 1 ) K P (t; T ) N (d ) d 1 = 1 F V P (t; s) ln + P (t; T ) K d = 1 ln F V P (t; s) P (t; T ) K

32 1.6. NO ARBITRAGE MODELS 31 h a (t) a i r t is the drift of the stochastic rocess of the short rate and it is mean reverting; a is the constant seed of mean reversion and (t) a is a function reresenting the long run level of the istantaneous short rate. This means that at the generic time t the instantaneous short rate goes to (t) =a with seed equal to a; f W t is a standard Brownian motion under the risk neutral robability measure P; is the instantaneous standard deviation of the short rate and it is constant. We have to observe that, as in the Vasicek model, also in the Hull and White model the short rate drift is mean reverting but in this seci cation the long run level of the instantaneous short rate is a deterministic function of the time. Analytically comuting the variable (t), under no-arbitrage condition the following equation holds: (t) = F t (0; t) + af (0; t) + a 1 e at (1.75) where F t (0; t) is the artial derivative with resect to t of the instantaneous forward rate F (0; t) observed at time zero for the maturity t: The rst two terms in the RHS of equation 1.75 show that, since (t) is a function of the initial term structure of istantaneous forward rates, the seci cation of the Hull and White is consistent with the initial term structure observed in the market. Moreover, the last term in equation 1.75 is negligible, so that the drift of the rocess r t at time t is aroximately equal to: F t (0; t) + af (0; t) : At this oint we have that, in average, the short rate follows aroximately the sloe of the initial instantaneous forward rate curve and, if it is faraway from that level, it will move towards it with a seed equal to a. Also we can observe that the seci cation in equation 1.73 include the Ho and Lee model as a articular case when the arameter a = 0: The stochastic integral corresonding to the stochastic di erential equation 1.73 can be exressed as : r t = x t + t (1.76) where x t is a Gaussian stochastic rocess described by the following stochastic di erential equation: dx t = ax t dt + d f W t (1.77) and t is the following deterministic function: t = F (0; t) + a 1 e at (1.78) See. Brigo, Mercurio, 006, Interest Rate Models - Theory and Practice, Sringer, ages. 7-74