Kees E. Bouwman. Essays on Financial Econometrics. Modeling the Term. Structure of Interest Rates. Kees E. Bouwman Essays on Financial Econometrics

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1 Uitnodiging som 2008 Modeling the Term Structure of Interest Rates Research school SOM Faculty of Economics and Business University of Groningen PO Box AV Groningen The Netherlands This dissertation bundles five studies in financial econometrics that are related to the theme of modeling the term structure of interest rates. The main contribution of this dissertation is a new arbitrage-free term structure model that is applied in an empirical analysis of the US term structure. The model has a simple but flexible structure and uses factors that have a clear economic interpretation. The empirical analysis indicates that time-varying risk premia are a dominating factor in the relation between the term structure and the business cycle. Furthermore, a comparison of the short rate implied by the model and the Federal Funds target rate shows that the implied short rate leads the target. Kees E. Bouwman som Kees E. Bouwman Essays on Financial Econometrics Essays on Financial Econometrics: Modeling the Term Structure of Interest Rates faculty of economics and business Hierbij nodigt Kees Bouwman Kees E. Bouwman Essays on Financial Econometrics Modeling the Term Structure of Interest Rates Essays on Financial Econometrics Modeling the Term Structure of Interest Rates De verdediging vindt plaats op donderdag 6 maart 2008 om 13:15 in de aula van het Academiegebouw van de Rijksuniversiteit Groningen, Broerstraat 5, Groningen. Aansluitend is een receptie in het Academiegebouw. De paranimfen: Richard Jong-a-Pin r.m.jong-a-pin@eco.rug.nl Lydian Medema l.medema@rug.nl ISBN u uit voor de bijwoning van de openbare verdediging van zijn proefschrift getiteld: Theses in Economics and Business

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3 Essays on Financial Econometrics Modeling the Term Structure of Interest Rates Kees Evert Bouwman

4 Publisher: PPI Publishers Postbus AH Enschede The Netherlands Printed by: PrintPartners Ipskamp ISBN: c 2008 Kees E. Bouwman All rights reserved. No part of this publication may be reproduced, stored in a retrieval system of any nature, or transmitted in any form or by any means, electronic, mechanical, now known or hereafter invented, including photocopying or recording, without prior written permission of the publisher.

5 Essays on Financial Econometrics Modeling the Term Structure of Interest Rates Proefschrift ter verkrijging van het doctoraat in de Economie en Bedrijfskunde aan de Rijksuniversiteit Groningen op gezag van de Rector Magnificus, dr. F. Zwarts, in het openbaar te verdedigen op donderdag 6 maart 2008 om uur door Kees Evert Bouwman geboren op 29 februari 1980 te Haren

6 Promotor: Prof. dr. P.A. Bekker Beoordelingscommissie: Prof. dr. T.K. Dijkstra Prof. dr. B. Melenberg Prof. dr. A.C.F. Vorst

7 Acknowledgements For me personally, this thesis not only represents the research I conducted as a PhD student at the University of Groningen, but it also marks the end of four great years in Groningen. Many people have been helpful and supportive and I like to take the opportunity to express my gratitude to them. First and foremost, I would like to thank my supervisor Paul Bekker for showing me how to do research by getting me actively involved in his challenging research. I remember our many intensive and stimulating discussions about research and other subjects. I learned from Paul that creativity, perseverance, a critical attitude and optimism are essential qualities for a good researcher. Special thanks go to Richard Jong-a-Pin and Lydian Medema for being great roommates. Furthermore, I thank my colleagues at the Department of Economics and Econometrics for providing a pleasant working atmosphere. I enjoyed working with Jan Jacobs on our paper about the use of real-time data in macroeconomic forecasting. Furthermore, Jan was always available for advice. Ward Romp gave advice on LATEX and provided a complete thesis class. I would like to thank the members of the reading committee, Theo Dijkstra, Bertrand Melenberg and Ton Vorst for reading the manuscript. At the Econometric Institute of the Erasmus University Rotterdam I found a new and stimulating working environment. I am grateful to Dick van Dijk for giving me the opportunity to complete this thesis and continue with my research. Fortunately these four years were not exclusively about research and I found the time for windsurfing and playing saxophone as well. Therefore, I thank my friends from windsurfing and music, in particular my band members Marcel, Philip, Theo and Donald from Fee-5-Fo-Fun for sharing so many enjoyable moments. A special word goes to my friends Bernardo Kok and Gerrit Starrenburg for the many interesting discussions. Furthermore, I am grateful for the help and support of my parents Luuk and Gien and my brother Ewout. Their advice and assistance in more practical matters proved invaluable, particularly in stressful times. Dieneke Cupido helped out numerous times with practical matters and she is gratefully acknowledged.

8 ii Finally I like to thank Charlotte for her love and support. Charlotte, your perseverance is inspiring and your support is invaluable to me. Together, we make such a great team! Kees Bouwman

9 Contents 1 Introduction Objective Outline Arbitrage-Free Modeling of the Term Structure in Continuous Time Introduction Bond preliminaries Bond price dynamics Asset pricing theory for the term structure of interest rates Replication and arbitrage Bond pricing relations Market prices of risk and market completeness Arbitrage-free pricing in dynamic factor models of the term structure Short rate models Vasicek model Cox, Ingersoll and Ross model Time-inhomogeneous models Affine term structure models Classification of ATSMs Market prices of risk in ATSMs Quadratic term structure models The Heath, Jarrow and Morton framework The setup of the HJM framework Dynamic factor models as HJM models Consistency problems and finite dimensional realizations in HJM models

10 iv Contents 2.8 Dynamic factor models with risky assets as factors The general setup A single-factor model with a detrended asset price as factor A model with multiple detrended assets Discussion A Unified Approach to Mean-Variance Portfolio Selection Introduction The Framework and Single-Period Results The market The frontiers MV efficient strategies and the value function The instantaneous frontier The one-period frontier Dynamic MV efficiency MV efficiency in continuous time MV efficiency in discrete time Duality and MV bounds for SDF s A dual market The discount factors Conclusion A Proof of Theorem B Proof of Theorem C Derivation of s 2 LSR and m LSR in Example D Proof of Theorem E Proof of Theorem Arbitrage Smoothing in Fitting a Sequence of Yield Curves Introduction Methodology Empirical Analysis The data The Nelson-Siegel model The estimation procedure The results Concluding Remarks

11 Contents v 5 Modeling Asset Volatility with GARCH Models in the Presence of Outliers Introduction GARCH models Univariate GARCH models Two-Component models Multivariate Models Estimation methods Maximum Likelihood estimation Least Absolute Deviation estimation S estimation Reparameterizations Empirical results The data A univariate analysis of the return volatility of the Wilshire A multivariate analysis of the joint return volatility of US stock and bond market Conclusion A Parsimonious Arbitrage-Free Model of the US Term Structure Introduction A general model with asset prices as factors The aggregate capital market The aggregate capital market frontier Dynamic trading and the aggregate capital market A reduction of the parameter space Equilibrium growth rates Factor volatilities Methodology The estimation procedure Empirical results Fit of the yield curve Factors Risk premia and the business cycle Expected returns Bond return volatilities Short rate

12 vi Contents 6.7 Concluding remarks A Derivation of the optimal Sharpe ratio B Data sources and data construction C Estimation of the covariance matrix of bond returns Bibliography 163 Samenvatting (Summary in Dutch) 171

13 Chapter 1 Introduction Over the past decades, the field of finance has undergone some major developments. One of the major developments is the introduction of Mean-Variance analysis by Markowitz (1952) to solve portfolio selection problems. Till then, portfolio selection was primarily based on a fundamental analysis of the individual securities, which is often referred to as stock-picking. With Mean-Variance analysis, the perspective shifted from fundamentals to the statistical properties of security prices, because Mean-Variance analysis defines optimal portfolios by a trade-off between expected return and risk. Mean-Variance analysis subsequently lead to the development of the Capital Asset Pricing Model (CAPM) by Sharpe (1964), Lintner (1965a,b) and Mossin (1966). A second important development is the generalization from a single-period to a multi-period treatment of the portfolio selection problem. An important limitation of the single-period treatment is that it does not recognize that investors can adjust their portfolios over time as more information comes available. Multi-period portfolio selection problems are however much harder to solve than single-period problems and require the use of dynamic optimization tools like dynamic programming. The multi-period portfolio selection problem was pioneered by Merton (1969, 1971) in a continuous-time expected utility framework by using dynamic programming techniques. One of the most important innovations in the field of finance is the path-breaking work of Black and Scholes on the pricing of options (Black and Scholes, 1973). The fundamental insight of Black and Scholes is that a stock option can be synthesized by a trading strategy that invests in the underlying stock and the risk-free rate. Hence, the value of the stock option is determined by the underlying stock and the

14 2 Chapter 1 risk-free rate. The asset pricing approach by Black and Scholes is characterized as relative pricing as opposed to absolute pricing. Absolute pricing the prevailing approach before Black and Scholes uses a general equilibrium framework to explain asset prices by economic fundamentals like the preferences and expectations of a representative agent. A strength of this approach is that it offers an explanation of asset prices in terms of economic fundamentals. However, this explanatory focus is also an important weakness as economic fundamentals like preferences are hard to measure and the approach requires many strong assumptions. The relative pricing approach gives up on the ambition to explain asset prices by economic fundamentals, but instead describes asset prices in relation to other asset prices. In return for giving up this ambition, relative pricing requires weaker assumptions and produces more accurate predictions of asset prices. These developments in financial economics are strongly influenced by advancements in econometric techniques. The work of Markowitz and Merton promoted interest in the time-series properties of asset prices. A key feature of asset prices is a time-varying volatility. Volatility refers to the variability of a time-series and plays a dominant role in finance as it measures risk. With the introduction of the class of AutoRegressive Conditional Heteroscedasticity (ARCH) models by Engle (1982), researchers are given a powerful tool to empirically model the volatility of asset prices. An important and challenging area of finance is the modeling of interest rates. Interest rate models share important commonalities with models of other financial markets and have been strongly influenced by the developments in the field of finance. On the other hand, the bond market displays some distinct features that present unique challenges to interest rate modeling. In particular, the bond market contains a wide variety of different but closely related assets, in particular bonds with a different time to maturity. This gives rise to the notion of the term structure of interest rates, which describes the relation between the price of a bond and its time to maturity. A typical interest rate model describes the evolution of the term structure of interest rates and is called a term structure model. 1.1 Objective This dissertation comprises a collection of five studies in financial econometrics. All five studies are linked, at least indirectly, to the theme of modeling the term

15 Introduction 3 structure of interest rates. The main contribution of this dissertation is presented in the last study, where a new term structure model is developed that is applied in an empirical analysis of the US term structure. Many term structure models are arbitrage-free models. These models are typically based on a relative pricing approach and impose a consistency condition on the shape and evolution of the term structure to rule out arbitrage opportunities. Most often these models are dynamic factor models that assume that the term structure is driven by a small number of factors. Absence of arbitrage then requires consistency between the factor dynamics and the shape of the yield curve. The complexity of this condition makes tractability a major concern and has motivated specifications of the factor dynamics that generate tractable expressions for the term structure. From an empirical perspective, the complex structure of arbitrage-free models poses a problem. In particular, it is difficult to find parsimonious, flexible arbitragefree models that accurately describe the yield curve. This has lead empirical researchers and practitioners to use models and practices that are inconsistent with absence of arbitrage. For example, the yield curve models by Nelson and Siegel (1987) and Svensson (1999) are inconsistent with absence of arbitrage, as observed by Björk and Christensen (1999) and Filipović (1999, 2000). Nevertheless, these models are widely used for their simple and flexible structure. An important question is whether or not deviations from no-arbitrage are bad. Arguably not to bad for purposes of interpretation, forecasting and risk management, because models like the Nelson-Siegel are sufficiently flexible to fit the term structure well when the true term structure is arbitrage-free. The question remains whether we can account for and control these deviations from no-arbitrage. In particular, the information content of the no-arbitrage assumptions is potentially valuable so that incorporating this information might enhance the performance of the model. These questions are investigated in Chapter 4, where an empirical device is introduced that allows us to partially incorporate no-arbitrage information in fitting a sequence of yield curves. An alternative approach is to develop arbitrage-free models that have a more parsimonious structure. This approach is followed in Section 2.8 and Chapter 6, where a new arbitrage-free term structure is developed that uses asset prices as factors. This model has two major advantages over other term structure models. First, the model has a simple but flexible structure where bond prices are linear functions of the factors. Secondly, the factors have a clear economic interpretation

16 4 Chapter 1 as asset prices. Chapter 6 follows this approach to derive a three-factor model that is used in an empirical analysis of the US term structure. Chapter 3 discusses Mean-Variance analysis in a multi-period framework. Though not directly related to term structure modeling, Mean-Variance analysis proves an invaluable modeling tool in the construction of our term structure model in Chapter 6. Mean-Variance analysis is around for more than half a century, but it has only recently been generalized to a multi-period framework by Li and Ng (2000) in discrete time and by Bajeux-Besnainou and Portait (1998) and Zhou and Li (2000) in continuous time. This is rather surprising as portfolio selection has been solved in a multi-period expected utility framework by Merton (1969, 1971) almost 40 years ago. An important void in the current multi-period MV literature is a uniform treatment of MV portfolio selection for a general incomplete market in both discrete and continuous time. For instance, we are unaware of results on the continuous-time generalization of the MV portfolio selection problem for markets with only risky assets. Therefore in Chapter 3 we derive a uniform solution to the multi-period MV portfolio selection problem in both discrete and continuous time, covering both complete and incomplete market. Furthermore, we derive the dynamic generalization of the pricing implications of incomplete markets as described by Hansen and Jagannathan (1991). Chapter 5 turns to the modeling of asset return volatility with GARCH models. Volatility is not directly observable and is therefore often estimated on the basis of a GARCH model. However, outliers in the returns like the 87 stock market crash can strongly affect the volatility estimates, typically producing unrealistically high volatility estimates. This makes these volatility estimates of limited use in e.g. asset pricing models. This issue is further explored in Chapter Outline Chapter 2 gives an overview of arbitrage-free term structure models in continuous time. First, it reviews the asset pricing theory that underlies these models. Next, the dynamic factor models are discussed with a particular emphasis on the widely used class of Affine Term Structure Models. The HJM framework of Heath, Jarrow and Morton (1992) provides a different perspective on term structure modeling and is subsequently reviewed. Finally we introduce a new approach to interest rate modeling that uses asset prices as factors. The idea is explored in a simple setting of a single-factor model where the factor is a discounted asset price.

17 Introduction 5 Chapter 3 turns to Mean-Variance portfolio selection in a multi-period context. We use dynamic programming to solve the multi-period Mean-Variance (MV) portfolio selection problem in both discrete and continuous time. Stochastic market parameters are incorporated using a vector of state variables. The joint process of the prices and the state variables is assumed Markov. Our solution applies to both complete and incomplete markets as well as to markets with or without a riskless asset. In particular for markets with deterministic parameters, we provide explicit solutions. In such markets, where no riskless asset needs to be present, we describe term-independent uniform MV efficient investment strategies. In addition to the MV efficient investment strategies in the primal market, we construct a dual market in which stochastic discount factors can be expressed as self-financing strategies. We establish duality between MV efficient strategies in the primal and dual markets, and obtain MV bounds on the space of stochastic discount factors. Chapter 4 introduces a simple method to incorporate arbitrage information when fitting a sequence of yield curves. Empirical modeling of the yield curve is often inconsistent with absence of arbitrage. In fact, many parsimonious models, like the popular Nelson-Siegel model, are inconsistent with absence of arbitrage. In other cases, arbitrage-free models are often used in inconsistent ways by recalibrating parameters that are assumed constant. For these cases, Chapter 4 introduces an arbitrage smoothing device to control arbitrage errors that arise in fitting a sequence of yield curves. The device is applied to the US term structure for the families of Nelson-Siegel curves. It is shown that the arbitrage smoothing device contributes to parameter stability and smoothness. Chapter 5 investigates different GARCH models and different estimation methods for their robustness to outliers in the returns like the 87 stock market crash. Empirical results for the daily returns of the Wilshire 5000 Composite Index reveal that volatility estimates based on robust estimation are comparable with estimates of a long-run volatility component based on Maximum Likelihood estimation. Subsequently, the effect of outliers is further explored with a multivariate GARCH model for the returns on the Wilshire and on bonds. Since robust estimation has not yet been generalized to multivariate models, we account for outliers in the Wilshire by using the long-run volatility component. Chapter 6 introduces a new three-factor model for the US term structure that uses prices of risky assets as factors. The model describes the yield and return volatility for bonds with a maturity over one year. The factors are modeled by using Mean-Variance analysis to describe the aggregate capital market. Bond prices in

18 6 Chapter 1 this model are affine functions of the factors, which contrasts the popular class of affine term structure models where bond prices are exponentially affine functions of the factors. An exploratory analysis of the estimation results indicates that timevarying risk premia are a dominating factor in the relation between the term structure and the business cycle. Furthermore, a comparison of the short rate implied by the model and the Federal Funds target rate shows that the implied short rate leads the target.

19 Chapter 2 Arbitrage-Free Modeling of the Term Structure of Interest Rates in Continuous Time 2.1 Introduction Capital markets play a crucial role in an economy. They allow consumers to substitute consumption over time, as well as investors to obtain the necessary funds for investment. The main question in financial economics is how to determine prices of the assets in these market. Historically, asset pricing was mainly considered in a equilibrium framework. Such a framework provides powerful insights in asset prices in relation with other aspects of an economy, like for instance the preference structure of consumers. Unfortunately, the equilibrium approach is often difficult to apply, because it involves a whole range of constructs that are not directly observable and difficult to measure, such as the preference structure. A major change in the field of asset pricing was caused by the seminal work of Black and Scholes (1973). They used a replication argument to derive their famous option pricing formula. Their model assumes an underlying market with a stock and a risk-free rate. They showed that an option on the stock can be perfectly replicated by trading continuously and without friction in the underlying market. Assuming absence of arbitrage then fixes the price of the option relative to the underlying market. This approach does not demand strong assumptions regarding the preference structure in the economy.

20 8 Chapter 2 The Black and Scholes approach can be characterized as a relative pricing approach. The assumption of absence of arbitrage plays a key role in the approach, therefore this approach is also referred to as arbitrage pricing. Since Black and Scholes seminal paper in 1973, the absence of arbitrage paradigm has dominated the field of finance in both the financial industry as well as in research. This holds to no lesser extent for the field of interest rate modeling. Much emphasis in asset pricing is put on stock market related assets, but the bond market is of equal importance. The bond market is markedly different from the stock market, since bonds generate a deterministic cash flow, while the revenues of stocks are inherently uncertain. Thus bonds generate a fixed stream of income, therefore the bond market is often called the fixed-income market. Despite their fixed stream of income, bonds are still risky assets, because the the valuation of the income stream in the future is still subject to much uncertainty. This uncertainty in bond prices has induced a demand for all kinds of derivative instruments like e.g. call options on bonds. Over the years, these fixed-income instruments have become increasingly more important and are now widely traded in the fixed-income market. Interest rate modeling is an important field in asset pricing and deals with pricing assets in the fixed-income market. The most basic asset in the fixed-income market is a zero-coupon bond. The complete set of zero-coupon bonds of all maturities results in the term structure of interest rates and forms the basis of the fixedincome market. A typical interest rate model describes the evolution of the term structure of interest rates, and is thus often referred to as a term structure model. The relative pricing approach has a marked impact on term structure modeling. This has lead to the arbitrage-free term structure models that require that the evolution of term structure is consistent with absence of arbitrage. The absence of arbitrage typically restricts the dynamics of the term structure and not necessarily its shape. However, most arbitrage-free term structure models are so-called dynamic factor models. They assume that bond prices are driven by a limited set of factors. The dynamics of the factors and the dependence between bond prices and the factors must be consistent with absence of arbitrage, which generally imposes restrictions on the attainable shapes of the term structure. It is important to realize that an arbitrage-free term structure can be supported by a general equilibrium. A general equilibrium framework can be utilized to derive factors for a dynamic factor model of the term structure as in Cox, Ingersoll and Ross (1985). However models are more often motivated by their ability to capture

21 Arbitrage-Free Modeling of the Term Structure in Continuous Time 9 observed characteristics of the term structure and factors are often specified on a rather ad hoc basis. Derivative pricing in arbitrage-free term structure models is relatively straightforward, because in general the derivative can be perfectly replicated by a dynamic bond strategy. Thus, given the evolution of the bond prices, the absence of arbitrage dictates the price of the derivative. Many term structure models are used for the purpose of derivative pricing, but they can also be used in managing bond portfolios and in explaining the term structure. The dynamic factor models in particular can potentially be used to summarize and interpret the information contained in the term structure. Interest rate models are often most conveniently stated in continuous time as we can rely on powerful toolbox of mathematical results in continuous time. In particular, Itô s Lemma is a powerful tool in continuous-time modeling that allows us derive the differential of a function of a stochastic process. Nevertheless, most models can be stated equally well in discrete time. This chapter gives an overview of arbitrage-free term structure modeling in continuous time. The outline of this chapter is as follows. Section 2.2 and 2.3 introduce the fundamentals of interest rate modeling and the related asset pricing methodology. Section 2.4 reflects on the the most widely used interest rate models, the Vasicek and the CIR model, and discusses time-inhomogeneous extensions of these models. Subsequently, two important classes of multi-factor models, the affine term structure models and the quadratic term structure models, are reviewed in Section 2.5 and 2.6. An alternative perspective on interest rate modeling is obtained by the HJM framework of Heath, Jarrow and Morton (1992), which is discussed in Section 2.7. Finally Section 2.8 introduces a new approach to interest rate modelling that uses asset prices as factors. The idea is explored in a simple setting of a single-factor model where the factor is a discounted asset price. 2.2 Bond preliminaries The zero-coupon bond is the most fundamental asset in the bond market. It pays for certain one monetary unit on a specified future date, the so-called maturity date. By combining different zero-coupon bonds, we can construct more complicated assets like coupon-bearing bonds. In fact, any deterministic stream of future payments can be replicated by a portfolio of different zero-coupon bonds.

22 10 Chapter 2 Definition 2.1. A zero coupon bond with maturity date T pays for certain one monetary unit at maturity date T. In period t T its maturity is defined by τ = T t and represents the remaining time to maturity. The price at time t T is denoted by P t (T). Note that the zero-coupon bond pays out for certain, thereby excluding the possibility of defaulting on behalf of the issuer. Therefore, we think of zero-coupon bonds as government bonds rather than corporate bonds. We will assume a market in which every zero-coupon bond is frictionlessly traded and that P t (T) is differentiable in T. This gives rise to a differentiable bond curve for every time t given by {P t (T) : T > t} that represents the term structure of interest rates at time t. The prices of different bonds are not directly comparable or meaningful. Therefore we often consider other important aspects of the term structure. Definition 2.2. The yield of a T-bond at t < T is given by y t (T) = log P t(t). T t The instantaneous forward rate with maturity T at t T is given by f t (T) = log P t(t). T The short rate at t is defined by r t = f t (t) = lim T t y t (T). The long rate at t (if it exists) is defined by l t = lim T y t (T) Bond price dynamics An interest rate model generally describes both the bond curve as well as its dynamic evolution. Bonds are risky assets and therefore a stochastic model is used. The probability structure and flow of information in a model is formally described by a filtered probability space given by ( Ω, F, {F t } t 0, P ). The filtration F t represents all information available at time t. We restrict ourselves to continuous-time

23 Arbitrage-Free Modeling of the Term Structure in Continuous Time 11 models that are driven by Brownian motions and therefore we introduce a multidimensional standard Brownian motion W t that is adapted to the filtration F t. The dynamics of the bond curve, the yield curve, the forward curve and the short rate are given by dp t (T) = P t (T) [ µ P,t (T) dt + σ P,t (T) ] dw t, (2.1) dy t (T) = µ y,t (T) dt + σ y,t (T) dw t, (2.2) d f t (T) = µ f,t (T) dt + σ f,t (T) dw t, (2.3) dr t = µ r,t dt + σ r,t dw t, (2.4) respectively, where the drifts and volatilities are adapted processes 1. The relation between the dynamics of the bond curve and other aspects of the term structure can directly be obtained by applying Itô s Lemma (see e.g. Björk, 2004). Regularity assumptions regarding the drift and volatility processes are needed though (see Heath et al. (1992)). The dynamics of the bond curve are stated above in terms of a fixed maturity date. These dynamics reflect the fact that maturity reduces as time passes. Alternatively, constant maturity dynamics represent the price change of a bond with a particular maturity, thereby ignoring the reduction in maturity. Constant maturity dynamics are useful for describing time series of bond prices, because most bond price data is available in constant form. The bond price dynamics can be converted to constant maturity dynamics by applying Itô s Lemma. If we define Pt (τ) = P t(t + τ), then ( ) dpt Pt (T) (τ) = dp t (t + τ) + T dt. T=t+τ Notice that changing to constant maturity dynamics only affects the drift of the dynamics. 2.3 Asset pricing theory for the term structure of interest rates Bonds are different from stocks, and so are interest rate models different from stock market models. Still, all these asset pricing models share the same foundations, that of the modern asset pricing theory. 1 i.e. adapted to the filtration {F t } t 0

24 12 Chapter 2 Asset pricing in continuous time heavily relies on advanced mathematical results from stochastic calculus, martingale theory and probability theory. It is beyond the scope of this chapter to give a rigorous mathematical treatment on these topics. Instead, an intuitive economic understanding of the modeling framework is emphasized. For a more extensive and formal treatment of arbitrage pricing the reader is referred to e.g. Björk (2004), Cochrane (2001), Etheridge (2002) or Duffie(2001). Cochrane (2001, p.xiv) identifies two main approaches in asset pricing. The first approach, known as absolute pricing, relates asset prices to fundamental economic factors like preferences and expectations of market participants, the uncertainty in the economy and the structure of the market. The second approach, known as relative pricing, relates asset prices to prices of other assets. The idea is that an asset can be replicated by trading in group of other assets. Ruling out arbitrage opportunities fixes the price of the asset to the value of the replicating strategy. The advantage of relative pricing is that it requires less stringent assumptions regarding preferences and expectations. Relative pricing uses replication and the assumption of an arbitrage-free market to establish relations between asset prices. The following common assumptions are made Assumption 2.1. The asset market is characterized by: All market participants only value the return and risk characteristics of their holdings. At all times it is possible to trade any amount a R of any asset. This includes the possibility of unlimited short selling. Trading is frictionless. All market participants are price takers. The market is arbitrage free. A direct consequence of these assumptions is that in market equilibrium any two portfolio strategies with the same risk and return characteristics should have the same price, otherwise there exist arbitrage opportunities. Absence of arbitrage corresponds with a market equilibrium, because if an arbitrage opportunity exists, then rational investors will fully exploit the opportunity. This will affect asset prices to the extend that the arbitrage opportunity will disappear.

25 Arbitrage-Free Modeling of the Term Structure in Continuous Time 13 The option pricing formula of Black and Scholes (1973) is a famous illustration of relative pricing. Black and Scholes consider a market consisting of a risk-free rate and a stock and construct a dynamic investment strategy that replicates the payoff of European option. Assuming absence of arbitrage fixed the option price to value of the replicating strategy. The Black-Scholes approach prices the option in terms of more fundamental assets like the risk-free rate and the stock. This is different for interest rate modeling, because bonds are rather fundamental assets themselves. Instead, bonds are priced relative to each other such that the joint evolution of the term structure is arbitrage free Replication and arbitrage For ease of exposition consider a set of bonds with arbitrary maturity dates T 1,..., T k. The prices of these bonds are collected in the vector P t = (P t (T 1 ),..., P t (T k )). Investors in the market trade these bonds continuously and without friction and form dynamic portfolios. A portfolio is represented by a k-dimensional adapted process θ t of portfolio weights, where each component represents the number of corresponding assets held at time t. The portfolio value equals the total amount that is invested and is given by V θ,t = θ t P t. Using Itô s Lemma, the value dynamics of the portfolio are given by dv θ,t = θ t dp t + P t dθ t + dp t dθ t. The first term on the right-hand side reflects the change in the value due to changes in asset prices. The second and third term reflect changes due rebalancing the portfolio. If the sum of the second and third term are positive then extra money is invested and vice versa. A portfolio strategy is called self-financing if no value is invested or withdrawn from the portfolio apart from an initial investment. Hence, a portfolio is selffinancing if dv θ,t = θ t dp t. The existence of an arbitrage opportunity gives rise to an arbitrage portfolio, as defined in Definition 2.3, that exploits such opportunity.

26 14 Chapter 2 Definition 2.3. A self-financing portfolio defined by θ is called an arbitrage portfolio if V θ,0 = 0, P(V θ,t 0) = 1, P(V θ,t > 0) > 0, for T > 0. The market is said to be arbitrage free if no arbitrage opportunities exist, i.e. there does not exist an arbitrage portfolio. The short rate represents the rate of return on a just-maturing bond. By continuously rolling over just-maturing bonds we obtain a portfolio strategy known as the bank account or the money market account. The bank account yields a deterministic instantaneous return equal to the short rate. With B t denoting the value of a bank account at time t, we have the following dynamics for the bank account db t = B t r t dt. The bank account is helpful in understanding the implications of arbitrage-free pricing for the term structure. If the short rate is deterministic then pricing zerocoupon bonds is straightforward. Buying a zero-coupon bond with maturity date T is then equivalent to investing B(t) in the bank account. Assuming the market is B(T) arbitrage free implies P t (T) = B t B T = e or equivalently T t r t dt f t (T) = r T, and the whole term structure is determined by the short rate process. The dynamics of term structure are deterministic, which makes a deterministic short rate less adequate as a model for the term structure Bond pricing relations A stochastic short rate process introduces risk in bond prices. This produces more realistic interest rate models for the term structure, but also complicates the charac-

27 Arbitrage-Free Modeling of the Term Structure in Continuous Time 15 terization of arbitrage-free bond prices. Modern asset pricing theory uses martingale theory to characterize arbitrage-free prices. Definition 2.4. An adapted process X t is a martingale if for all T > 0 E( X T ) < and satisfies the martingale property E t (X T ) = X t t T. The symbol E t denotes conditional expectation conditioned on all information at time t, i.e. E t (X T ) = E (X T F t ). The main idea in arbitrage-free pricing is to transform price processes to martingales and then use the martingale property to obtain arbitrage-free prices of contingent claims. An important tool in transforming price processes to martingales is the change of probability measure. Instead of using the natural probability measure P, processes can be represented under alternative probability measures that are equivalent to P. Definition 2.5. Two probability measures P 1 and P 2 defined on (Ω, F) are equivalent if for all A F P 1 (A) = 0 P 2 (A) = 0. There are two equivalent characterizations of arbitrage-free prices as summarized by the following proposition. Proposition 2.1. The following three statements are equivalent: 1. The market is arbitrage free. 2. There exists a positive adapted process Λ t such that for all T > t and for every asset price process S t in the market, the process Λ t S t is a P-martingale. 3. There exists a probability measure Q, equivalent to P, such that for all T > t and for every asset price process S t in the market, the process S t /B t is a Q-martingale. A formal derivation of the proposition is beyond the scope of this chapter and the reader is referred to texts on asset pricing like e.g. Björk (2004), Etheridge (2002), Cochrane (2001) and Duffie(2001).

28 16 Chapter 2 The first characterization the stochastic discount factor approach uses a process Λ t, called the stochastic discount factor (SDF), to discount asset prices. If the market is arbitrage-free, then the discounted asset prices are martingales under the natural probability measure P. The second characterization the risk neutral approach uses an equivalent probability measure Q, called the risk-neutral probability measure, such that all asset prices discounted by the bank account are martingales under Q. It is straightforward to obtain arbitrage-free bond prices with Proposition 2.1. Using the martingale property for the discounted assets and the fact that P T (T) = 1, we obtain ( ) ΛT P t (T) = E t Λ t (2.5) and P t (T) = E Q t ( Bt B T ) = E Q t (e T t r s ds ). The stochastic discount factor approach can be compactly expressed in differential form by applying Itô s Lemma to Λ t S t as d(λ t S t ) = Λ t ds t + dλ t S t + dλ t ds t. (2.6) Because Λ t S t is a P-martingale, it should have no drift, i.e. E t ((d(λ t S t )) = 0. Substituting the bank account B t for S t in the above expression and using the martingale property, we see that E t ( dλt therefore be expressed as Λ t ) = r t dt. The dynamics of the SDF can dλ t = Λ t ( r t dt λ t dw t ), (2.7) where λ t is a n-dimensional adapted process. The process λ t is known as the market prices of risk as will be discussed below (see also Cochrane (2001)). For positive asset price processes, the condition that Λ t S t is a P-martingale can be expressed as a restriction on the expected returns (Cochrane, 2001). Combining (2.6) and (2.7) and using the martingale property, we have ( ) ( dst dλt E t r t dt = E t S t Λ t ) ds t. (2.8) S t

29 Arbitrage-Free Modeling of the Term Structure in Continuous Time 17 The differential form in (2.8) clearly illustrates the implications of arbitrage-free pricing. The left-hand side of (2.8) represents the expected instantaneous excess return. This generally differs from zero by a risk premium that is represented by the right-hand side. The risk premium is determined by the covariation between the asset and the stochastic discount factor. For bonds with price dynamics given by (2.1), this no-arbitrage restriction (2.8) reduces to µ P,t (T) = r t + σ P,t (T) λ t. This relation provides a clear interpretation for the market prices of risk λ t. Each component of λ t measures the impact of exposure to an extra unit risk on the excess return, where the unit of risk is given by the corresponding component in W t. The risk-neutral approach uses a risk-neutral probability measure Q to obtain arbitrage-free prices. Pricing is relatively easy under measure Q, because the asset price discounted by the short rate is Q-martingale. This means that the price of a contingent claim equals the expected payoff discounted by the short rate under Q, which corresponds to pricing in a risk-neutral world. Analogous to the stochastic discount factor approach, we can express the riskneutral pricing relation in a differential form. Assuming a positive asset price process S t and applying Itô s Lemma and the martingale property, we have the following no-arbitrage condition E Q t ( ) dst = r t dt. (2.9) S t So the expected instantaneous bond return under the risk neutral measure Q equals the short rate. The risk-neutral measure Q can be obtained from the physical measure P by referring to Girsanov s theorem. Starting with the physical measure P and choosing an n-dimensional adapted process, known as the Girsanov kernel, the application of Girsanov s theorem yields an equivalent measure P. It will be shown below that choosing a Girsanov kernel equal to the the market prices of risk λ t will lead to the risk-neutral measure Q. First, fix T > 0 and assume that λ t satisfies the Novikov condition ( ( 1 T )) E exp λ 2 tλ t dt <. (2.10) 0

30 18 Chapter 2 Next, define the process L t on t [0, T] as ( L t = exp 1 t t ) λ 2 sλ s ds λ s dw s. 0 0 This process defines an equivalent probability measure P by P(A) = A L T (ω) dp(ω), A F T and L T is therefore the Radon-Nikodym derivative of P with respect to P. The process λ t is known as the Girsanov kernel for the change of measure from P to P. Girsanov s theorem explains how the dynamics of all processes change by changing the probability measure. For the change of measure from P to P, the theorem states that t W t = W t + λ s ds (2.11) 0 is standard P-Brownian motion. As a consequence, the change of measure only affects the drift of a process. A comparison of the process L t with the stochastic discount factor reveals why choosing a Girsanov kernel equal to the market prices of risk λ t leads to the riskneutral measure Q. Solving (2.7) gives ( Λ t = exp Λ 0 t 0 [r s + 12 ] t ) λ s λ s ds λ s dw s = L t B 0. 0 L 0 B t Consequently, Λ t S t is P-martingale if and only if L ts t B t s > t, we have ( ) S t Ls S s = E t B t L t B s ( ) = E P Ss t, B s is a P-martingale. For all so that S t B t is a P-martingale and we find that Q = P is a risk-neutral measure Market prices of risk and market completeness Market prices of risk play an interesting role in term structure modeling. The interpretation of the market prices of risk is evident from the above. In a risk-averse world, expected asset returns generally differ from the risk-free rate by a risk pre- 2 For the last equality in this expression see Etheridge (2002, p. 98)

31 mium. This risk premium depends on the exposure to the various sources of risk. The market prices of risk represent the compensation in the expected return for exposure to one unit of risk induced by the Brownian motion W t. Each component of W t has its own market price of risk represented by the corresponding component of λ t. As mentioned by Björk (2004), the name market price of risk can be misleading because it is not a price in an economic sense. The market price of risk only reflects the compensation for the exposure to risk, and can equally well be negative. 3 Market completeness is a concept closely related with the market prices of risk. Arbitrage-Free Modeling of the Term Structure in Continuous Time 19 A market is said to be complete if every contingent claim can be replicated by a dynamic portfolio strategy in the assets of the market. 4 Hence market completeness plays an important role in derivative pricing. For instance Black and Scholes assumed a market consisting of a bank account and a stock. This market is complete, so that any contingent claim can be replicated by investing in the underlying assets (see e.g. Björk(2004)). Hence, prices of all contingent claims are uniquely determined in terms of the prices of the underlying assets. This is different in interest rate modeling, because there is no underlying market assumed to replicate bonds. Arbitrage pricing does in this case not mean that bonds are priced relative to an exogenously specified underlying market, but rather priced relative to each other. Bond prices must be internally consistent to rule out arbitrage opportunities (Björk, 2004). This requirement of internal consistency is represented by the pricing relations discussed above. Absence of arbitrage does not always uniquely determine prices, because the stochastic discount factor Λ t or the risk-neutral measure Q are not necessarily unique. In fact, market completeness corresponds with a unique stochastic discount factor and risk-neutral measure. A complete market can be obtained by successively adding assets to the market. Generally speaking, given n-random sources we need n independent bonds plus the bank account to complete the market. All other assets are then uniquely priced relative to these basis assets (see Björk, 2004). The risk-neutral approach incorporates the market prices of risk in the riskneutral measure Q. This gives rise to two different forms of a model, the extensive 3 In interest rate models, market prices of risk are often negative, because these models are generally specified in terms of the short rate or the forward curve for which the volatilities are normalized to be positive. Consequently, the volatility of the bond prices is negative, implying negative market prices of risk to ensure a positive risk premia. 4 A contingent claim with maturity T is an F T measurable function.