The Valuation of Interest Rate Derivative Securities

Jeroen F. J. De Munnik

• Publisher: Routledge
• Year: 2005
• eBook ISBN: 9781134775910
• Edition: 1
• Topic: Negocios y empresa
• Subtopic: Negocios en general

The increased volatility of interest rates during recent years and the corresponding introduction of a variety of interest rate derivative securities like bond options, futures and embedded options in mortgages, underlines the need for a comprehensive financial theory to determine values of fixed income instruments and derivative securities consistently. This book provides: * a detailed overview and classification of the different approaches to value interest rate dependent securities * a comparison of the numerical approaches to value complex securities * an empirical examination for the Dutch Fixed Income Market of some well-known interest rate models which demonstrates recent improvements to describe interest rate movements in relation to contingent claim valuation.

1

INTRODUCTION

If Aristotle is to be believed, Thales started it all in the 6th century BC with an option on olives. But it was only after academic options-pricing theory of the 1960s and 1970s met the volatile financial markets of the deregulating 1980s that options took off. So did futures, warrants, swaps, swaptions, collars, caps, floors, circuses and scores of other products known collectively as derivatives. Nothing today is transforming financial markets as rapidly and completely as what their inventors like to call tools for the management of financial risk. Nothing now gives so many financial regulators so many nightmares.[…] What especially worries most of them is that banks are the biggest traders and counterparts, and neither banks (nor anyone else) understand the risks well enough to price them properly. A derivatives disaster could overwhelm the world’s financial system, as third-world debt, highly leveraged transactions and property lending have not managed to do.¹

This quotation is but one of many examples of the caution and sceptical attitudes existing regarding the development and further evolution of the use of financial derivative securities. At the same time, it stresses the importance and need of an accurate assessment of the values of the different contingent claims and the risks involved.

During the last two decades a lot of academic research has concentrated on the theoretical valuation and the associated empirical validity of commonly known contingent claims like call and put options on stocks. The relatively high risk of stocks in comparison to other alternative financial assets like bonds and the corresponding popularity of these instruments for portfolio management easily explains the primary focus of contingent claim research during this period.²

In the last decade, however, increased attention has been paid to the valuation of contingent claims whose values depend on the term structure of interest rates and its subsequent movement over time. Although the level of price risk of Traded Government Bonds³ may give the impression at first sight of a relatively unimportant problem, the variety of financial instruments with complex option characteristics such as callable bonds, different types of mortgages and the delivery option embedded in a futures contract and the size of the different markets in which these instruments are traded,⁴ definitely leads to an opposite conclusion. In addition, the modelling and estimation of the stochastic dynamics of the yield curve not only enables an assessment of the interest rate risk of the above-mentioned instruments but also allows for a general interest rate risk management of fixed income portfolios.

Regarding the strong attention that has been paid to the theoretical valuation problem of ordinary options on stocks, it is important to explain the institutional differences between a stock and a bond to understand and justify the separate treatment of the valuation of interest rate derivative securities.

The main difference between a stock and an ordinary coupon paying bond is the certainty at some valuation date of the amounts and corresponding payment dates of the different coupons and face value. Obviously, this affects the possible price movements of bonds in comparison to those of stocks. Near the final maturity date of a bond, for example, the probability of an increase in value of a par bond is much smaller than it is at some other valuation date, all else being equal. In the case of stocks, however, there is no reason why such particular stochastic behavior can be assumed or derived from institutional characteristics. As another result of this price effect, the corresponding volatility of possible price movements decreases as the maturity of the bond decreases. The range of possible bond prices that can be attained with some probability narrows when the final payment date is reached.

Although not generally empirically justified, one of the basic assumptions in the classical stock options valuation problem is a constant interest rate at which long and short asset positions can be financed. In the case of interest rate derivative securities, however, it is clearly theoretically inconsistent to adopt this assumption. The relationship between bond values and the term structure of interest rates implied by the familiar discounting of future payments, necessitates a formulation of the stochastic movement of the yield curve over time. As will be seen in the remainder of this thesis, this difference results in an increased theoretical and empirical complexity of the valuation problem of these contingent claims and leads to an important distinction between the different valuation methods.

RESEARCH OBJECTIVES

Having explained and justified a separate and extensive treatment of the general valuation problem of interest rate derivative securities, it is important to formulate and discuss some more specific research objectives.

Given the above-mentioned institutional and theoretical differences between the stochastic dynamics of stocks and bonds affecting the valuation problem of interest rate derivative securities, it is both necessary and important to investigate and to give an overview of the different conditions under which derivative securities can be valued. Given the characteristics of some contingent claims, for example, and given the stochastic properties of the underlying values of these claims, is it possible to formulate general conditions regarding the possible values of the claims that exclude riskless arbitrage opportunities between the underlying values and the derivative securities?

The increased academic interest that has been paid during the last few years to the valuation of interest rate derivative securities has resulted in a variety of different theoretical models. To be able to investigate the possible advantages and drawbacks of these approaches with respect to each other and to decide under which theoretical and empirical circumstances a particular model should be preferred, it is necessary to develop a basic classification scheme. Is it possible, therefore, to classify the different interest rate models according to some basic or general characteristics?

The first paragraph of this chapter mentioned that during the last few years a lot of highly complex derivative securities have been developed and introduced. Because of the possible complicated payouts of some of these securities, the actual valuation given some interest rate model relies heavily on the use of numerical methods. Is it possible to classify these numerical approaches also and to develop some decision rules to be able to decide under which conditions a particular method should be preferred?

The principal reason for the increased theoretical attention to the valuation of interest rate derivative securities has been the aim to incorporate the institutional characteristics and the observed empirical properties of interest rate dynamics as much as possible into the derivative securities models. Less emphasis has been paid, however, to an empirical evaluation of the different models and to an assessment of the actual need to incorporate these characteristics. As the resulting valuation complexity and estimation difficulties generally rapidly increase as more properties are built into the model, it is obvious that this comparison should be carried out. Is it possible to distinguish different interest rate models with respect to their empirical validity?

OVERVIEW

The recent extensive theoretical developments within the field of the valuation of interest rate derivative securities implies the necessity of an accurate description of the various models and corresponding numerical approaches according to some basic characteristics. In addition, an empirical investigation has to be carried out to compare different interest rate models and to assess the trade-off between the desire to incorporate as many institutional characteristics as possible and the resulting theoretical and numerical complexity. The different research objectives of this thesis discussed in the previous section are, therefore, either theoretical or empirical in nature and justify a separation of the theoretical and empirical issues into two parts. The distinction further clarifies the clear emphasis on theoretical research during the past and the only recent empirical developments.

The first chapter of the theoretical part (Chapter 2) derives and formulates conditions under which riskless arbitrage opportunities between contingent claims and the corresponding underlying assets are excluded and under which it is possible to determine a unique arbitrage-free value of a derivative security. It is shown that every contingent claim can uniquely be valued in this way if there exists a unique equivalent probability measure such that the stochastic process of the underlying values of this security in terms of a short-term money market account is a martingale. Furthermore, the unique arbitragefree value of the claim is shown to be equal to the discounted expected value of the payout of this claim under the equivalent martingale measure.

To proceed with the theoretical investigation of the valuation of interest rate derivative securities, Chapter 3 presents a general overview and classification of the different theoretical approaches. The major distinction between the different approaches can be made with respect to the modelling of the underlying values of the different securities. Just as in the case of the classical stock options valuation problem, the underlying values are explicitly modelled in the direct approach. Given the stochastic dynamics of these values, contingent claims can be valued according to the unique equivalent martingale measure approach. The indirect approach, however, starts with a description of the stochastic dynamics of interest rates. The exclusion of arbitrage opportunities between different bonds, then, results in a description of the specific shape of the term structure of interest rates at a given date and the corresponding stochastic movement over time. The equivalent martingale approach and the derived yield curve then enable the arbitrage-free valuation of any interest rate contingent claim.

Chapter 4 discusses the approach based on the explicit modelling of bond prices. Given the above-mentioned academic research regarding the valuation of options on stocks, it is natural to extend these approaches to incorporate the specific characteristics of bonds and to value contingent claims analogously. Because the model presented basically combines and extends two existing models with respect to the theoretical validity of the proposed stochastic processes, a lot of attention is paid in this chapter to regularity of the processes and the existence of a unique equivalent martingale measure.

The different models within the indirect approach are discussed and illustrated in Chapter 5. Within this class, the different models can be further distinguished according to the term structure of interest rates at the valuation date. The first part of this chapter discusses those models in which the yield curve is endogenously implied by the stochastic characteristics of the interest rate processes and the no-arbitrage relationships. The second part then presents the models in which this term structure is exogenously specified at the given valuation date. In this part, a similar distinction is made between the endogenous and exogenous term structure of interest rate volatilities at some date.

As mentioned above, the complexity of the different valuation models and of the characteristics of the different contingent claims results in the use of numerical valuation methods. In Chapter 6, three approaches are discussed to value a contingent claim numerically given a general stochastic process of the underlying state variable. In addition to this overview, general decision rules are developed to assess and distinguish the numerical accuracy of the different methods in terms of the numerical complexity. Although these methods allow for a numerical approximation of rather general interest rate processes, some interest rate models enable a significant simplification of the original stochastic process with respect to the numerical applicability or can only be approximated by different approaches. At the end of this chapter, some specific numerical methods to value interest rate derivative securities in the case of these interest rate models are presented and further developed.

In the first two chapters of the empirical part of this thesis, the estimation and corresponding results of two models within the class of the endogenous term structure of interest rate models are discussed. The reasons for concentrating on these two models within this particular class are two-fold. The endogenous yield curve at some valuation date is a result of the stochastic characteristics of the underlying interest rate process and the no-arbitrage conditions. A cross-sectional estimation of this yield curve and a time series estimation of the corresponding interest rate process, therefore, enables an interesting comparison between the implicitly and explicitly estimated interest rate processes. In addition, because the two models basically differ with respect to the assumed interest rate process, an empirical comparison allows for an assessment of the increased complexity resulting from the exclusion of negative nominal interest rates.

Chapter 7, therefore, discusses the time series estimation of the two models and the corresponding results. Chapter 8 follows with the estimation method and results of the cross-sectional approach. In addition to an actual comparision of the estimated interest rate processes, these two chapters compare the implications of the estimations for the valuations of European call options on discount bonds.

In the last chapter of this part, some serious problems resulting from the use of principal component analysis to value interest rate derivative securities are discussed. Because this estimation technique is used both by practitioners and academics, Chapter 10, finally, provides an important and illustrative example of taking a cautious approach to the practical implementation of valuation models.

NOTES

1  “Taming the Derivative Beast”, 23 May 1992, The Economist, pp. 85–6.

2  See, for example, Malkiel (1990, p. 221).

3  These bonds are assumed to be default-free and as such, the risk of an actual default may be ignored. Throughout this thesis, only this particular type of bonds is considered.

4  For an example in the case of the Dutch Fixed Income Market, see de Munnik and Vorst (1988).

Part I

THE THEORETICAL VALUATION OF INTEREST RATE DERIVATIVE SECURITIES

2

ARBITRAGE OPPORTUNITIES AND THE VALUATION OF CONTINGENT CLAIMS

The notion of the existence of arbitrage opportunities in financial markets constitutes a significant area of continuing research in financial economics. Given the current values at which traded securities, such as stocks and bonds, can be bought and sold, is it possible, for example, to construct so-called arbitrage portfolios containing some or all of these securities with a current total value of zero and a strict positive value, with certai...

Table of contents

• Cover
• Half Title
• Title Page
• Copyright Page
• Table of Contents
• List of Figures
• List of Tables
• List of Symbols
• Preface
• 1 Introduction
• I The Theoretical Valuation of Interest Rate Derivative Securities
• II Empirical Results of the Estimation of Interest Rate Dynamics
• References
• Index