Finite Dimensional Affine Realisations of HJM Models in Terms of Forward Rates and Yields

Finite dimensional Markovian HJM term structure models provide ideal settings for the study of term structure dynamics and interest rate derivatives where the flexibility of the HJM framework and the tractability of Markovian models coexist. Consequently, these models became the focus of a series of papers including Carverhill (1994), Ritchken and Sankarasubramanian (1995), Bhar and Chiarella (1997), Inui and Kijima (1998), de Jong and Santa-Clara (1999), Björk and Svensson (2001) and Chiarella and Kwon (2001a). However, these models usually required the introduction of a large number of state variables which, at first sight, did not appear to have clear links to the market observed quantities, and the explicit realisations of the forward rate curve in terms of the state variables were unclear. In this paper, it is shown that the forward rate curves for these models are affine functions of the state variables, and conversely that the state variables in these models can be expressed as affine functions of a finite number of forward rates or yields. This property is useful, for example, in the estimation of model parameters. The paper also provides explicit formulae for the bond prices in terms of the state variables that generalise the formulae given in Inui and Kijima (1998), and applies the framework to obtain affine representations for a number of popular interest rate models.     

A generalised arbitrage-free Nelson–Siegel model: The impact of unspanned stochastic volatility

Although statistical term structure models provide exceptional in-sample fitting and out-of-sample forecasting of interest rates, the lack of theoretical background is criticized by academics and practitioners, such as the absent of arbitrage free. In this paper we develop a general Arbitrage-Free Nelson–Siegel model under the HJM framework. It features unspanned stochastic volatility factors while maintaining a Nelson–Siegel factor loading structure. This paper also exploits the potential to jointly model the interest rates and their derivatives.     

Alternative Defaultable Term Structure Models

The objective of this paper is to consider defaultable term structure models in a general setting beyond standard risk-neutral models. Using as numeraire the growth optimal portfolio, defaultable interest rate derivatives are priced under the real-world probability measure. Therefore, the existence of an equivalent risk-neutral probability measure is not required. In particular, the real-world dynamics of the instantaneous defaultable forward rates under a jump-diffusion extension of a HJM type framework are derived. Thus, by establishing a modelling framework fully under the real-world probability measure, the challenge of reconciling real-world and risk-neutral probabilities of default is deliberately avoided, which provides significant extra modelling freedom. In addition, for certain volatility specifications, finite dimensional Markovian defaultable term structure models are derived. The paper also demonstrates an alternative defaultable term structure model. It provides tractable expressions for the prices of defaultable derivatives under the assumption of independence between the discounted growth optimal portfolio and the default-adjusted short rate. These expressions are then used in a more general model as control variates for Monte Carlo simulations of credit derivatives. Nicola Bruti-Liberati: In memory of our beloved friend and colleague.     

On the independence of assets and liabilities: Evidence from U.S. commercial banks, 1990–2005

Traditional asset–liability management techniques limit banks’ abilities to structure their balance sheets—but more recently, financial innovations have allowed banks the chance to manage interest rate risk without constraining their asset–liability choices. Using canonical correlation analysis, we examine how the relationships between asset and liability accounts at U.S. commercial banks changed between 1990 and 2005. Importantly, we show that asset–liability linkages are weaker for banks that are intensive users of risk-mitigation strategies such as interest rate swaps and adjustable loans. Perhaps surprisingly, we find that asset–liability linkages are stronger at large banks than at small banks, although these size-based differences have diminished over time, both because of increased asset–liability linkages at small banks and decreased linkages at large banks.     

Traffic light options

This paper introduces, prices, and analyzes traffic light options. The traffic light option is an innovative structured OTC derivative developed independently by several London-based investment banks to suit the needs of Danish life and pension (L&P) companies, which must comply with the traffic light solvency stress test system introduced by the Danish Financial Supervisory Authority (DFSA) in June 2001. This monitoring system requires L&P companies to submit regular reports documenting the sensitivity of the companies’ base capital to certain pre-defined market shocks – the red and yellow light scenarios. These stress scenarios entail drops in interest rates as well as in stock prices, and traffic light options are thus designed to pay off and preserve sufficient capital when interest rates and stock prices fall simultaneously. Sweden’s FSA implemented a traffic light system in January 2006, and supervisory authorities in many other European countries have implemented similar regulation. Traffic light options are therefore likely to attract the attention of a wider audience of pension fund managers in the future. Focusing on the valuation of the traffic light option we set up a Black–Scholes/Hull–White model to describe stock market and interest rate dynamics, and analyze the traffic light option in this framework.     

On pricing kernels and finite-state variable Heath Jarrow Morton models

Once a pricing kernel is established, bond prices and all other interest rate claims can be computed. Alternatively, the pricing kernel can be deduced from observed prices of bonds and selected interest rate claims. Examples of the former approach include the celebrated Cox, Ingersoll, and Ross (1985b) model and the more recent model of Constantinides (1992). Examples of the latter include the Black, Derman, and Toy (1990) model and the Heath, Jarrow, and Morton paradigm (1992) (hereafter HJM). In general, these latter models are not Markov. Fortunately, when suitable restrictions are imposed on the class of volatility structures of forward rates, then finite-state variable HJM models do emerge. This article provides a linkage between the finite-state variable HJM models, which use observables to induce a pricing kernel, and the alternative approach, which proceeds directly to price after a complete specification of a pricing kernel. Given such linkages, we are able to explicitly reveal the relationship between state-variable models, such as Cox, Ingersoll, and Ross, and the finite-state variable HJM models. In particular, our analysis identifies the unique map between the set of investor forecasts about future levels of the drift of the pricing kernel and the manner by which these forecasts are revised, to the shape of the term structure and its volatility. For an economy with square root innovations, the exact mapping is made transparent.     

A multi-factor Markovian HJM model for pricing American interest rate derivatives

This article presents a numerically efficient approach for constructing an interest rate lattice for multi-state variable multi-factor term structure models in the Makovian HJM [Econometrica 70 (1992) 77] framework based on Monte Carlo simulation and an advanced extension to the Markov Chain Approximation technique. The proposed method is a mix of Monte Carlo and lattice-based methods and combines the best from both of them. It provides significant computational advantages and flexibility with respect to many existing multi-factor model implementations for interest rates derivatives valuation and hedging in the HJM framework.

A Class of Jump-Diffusion Bond Pricing Models within the HJM Framework

This paper considers a class of term structure models that is a parameterisation of the Shirakawa (1991) extension of the Heath et al. (1992) model to the case of jump-diffusions. We consider specific forward rate volatility structures that incorporate state dependent Wiener volatility functions and time dependent Poisson volatility functions. Within this framework, we discuss the Markovianisation issue, and obtain the corresponding affine term structure of interest rates. As a result we are able to obtain a broad tractable class of jump-diffusion term structure models. We relate our approach to the existing class of jump-diffusion term structure models whose starting point is a jump-diffusion process for the spot rate. In particular we obtain natural jump-diffusion versions of the Hull and White (1990, 1994) one-factor and two-factor models and the Ritchken and Sankarasubramanian (1995) model within the HJM framework. We also give some numerical simulations to gauge the effect of the jump-component on yield curves and the implications of various volatility specifications for the spot rate distribution.     

Inferring future volatility from the information in implied volatility in Eurodollar options: a new approach

We study the information content of implied volatility fromseveral volatility specifications of the Heath-Jarrow-Morton(1992) (HJM) models relative to popular historical volatilitymodels in the Eurodollar options market. The implied volatilityfrom the HJM models explains much of the variation of realizedinterest rate volatility over both daily and monthly horizons.The implied volatility dominates the GARCH terms, the Glostenet al. (1993) type asymmetric volatility terms, and the interestrate level. However, it cannot explain that the impact of interestrate shocks on the volatility is lower when interest rates arelow than when they are high.     

Explosion in the quasi-Gaussian HJM model

We study the explosion of the solutions of the SDE in the quasi-Gaussian HJM model with a CEV-type volatility. The quasi-Gaussian HJM models are a popular approach for modeling the dynamics of the yield curve. This is due to their low-dimensional Markovian representation which simplifies their numerical implementation and simulation. We show rigorously that the short rate in these models explodes in finite time with positive probability, under certain assumptions for the model parameters, and that the explosion occurs in finite time with probability one under some stronger assumptions. We discuss the implications of these results for the pricing of the zero coupon bonds and Eurodollar futures under this model.     

Valuation of quanto options in a Markovian regime-switching market: A Markov-modulated Gaussian HJM model

We consider the valuation of European quanto call options in an incomplete market where the domestic and foreign forward interest rates are allowed to exhibit regime shifts under the Heath–Jarrow–Morton (HJM) framework, and the foreign price dynamics is exogenously driven by a regime switching jump-diffusion model with Markov-modulated Poisson processes. We derive closed-form solutions for four different types of quanto call options, which include: options struck in a foreign currency, a foreign equity call struck in domestic currency, a foreign equity call option with a guaranteed exchange rate, and an equity-linked foreign exchange-rate call.     

Term structure modelling of defaultable bonds

In this paper we present a model of the development of the term structure of defaultable interest rates that is based on a multiple-defaults model. Instead of modelling a cash payoff in default we assume that defaulted debt is restructured and continues to be traded.The term structure of defaultable bond prices is represented in terms of defaultable forward rates similar to the Heath-Jarrow-Morton (HJM) (Heath et al., 1992) approach, and conditions are given under which the dynamics of these rates are arbitrage-free. These conditions are a drift restriction that is closely related to the HJM drift restriction for risk-free bonds, and the restriction that the defaultable short rate must always be not below the risk-free short rate. In its most general version the model is set in a marked point process framework, to allow for jumps in the defaultable rates at times of default.Financial Assistance by Deutsche Forschungsgemeinschaft, Sonderforschungsbereich 303, at the University of Bonn and the DAAD is gratefully acknowledged.I thank Pierre Mella-Barral, David Lando and David Webb for helpful conversations, and the participants of the FMG Conference on Defaultable Bonds (March 1997) in London and the QMF 97 conference in Cairns for helpful comments. All errors are of course my own.     

VaR limits for pension funds: an evaluation

In a series of papers during the last ten years an interest rate theory with models which are driven by Lévy or more general processes has been developed. In this paper we derive explicit formulas for the correlations of interest rates as well as zero coupon bonds with different maturities. The models considered in this general setting are the forward rate (HJM), the forward process and the LIBOR model as well as the multicurrency extension of the latter. Specific subclasses of the class of generalized hyperbolic Lévy motions are studied as driving processes. Based on a data set of parametrized yield curves derived from German government bond prices we estimate correlations. In a second step the empirical correlations are used to calibrate the Lévy forward rate model. The superior performance of the Lévy driven models becomes obvious from the graphs.     

Differences in the Cost of Mortgage Credit Implications for Discrimination

This paper estimates the mortgage interest rate differences paid by Asian, Hispanic, and African–American borrowers to a national home mortgage lender in the years 1988–1989. Controlling for differences in market rates, rate lock protection, and borrower risk factors, conventional loan interest rates are almost perfectly race-neutral. The single deviation from race-neutrality is that when interest rates fall during the borrower's rate-lock period, only African–American borrowers are unable to capture a share of this decline. Government (FHA and VA) credit models show small premia paid by African–American borrowers of about $1.80 per month on average. In government lending, Hispanic borrowers alone are unable to capture rate declines occurring during the borrower's rate-lock period.     

Pricing inflation-indexed derivatives with default risk

Inflation-indexed derivatives with default risk are modeled using the jump-diffusion processes in the Heath–Jarrow–Morton’s (HJM) [(1992). “Bond Pricing and the Term Structure of Interest Rates: A New Methodology for Contingent Claim Valuation.” Econometrica 60: 77–105] framework. A four-factor HJM model is proposed by incorporating an exogenous intensity function into a foreign currency analogy under the three-factor HJM model proposed by Jarrow and Yildirim [(2003). “Pricing Treasury Inflation Protected Securities and Related Derivatives Using a HJM Model.” Journal of Financial and Quantitative Analysis 38: 337–358]. The proposed model improves the valuation accuracy of zero-coupon inflation-indexed swaps (IIS) through calibrating the model to swap market data. In addition, the valuation formulas of year-on-year IIS and caps with default risk are derived.     

An event study analysis of international ventures between banks and insurance firms

The current study examines the effects on shareholders wealth as a result of the bank–insurance interface. Using a global sample of financial intermediaries and an event-study framework the findings reveal significant abnormal returns surrounding the announcement of bank–insurance ventures. A control sample using financial institutions that do not pursue bank–insurance deals shows negative abnormal returns with much higher magnitude in absolute terms. When the sample is separated on the basis of the bidder's nature, then bank-bidders earn significant positive returns, while the insurance-bidders experience significant losses. The analysis further unveils either statistically significant negative returns or insignificant values for bank–insurance divestments. Finally, profitability, size and functional diversification are all found significant in determining abnormal returns over various intervals.     

Pricing and Hedging Discount Bond Options in the Presence of Model Risk

This paper focuses on pricing and hedging options on a zero-coupon bond in a Heath–Jarrow–Morton (1992) framework when the value and/or functional form of forward interest rates volatility is unknown, but is assumed to lie between two fixed values. Due to the link existing between the drift and the diffusion coefficients of the forward rates in the Heath, Jarrow and Morton framework, this is equivalent to hedging and pricing the option when the underlying interest rate model is unknown. We show that a continuous rangeof option prices consistent with no arbitrage exist. This range is bounded by the smallest upper-hedging strategy and the largest lower-hedging strategy prices, which are characterized as the solutions of two non-linear partial differential equations. We also discuss several pricing and hedging illustrations.     

Regime switches between dividend and bond yields

The relation between bond and equity returns serves as a proxy for estimating the premia investors' demand on their equity portfolio holdings and assessing the substitution effects between the two markets. With this in mind, we examine empirically the co-movements and the underlying information between equities and bonds. Our approach relies on the comparison between bond and dividend yields — a relation better known as the gilt-equity yield ratio–GEYR — by examining the characteristics of the cointegration relation between the bond and equity yields. In this context, this paper's contribution is that it lifts the restrictions of linearity both in the long-run cointegration relations and in the underlying short-run relations presented in the VECM. Specifically, we apply the regime-switching framework of Gregory and Hansen (Gregory, A. W. & Hansen, B. E. (1996). Residual-based tests for cointegration in models with regime shifts. Journal of Econometrics 70, 99–126) for the long-run equilibriums and the Markov Switching VECM, established by Krolzig (Krolzig, H.M., 1997. Markov switching vector autoregressions. Modelling statistical inference and application to business cycle analysis. Springer, Verlag) for the short run ones. Our aim is to examine the allocation of capital among the UK bond (or else, gilt) and stock markets for the period of 01-1987 to 01-2007, in a fashion that better reflects the structural breaks and regime shifts of the underlying market conditions. Our findings confirm the substitution effects among stocks and bonds in the long run and highlight the importance of market conditions for the allocation of capital among stocks and bonds.     

Extending the universality of the Heath-Jarrow-Morton model

Heath, Jarrow, and Morton (HJM) developed an important model of the evolution of interest rates. A key assumption of the model is that interest rate changes are normally distributed in continuous time. Implementing the HJM-method of evolution of interest rates in discrete time for more complex volatility functions remains a significant challenge. In this article, we present a relatively simple and flexible method of implementation, that extends the usefulness of the HJM model. The derivation assumes that the distribution of interest rates is stable, but not necessarily identical, for each discrete time period. This allows us to identify the drift-adjustment terms necessary to build interest rate lattices and trees and Monte Carlo simulations that satisfy exactly the no-arbitrage and volatility conditions, even complex ones, of the model. The much more difficult discrete-time implementation methods suggested in the literature (Heath, Jarrow, and Morton (1991) [Heath, D., Jarrow, R. & Morton, A. (1991). Contingent claim valuation with a random evolution of interest rates. Review of Futures Markets, 54-76.] and Jarrow (1996) [Jarrow, R. (1996). Modeling fixed income securities and interest rate options. New York, NY: McGraw-Hill Companies Inc.]) do not accomplish that. We illustrate our analytical implementation with three examples of volatility functions and demonstrate its superiority to other methods of implementation.