On the accuracy of the local linear approximation for the term structure of interest rates

We examine by numerical experiments the accuracy of an analytical approximation for the nonlinear term structure of interest rates, which is obtained by applying the local linear approximation to a generally specified process of the short rate. Under various short-rate models, we compare discount-bond prices computed by the approximation with those calculated by the Monte Carlo method as the benchmark, which shows that deviations are small. Also in this paper, we show that the approximation originally derived in single-factor framework can be easily extended to a multifactor counterpart. We examine the accuracy using an illustrative two-factor model, which also shows the approximation is accurate.     

A Simple Measure for Examining the Proxy Problem of the Short-Rate

The behavior of a finite-maturity yield used as a proxy for the short-rate can deviate substantially from that of the short-rate, which causes estimation biases of model parameters and pricing errors of interest-rate claims. This study proposes a simple measure that visualizes this deviation based on an analytical approximation of the term structure of interest rates. The computation of the measure is almost as easy as that of an affine model, so the adequacy of proxy can be readily checked even for short-rate models that do not admit closed-forms of bond prices.     

Risk management implications of time-inconsistency: Model updating and recalibration of no-arbitrage models

A widespread approach in the implementation of asset pricing models is based on the periodic recalibration of its parameters and initial conditions to eliminate any conflict between model-implied and market prices. Modern no-arbitrage market models facilitate this procedure since their solution can usually be written in terms of the entire initial yield curve. As a result, the model fits (by construction) the interest rate term structure. This procedure is, however, generally time inconsistent since the model at time t = 0 completely specifies the set of possible term structures for any t > 0. In this paper, we analyze the pros and cons of this widespread approach in pricing and hedging, both theoretically and empirically. The theoretical section of the paper shows (a) under which conditions recalibration improves the hedging errors by limiting the propagation of an initial error, (b) that recalibration introduces time-inconsistent errors that violate the self-financing argument of the standard replication strategy. The empirical section of the paper quantifies the trade-off between (a) and (b) under several scenarios. First, we compare this trade-off for two economies with and without model specification error. Then, we discuss the trade-off when the underlying economy is not Markovian.     

Optimal excess-of-loss reinsurance contract with ambiguity aversion in the principal-agent model

ABSTRACT

We discuss an optimal excess-of-loss reinsurance contract in a continuous-time principal-agent framework where the surplus of the insurer (agent/he) is described by a classical Cramér-Lundberg (C-L) model. In addition to reinsurance, the insurer and the reinsurer (principal/she) are both allowed to invest their surpluses into a financial market containing one risk-free asset (e.g. a short-rate account) and one risky asset (e.g. a market index). In this paper, the insurer and the reinsurer are ambiguity averse and have specific modeling risk aversion preferences for the insurance claims (this relates to the jump term in the stochastic models) and the financial market's risk (this encompasses the models' diffusion term). The reinsurer designs a reinsurance contract that maximizes the exponential utility of her terminal wealth under a worst-case scenario which depends on the retention level of the insurer. By employing the dynamic programming approach, we derive the optimal robust reinsurance contract, and the value functions for the reinsurer and the insurer under this contract. In order to provide a more explicit reinsurance contract and to facilitate our quantitative analysis, we discuss the case when the claims follow an exponential distribution; it is then possible to show explicitly the impact of ambiguity aversion on the optimal reinsurance.     

A Note on Gaussian Estimation of the CKLS and CIR Models with Feedback Effects for Japan

In this note we extend the Gaussian estimation of two factor CKLS and CIR models recently considered in Nowman, K. B. (2001, Gaussian estimation and forecasting of multi-factor term structure models with an application to Japan and the United Kingdom, Asia Pacif. Financ. Markets 8, 23–34) to include feedback effects in the conditional mean as was originally formulated in general continuous time models by Bergstrom, A. R. (1966, Non-recursive models as discrete approximations to systems of stochastic differential equations, Econometrica 34, 173–182) with constant volatility. We use the exact discrete model of Bergstrom, A. R. (1966, Non-recursive models as discrete approximations to systems of stochastic differential equations, Econometrica 34, 173–182) to estimate the parameters which was first used by Brennan, M. J. and Schwartz, E. S. (1979, A continuous time approach to the pricing of bonds, J. Bank. Financ. 3, 133–155) to estimate their two factor interest model but incorporating the assumption of Nowman, K. B. (1997, Gaussian estimation of single-factor continuous time models of the term structure of interest rates, J. Financ. 52, 1695–1706; 2001, Gaussian estimation and forecasting of multi-factor term structure models with an application to Japan and the United Kingdom, Asia Pacif. Financ. Markets 8, 23–34). An application to monthly Japanese Euro currency rates indicates some evidence of feedback from the 1-year rate to the 1-month rate in both the CKLS and CIR models. We also find a low level-volatility effect supporting Nowman, K. B. (2001, Gaussian estimation and forecasting of multi-factor term structure models with an application to Japan and the United Kingdom, Asia Pacif. Financ. Markets 8, 23–34).     

Calibration of Gaussian heath, Jarrow and Morton and random field interest rate term structure models

We show how the recently introduced Gaussian random field interest rate term structure models can be calibrated accurately and quickly to market caps and swaptions prices. We show how the calibrated random field model can be approximated arbitrarily closely with a multi-factor Gaussian Heath, Jarrow and Morton model. We argue that the Gaussian random field model is easier to calibrate and can be used as an indirect way to calibrate multi-factor Gaussian Heath, Jarrow and Morton interest rate models.This work was carried out when the author was at the Financial Options Research Centre, Warwick Business School, University of Warwick. I wish to thank Stewart Hodges for many helpful discussions and comments and Martin Cooper of Tokai Bank, Europe, for supplying the data used in this paper. I also wish to thank the Economic and Social Research Council and FORC for funding. An earlier version of this paper entitled Multi-Factor Gaussian HJM Approximation to Kennedy and Calibration to Caps and Swaptions Prices was presented at the 9th Annual Conference, FORC., Warwick Business School, September 1996. Another version also appears in the author's Ph.D. thesis. I am grateful to the helpful comments provided by Marti Subrahmanyam and the two anonymous referees. All errors are my own and the views expressed in no way reflect the opinion of my employer.     

New No-arbitrage Conditions and the Term Structure of Interest Rate Futures

Interest rate futures are basic securities and at the same time highly liquid traded objects. Despite this observation, most models of the term structure of interest rate assume forward rates as primary elements. The processes of futures prices are therefore endogenously determined in these models. In addition, in these models hedging strategies are based on forward and/or spot contracts and only to a limited extent on futures contracts. Inspired by the market model approach of forward rates by Miltersen, Sandmann, and Sondermann (J Finance 52(1); 409–430, 1997), the starting point of this paper is a model of futures prices. Using, as the input to the model, the prices of futures on interest related assets new no-arbitrage restrictions on the volatility structure are derived. Moreover, these restrictions turn out to prevent an application of a market model based on futures prices.     

On the estimation and comparison of short-rate models using the generalised method of moments

Subsequent to the influential paper of [Chan, K.C., Karolyi, G.A., Longstaff, F.A., Sanders, A.B., 1992. An empirical comparison of alternative models of the short-term interest rate. Journal of Finance 47, 1209–1227], the generalised method of moments (GMM) has been a popular technique for estimation and inference relating to continuous-time models of the short-term interest rate. GMM has been widely employed to estimate model parameters and to assess the goodness-of-fit of competing short-rate specifications. The current paper conducts a series of simulation experiments to document the bias and precision of GMM estimates of short-rate parameters, as well as the size and power of [Hansen, L.P., 1982. Large sample properties of generalised method of moments estimators. Econometrica 50, 1029–1054], J-test of over-identifying restrictions. While the J-test appears to have appropriate size and good power in sample sizes commonly encountered in the short-rate literature, GMM estimates of the speed of mean reversion are shown to be severely biased. Consequently, it is dangerous to draw strong conclusions about the strength of mean reversion using GMM. In contrast, the parameter capturing the levels effect, which is important in differentiating between competing short-rate specifications, is estimated with little bias.     

A general characterization of one factor affine term structure models

We give a complete characterization of affine term structure models based on a general nonnegative Markov short rate process. This applies to the classical CIR model but includes as well short rate processes with jumps. We provide a link to the theory of branching processes and show how CBI-processes naturally enter the field of term structure modelling. Using Markov semigroup theory we exploit the full structure behind an affine term structure model and provide a deeper understanding of some well-known properties of the CIR model. Based on these fundamental results we construct a new short rate model with jumps, which extends the CIR model and still gives closed form expressions for bond options. Manusript received: June 2000, final version received: October 2000