Ren-Raw Chen

Maximum Likelihood Estimation for a Multifactor Equilibrium Model of the Term Structure of Interest Rates

LOUIS SCOTT is associate professor of finance at the University of Georgia in Athens. model of the term structure of interest necessary for the valuation of bonds and interest rate options; and parameter valestimates, are necessary for the implementation of a specific model. This article presents a method for estimating the parameters of a particular class of continuous-time equilibrium models of the term structure. The theoretical framework for the analysis is the model of Cox, Ingersoll, and Ross [1985a, 1985b1, where a general equilibrium model of asset pricing is used to examine the behavior of the term structure and related issues such as the valuation of interest ratecontingent claims. The Cox, Ingersoll, Ross model, hereafter the CIR model, is a single-factor equilibrium model of the term structure that is consistent with an asset pricing equilibrium, free of arbitrage opportunities, and retains the feature that the interest rate must be non-negative. The one-factor model, however, has the undesirable property that all bond returns are perfectly correlated, and it may not be adequate to characterize the term structure of interest rates and its changing shape over time. The advantage of this one-factor model is the relatively simple closed-form solution for bond prices. As CIR show, the model can be extended to a multifactor setting, with closed-form solutions for bond prices. We follow suggestions in CIR and develop a multifactor equilibrium model of the term structure. The primary objective is to estimate the parameters of the processes that drive interest rate changes and determine the number of factors necessary to characterize the term structure adequately over time. This analysis

Multi-Factor Cox-Ingersoll-Ross Models of the Term Structure: Estimates and Tests from a Kalman Filter Model

This paper presents a method for estimating multi-factor versions of the Cox-Ingersoll-Ross (1985b) model of the term structure of interest rates. The fixed parameters in one, two, and three factor models are estimated by applying an approximate maximum likelihood estimator in a state-space model using data for the U.S. treasury market. A nonlinear Kalman filter is used to estimate the unobservable factors. Multi-factor models are necessary to characterize the changing shape of the yield curve over time, and the statistical tests support the case for two and three factor models. A three factor model would be able to incorporate random variation in short term interest rates, long term rates, and interest rate volatility.

An Explicit, Multi-Factor Credit Default Swap Pricing Model with Correlated Factors

With the recent significant growth in the single-name credit default swap (CDS) market has come the need for accurate and computationally efficient models to value these instruments. While the model developed by Duffie, Pan, and Singleton (2000) can be used, the solution is numerical (solving a series of ordinary differential equations) rather than explicit. In this paper, we provide an explicit solution to the valuation of a credit default swap when the interest rate and the hazard rate are correlated by using the “change of measure” approach and solving a bivariate Riccati equation. CDS transaction data for the period 2/15/2000 through 4/8/2003 for 60 firms are used to test both the goodness of fit of the model and provide estimates of the influence of economic variables in the market for credit-risky bonds.

Option Pricing in a Multi-Asset, Complete Market Economy

This paper extends the seminal Cox-Ross-Rubinstein ((1979), CRR hereafter) binomial model to multiple assets. It differs from previous models in that it is derived under the complete market environment specified by Duffie and Huang (1985) and He (1990).

Exact Solutions for Futures and European Futures Options on Pure Discount Bonds

This paper provides closed form solutions for futures and European futures options on pure discount bonds under the Ornstein-Uhlenbeck (normal) process. A significant difference between Blacks model (1976) and the model in this paper for futures options is discussed.

The Valuation of Default-Triggered Credit Derivatives

Credit derivatives are among the fastest growing contracts in the derivatives market. We present a simple, easily implementable model to study the pricing and hedging of two widely traded default-triggered claims: default swaps and default baskets. In particular, we demonstrate how default correlation (the correlation between two default processes) impacts the prices of these claims. When we extend our model to continuous time, we find that, once default correlation has been taken into consideration, the spread dynamics have very little explanatory power.

Analytical Upper Bounds for American Option Prices

American options require numerical methods, namely lattice models, to provide accurate price estimates. The computations can become expensive when more than one state variable is involved. Analytical upper bounds can therefore provide a useful guideline for how high American values can reach. In this paper, we derive analytical (closed-form) upper bounds for American option prices under stochastic interest rates, stochastic volatility, and jumps where American option prices are difficult to compute with accuracy. In a stochastic volatility model (Heston (1993) and Scott (1997)) that has two random factors, we demonstrate that the upper bound only takes a very small fraction of the time that the American option needs to compute.

A Simple Multi-factor, Time-Dependent-Parameter Model for the Term Structure of Interest Rates

In this paper, we present a simple version of the Duffie and Kan model (1996). Our model can perfectly fit the yield curve and the volatility curve and further provide true closed form solutions to the pure discount bond price and its European contingent claims. Due to the specific factor structure in our model, the calibration exercise is easy to implement. This advantage will improve the computational efficiency in pricing American style claims.

A Note on Forward Price and Forward Measure

The forward measure is convenient in calculating various contingent claim prices under stochastic interest rates. We demonstrate that caution needs to be drawn when the forward measure is used to price contingent claims that involve multiple cash flows. We also derive partial different equations for the forward price to demonstrate how forward contracts can be used for dynamic hedging and how hedges can be conducted if the payoff of a contingent claim depends on the forward price.

Valuing Fixed‐Income Options and Mortgage‐Backed Securities with Alternative Term Structure Models

To value mortgage-backed securities and options on fixed-income securities, it is necessary to make assumptions regarding the term structure of interest rates. We assume that the multi-factor fixed parameter term structure model accurately represents the actual term structure of interest rates, and that the values of mortgage-backed securities and discount bond options derived from such a term structure model are correct. Differences in the prices of interest rate derivative securities based on single-factor term structure models are therefore due to pricing bias resulting from the term structure model. The price biases that result from the use of single-factor models are compared and attributed to differences in the underlying models and implications for the selection of alternative term structure models are considered. Copyright Blackwell Publishers Ltd 1999.