Pricing UK and US securities within the CKLS model Further results

In a recent article, Byers and Nowman obtained estimates of the CKLS interest rate model based on weekly Euro-currency data for the UK and US over a range of maturities. In this article we apply the Box numerical method for valuing default free bonds and cointegent claims using these historical UK and US estimates to compare implied bond and contingent claim prices. Our results indicate that default free bond prices and contingent claim prices are sensitive to the underlying interest rate model used.     

Derivative prices from interest rate models: results for Canada, Hong Kong, and United States

In this paper, we compute implied bond and contingent claim prices from the CKLS, Vasicek, CIR, and BS interest rate models using historical estimates for Canada, Hong Kong, and the United States. We find that default-free bond prices and contingent claim prices are sensitive to the assumed model used for these currencies, and that for Canada the CIR is the best, for Hong Kong the Vasicek and CIR models, and for the US the BS model.     

Valuation of defaultable bonds and debt restructuring

In this paper we develop a contingent valuation model for zero-coupon bonds with default. In order to emphasize the role of maturity time and place of the lender's claim in a firm's debt hierarchy, we consider a firm that issues two bonds with different maturities and different seniorage. The model allows us to analyze the implications of both debt renegotiation and capital structure of a firm on the prices of bonds. We obtain that renegotiation brings about a significant change in the bond prices and that the effect is dispersed through various channels: increasing the value of the firm, reallocating payments, and avoiding costly liquidation. Moreover, the presence of two creditors leads to qualitatively different implications for pricing, while emphasizing the importance of bond covenants and renegotiation of the entire debt.     

Mortgage Termination: An Empirical Hazard Model with a Stochastic Term Structure

Pricing for mortgage and mortgage-backed securities is complicated due to the stochastic and interdependent nature of prepayment and default risks. This paper presents a unified economic model of the contingent claims and competing risks of mortgage termination by prepayment and default. I adopt a proportional hazard framework to analyze these competing and interdependent risks in a model with time-varying covariates. The paper incorporates a stochastic interest rate model into the hazard function for prepayment. The empirical results reported in the paper provide new evidence about the ruthlessness of default and prepayment behavior and the sensitivity of these decisions to demographic as well as financial phenomena. The results also illustrate that evaluating the interest rate contingent claims with a stochastic term structure has effects on predicting not only the mortgage prepayment behavior but also the mortgage default behavior.     

Interest Rate Futures Options and Interest Rate Options

This paper derives pricing models of interest rate options and interest rate futures options. The models utilize the arbitrage-free interest rate movements model of Ho and Lee. In their model, they take the initial term structure as given, and for the subsequent periods, they only require that the bond prices move relative to each other in an arbitrage-free manner. Viewing the interest rate options as contingent claims to the underlying bonds, we derive the closed-form solutions to the options. Since these models are sufficiently simple, they can be used to investigate empirically the pricing of bond options. We also empirically examine the pricing of Eurodollar futures options. The results show that the model has significant explanatory power and, on average, has smaller estimation errors than Black's model. The results suggest that the model can be used to price options relative to each other, even though they may have different expiration dates and strike prices.     

DURATION,DEFAULT RISK,AND THE TERM STRUCTURE OF INTEREST RATES

We examine the interactive effect of default and interest rate risk on duration of defaultable bonds. We show that duration for defaultable bonds can be longer or shorter than default‐free bonds depending on the relation between default intensity and interest rates. Empirical evidence indicates that in most cases duration for defaultable bonds is much shorter than for their default‐free counterparts because of the negative relation between default risk and interest rates. Results suggest that the duration measure must be adjusted for the effects of default risk and stochastic interest rates to achieve an effective bond portfolio immunization.     

Applications of Gram–Charlier expansion and bond moments for pricing of interest rates and credit risk

The purpose of this paper is to demonstrate the powerful and flexible applicability of the Gram–Charlier expansion to pricing of a wide variety of interest rate related products involving interest rate risk and credit risk. In this paper, we develop easily implemented approximations of the prices of several derivatives; swaptions, CMS, CMS options, and vulnerable options. Associated with the default risk, a survival contingent forward measure is constructed.     

Savings bonds,retractable bonds and callable bonds

Savings bonds, retractable bonds and callable bonds are each equivalent to a straight bond with an option. Neglecting default risk the value of these contingent claims depends upon the riskless interest rate. This paper employs the option pricing framework to value these bonds, under the assumptions that the interest rate follows a Gauss-Wiener process and that the pure expectations hypothesis holds.     

An Asymptotic Expansion Approach to Pricing Financial Contingent Claims

We propose a new methodology for the valuation problem of financial contingent claims when the underlying asset prices follow a general class of continuous Itô processes. Our method can be applicable to a wide range of valuation problems including contingent claims associated with stocks, foreign exchange rates, the term structure of interest rates, and even their combinations. We illustrate our method by discussing the Black-Scholes economy when the underlying asset prices follow the continuous diffusion processes, which are not necessarily time-homogeneous. The standard Black-Scholes model on stocks and the Cox-Ingersoll-Ross model on the spot interest rate are simple examples. Then we shall give a series of examples on the valuation formulae including plain vanilla options, average options, and other contingent claims. We shall also give some numerical evidence of the accuracy of the approximations we have obtained for practical purposes. Our approach can be rigorously justified by an infinite dimensional mathematics, the Malliavin-Watanabe-Yoshida theory recently developed in stochastic analysis.     

Credit ratings and convertible bond prices: a simulation-based valuation

This study presents a simulation-based model of convertible bond prices under the assumption of stochastic interest rates. The model is developed such that the convertible bond price explicitly depends on the credit rating at the time of issuance. Key ideas explored in this study include terminating the simulated sample path immediately when the issuer defaults on the bond at time t, which is the same as the investor and the issuer optimally exercising their options and discounting the resulting cash flows at a risk-free rate. In turn, the defaulted group of sample paths belongs to the bottom xth percentile of the realized stock prices at each time, which is exogenously given by the cumulative or marginal default probability of a firm that has the same rating as the issuer. Upon calibrating the model, we can see that the moneyness of convertible bonds is strongly responsible for influencing the convertible bond price when the rating changes. Furthermore, the effects of stochastic interest rates are shown to be possibly significant when the interest rate risk’s market price is not zero.     

Fundamental Properties of Bond Prices in Models of the Short-Term Rate

This article develops restrictions that arbitrage-constrainedbond prices impose on the short-term rate process in order tobe consistent with given dynamic properties of the term structureof interest rates. The central focus is the relationship betweenbond prices and the short-term rate volatility. In both scalarand multidimensional diffusion settings, typical relationshipsbetween bond prices and volatility are generated by joint restrictionson the risk-neutralized drift functions of the state variablesand convexity of bond prices with respect to the short-termrate. The theory is illustrated by several examples and is partiallyextended to accommodate the occurrence of jumps and default.     

Taxes,Default Risk,and Yield Spreads

This paper develops a model of bond prices and yield spreads that incorporates the effect of both taxes and differences in default probabilities. The tax loss consequences of default are recognized. Traditionally, tax-free (municipal) bond yields have been viewed as linearly related to taxable yields with a slope coefficient equal to one minus the tax rate and the intercept representing differences in default risk. While our model supports the linearity assumption, it implies that the slope and intercept are both functions of both the break-even tax rate and the default probability(ies). Clientele effects among both municipal and taxable bonds are demonstrated. Finally, the implied marginal tax rates and the implied default probabilities are estimated for different categories of municipal bonds.     

The Pricing of Default-free Interest Rate Cap,Floor, and Collar Agreements

The paper focuses on the valuation of caps, floors, and collars in a contingent claim framework under continuous time. These instruments are interpreted as options on traded zero coupon bonds. The bond prices themselves are used as the underlying stochastic variables. This has the advantage that we end up with closed form solutions which are easy to compute. Special attention is devoted to the choice of the stochastic process appropriate for the price dynamics of the underlying zero coupon bonds.     

Pricing default swaps: Empirical evidence

In this paper we compare market prices of credit default swaps with model prices. We show that a simple reduced form model outperforms directly comparing bonds' credit spreads to default swap premiums. We find that the model yields unbiased premium estimates for default swaps on investment grade issuers, but only if we use swap or repo rates as proxy for default-free interest rates. This indicates that the government curve is no longer seen as the reference default-free curve. We also show that the model is relatively insensitive to the value of the assumed recovery rate.     

An arbitrage model of the term structure of interest rates

A formula for the price of default-free discount bonds of all maturities is found using a Black- Scholes type of arbitrage model which is based on the assumption that a portfolio of three default-free discount bonds of distinct maturities can be managed to be a perfect substitute for any other default-free discount bond. The formula relates the price of bonds to the real rate of interest, the anticipated rate of inflation and the equilibrium prices of interest rate and inflation risks. Bond prices are shown to be the expected value of the sure nominal proceeds of the bond discounted to the present at a random discount rate. It is shown that the unbiased expectations hypothesis is in general inconsistent with this model.     

Term Structure Movements and Pricing Interest Rate Contingent Claims

This paper derives an arbitrage-free interest rate movements model (AR model). This model takes the complete term structure as given and derives the subsequent stochastic movement of the term structure such that the movement is arbitrage free. We then show that the AR model can be used to price interest rate contingent claims relative to the observed complete term structure of interest rates. This paper also studies the behavior and the economics of the model. Our approach can be used to price a broad range of interest rate contingent claims, including bond options and callable bonds.     

中国利率市场的价格发现——对国债现货、期货以及利率互换市场的研究

本文采用信息份额模型和基于向量自回归(VAR)模型的格兰杰因果检验,研究了国债现货、国债期货和利率互换三个市场之间的价格发现机制。信息份额模型表明,从整体来看利率互换相对于国债期货和国债现货都具有信息优势,而国债期货相对于国债现货具有信息优势。另外,国债期货的价格发现能力相对于另外两个市场都在随时间增强。格兰杰因果检验结果显示,利率互换在价格发现中单向引领国债期货以及国债现货,国债期货单向引领国债现货。所有结果一致表明, 利率互换和国债期货这两种利率衍生产品在引导中国利率市场价格发现中发挥了重要作用。     

Credit derivatives with multiple debt issues

We evaluate the most actively traded types of credit derivatives within a unified pricing framework that allows for multiple debt issues. Since firms default on all of their obligations, total debt is instrumental in the likelihood of default and therefore in credit derivatives valuation. We use a single factor interest rate model where the exponential default frontier is based on total debt and is made coherent with observed bond prices. Analytical formulae are derived for credit default swaps, total return swaps (both fixed-for-fixed and fixed-for-floating), and credit risk options (CROs). Price behaviors and hedging properties of all these credit derivatives are investigated. Simulations document that credit derivatives prices may be significantly affected by terms of debt other than those of the reference obligation. The analysis of CROs indicates their superior ability to fine-tune the hedging of magnitude and arrival risks of default.     

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.     

Pricing of swaps with default risk

In this paper, I study the valuation of interest rate and currency swaps with default risk under the contingent claim analysis framework. I demonstrate that the traditional approach of pricing swap contracts as exchanges of loans underestimates the value of such contracts to the counterparty with higher credit rating and exaggerates the credit spread required to guard against default risk. Numerical simulations show that the swap rate is not sensitive to counterparty credit rating: for a ten year interest rate swap, a one hundred basis point increase in counterparty bond yield spread results in only about one basis point increase in the swap rate. (JEL G10, G12, G13)This paper is based on Chapter 2 of my Ph.D. dissertation at Yale University. I would like to thank my dissertation committee, Kenneth French, Roger Ibbotson, and Jonathan Ingersoll, Jr. (chairman), for helpful advice and guidance. I would also like to thank Keny Back, Richard Lindsey, N. R. Prabhala, Ming Huang, Marti Subrahmanyam, three anonymous referees and especially Bob Jarrow, the editor, for helpful comments and suggestions. Any errors that remain are solely mine. This paper won the 1996 Trefftzs Award for best student paper from the Western Finance Association.