Default-free Bonds Research Articles

Interest Rate Models

In this paper, we review recent developments in modeling term structures of market yields on default-free bonds. Our discussion is restricted to continuous-time dynamic term structure models (DTSMs). We derive joint conditional moment generating functions (CMGFs) of state variables for DTSMs in which state variables follow multivariate affine diffusions and jump-diffusion processes with random intensity. As an illustration of the pricing methods, we provide special cases of the general formulations as examples. The examples span a wide cross-section of models from early one-factor models of Vasicek to more recent interest rate models with stochastic volatility, random intensity jump-diffusions and quadratic-Gaussian DTSMs. We also derive the European call option price on a zero-coupon bond for linear quadratic term structure models.

General Acceptance Sets, Risk Measures and Optimal Capital Injections

We study capital requirements for bounded financial positions defined as the minimum amount of capital to invest in a chosen eligible asset targeting a pre-specified acceptability test. We allow for general acceptance sets and general eligible assets, including defaultable bonds. Since the payoff of these assets is not necessarily bounded away from zero the resulting risk measures cannot be transformed into cash-additive risk measures by a change of numeraire. However, extending the range of eligible assets is important because, as exemplified by the recent financial crisis, assuming the existence of default-free bonds may be unrealistic. We focus on finiteness and continuity properties of these general risk measures. As an application, we discuss capital requirements based on Value-at-Risk and Tail-Value-at-Risk acceptability, the two most important acceptability criteria in practice. Finally, we prove that there is no optimal choice of the eligible asset. Our results and our examples show that a theory of capital requirements allowing for general eligible assets is richer than the standard theory of cash-additive risk measures.

Interest Rate Models

In this paper, we review recent developments in modeling term structures of market yields on default-free bonds. Our discussion is restricted to continuous-time dynamic term structure models (DTSMs). We derive joint conditional moment generating functions (CMGFs) of state variables for DTSMs in which state variables follow multivariate affine diffusions and jump-diffusion processes with random intensity. As an illustration of the pricing methods, we provide special cases of the general formulations as examples. The examples span a wide cross-section of models from early one-factor models of Vasicek to more recent interest rate models with stochastic volatility, random intensity jump-diffusions and quadratic-Gaussian DTSMs. We also derive the European call option price on a zero-coupon bond for linear quadratic term structure models.

Discrete time linear-quadratic pricing of bonds and options

This article presents a discrete time pricing model whereby prices are either exponential linear-quadratic functions of stochastic factors or transforms of such exponential linear-quadratic functions. The model is applied to price default-free bonds and stock options under stochastic volatility and is the discrete time counterpart of the continuous time Linear Quadratic (LQ) model of Cheng and Scaillet (2007). In discrete time, the factors are conditionally Gaussian and market prices of risk can be specified with much freedom.

Yield Expectations and Monetary Policy

This paper applies a yield curve model that separates expectations and volatility components of market yields on default-free bonds. Expected future riskless rates derived from the model are unbiased, reasonably accurate indicators of subsequent actual riskless rates for periods up to three years. Variances between Fed Funds and expectations of future riskless rates derived from the model correspond with incidences of tight money and easy money in the past and are reflected in subsequent inflation rates three years out. Ex-post inflation forecasts based upon these variances have accuracy within the range of alternative forecasts. Given this direct relationship with future inflation, expectations of future riskless rates may be useful as an indicator or policy target for inflation-targeting monetary policy. For this purpose, robust “headline” inflation measures produce the strongest relationship and may be more useful than “core” measures. Difficulties stimulating the Japanese economy and, recently, the U.S. and other economies might arise from limits on the magnitude of potential monetary stimulus when yield expectations are at extremely low levels and policy rates are zero bound.

Implied Bond and Derivative Prices Based on Non-Linear Stochastic Interest Rate Models

In this paper we expand the Box Method of Sorwar et al. (2007) to value both default free bonds and interest rate contingent claims based on one factor non-linear interest rate models. Further we propose a one-factor non-linear interest rate model that incorporates features suggested by recent research. An example shows the extended Box Method works well in practice.

A Note to Correct Modeling Term Structures of Defaultable Bonds

This paper briefly identifies and corrects an error in Duffie and Singleton (1999). The error to omit a variable casts doubt on the arguments of Duffie and Singleton to constitute Heath-Jarrow-Morton (1992) type term structures on defaultable bonds. The error reveals inconsistency through their model in the sense that Heath, Jarrow, and Morton construct no-arbitrage term structures composed of forward rate dynamics explicitly not depending on the market price of risk with default-free bonds. Recent studies for change methods of measures to constitute alternative fundamental theorems of asset pricing shed some light on their arguments in the Heath-Jarrow-Morton framework.

A Stochastic Model for Default-Free Bond Prices and Continuously Compounding Yield-to-Maturity

The following article examines a stochastic, log-normal model for the continuously compounding yield-to-maturity and a corresponding price model for default-free zero coupon bonds. This article sets conditions for the validity of the model and goes on to show that this model is a special case of the Heath, Jarrow and Morton (HJM) model for forward interest rates, and it examines the conditions for the equivalence. We show that given these conditions, the HJM model and the continuously compounding yield-to-maturity model have the same market price of risk and thus, must satisfy identical conditions for the existence and uniqueness of a risk-neutral measure.

An Option Theory Based Yield Curve Model

The Black-Scholes option theory provides a simple analytical model about the yields of corporate bonds. We extend the theory to model the yields of default free government bonds from a simple observation. From the purchasing power perspective, the values of corporate and government bonds follow similar patterns. The option theory based model of the term structure of interest rates explains major empirical patterns on the shapes and dynamics of yield curves. Compared with existing theories on yield curves, this theory provides a simple analytical theory without additional assumptions about risk, liquidity and preference. It greatly simplifies the understanding and teaching of yield curves.

The effects of default and call risk on bond duration

This paper examines the effects of default risk, call risk, and their interactions on bond duration. We find that call risk decreases durations of default-free bonds, while default risk alone generally decreases durations for risky bonds with only a few exceptions. The joint effect of default and call risk always results in shorter durations for corporate bonds. Controlling for the effect of default risk, call risk has a negative effect on duration, which diminishes as bond ratings decline. Finally, the effect of call risk on duration depends on bond characteristics. Empirical evidence shows that the effect of call risk is smaller for discount bonds and for deep-discount fallen angels.

Estimate of Term Structure Using DQTSM's with Non-Linear MPR

The main subject of this paper is to estimate the term structure of default-free zero-coupon bond yields using DQTSMs and to investigate the interplay between the models' theoretical specifications and their empirical performance. The research relies on the Extended Kalman Filter and the Quasi-Maximum-Likelihood method. I assume the market price of risk to be a function of state variables that can be affine or nonlinear. The results suggest that, ... firstly, discrete-time quadratic term structure models, like their continuous-time counterparts, can overcome the drawbacks of the affine term structure models. Secondly, the nonlinear market price of risk model, especially, the cubic market price of risk model outperforms the affine, constant and zero market price of risk models in explaining historical bond price behaviour in the United States.

Pricing Interest Rate-Sensitive Credit Portfolio Derivatives

In this paper we present a modelling framework for portfolio credit risk which incorporates the dependence between risk-free interest-rates and the default loss process. The contribution in this approach is that - besides the traditional diffusion based covariation between loss intensities and interest-rates - a direct dependence between interest-rates and the loss process is allowed, in particular default-free interest-rates can also depend on the loss history of the credit portfolio. Amongst other things this enables us to capture the effect that economy-wide default events are likely to have on government bond markets and/or central banks' interest-rate policies. Similar to Schonbucher (2005), the model is set up using a set of loss contingent forward interest-rates fn(t; T) and loss-contingent forward credit protection rates Fn(t; T) to parameterize the market prices of default-free bonds and credit-sensitive assets such as CDOs. We show that (up to weak regularity conditions), existence of such a parametrization is necessary and sufficient for the absence of static arbitrage opportunities in the underlying assets. We also give necessary conditions and sufficient conditions on the dynamics of the parametrization which ensure absence of dynamic arbitrage opportunities in the model. Similar to the HJM drift restrictions for default-free interest-rates, these conditions take the form of restrictions on the drifts of fn(t; T) and Fn(t; T), together with a set of regularity conditions.

Valuation of derivatives based on single-factor interest rate models

The CKLS [Chan, K.C., Karolyi, G.A., Longstaff, F.A., & Sanders, A.B. (1992). An empirical comparison of the short-term interest rate, Journal of Finance, 1(7), 1209–1227] short-term risk-free interest rate process leads to a valuation model for both default-free bonds and contingent claims that can only be solved numerically for the general case. Valuation equations of this nature in the past have been solved using the Crank Nicolson scheme. In this paper, we introduce a new numerical scheme — the Box method, and compare it with the traditional Crank Nicolson scheme. We find that in specific cases of the CKLS process where analytical prices are available, the new scheme leads to more accurate results than the Crank Nicolson scheme.

Arbitrage-Free Bond Pricing with Dynamic Macroeconomic Models

The authors examine the relationship between changes in short-term interest rates induced by monetary policy and the yields on long-maturity default-free bonds. The volatility of the long end of the term structure and its relationship with monetary policy are puzzling from the perspective of simple structural macroeconomic models.

The Valuation of Options on Bonds with Default Risk

In this paper we present a model for valuing European and American options, which incorporates both default and interest rate risks. We develop a framework that permits evaluation of three kinds of options: (i) options issued by default-free counterparties on risky bonds, (ii) options issued by risky counterparties on default-free bonds and (iii) options issued by risky counterparties on risky bonds — a case where default risk enters at both levels. We show that the price of a put option on a risky discount bond is hump shaped for a European put and monotone increasing for an American put. We also find that the price impact of default risk is less for an American put option than for a European one (JEL: G13).

The Valuation of Deposit Insurance in an Arbitrage-free Basel II Consistent Framework

This paper analyzes the joint influence of the quality of a bank's loan portfolio, the bank's maturity gap and its deposit rate policy on the value of deposit insurance in an arbitrage-free Basel II consistent framework. We develop and apply a two-stage structural model of a bank where deposit insurance is a European put option on the loan portfolio, the default of each loan is driven by the borrower's asset value and interest rates are stochastic. Modeling the firms' asset values by conditional independent three-factor geometric Brownian motions and applying forward measure techniques, we obtain semi-analytical solutions of arbitrage-free deposit insurance premiums for sufficiently large and homogenous loan portfolios. We show for realistic parameter combinations that the correlation within the loan portfolio and the bank's maturity gap can have a material impact on fair deposit insurance premiums. Furthermore, we find that fair deposit insurance premiums can depend negatively on the borrowers' default risk when the bank faces a maturity gap. The fair deposit insurance premium for a bank holding defaultable loans can even be lower than for a bank investing in default-free bonds.

Asset Liability Management in Insurance Company

The model, by using the option theory, determines the fair value of the policies life with different time of maturity and shows that the effective liabilities duration of an Insurance Company exposed to the default-risk is different from the duration of a default-free zero coupon bond with the same time of maturity. Furthermore, it shows that the value of equity can be immunized in a dynamic way with respect to the movement of the spot-rate by selling and purchasing the default-free bonds in the firm asset. Moreover, the equity value, by the right bond allocation, can be immunized without varying continually the weight of the bonds on the firm asset. Furthermore, it considers the surrender option and the mortally issue such that it corrects some pitfalls that are commonly encountered in the insurance industry.

SOME MATHEMATICAL PROPERTIES OF THE PRICE OF AN ARBITRAGE PROFITABLE RISKLESS BOND

It is well known that a discounting function describes the price of a default free bond. In this paper, some mathematical properties of this function are studied when it is subadditive. Subadditive discounting means that the total discount is minor when the year is divided into months and implies that the more divided the interval is into subintervals, the smaller the total discount will be. First, two sufficient conditions for a discounting function being subadditive (resp. superadditive) are shown; more precisely, the increase (resp. decrease) of the instantaneous discount rate and certain integral inequality are proposed as such sufficient conditions. On the other hand, the convexity (resp. concavity) of the Napierian logarithm of the discounting function is shown as a particular case of another more general sufficient condition. Finally, under certain conditions, subadditive discounting functions are necessarily regular.

A risk hedging strategy under the nonparallel-shift yield curve

Under the assumption of the movement of rigid, a nonparallel-shift model in the term structure of interest rates is developed by introducing Fisher & Weil duration which is a well-known concept in the area of interest risk management. This paper has studied the hedge and replication for portfolio immunization to minimize the risk exposure. Throughout the experiment of numerical simulation, the risk exposures of the portfolio under the different risk hedging strategies are quantitatively evaluated by the method of value at risk (VaR) order statistics (OS) estimation. The results show that the risk hedging strategy proposed in this paper is very effective for the interest risk management of the default-free bond.

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.