Zero-coupon Bond Research Articles

The valuation of convertible bonds with numeraire changes

The changes of numeraire can be used as a very powerful mean in pricing contingent claims in the context of a complete market. We apply the method of nurmeraire changes to evaluate convertible bonds when the instantaneous growth and variance of the value of issuer and those of zero-coupon bonds follow a general adapted stochastic process in this paper. A closed-form solution is derived when the instantaneous growth and variance of the value of issuer and those of zero-coupon bonds are deterministic function of time. We also consider a special case when the asset price follows GBM (Geometric Brownian Motion) and interest rate follows Vasicek’s model.

Dimension-Invariant Dynamic Term Structures

We model term structure dynamics using a recursive cascade of heterogeneously persistent factors. The cascade naturally orders the factors by their adjustment speeds, and generates smooth zero-coupon bond prices and forward curves in closed form. For a class of specifications, the number of parameters is invariant to the size of the state space, and the term structure converges to a stochastic limit as the number of factors goes to infinity. The cascade construction thereby defeats the curse of dimensionality associated with general affine models. Using a panel of interest rates, we estimate specifications with from one to 15 factors and only five parameters. High-dimensional versions of the model fit empirical yield curves almost perfectly in sample, with root mean squared errors of less than one basis point, while maintaining reasonable maximal Sharpe ratios. Out-of-sample interest rate forecasts significantly outperform prior benchmarks.

The Equity-Smile and Credit-Spread Puzzles: Are They One and the Same?

We examine whether the puzzles of high prices for out-of-the-money put options on equity indices (the equity-smile puzzle) and large sizes of credit spreads on corporate bonds (the credit-spread puzzle) are related. To do this we compare the implied volatilities of long-term equity-index puts with the implied volatilities of puts within corporate bonds, at the bonds’ moneyness and maturity levels, in each month over 1992 to 2004. The data used are short-term index options, a proprietary set of long-term index options, and a set of zero-coupon bonds which are issued by investment companies. We find that the equity smile and the bond smile show remarkable similarity in all periods, not only in how volatility relates to moneyness but also in how it relates to maturity. This indicates that the same pricing kernel, with a high risk-premium on the extreme downside, is likely to explain them both.

Discrete-Time AffineℚTerm Structure Models with Generalized Market Prices of Risk

This article develops a rich class of discrete-time, nonlinear dynamic term structure models (DTSMs). Under the risk-neutral measure, the distribution of the state vector Xt resides within a family of discrete-time affine processes that nests the exact discrete-time counterparts of the entire class of continuous-time models in Duffie and Kan (1996) and Dai and Singleton (2000). Under the historical distribution, our approach accommodates nonlinear (nonaffine) processes while leading to closed-form expressions for the conditional likelihood functions for zero-coupon bond yields. As motivation for our framework, we show that it encompasses many of the equilibrium models with habit-based preferences or recursive preferences and long-run risks. We illustrate our methods by constructing maximum likelihood estimates of a nonlinear discrete-time DTSM with habit-based preferences in which bond prices are known in closed form. We conclude that habit-based models, as typically parameterized in the literature, do not match key features of the conditional distribution of bond yields. (JEL G12, C50, E13, E21)

Interest Rate Risk Hedging Demand Under a Gaussian Framework

This article analyzes the state variables Merton-Breeden hedging demand for an investor endowed with a utility function over both intermediate consumption and terminal wealth. Based on the three-factor model of Babbs and Nowman (1999), we show that this demand can be simply expressed as weighted average zero-coupon bonds sensitivities to these factors. The weighting parameter is actually the proportion of wealth our investor sets aside for future consumption rather than for terminal wealth.

Real world pricing and affine representation for forward contracts

AbstractThis paper makes use of an integrated benchmark modeling framework that allows us to derive a term structure for the forward contract under real world probability measure. The benchmark or numeraire is chosen to be the growth optimal portfolio (GOP). In general forward prices are not martingales under the real world probability measures. To overcome this problem, we introduce the benchmarked forward measure under which we recover the martingale property for forward prices. Under the benchmarked forward measure we establish an actuarial pricing formula for contingent claims that are not necessarily independent of the GOP. Thereafter, the issue of relating state variables in forward price models to market observed quantities is addressed. The paper links state variables directly, and explicitly, to forward prices, and establishes that models are, in fact, affine with respect to a finite number of forward prices. Applying the methodology to Oil forward models we remark that the forward price is independent of fluctuations of the zero‐coupon bond price and therefore of those of the interest rate. The paper shows also that under the benchmark setting the contribution of the market price of risk is naturally included in the modeling of the zero‐coupon bond and the forward price, which is not systematic in the risk neutral setting. Copyright © 2010 Wilmott Magazine Ltd.

A Practical Guide to Implied and Local Volatility

We consider a stochastic local volatility model with domestic and foreign stochastic interest rates such that the volatility decomposes into a deterministic local volatility plus some bias terms. Assuming a collapse process for the variance with the same random variable for all time and deterministic zero-coupon bond volatility functions, we are going to describe in detail the implementation of that model, focusing on the computation of a proper deterministic local volatility. To do so, we choose to generate an implied volatility surface without arbitrage in space and in time by parametrising a mixture of shifted lognormal densities under constraints and we use a Differential Evolution algorithm to calibrate the model's parameters to a finite set of option prices. We will therefore need to devise an evolutionary algorithm that handle constraints in a simple and efficient way. Using some of the improvements made to the DE algorithm combined with simple and robust constraints handling mechanisms we will propose a modified algorithm for solving our optimisation problem under constraints which greatly improves its performances.

A regime-switching term structure model with observable state variables

The paper proposes in a regime-shift framework, an arbitrage-free term structure model based on the target and Fed Funds Rates. Empirical observations suggest that a three-state regime-shift environment associated with FOMC monetary actions is justified. Then, a closed-form solution for zero-coupon bonds is derived where regime-shift risk is priced. The solution is flexible enough to incorporate additional state variables.

A Reduced Form Model of Default Spreads with Markov-Switching Macroeconomic Factors

An important research area of the corporate yield spread literature seeks to measure the proportion of the spread that can be explained by factors such as the possibility of default, liquidity, tax differentials and market risk. We contribute to this literature by assessing the ability of observed macroeconomic factors and the possibility of changes in regime to explain the proportion of yield spreads caused by the risk of default in the context of a reduced form model. For this purpose, we extend the Markov Switching risk-free term structure model of Bansal and Zhou (2002) to the corporate bond setting and develop recursive formulas for default probabilities, risk-free and risky zero-coupon bond yields as well as credit default swap premia. The model is calibrated with consumption, inflation, risk-free yields and default data for Aa, A and Baa bonds from the 1987-2008 period. We find that our macroeconomic factors are linked with two out of three sharp increases in the spreads during this sample period, indicating that the variations can be related to macroeconomic undiversifiable risk. The estimated default spreads can explain almost half of the 10 years to maturity industrial Baa zero-coupon yields in some regime. Much smaller proportions are found for Aa and A bonds with numbers around 10%. The proportions of default estimated with credit default swaps are higher, in many cases doubling those found with corporate yield spreads.

Fair valuation of universal life policies via a replicating portfolio

In this paper we develop a new method for valuing universal life policies via a zero-coupon bond replicating portfolio. The method combines the idea of replicating portfolio with that of a fixed point. In addition, we accompany the proposed method with an equation relating account value, reserve and present value of future profits.

Higher order asymptotic bond price valuation for interest rates with non-Gaussian dependent innovations

This paper considers the effect on zero-coupon bond price valuation when short rate model has non-Gaussian dependent innovations. Higher order asymptotic theory enables us to obtain the approximate bond price formula. Some numerical examples are presented, where the process of innovations follows particular model. These examples indicate non-Gaussianity and dependency of innovations have a great influence on zero-coupon bond price.

Use of Mathematical Models of the Interest Rate Processes for the Analysis of Yield Time Series

As it can be seen from the literature the affine yield class of the term structure is very important for obtaining the fundamental properties of the zero-coupon bond yield. Therefore any new results about the properties the affine yield class are very desirable. The present paper is devoted to this problem. In framework of the affine yield class of the term structure the price of the zero-coupon bonds is determined in terms of the short interest rate. However the short interest rate process is non-observed on practice. Hence the values of short interest rate process cannot be used for the identification of the adequate model bond price process. At the same time it is known that on market bid quotations on the most actively traded securities are base to issue the yield curves. Therefore from the practice point of view it is more natural to use the yield interest rate instead the short interest rate. Thus it is interesting the problem of the identification of the adequate model bond price process by handling of the yield rate samples. The solution of this problem will allow understand which from known models is more relevant to describe the bond price process. In the paper the analysis is based on three models: V-model (Vasicek, 1977), CIR-model (Cox, Ingersoll and Ross, 1985), MC-model (Medvedev and Cox, 1996). The difference versions of these models are constructed as the approximations of stochastic differential equations for the yield interest rate. Then the maximum likelihood method is applied for estimation of the difference model parameters. This procedure is used for statistical analysis of the US Treasury securities.

ARBITRAGE-FREE INTERPOLATION OF THE SWAP CURVE

We suggest an arbitrage free interpolation method for pricing zero-coupon bonds of arbitrary maturities from a model of the market data that typically underlies the swap curve; that is short term, future and swap rates. This is done first within the context of the Libor or the swap market model. We do so by introducing an independent stochastic process which plays the role of a short term yield, in which case we obtain an approximate closed-form solution to the term structure while preserving a stochastic implied short rate. This will be discontinuous but it can be turned into a continuous process (however at the expense of closed-form solutions to bond prices). We then relax the assumption of a complete set of initial swap rates and look at the more realistic case where the initial data consists of fewer swap rates than tenor dates and show that a particular interpolation of the missing swaps in the tenor structure will determine the volatility of the resulting interpolated swaps. We give conditions under which the problem can be solved in closed-form therefore providing a consistent arbitrage-free method for yield curve generation.

Multi-Currency Fast Stochastic Local Volatility Model

We consider a stochastic local volatility model with domestic and foreign stochastic interest rates and identify a bias with respect to the deterministic local volatility with deterministic rates. Relating the local volatility of our model to that of the forward price, we quantify that bias by equating the variance swap contract under the risk-neutral measure with that under the forward probability measure. Assuming a collapse process for the variance with the same random variable for all time and deterministic zero-coupon bond volatility functions, the bias term simplifies and can easily be computed. We then describe a simple implementation of our model. In the discretised dynamics, the local volatility function being piecewise constant, we apply the bootstrapping technique to calculate its value by solving a quadratic equation at each maturity and strike. As a result, we get a fast and robust way of calibrating a stochastic local volatility model with stochastic rates to market prices.

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.

Valuation for an American Continuous-Installment Put Option on Bond under Vasicek Interest Rate Model

The valuation for an American continuous-installment put option on zero-coupon bond is considered by Kim's equations under a single factor model of the short-term interest rate, which follows the famous Vasicek model. In term of the price of this option, integral representations of both the optimal stopping and exercise boundaries are derived. A numerical method is used to approximate the optimal stopping and exercise boundaries by quadrature formulas. Numerical results and discussions are provided.

Basket Options on Heterogeneous Underlying Assets

In this article, we make the design of a basket option on commodity prices with stochastic convenience yields, exchange rates and zero-coupon bonds. We carry out Monte Carlo simulations to value this option. We use maximum likelihood method to estimate the different parameters of the theoretical model proposed as well as the correlations between these variables. As done in Duan (1994), we apply maximum likelihood approach adapted to unobservable variables to estimate the temporal series of the convenience yield as well as its parameters. Monte Carlo studies are conducted to examine the performance of the maximum likelihood technique in finite samples of simulated data.

Stochastic Models for Actuarial Use: The Equilibrium Modelling of Local Markets

AbstractIn this paper, a long-term equilibrium model of a local market is developed. Subject to minor qualifications, the model is arbitrage-free. The variables modelled are the prices of risk-free zero-coupon bonds – both index-linked and conventional – and of equities, as well as the inflation rate. The model is developed in discrete (nominally annual) time, but allowance is made for processes in continuous time subject to continuous rebalancing. It is based on a model of the market portfolio comprising all the above-mentioned asset categories. The risk-free asset is taken to be the one-year index-linked bond. It is assumed that, conditionally upon information at the beginning of a year, market participants have homogeneous expectations with regard to the forthcoming year and make their decisions in mean-variance space. For the purposes of illustration, a descriptive version of the model is developed with reference to UK data. The parameters produced by that process may be used to inform the determination of those required for the use of the model as a predictive model. Illustrative results of simulations of the model are given.

A Simple Timing Strategy for Canadian Fixed Income Portfolios

Abstract Fixed income return enhancement can be obtained in the Canadian marketplace when one conditions on past stock market movements. A simple timing rule is demonstrated to increase holding period return significantly. It is likely that this strategy is capitalizing on increases in ex ante return due to higher levels of risk aversion induced by lower societal wealth. Nevertheless, although these opportunities were available to all, it is shown that Canadian fixed income portfolio managers failed to seize them. Resume On peu ameliorer le rendement des instruments a taux fixe sur le marche canadien en se basant sur les mouvements passes du marche boursier II est prouve que le simple choix du bon moment peut augmenter de fa,con significative le rendement enregistre au cours de la periode d'investissement. Cette strategie tire probablement parti des augmentations du rendement ex ante imputable i des niveaux eleves d'aversion pour le risque induits par une richesse sociale amoindrie. Toutefois, bien que tous aient eu acces rt ces possibilites, l'etude revele que les gestionnaires canadiens de portefeuille d'instruments a taux fixe n'en ont pas profite. Portfolio managers are forever seeking ways to enhance return. For active fixed income managers this amounts in large part to searching for tools that allow one to occasionally time interest rate movements and make anticipatory duration-switching adjustments. Nevertheless, conventional wisdom and the empirical evidence tell us that interest rate forecasting is a very difficult undertaking.1 Nevertheless, the unpredictability of interest rate changes does not necessarily close the door on return enhancement through holding-period-return prediction. To see this most simply, suppose we have only 1-year and 2-year zero-coupon bonds. Further assume that the best prediction for the future 1-year interest rate is always the current 1-year interest rate. If managers wish to maximize expected return on their portfolio over a 1year holding period, the correct strategy will always be to buy two-period bonds when the yield curve is upwardsloping, but to otherwise stay short. Of course, in this hypothetical world, the long-short spread is entirely accounted for by what is called the term premiunm. Its two principal definitions, which coalesce in this twoperiod example, are a) the gap between the implicit forward interest rate, as calculated from the current yield curve and the corresponding expected future spot rate, and b) the expected holding period return of a bond net of the rate whose length corresponds to the measurement horizon.2 In reality, the slope of the term structure is also partly the result of anticipated changes in yields. Still, if one could estimate term premia (however defined), scope would be indeed opened up for duration-switching techniques. A technique, based on what researchers have recently learned about the nature of term premia and what factors have an impact on their magnitude, is tested here. This is at the heart of current term structure research. The expectations model of the term structure of interest rates, which argues that the term premia embedded in the yield curve are time-invariant, has now been discredited by abundant evidence. It is apparent that premia are positively related to the long-short spread, various sources of uncertainty (whether in the bond market or throughout the economy), and the current interest rate level (though the record is far from perfect on the latter point).3 Most striking has been the recent finding that holding period bond returns net of the short-term interest rate (hereafter excess returns) are related to lagged movements in the aggregate stock market.4 Ilmanen (1995) calls the ratio of the past level of the stock market to today's level, which is just the inverse of the most recent gross aggregate stock market return (ignoring dividends), inverse relative wealth. …

Axiomatization of Residual Income and Generation of Financial Securities

This paper presents an axiomatization of residual income, aka excess profit, and illustrates how it may univocally engenders fixed-income or variable-income assets. In the first part it is shown that, depending on the relations between excess profit and the investor's excess wealth, a well-specified theory of residual income is generated: one is the standard theory, which historically traces back to Hamilton (1777) and Marshall (1890) and is a deep-rooted notion in economic theory, finance, and accounting. Another one is the systemic value added or lost-capital paradigm: introduced in Magni (2000, 2003), the theory is enfolded in Keynes's (1936) notion of user cost and is naturally generated by an arbitrage-theory perspective. In the second part, the paper reverts the usual analysis: instead of computing residual incomes profits from a pattern of cash flows, residual incomes are fixed first to derive vectors of cash flows. It is shown that variable- or fixed-income assets may be constructed on the basis of either theory starting from pre-determined growth rates for excess profit. In particular, zero-coupon bonds and coupon bonds traded in a capital market are shown to be deducted as equilibrium vectors of residual-income-based assets.