Stochastic duration and fast coupon bond option pricing in multi-factor models

Generalizing Cox, Ingersoll, and Ross (1979), this paper defines the stochastic duration of a bond in a general multi-factor diffusion model as the time to maturity of the zero-coupon bond with the same relative volatility as the bond. Important general properties of the stochastic duration measure are derived analytically, and the stochastic duration is studied in detail in various well-known models. It is also demonstrated by analytical arguments and numerical examples that the price of a European option on a coupon bond (and, hence, of a European swaption) can be approximated very accurately by a multiple of the price of a European option on a zero-coupon bond with a time to maturity equal to the stochastic duration of the coupon bond. This revised version was published online in June 2006 with corrections to the Cover Date.     

The analysis and valuation of interest rate options

This paper provides a simple, alternative model for the valuation of European-style interest rate options. The assumption that drives the hedging argument in the model is that the forward prices of bonds follow an arbitrary two-state process. Later, this assumption is made more specific by postulating that the discount on a zero-coupon bond follows a multiplicative binomial process. In contrast to the Black-Scholes assumption applied to zero-coupon bonds, the limiting distribution of this process has the attractive features that the zero-bond price has a natural barrier at unity (thus precluding negative interest rates), and that the bond price is negatively skewed. The model is used to price interest rate options in general, and interest rate caps and floors in particular. The model is then generalized and applied to European-style options on bonds. A relationship is established between options on swaps and options on coupon bonds. The generalized model then provides a computationally simple formula, closely related to the Black-Scholes formula, for the valuation of European-style options on swaps.     

The nominal duration of TIPS bonds

This paper studies the price responsiveness (effective duration) of U.S. government issued inflation-indexed bonds, known by the acronym TIPS (Treasury Inflation-Protected Securities), to changes in nominal interest rates, real interest rates, and expected inflation. Using the TIPS pricing formula derived by Laatsch and Klein [Q. Rev. Econ. Finance 43 (2002) 405], we first confirm that TIPS bonds have zero sensitivity to changes solely in expected inflation. By changes solely in expected inflation, we mean that the real rate remains unchanged and the nominal rate changes in accordance with the established Fisher [Publ. Am. Econ. Assoc. 11 (1896)] effect. We show that the first derivative of the TIPS price is zero whenever the real rate is held constant. Thus, the first partial derivative of the TIPS bond pricing formula with respect to expected inflation is zero and the first partial derivative of the TIPS bond price with respect to nominal rates is also zero, given, in each case, that we hold the real rate constant. We then temporarily shift the analysis to zero-coupon TIPS bonds and zero-coupon ordinary Treasury bonds. We prove that the nominal duration of zero-coupon TIPS bonds equals that of zero-coupon ordinary Treasury bonds when the real rate changes but expected inflation is held constant.However, if expected inflation changes and the change in the nominal rate does not yield a constant real rate, zero-coupon TIPS prices will change and they will change by a smaller percentage than will zero-coupon ordinary Treasury bonds. We analyze TIPS responsiveness to changes in nominal rates under such conditions. We derive an approximation to effective duration that demonstrates that the effective durations of various maturity zero-coupon TIPS bonds are approximately linear functions in time to maturity of the effective duration of the one-year zero-coupon TIPS bond, ceteris paribus.Nominal effective duration of TIPS bonds is certainly of interest to fixed income portfolio managers that might have a desire to include such bonds in their portfolio. After all, the greater portion of a typical fixed income portfolio is in traditional, noninflation protected bonds whose major risk exposure is to changes in nominal rates. To properly assess the role of TIPS bonds in the portfolio, portfolio managers need information as to how TIPS bonds respond to the changes in nominal rates that are driving the price behavior of the bulk of the portfolio's assets. Prior to concluding the paper, we demonstrate how portfolio managers can calculate the nominal durations of coupon TIPS bonds using the zero-coupon duration formula we derive.     

A Gaussian Process of Yield Rates Calibrated with Strips

Abstract

This paper presents a Gaussian multivariate factor model of the term structure of interest rates. It shows that there exists a martingale valuation law of the factors so that the price function of a zero-coupon bond is an exponential spline. The model’s linear and Gaussian structure yields a simple model where estimation and calibration are relatively easy to do. Using yield data on stripped bonds, the spline model gives a very good approximation of the yield curve at all times. Moreover, the crucial Gaussian assumption is reasonable when modeling the dynamics for short periods like one year.     

Asymptotics of bond yields and volatilities for extended CIR models under the real-world measure

ABSTRACT

The Cox–Ingersoll–Ross CIR short rate model is a mean-reverting model of the short rate which, for suitably chosen parameters, permits closed-form valuation formulae of zero-coupon bonds and options on zero-coupon bonds. This article supplies proofs of the formulae for the expected present value of payoffs under the real-world probability measure, known as actuarial valuation. Importantly, we give formulae for asymptotic levels of bond yields and volatilities for extended CIR models when suitable conditions are imposed on the model parameters.     

LIBOR and swap market models and measures

A self-contained theory is presented for pricing and hedging LIBOR and swap derivatives by arbitrage. Appropriate payoff homogeneity and measurability conditions are identified which guarantee that a given payoff can be attained by a self-financing trading strategy. LIBOR and swap derivatives satisfy this condition, implying they can be priced and hedged with a finite number of zero-coupon bonds, even when there is no instantaneous saving bond. Notion of locally arbitrage-free price system is introduced and equivalent criteria established. Stochastic differential equations are derived for term structures of forward libor and swap rates, and shown to have a unique positive solution when the percentage volatility function is bounded, implying existence of an arbitrage-free model with such volatility specification. The construction is explicit for the lognormal LIBOR and swap “market models”, the former following Musiela and Rutkowski (1995). Primary examples of LIBOR and swap derivatives are discussed and appropriate practical models suggested for each.     

An Econometric Model of the Term Structure of Interest-Rate Swap Yields

This article develops a multi-factor econometric model of the term structure of interest-rate swap yields. The model accommodates the possibility of counterparty default, and any differences in the liquidities of the Treasury and Swap markets. By parameterizing a model of swap rates directly, we are able to compute model-based estimates of the defaultable zero-coupon bond rates implicit in the swap market without having to specify a priori the dependence of these rates on default hazard or recovery rates. The time series analysis of spreads between zero-coupon swap and treasury yields reveals that both credit and liquidity factors were important sources of variation in swap spreads over the past decade.     

Bond and option pricing for interest rate model with clustering effects

This paper analyzes an interest rate model with self-exciting jumps, in which a jump in the interest rate model increases the intensity of jumps in the same model. This self-exciting property leads to clustering effects in the interest rate model. We obtain a closed-form expression for the conditional moment-generating function when the model coefficients have affine structures. Based on the Girsanov-type measure transformation for general jump-diffusion processes, we derive the evolution of the interest rate under the equivalent martingale measure and an explicit expression of the zero-coupon bond pricing formula. Furthermore, we give a pricing formula for the European call option written on zero-coupon bonds. Finally, we provide an interpretation for the clustering effects in the interest rate model within a simple framework of general equilibrium. Indeed, we construct an interest rate model, the equilibrium state of which coincides with the interest rate model with clustering effects proposed in this paper.     

Closed Form Solutions for Term Structure Derivatives with Log-Normal Interest Rates

We derive a unified model that gives closed form solutions for caps and floors written on interest rates as well as puts and calls written on zero-coupon bonds. The crucial assumption is that simple interest rates over a fixed finite period that matches the contract, which we want to price, are log-normally distributed. Moreover, this assumption is shown to be consistent with the Heath-Jarrow-Morton model for a specific choice of volatility.     

Regime Shifts in a Dynamic Term Structure Model of U.S. Treasury Bond Yields

This article develops and empirically implements an arbitrage-free,dynamic term structure model with "priced" factor and regime-shiftrisks. The risk factors are assumed to follow a discrete-timeGaussian process, and regime shifts are governed by a discrete-timeMarkov process with state-dependent transition probabilities.This model gives closed-form solutions for zero-coupon bondprices, an analytic representation of the likelihood functionfor bond yields, and a natural decomposition of expected excessreturns to components corresponding to regime-shift and factorrisks. Using monthly data on U.S. Treasury zero-coupon bondyields, we show a critical role of priced, state-dependent regime-shiftrisks in capturing the time variations in expected excess returns,and document notable differences in the behaviors of the factorrisk component of the expected returns across high and low volatilityregimes. Additionally, the state dependence of the regime-switchingprobabilities is shown to capture an interesting asymmetry inthe cyclical behavior of interest rates. The shapes of the termstructure of volatility of bond yield changes are also verydifferent across regimes, with the well-known hump being largelya low-volatility regime phenomenon.     

Liquidity,Reconstitution, and the Value of U.S. Treasury Strips

An apparent pricing anomaly exists in the market for U.S. Treasury strips: zero-coupon strips created from principal payments typically trade at significantly higher prices than otherwise identical zero-coupon strips created from coupon payments. In addition to documenting this phenomenon, this study demonstrates that differences in liquidity and differences in reconstitution characteristics explain much of this price variation.     

Macroeconomic fundamentals and the exchange rate dynamics: A no-arbitrage macro-finance approach

In this paper, we propose an arbitrage-free international macro-finance model that links the exchange rate dynamics to macroeconomic fundamentals. Jointly using data on exchange rates, yields of zero-coupon bonds, and macroeconomic variables of the US and the Euro area, we find a close link between macroeconomic fundamentals and the exchange rate dynamics. The model-implied monthly exchange rate changes can explain about 57% variation of the observed data. The macroeconomic innovations can help capture large variation of exchange rate changes. Robustness checks show that the results also hold for other major exchange rates.     

Are interest rate options important for the assessment of interest rate risk?

Fixed income options contain substantial information on the price of interest rate volatility risk. In this paper, we ask if those options will also provide information related to other moments of the objective distribution of interest rates. Based on dynamic term structure models within the class of affine models, we find that interest rate options are useful for the identification of interest rate quantiles. Two three-factor models are adopted and their adequacy to estimate Value at Risk of zero-coupon bonds is tested. We find significant difference on the quantitative assessment of risk when options are (or not) included in the estimation process of each of these dynamic models. Statistical backtests indicate that bond estimated risk is clearly more adequate when options are adopted, although not yet completely satisfactory.     

Derivatives pricing with marked point processes using tick-by-tick data

I propose to model stock price tick-by-tick data via a non-explosive marked point process. The arrival of trades is driven by a counting process in which the waiting time between trades possesses a Mittag–Leffler survival function and price revisions have an infinitely divisible distribution. I show that the partial-integro-differential equation satisfied by the value of European-style derivatives contains a non-local operator in time-to-maturity known as the Caputo fractional derivative. Numerical examples are provided for a marked point process with conditionally Gaussian and with conditionally CGMY price innovations. Furthermore, the infinitesimal generator of the marked point process derived to price derivatives coincides with that of a Lévy process of either finite or infinite activity.     

Pricing Bonds and Bond Options with Default Risk

The pricing of bonds and bond options with default risk is analysed in the general equilibrium model of Cox, Ingersoll, and Ross (1985). This model is extended by means of an additional parameter in order to deal with financial and credit risk simultaneously. The estimation of such a parameter, which can be considered as the market equivalent of an agencies' bond rating, allows to extract from current quotes the market perceptions of firm's credit risk. The general pricing model for defaultable zero-coupon bond is first derived in a simple discrete-time setting and then in continuous-time. The availability of an integrated model allows for the pricing of default-free options written on defaultable bonds and of vulnerable options written either on default-free bonds or defaultable bonds. A comparison between our results and those given by Jarrow and Turnbull (1995) is also presented.     

Optimal Refunding Strategies, Transaction Costs, and the Market Value of Corporate Debt

The primary purpose of this paper is to consider both qualitatively and quantitatively the effects of refunding transaction costs and interest rate uncertainty on optimal refunding strategies and the market value of corporate debt. A dynamic model of corporate bond refunding with transaction costs is developed that simultaneously solves for the optimal refunding strategy, the value of the refunding call option, the value of the bond liability to the firm, and the market (investor) value of the fixed-rate contract. We provide examples in which the price of the callable bond does exceed the call price, and we examine whether or not typical levels of refunding costs are sufficient to explain the magnitude and duration of frequently observed premiums on callable corporate bonds.     

Pricing the term structure with linear regressions

We show how to price the time series and cross section of the term structure of interest rates using a three-step linear regression approach. Our method allows computationally fast estimation of term structure models with a large number of pricing factors. We present specification tests favoring a model using five principal components of yields as factors. We demonstrate that this model outperforms the Cochrane and Piazzesi (2008) four-factor specification in out-of-sample exercises but generates similar in-sample term premium dynamics. Our regression approach can also incorporate unspanned factors and allows estimation of term structure models without observing a zero-coupon yield curve.     

A positive interest rate model with sticky barrier

This paper proposes an efficient model for the term structure of interest rates when the interest rate takes very small values. We make the following choices: (i) we model the short-term interest rate, (ii) we assume that once the interest rate reaches zero, it stays there and we have to wait for a random time until the rate is reinitialized to a (possibly random) strictly positive value. This setting ensures that all term rates are strictly positive.

Our objective is to provide a simple method to price zero-coupon bonds. A basic statistical study of the data at hand indeed suggests a switch to a different mode of behaviour when we get to a low level of interest rates. We introduce a variable for the time already spent at 0 (during the last stay) and derive the pricing equation for the bond. We then solve this partial integro-differential equation (PIDE) on its entire domain using a finite difference method (Cranck–Nicholson scheme), a method of characteristics and a fixed point algorithm. Resulting yield curves can exhibit many different shapes, including the S shape observed on the recent Japanese market.     

Information reduction via level crossings in a credit risk model

This paper provides an alternative credit risk model based on information reduction where the market only observes the firm’s asset value when it crosses certain levels, interpreted as changes significant enough for the firm’s management to make a public announcement. For a class of diffusion processes we are able to provide explicit expressions for the firm’s default intensity process and its zero-coupon bond prices.      

A SIMPLE FORMULA FOR DURATION

This paper presents a simplification of Macaulay's formula for duration. The derivation treats any bond as a portfolio of a coupon bearing bond at par and a zero-coupon bond. The resultant equations are simple to use and show how coupon payments, premiums, and discounts affect duration. The equations are easily modified without loss of simplicity for various coupon payment intervals and for calculations between payment periods.