Coupon-bearing Bonds Research Articles

Pricing and hedging bond options and sinking-fund bonds under the CIR model

<abstract><p>This article derives simple closed-form solutions for computing Greeks of zero-coupon and coupon-bearing bond options under the CIR interest rate model, which are shown to be accurate, easy to implement, and computationally highly efficient. These novel analytical solutions allow us to extend the literature in two other directions. First, the static hedging portfolio approach is used for pricing and hedging American-style plain-vanilla zero-coupon bond options under the CIR model. Second, we derive analytically the comparative static properties of sinking-fund bonds under the same interest rate modeling setup.</p></abstract>

Price impact on term structure

We introduce a first theory of price impact in the presence of an interest rate term structure. We explain how one can formulate instantaneous and transient price impact on zero-coupon bonds with different maturities, including a cross price impact that is endogenous to the term structure. We connect the introduced impact to classic no-arbitrage theory for interest rate markets, showing that impact can be embedded in the pricing measure and that no arbitrage can be preserved. We extend the price impact setup to coupon-bearing bonds and further show how to implement price impact in an HJM framework. We present pricing examples in the presence of price impact and numerical examples of how impact changes the shape of the term structure. Finally, we show that our approach is applicable by solving an optimal execution problem in interest rate markets with the type of price impact we developed in the paper.

Convertible bond valuation with regime switching

We present a valuation formula for convertible bonds with regime-switching market conditions by decomposing the convertible bond into a coupon-bearing bond and the American-type exchange option. A coupon-bearing bond component is modeled with a four-factor model: a two-factor affine model for the risk-free rate and a two-factor affine model with stochastic volatility for the credit spreads on the coupon-bearing bond component. We also derive a new valuation formula for the American-type exchange option component.

PDE models for the pricing of a defaultable coupon-bearing bond under an extended JDCEV model

We consider a two-factor model for the pricing of a non callable defaultable bond which pays coupons at certain given dates. The model under consideration is the Jump to Default Constant Elasticity of Variance (JDCEV) model. The JDCEV model is an improvement of the reduced form approach, which unifies credit and equity models into a single framework allowing for stochastic and possible negative interest rates. From the mathematical point of view, the valuation involves two partial differential equation (PDE) problems for each coupon. First, we obtain the existence of solution for these PDE problems. In order to solve them, we propose appropriate numerical schemes based on a Crank-Nicolson semi-Lagrangian method for time discretization combined with quadratic Lagrange finite elements for space discretization. Once the numerical solutions of the PDEs are obtained, a post-processing procedure is carried out in order to achieve the value of the bond. This post-processing includes the computation of an integral term which is approximated by using the composite trapezoidal rule. Finally, we present some numerical results for real market bonds issued by different firms in order to illustrate the proper behaviour of the numerical schemes. Moreover, we obtain an agreement between the numerical results from the PDE approach and those ones obtained by applying a Monte Carlo technique and an asymptotic aproximation method.

Convertible Bond Valuation with Regime Switching

We present a valuation formula for convertible bonds with regime-switching market conditions by decomposing the convertible bond into a coupon-bearing bond and the American-type exchange option. A coupon-bearing bond component is modeled with a four-factor model: a two-factor affine model for the risk-free rate and a two-factor affine model with stochastic volatility for the credit spreads on the coupon-bearing bond component. We also derive a new valuation formula for the American-type exchange option component.

Risk Management for Bonds with Embedded Options

This paper examines the behavior of the interest rate risk management measures for bonds with embedded options and studies factors it depends on. The contingent option exercise implies that both the pricing and the risk management of bonds requires modelling future interest rates. We use the Ho and Lee (HL) and Black, Derman, and Toy (BDT) consistent interest rate models. In addition, specific interest rate measures that consider the contingent cash-flow structure of these coupon-bearing bonds must be computed. In our empirical analysis, we obtained evidence that effective duration and effective convexity depend primarily on the level of the forward interest rate and volatility. In addition, the higher the interest rate change and the lower the volatility, the greater the differences in pricing of these bonds when using the HL or BDT models.

Closed-Form Solution for Defaultable Bond Options under a Two-Factor Gaussian Model for Risky Rates Modeling

In this article, the authors provide a closed-form solution for defaultable bond options under a two-factor Gaussian model for risky rates. The key feature of the proposed stochastic model is the introduction of two stochastic dynamics to address the behavior of both risk-free interest rates and credit spreads where the two sources of risk are correlated. Moreover, the model can match exactly the term structure of defaultable interest rates. In order to model credit-sensitive bonds, the authors assume an intensity-based reduced-form framework where the approach based on the recovery of market value is considered and where the recovery rate is one of the model’s input. Under this framework, the authors derive closed formulas for the price of European options having as their underlying both defaultable zero-coupon bonds and coupon-bearing bonds, which the authors assume are approximately log-normally distributed. Moreover, the options are assumed to be knocked out upon a default event of the bond, with zero value to the option holder. An empirical analysis is performed to illustrate the calibration process of the proposed model using financial market data to price call and put options across different credit ratings and different levels of the loss given default. TOPICS:Factor-based models, derivatives, options Key Findings • An option pricing model that considers the dynamic of risky rates under a two-factor Gaussian model where risky rates are the sum of risk-free rates and credit spreads is proposed. This distributional hypothesis is consistent with negative interest rates and benefits from analytical tractability. • Under this framework, closed formulas for the price of European options having both defaultable zero-coupon bonds and coupon-bearing bonds as their underlying are derived under an approximately log-normal distribution assumption. • A detailed two-step calibration procedure is provided. The first step estimates the parameters of the one-factor Hull-White model for the risk-free rates and the second step estimates the parameters of the model’s credit component under a grid of correlation values.

Sensitivities under G2++ model of the yield curve

The two-additive-factor Gaussian model G2[Formula: see text] is a famous stochastic model for the instantaneous short rate. It has functional qualities required in various practical purposes, as in Asset Liability Management and in Trading of interest rate derivatives. Though closed formulas for the prices of various main interest-rate instruments are known and used under the G2[Formula: see text] model, it seems that references for the corresponding sensitivities are not clearly presented over the financial literature. To fill this gap is one of our purposes in the present work. We derive here analytic expressions for the sensitivities of zero-coupon bond, coupon-bearing bonds, portfolio of coupon bearing bonds. The sensitivities under consideration here are those with respect to the shocks linked to the unobservable two-uncertainty shock risk/opportunity factors underlying the G2[Formula: see text] model. As a such, the hedging of a position sensitive to the interest rate by means of a portfolio (in accordance with the market participants practice) becomes easily transparent as just resulting from the balance between the various involved sensitivities.

Maximally Smooth Forward Rate Curves for Coupon Bearing Bonds

Maximally Smooth Forward Rate Curves for Coupon Bearing Bonds

Hedging with Portfolio of Bonds and Swaps Under the G2 Model

The two-additive-factor Gaussian model G2 (which encompasses the famous twofactor Hull-White model) is a stochastic model which describes the instantaneous short rate dynamic. It has functional qualities required in various practical purposes as in Asset Liability Management and in Trading of interest rate derivatives. Recently the second author, has derived analytic expressions for the price sensitivities of zero-coupon bonds, coupon-bearing bonds and interest rate swap contracts. These sensitivities are those with respect to the shocks linked to the unobservable two-uncertainty factors underlying the G2 model. As a such, the hedging of a position sensitive to the interest rate by means of a portfolio by bonds or swaps (in accordance with the market participants practice) becomes easily transparent as resulting from the balance between the various sensitivities involved.Our main purpose here is to fully document this hedging approach portfolio oriented. This is done first by the presentation of the rationale behind the methodology, second by the derivation of the involved analytic tools (i.e. sensitivities), third by the proofs of theoretical foundations, and last by the display of various numerical illustrations showing the effectiveness of our approach. In contrast with various hedging results in the literature, our approach is optimal in the sense that the hedger may know or fix in advance the maximal loss amount which can occur from the hedging approach itself.

Direct Extracting the Implied Forward Yield Curve from the Coupon Bond Prices

It is developed the analytical approach based on the future value concept to directly extract the forward yield curve from the fixed coupon-bearing bond prices. It meets the next no-arbitrage condition: the rates of return on zero-coupon and coupon bonds with the same maturities should be equal. It is shown that there are the essential differences between the estimates of yield curves obtained by the direct and indirect methods. It is suggested to employ only direct methods to extract the spot and forward yield curves: the spot yield curve is directly extracted using the present value concept, and at the same time the forward yield curve is directly extracted using the future value one. It is shown that there is a complementary set of forward rates meeting the no-arbitrage condition side by side with a set of forward rates defined by standard approach.

Interest Rate Sensitivities Under the G2&#43&#43 Model

The two-factor Hull-White (2-HW) model is a famous stochastic model that describes the instantaneous short rate. It has functional qualities required in various practical purposes as in Asset Liability Management and in Trading of interest rate derivatives. The 2-HW is actually a special case of a two-additive-factor Gaussian model G2&#43&#43, which despite of its unpleasant feature of theoretical possibility of negative rates, is quite analytically tractable in that explicit formulas for basic instruments can be readily available.Though closed-formulas for the prices of various main interest-rate instruments are known and used under the G2&#43&#43 model, it seems that references for the corresponding sensitivities are not clearly presented over the financial literature. To fill this gap is among of our purposes in the present work.So we derive here analytic expressions for the sensitivities of zero-coupon bond, coupon-bearing bonds, forward rate agreement and interest rate swap contracts. The sensitivities under consideration here are those with respect to the shocks linked to the unobservable two-uncertainty shocks risk/opportunity factors underlying the G2&#43&#43 model. As a such, the hedging of a position sensitive to the interest rate by means of a portfolio (in accordance with the market participants practice) becomes easily transparent as just resulting from the balance between the various involved sensitivities. MatLab codes of our results are also provided here.

VALUATION OF GUARANTEED ANNUITY OPTIONS IN AFFINE TERM STRUCTURE MODELS

We propose three analytic approximation methods for numerical valuation of the guaranteed annuity options in deferred annuity pension policies. The approximation methods include the stochastic duration approach, Edgeworth expansion, and analytic approximation in affine diffusions. The payoff structure in the annuity policies is similar to a quanto call option written on a coupon-bearing bond. To circumvent the limitations of the one-factor interest rate model, we model the interest rate dynamics by a two-factor affine interest rate term structure model. The numerical accuracy and the computational efficiency of these approximation methods are analyzed. We also investigate the value sensitivity of the guaranteed annuity option with respect to different parameters in the pricing model.

On the term structure of lending interest rates when a fraction of collateral is recovered upon default

This article provides an arbitrage-free model to evaluate the term structure of lending interest rates when a fraction of collateral is recovered upon default of the borrower. Unlike the previous literature, we assume that the value of the collateral asset fluctuates over time with certain correlation to the risk-free interest rate as well as the default hazard rate of the borrower. It is shown that the bank loan is a sum of holding a coupon-bearing bond and selling a put option, both being callable upon default. A Gaussian model is considered, as a special case, to derive an analytic expression of the appropriate lending interest rates, and some numerical example is given to demonstrate that bank loans exhibit different properties from corporate bonds.

Valuation of guaranteed annuity conversion options

In this note we introduce a theoretical model for the pricing and valuation of guaranteed annuity conversion options associated with certain deferred annuity pension-type contracts in the UK. The valuation approach is based on the similarity between the payoff structure of the contract and a call option written on a coupon-bearing bond. The model makes use of a one-factor Heath–Jarrow–Morton framework for the term structure of interest rates. Numerical results are investigated and the sensitivity of the price of the option to changes in the key parameters is also analyzed.

Mortality derivatives and the option to annuitise

Most US-based insurance companies offer holders of their tax-sheltered savings plans (VAs), the long-term option to annuitise their policy at a pre-determined rate over a pre-specified period of time. Currently, there is approximately one trillion dollars invested in such policies, with guaranteed annuitisation rates, in addition to any guaranteed minimum death benefit. The insurance company has essentially granted the policyholder an option on two underlying stochastic variables; future interest rates and future mortality rates. Although the (put) option on interest rates is obvious, the (put) option on mortality rates is not. Motivated by this product, this paper attempts to value (options on) mortality-contingent claims, by stochastically modelling the future hazard-plus-interest rate. Heuristically, we treat the underlying life annuity as a defaultable coupon-bearing bond, where the default occurs at the exogenous time of death. From an actuarial perspective, rather than considering the force of mortality (hazard rate) at time t for a person now age x, as a number μx(t), we view it as a random variable forward rate μ˜x(t), whose expectation is the force of mortality in the classical sense ( μx(t)=E[μ˜x(t)]). Our main qualitative observation is that both mortality and interest rate risk can be hedged, and the option to annuitise can be priced by locating a replicating portfolio involving insurance, annuities and default-free bonds. We provide both a discrete and continuous-time pricing framework.

Mortality Derivatives and the Option to Annuitize

Most U.S.-based insurance companies offer holders of their tax-sheltered savings plans (VAs), the long-term option to annuitize their policy at a pre-determined rate over a pre-specified period of time. Currently, there is approximately one trillion dollars invested in such policies, with guaranteed annuitization rates, in addition to any guaranteed minimum death benefit. The insurance company has essentially granted the policyholder an option on two underlying stochastic variables; future interest rates and future mortality rates. Although the (put) option on interest rates is obvious, the (put) option on mortality rates is not. Motivated by this product, this paper attempts to value (options on) mortality-contingent claims, by stochastically modelling the future hazard-plus-interest rate. Heuristically, we treat the underlying life annuity as a defaultable coupon-bearing bond, where the default occurs at the exogenous time of death. From an actuarial perspective, rather than considering the force of mortality (hazard rate) at time-t for a person now age-x, as a number, we view it as a random variable forward rate,whose expectation is the force of mortality in the classical sense. Our main qualitative observation is that both mortality and interest rate risk can be hedged, and the option to annuitize can be priced by locating a replicating portfolio involving insurance, annuities and default-free bonds. We provide both a discrete and continuous-time pricing framework.

Estimating and Testing an Exponential-Affine Term Structure Model by Nonlinear Filtering

In general, the interest rate is not directly observable in the financial market. Short term interest rates are quoted in the money market for maturities up to approximately one year, but longer term interest rates are traded only indirectly through the bond market. In this paper the spot interest rate is described by a bivariate stochastic differential equation state space model that gives rise to an exponential-affine term structure model. We propose a new maximum likelihood method for estimating parameters and interest rates in stochastic differential equations from observed coupon-bearing bond prices. The method utilizes continuous discrete second order nonlinear filtering techniques. As a preliminary analysis the method is applied to a cross-section of time series of Danish government bond prices.

American option pricing in Gauss–Markov interest rate models

In the context of Gaussian non-homogeneous interest-rate models, we study the problem of American bond option pricing. In particular, we show how to efficiently compute the exercise boundary in these models in order to decompose the price as a sum of a European option and an American premium. Generalizations to coupon-bearing bonds and jump-diffusion processes for the interest rates are also discussed.