Coupon Bonds Research Articles

Method of Reducing Dimension of Space Variables in Multi-dimensional Black-Scholes Equations

We study a method of reducing space dimension in multi-dimensionalBlack-Scholes partial differential equations as well as in multi-dimensional parabolic equations. We prove that a multiplicative transformation of space variables in the Black-Scholes partial differential equation reserves the form of Black-Scholes partial differential equation and reduces the space dimension. We show that this transformation can reduce the number of sources of risks by two or more in some cases by giving remarks and several examples of financial pricing problems. We also present that the invariance of the form of Black-Scholes equations is based on the invariance of the form of parabolic equation under a change of variables with the linear combination of variables. Keywords : Black-Scholes equations;Multi-dimensional; Reducing dimension; Options; Foreign currency strike price; Basket option; Foreign currency option; Zero coupon bond derivative. 2010 Mathematics Subject Classification: 35K15, 91B24.

Analytical pricing of American Put options on a Zero Coupon Bond in the Heath–Jarrow–Morton model

We study the optimal stopping problem of pricing an American Put option on a Zero Coupon Bond (ZCB) in Musiela’s parametrization of the Heath–Jarrow–Morton (HJM) model for forward interest rates.First we show regularity properties of the price function by probabilistic methods. Then we find an infinite dimensional variational formulation of the pricing problem by approximating the original optimal stopping problem by finite dimensional ones, after a suitable smoothing of the payoff. As expected, the first time the price of the American bond option equals the payoff is shown to be optimal.

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.

Pricing currency options under double exponential jump diffusion in a Markov-modulated HJM economy

Extending the framework of Amin and Jarrow (J Int Money Financ 10:310–329, 1991) and Bo et al. (Insur Math Econ 46:461–469, 2010), this study provides a theoretical exploration of currency options pricing under the presence of interest-rate regime shifts and exchange-rate asymmetric jumps. Evidence of interest-rate regime shifts inferred from UK and US zero coupon bond yields provides support for the regime-switching specifications which we reflect upon the domestic and foreign forward rates. Results of statistical tests conducted on JPY/USD and EUR/USD FX rates provide further support the rationale behind using a double exponential jump diffusion process within a Markov modulated Heath–Jarrow–Morton economy. Our numerical results suggest that, the pricing performance of our model is closely comparable to the Bo-Wang-Yang model for at-the-money options, yet yields improvements in percentage root mean errors for in-the-money options.

Direct Estimating Price of a Defaultable Zero-Coupon Bond Using Conception of Continuous Coupon Bond

In this paper, using a conception of continuous coupon bond with continuous accrual of coupons on simple fixed rate for pricing a risky zero-coupon bond is considered. It is shown that only employing this conception allows obtaining explicit equation for price of risky zero-coupon bond from fixed coupon bond prices without any assumption about default process and providing continuous monitoring default events despite periodicity of bond cash flows that improves estimation of credit risk. To apply the conception, it is proposed a simple condition for inversion of discrete bond into continuous coupon one. It is shown that this inversion is possible only for the following recovery assumption: a recovery rate is a part of the present value of remaining cash flows that will not be paid due to default but not for the fractional recovery of par assumption. Examples of calculating the implied survival probabilities and the credit spreads for Ukrainian Eurobonds are given.

Bond Option Pricing using the Vasicek Short Rate Model

An option is a financial instrument that allows the holder to buy or sell an underlying security in the future at an agreed strike or price set today. Many options are priced under the assumption of constant interest rates as seen in the Black-Scholes (1973) model. In interest rate markets however the underlying security is an interest rate, which cannot be assumed constant. Likewise bond markets have a similar requirement.In what follows the assumption of a constant interest rate is relaxed. Bond option pricing using the Vasicek short rate model is examined in such a way that the methodology could be applied to any short rate model such as the classical Hull-White model (1990a).Firstly we discuss the preliminaries, namely numeraires and measures, where it can be seen that a careful choice of numeraire can simplify option calculations. Secondly we summarize the Vasicek short rate process and a change of numeraire to the terminal-forward measure is outlined, which simplifies bond option pricing calculations. Thirdly we review both pure discount and coupon bond pricing. Fourthly bond option pricing formulae are derived and Jamshidian's Trick outlined.Finally in conclusion practical implementation considerations and model extensions are discussed. The aim of this paper is to provide a general overview of option pricing using short rate models, using the Vasicek model as an important case study.

International Yield Curves and Principal Components Selection Techniques: An Empirical Assessment

Using a common database, we provide a controlled empirical comparison of recently-proposed principal component (PC) methods for selecting a combination of common and local factors that characterize the joint dynamics of multi-country term structures. We build a database of daily Treasury yield curves for U.S., Germany, U.K. and Japan, using common criteria to filter coupon bond data, to ensure liquidity, and to interpolate the discount function. We then estimate each proposed PC method for all subgroups of these countries, using both yield levels and yield differences at weekly frequency. We find, in general, that the proposed methods do not agree with one another on the preferred combination of common and/or local factors. We identify the explained variability decision criterion as an important source of this lack of agreement and recommend consideration of alternative statistical model selection techniques for the purpose of identifying common and local yield curve factors in international data.

Historical VaR for Bonds - A New Approach

Bonds historical returns cannot be used directly to compute VaR because the maturities of returns implied by the historical prices do not have the relevant maturities to compute VaR. Given the so-called pull-to-par in bonds, with return volatilities necessarily decreasing with diminishing time-to-maturity,direct use of historical returns would lead to overestimation of the true VaR. Market practice deals with the problem of computing VaR for portfolios of bonds or mixed portfolios with cumbersome methods of cash-ow mappings. In this paper we propose a new approach. We develop a technique to adjust bonds historical prices and extract from them an adjusted history of returns, that can be used directly to compute historical VaR for bonds or bond portfolios. We illustrate the method using one concrete traded market zero-coupon bond, but the simplicity of the method makes this enough to the reader to understand how it would work with coupon bonds, portfolios of bonds or mixed portfolios.

Analytical pricing of defaultable discrete coupon bonds in unified two-factor model of structural and reduced form models

Pricing formulae for defaultable corporate bonds with discrete coupons (under consideration of the government taxes) in the united two-factor model of structural and reduced form models are provided. The aim of this paper is to generalize the two-factor structural model for defaultable corporate discrete coupon bonds (considered in [1]) into the unified model of structural and reduced form models. In our model the bond holders receive the stochastic coupon (which is the discounted value of a predetermined value at the maturity) at predetermined coupon dates and the face value (debt) and the coupon at the maturity as well as the effect of government taxes which are paid on the proceeds of an investment in bonds is considered. The expected default event occurs when the equity value is not sufficient to pay coupon or debt at the coupon dates or maturity and the unexpected default event can occur at the first jump time of a Poisson process with the given default intensity provided by a step function of time variable. We provide the model and pricing formula for equity value and using it calculate expected default barrier. Then we provide pricing model and formula for defaultable corporate bonds with discrete coupons and consider its duration.

Nonlinear Functional Analysis of Boundary Value Problems 2013

and Applied Analysis 3 solve a class of boundary value problems for second-order singular differential equations. (iii) In the paper titled “Existence and uniqueness of solution to nonlinear boundary value problems with sign-changing Green’s function,” by using the cone theory and the Banach contraction mapping principle, some existence and uniqueness results are established for a nonlinear higher-order differential equation boundary value problem with signchanging Green’s function. (iv) In the paper titled “Onboundedness and attractiveness of nonlinear switched delay systems,” the authors study the boundedness and attractiveness problems for a class of nonlinear switched delay systems. Some sufficient conditions are established to guarantee the system’s boundedness. The authors also work out the region where the solution will remain and the relationship between the initial function and the bounded region. (v) In the paper titled “Unbounded solutions of asymmetric oscillator,” the authors establish sufficient conditions for the existence of unbounded solutions of a nonlinear differential equation with a continuous, bounded, and periodic source function. (6) Stochastic Differential Equation Models (i) In the paper titled “A stochastic string with a compound poisson process,” the authors introduce a compound Poisson process with a constant jump intensity and random jump-size to capture information burst and the resulting discontinuous path and derive the no-arbitrage condition on the drift of instantaneous forward rates in the compound model. With the proposedmodel, the impact of random jump on the price of the zero coupon bond is shown in the paper. (ii) In the paper titled “Equilibrium asset and option pricing under jump diffusion model with stochastic volatility,” the authors study the equity premium and option pricing under a jump-diffusion model with stochastic volatility. The pricing kernel is established and the exact expression for the option price is derived by using the Fourier transformation method. (7) Fixed-Point Theory (i) In the paper titled “Generalizations of fixed-point theorems of Altman and Rothe types,” the authors present some extensions of the Altman and Rothe type fixed-point theorems. Some new fixed-point theorems for continuous operators are established by use of the theory of topological degree. (ii) In the paper titled “Commonfixed-points for weak contractive mappings in ordered metric space with applications,” the authors establish some new common fixed-point theorems satisfying a weak contractive condition in the framework of partially ordered metric spaces. Yong Hong Wu Lishan Liu Benchawan Wiwatanapataphee Shaoyong Lai Submit your manuscripts at http://www.hindawi.com Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Mathematics Journal of Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Mathematical Problems in Engineering Hindawi Publishing Corporation http://www.hindawi.com Differential Equations International Journal of Volume 2014 Applied Mathematics Journal of Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Probability and Statistics Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Journal of Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Mathematical Physics Advances in Complex Analysis Journal of Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Optimization Journal of Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Combinatorics Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 International Journal of Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Operations Research Advances in

An Option Pricing Analysis of Exotic Bonus Certificates—The Case of Bonus Certificates PLUS

In this paper we introduce a new financial product named Bonus Certificates PLUS and we provide detailed descriptions of the product specifications. We show that the payoff of a Bonus Certificates PLUS can be duplicated by the combination of a long position in the underlying asset, a short position on zero coupon bonds, a long position in down-and-out call options on the underlying asset, a long position in down-and-in call options on the underlying asset, and a long position in the down-and-out put options. We develop a pricing formula to price the certificates. A sample of ten Bonus Certificates PLUS outstanding on December 2007 is presented as an example to examine how well the model fits empirical data. The profitability in the primary market is examined and the results show that issuing Bonus Certificates PLUS is a profitable business. The results are in line with previous studies pricing other structured products. Finally, we simulate and present the sensitivities of the certificates to changes in different pricing inputs.

Analytical Pricing of Defaultable Discrete Coupon Bonds in Unified Model of Structural and Reduced Form Models (Two-Factor Model)

Pricing formulae for defaultable corporate bonds with discrete coupons under consideration of the government taxes in the united model of structural and reduced form models are provided. The aim of this paper is to generalize the comprehensive structural model for defaultable fixed income bonds (considered in [1]) into a comprehensive unified model of structural and reduced form models. Here we consider the one factor model and the two factor model. In the one factor model the bond holders receive the deterministic coupon at predetermined coupon dates and the face value (debt) and the coupon at the maturity as well as the effect of government taxes which are paid on the proceeds of an investment in bonds is considered under constant short rate. In the two factor model the bond holders receive the stochastic coupon (discounted value of that at the maturity) at predetermined coupon dates and the face value (debt) and the coupon at the maturity as well as the effect of government taxes which are paid on the proceeds of an investment in bonds is considered under stochastic short rate. The expected default event occurs when the equity value is not enough to pay coupon or debt at the coupon dates or maturity and unexpected default event can occur at the first jump time of a Poisson process with the given default intensity provided by a step function of time variable. We consider the model and pricing formula for equity value and using it calculate expected default barrier. Then we provide pricing model and formula for defaultable corporate bonds with discrete coupons and consider its duration and the effect of the government taxes.

General Properties of Solutions to Inhomogeneous Black-Scholes Equations with Discontinuous Maturity Payoffs and Application

We provide representations of solutions to terminal value problems of inhomogeneous Black-Scholes equations and studied such general properties as min-max estimates, gradient estimates, monotonicity and convexity of the solutions with respect to the stock price variable, which are important for financial security pricing. In particular, we focus on finding representation of the gradient (with respect to the stock price variable) of solutions to the terminal value problems with discontinuous terminal payoffs or inhomogeneous terms. Such terminal value problems are often encountered in pricing problems of compound-like options such as Bermudan options or defaultable bonds with discrete default barrier, default intensity and endogenous default recovery. Our results are applied in pricing defaultable discrete coupon bonds.

The Value of Gold as a Super-Sovereign Zero-Coupon Bond

In this paper, gold is investigated as a super- sovereign zero- coupon bond. A structural relationship between the gold and the observed yields on long maturity inflation indexed bonds is derived under the assumption that the gold price reflects the long run inflation level. This relationship can be interpreted as defining the fair value of the gold in an economy model where inflation indexed bonds are traded. A study of monthly real time decompositions of gold price into their fair value and risk premium components is directed by taking the theory to a data set spanning the period from April 2007 to June 2013. It is shown empirically that a rise of the gold price is contemporaneously associated with a drop of the inflation indexed bond yields and a rise of risk premium in appreciation of the gold.

Calculations in the Hull-White Model

In this note many trivial (and some advanced) calculations in the Hull-White interest rate model are performed. Especially the theta function is derived. This function ensures that prices of zero coupon bonds in the model fit market values of zero coupon bonds. In addition, a section explaining how to estimate the model using maximum likelihood is included.

A Review of the Dynamic Default-Free Term Structure Models

Enormous progress has been made by academics and fipractitioners alike in modeling the dynamics of the term structure of inte rest rates in the past 35 years. This paper extends Yan (2001). The dynamics of the term structure of interest rates are critical in assessing prices and hedging portfolios of fixed-income derivative instruments. This paper reviews the theoretical development of t he dynamic models of the default-free term structure and their applications in pricing in terest rate options. Equilibrium models and their multifactor extensions are examined. Thes e models offer the economic intuition linking the term structure to economic fundamentals . These models also constitute a framework of the arbitrage models which price inter est rate derivatives using the market prices of bonds. Recent studies expand this framewo rk, directly model observable market rates, incorporate an internally consistent correla tion structure, use pricing factors as observable portfolios of zero-coupon yields acting as state variables, offer a discrete time setting with recursive closed form solutions for ze ro coupon bonds, have unconstrained state dependence of the market prices of risk, buil d in smoothness restrictions on the factor loadings, and are driven by state vectors wi th unspanned macro risks.

Stochastic durations, the convexity effect, and the impact of interest rate changes

This article shows that the equilibrium models of bond pricing do not preclude arbitrage opportunities caused by convexity. Consequently, stochastic durations derived from these models are limited in their ability to act as interest rate risk measures. The research of the present article makes use of an intertemporal utility maximization framework to determine the conditions under which duration is an adequate interest rate risk measure. Additionally, we show that zero coupon bonds satisfy those equilibrium conditions, whereas coupon bonds or bond portfolios do not as a result of the convexity effect. The results are supported by empirical evidence, which confirms the influence of convexity on the deviation of coupon bond returns from equilibrium.

Libor Market Model with Stochastic Basis - Calibration Using OIS Yield and Money Market Basis Spreads

Usually a Libor Market model with a stochastic basis as speci ed for instance by Mercurio, F. (2009) lacks of a suitable calibration since there are not enough market quotes available. To this end we suggest to take a low parametric model which essentially is calibrated to the current OIS curve. Then, for the forwards we use a Libor Market model for which enough quoted instruments such as caps, swaptions, CMS or CMS spread options are available. The dependency of the OIS curve and the Libor model is given by the money market basis spread.The idea for taking a low factor model for the OIS dynamic is presented in Mercurio, F. and Xie, Z. (2012). But they also propose to use a low factor model for the forwards. In this note we combine a low factor dynamic for the OIS zero coupon bonds with the framework presented in Mercurio, F. (2009) for a multi-curve Libor Market model by a somewhat di erent modelling approach.We show how to obtain all model parameters and for the rst time an example of a calibration of a multi-curve Libor Market model using the current market quotes.

Risky Coupon Bond Option Pricing: Intensity Approach

In this paper we revisit Jarrow and Turnbull’s seminal paper on risky zero-bond option pricing and present two its extensions. Firstly, we are generalizing the zero-coupon bond option formula to coupon bond option formula while still preserving the analytical tractability. Secondly, we allow for intensity term-structure instead of just constant default intensity parameter. These two extensions result in comprehensive pricing framework for options linked on defaultable bonds.

Sovereign Recovery Schemes: Discounting and Risk Management Issues

We consider some pricing and risk management issues related to defaultable bonds, in the context of sovereign debt default and restructuring. Standard recovery schemes such as fractional recovery of market value, of Treasury and of face value are investigated: we discuss their consistency with market practice both from a pricing and a risk management perspective. We also pay attention to the tradable basic instruments such as defaultable discount bonds or IOs/POs that are the building blocks of traded level coupon bonds. Model-free pricing formulas are provided. Whatever the recovery framework, bond pricing formulas involve similar ingredients, such as par rates and defaultable level annuities. We also show that the fractional recovery of par, our preferred approach from an economical point of view, involves two discount curves, one for principal payments and one for coupon payments, a departure from the simplest bootstrapping and pricing engines.