Bond Option Pricing Research Articles

Pricing bond options under a Markovian regime-switching Hull–White model

In this paper, we investigate the valuation of bond options under a Markovian regime-switching Hull–White model, where both the mean-reverting level and the volatility of the interest rate are modulated by a continuous-time, finite-state Markov chain. Using techniques of measure changes and the inverse Fourier transform, we obtain an integral representation for the pricing formula of a standard European option on a zero-coupon bond. Numerical results for the prices and implied volatilities of bond options arising in our model are given in a two-regime case.

Bonds and Options in Exponentially Affine Bond Models

In this article we apply the Flesaker–Hughston approach to invert the yield curve and to price various options by letting the randomness in the economy be driven by a process closely related to the short rate, called the abstract short rate. This process is a pure deterministic translation of the short rate itself, and we use the deterministic shift to calibrate the models to the initial yield curve. We show that we can solve for the shift needed in closed form by transforming the problem to a new probability measure. Furthermore, when the abstract short rate follows a Cox–Ingersoll–Ross (CIR) process we compute bond option and swaption prices in closed form. We also propose a short-rate specification under the risk-neutral measure that allows the yield curve to be inverted and is consistent with the CIR dynamics for the abstract short rate, thus giving rise to closed form bond option and swaption prices.

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.

Convergence analysis of power penalty method for American bond option pricing

This paper is concerned with the convergence analysis of power penalty method to pricing American options on discount bond, where the single factor Cox---Ingrosll---Ross model is adopted for the short interest rate. The valuation of American bond option is usually formulated as a partial differential complementarity problem. We first develop a power penalty method to solve this partial differential complementarity problem, which produces a nonlinear degenerated parabolic PDE. Within the framework of variational inequalities, the solvability and convergence properties of this penalty approach are explored in a proper infinite dimensional space. Moreover, a sharp rate of convergence of the power penalty method is obtained. Finally, we show that the power penalty approach is monotonically convergent with the penalty parameter.

Numerical pricing of financial derivatives using Jain’s high-order compact scheme

Purpose: This paper develops new fast and accurate computational schemes for pricing European and American bond options under generalised Chan-Karoyli-Longstaff-Sanders term structure models. Methods: We use high-order compact discretisations of the pricing equations and an operator splitting method for American options. Results: Highly accurate numerical solutions can be computed using relatively coarse grid sizes and numerical solutions exhibiting fourth-order convergence are obtained for bond and bond option prices. The scheme is also stable and efficient for pricing financial problems with time dependent parameters. Conclusions: The new schemes are efficient alternatives to schemes based on the Crank-Nicolson discretisation for the pricing of interest rate derivatives.

Enhancing Quasi-Monte Carlo Methods by Exploiting Additive Approximation for Problems in Finance

The valuation of many financial securities can be formulated as high-dimensional integrals for which quasi-Monte Carlo (QMC) methods are becoming indispensable numerical tools. It is known that some typical high-dimensional problems in finance and in many other fields have a strong additive nature. This paper investigates the enhancement of QMC methods by exploiting the additive approximation in combination with suitable transformation methods. To this end, we study (a) the approximation of a multivariate function by a sum of one-dimensional functions; and (b) the effective use of the possible strong additive property inherent in finance problems to enhance QMC. For problem (a), we establish a relationship between the approximation quality resulting from anchored decomposition and the global sensitivity indices. For problem (b), we propose new methods for efficiency improvement—the additive weighted uniform sampling and the additive control variate, which exactly exploit the strong additive structure. The required importance distribution and control variate are found constructively based on the anchored additive approximation with a suitably chosen anchor. Numerical examples on bond valuation and option pricing illustrate that the proposed methods reduce the variance by huge factors in QMC. Transformation methods play a crucial role, since they may enhance the degree of additivity of a function. Traditional methods, such as Brownian bridge and principal component analysis, do not necessarily work well, whereas methods that take into account the underlying functions are robust and powerful.

Efficient calibration of the Hull White model

SUMMARY In this paper we study the calibration of the Hull White interest rate model for the pricing of Bermudan swaptions and Bermudan bond options. A common modelling approach for the calibration of the Hull White model is to choose the model parameters such that market prices of corresponding European derivatives are replicated by the model. This requires that a multidimensional non-linear optimization problem has to be solved. The focus of this paper lies in the efficient formulation and numerical solution of this optimization problem. We investigate the numerical properties of the iterative solution of the optimization problem by means of a Gauss Newton and an Adjoint Broyden quasi-Newton method. Required derivatives of the objective function are evaluated by techniques of Automatic Differentiation. The algorithms are benchmarked to Minpack's general purpose solver HYBRD. Numerical results for the calibration of long-term Bermudan swaptions are reported. Copyright © 2011 John Wiley & Sons, Ltd.

Fast approximations of bond option prices under CKLS models

A new computational method for approximating prices of zero-coupon bonds and bond option prices under general Chan–Karolyi–Longstaff–Schwartz models is proposed. The pricing partial differential equations are discretized using second-order finite difference approximations and an exponential time integration scheme combined with best rational approximations based on the Carathéodory–Fejér procedure is employed for solving the resulting semi-discrete equations. The algorithm has a linear computational complexity and provides accurate bond and European bond option prices. We give several numerical results which illustrate the computational efficiency of the algorithm and uniform second-order convergence rates for the computed bond and bond option prices.

The sensitivity analysis of propagator for path independent quantum finance model

Quantum finance successfully implements the imperfectly correlated fluctuation of forward interest rates at different maturities, by replacing the Wiener process with a two-dimensional quantum field. Interest rate derivatives can be priced at a more realistic value under this new framework. The quantum finance model requires three main ingredients for pricing: the initial forward interest rates, the volatility of forward interest rates, and the correlation of forward interest rates at different maturities. However, the hedging strategy only focused on fluctuation of forward interest rates. This hedging method is based on the assumption that the propagator, the covariance of forward interest rates, has an ergodic property. Since inserting the propagator is the main characteristic that distinguishes quantum finance from the Libor market model (LMM) and the Heath, Jarrow and Morton (HJM) model, understanding the impact of propagator dynamics on the price of interest rate derivatives is crucial. This research is the first step in developing a hedge strategy with respect to the evolution of the propagator. We analyze the dynamics of the propagator from Libor futures data and the integrated propagator from zero-coupon bond rate data. Then we study the sensitivity of the implied volatility of caplets and swaptions according to the three dominant dynamics of the propagator, and the change of the zero-coupon bond option price according to the two dominant dynamics of the integrated propagator.

The Normal Class of Arbitrage-Free Spot-Rate Models - Implication and Implementation

In this paper we first show how to determine the T-forward adjusted risk-measure using the concept of fundamental solution to linear PDE’s. After that, relying on Fourier transformation we derive bond-and bond-option prices for the Extended Vasicek model from Hull and White (1990) and the Quadratic Interest Rate model. With respect to the Quadratic Interest Rate model we succeed in carrying the analysis much further than Jamshidian (1996).A special discrete time model - which in some cases is appropriate for the Quadratic Interest Rate model - is also derived.The last part of the paper analyse Monte Carlo techniques in connection with spot-rate models with a time-dependent drift. We also introduce a method - using the concept of forward induction - that constrain the Monte Carlo simulated spot-rate process for the matching of the initial yield-curve. For the pricing of path-dependent contingent claim, we deduce that, even though Monte Carlo is the natural method to use, it might not be the most efficient one - at least not when the spot-rate is Markovian.

Face Value Convergence for Stochastic Bond Price Processes: A Note on Merton's Partial Equilibrium Option Pricing Model

This paper considers Merton's (1973) model for partial equilibrium bond option pricing when stochastic bond price processes are involved. A log-normal process with a stochastic drift is suggested that allows the price of a pure discount bond to converge to its face value upon maturity. The stochastic process for the instantaneous short rate implicit in the bond price dynamics is identified. A necessary condition for our approach to be consistent with the arbitrage-free martingale pricing results of Harrison and Kreps (1979) is that the excess holding period bond return and the standard deviation of the bond return are both continuously differentiable with respect to time.

Simulation-Based Estimation of Contingent-Claims Prices

A new methodology is proposed to estimate theoretical prices of financial contingent claims whose values are dependent on some other underlying financial assets. In the literature, the preferred choice of estimator is usually maximum likelihood (ML). ML has strong asymptotic justification but is not necessarily the best method in finite samples. This paper proposes a simulation-based method. When it is used in connection with ML, it can improve the finite-sample performance of the ML estimator while maintaining its good asymptotic properties. The method is implemented and evaluated here in the Black-Scholes option pricing model and in the Vasicek bond and bond option pricing model. It is especially favored when the bias in ML is large due to strong persistence in the data or strong nonlinearity in pricing functions. Monte Carlo studies show that the proposed procedures achieve bias reductions over ML estimation in pricing contingent claims when ML is biased. The bias reductions are sometimes accompanied by reductions in variance. Empirical applications to U.S. Treasury bills highlight the differences between thebondpricesimpliedbythesimulation-basedapproachandthosedeliveredbyML.Some consequences for the statistical testing of contingent-claim pricing models are discussed. (JEL C11, C15, G12)

Pricing Stock and Bond Options in Incomplete Markets

Modern options theory assumes that securities markets are complete and processes are known. But how should options be priced when markets are not complete and price processes are not known? The maximum-entropy framework, which formalizes the efficient market hypothesis, offers process-free option valuation in incomplete markets. This article reviews the gamma model for pricing stock options and the beta model for pricing bond options, both obtained in the maximum-entropy framework. Both models are as convenient as the Black–Scholes model, but unlike the Black–Scholes model they are valid in incomplete markets and do not restrict prices, interest rates, or volatilities to any specific paths. The gamma and beta models are both less restrictive and more accurate than alternative option valuation models. <b>TOPICS:</b>Portfolio theory, factor-based models, statistical methods

The CARMA Interest Rate Model

In this paper we present a multi-factor continuous-time autoregressive moving-average (CARMA) model for the short and forward interest rates. This models is able to present an adequate statistical description of the short and forward rate dynamics. We show that this is a tractable term structure model and provide closed-form solutions to bond prices, yields, bond option prices, and the term structure of forward rate volatility. We demonstrate the capabilities of our model by calibrating it to a panel of spot rates and the empirical volatility of forward rates simultaneously, making the model consistent with both the spot rate dynamics and forward rate volatility structure.

Spanned Stochastic Volatility in Bond Markets: A Reexamination of the Relative Pricing between Bonds and Bond Options

This paper reexamines the issue of unspanned stochastic volatility (USV) in bond markets and the puzzle of the poor relative pricing between bonds and bond options. I make a distinction between the weak USV and the strong USV scenarios, and analyze the evidence for each of them. I argue that the poor bonds/options relative pricing in the extant literature is not necessarily evidence for the strong USV scenario, and show that a maximally flexible 2-factor quadratic-Gaussian model (a non-USV model) estimated without bond options data can capture much of the movement in bond option prices. Dropping the positive-definiteness requirement for nominal interest rates and adopting regularized estimations turn out to be important in obtaining sensible results.

Quantum finance Hamiltonian for coupon bond European and barrier options

Coupon bond European and barrier options are financial derivatives that can be analyzed in the Hamiltonian formulation of quantum finance. Forward interest rates are modeled as a two-dimensional quantum field theory and its Hamiltonian and state space is defined. European and barrier options are realized as transition amplitudes of the time integrated Hamiltonian operator. The double barrier option for a financial instrument is "knocked out" (terminated with zero value) if the price of the underlying instrument exceeds or falls below preset limits; the barrier option is realized by imposing boundary conditions on the eigenfunctions of the forward interest rates' Hamiltonian. The price of the European coupon bond option and the zero coupon bond barrier option are calculated. It is shown that, is general, the constraint function for a coupon bond barrier option can -- to a good approximation -- be linearized. A calculation using an overcomplete set of eigenfunctions yields an approximate price for the coupon bond barrier option, which is given in the form of an integral of a factor that results from the barrier condition times another factor that arises from the payoff function.

A Comparison of Fixed Income Valuation Models: Pricing and Econometric Analysis of Interest Rate Derivatives

This study compares continuous-time stochastic interest rate and stochastic volatility models of interest rate derivatives, examining these models across several dimensions: different classes of models, factor structures, and pricing algorithms. We consider a broader universe of pricing models, using improved econometric and numerical methodologies. We establish several criteria for model quality that are motivated by financial theory as well as practice: realism of the assumed stochastic process for the term structure, consistency with no-arbitrage or financial market equilibrium, consistency with financial practice, parsimony, as well as computational efficiency. A model which scores well along these grounds will also exhibit superior pricing performance with regard to traded interest rate options. This helps resolve the controversies over the stochastic process for yield curve dynamics, the models that best manage and measure interest rate risk, and theories of the term structure that are supported by empirical results. We perform econometric experiments at three levels: the short rate, bond prices, as well as interest rate derivatives. We extend CKLS (1992) to a broader class of single factor spot rate models and international interest rates. We find that a single-factor general parametric model (1FGPM) of the term structure, with non-linearity in the drift function, better captures the time series dynamics of US 30 Day T-Bill rates. The 1FGPM not only forecasts interest rate changes out-of-sample better relative to other parametric models, but also relative to the non-parametric model of Jiang (1998). Finally, our results vary greatly across international markets. Building upon the work of Longstaff and Schwartz (1992), we perform a statistical analysis of the U.S. default-free term structure over the period 4:1964 to 10:1997. We utilize a constant correlation multivariate GARCH principal components analysis (CCM-PCA), and identify at least three factors associated with traditional measures of risk in the fixed income literature (level, slope, and curvature) that capture 98% of the variation in the default-free term structure. We perform tests of various term structure models on US Treasury bonds, comparing a two factor Cox-Ingersoll-Ross (2FCIR) model with a multi-layer perceptron neural network approach (MLP-ANN), in pricing and hedging discount bonds. We find that while the MLP-ANN can better fit bond prices in-sample, the 2F-CIR model is superior in hedging against unanticipated changes in the short rate and its volatility. Furthermore, we find the 2FCIR model to perform favorably in comparison to the CCM-PCA, MLP-ANN, as well as the 1FGPM in forecasting bond yield changes. Finally, we compare various interest rate bond option pricing models, in their ability to price interest rate derivatives and manage and interest rate risk. We compare three approaches to pricing interest rate derivatives: spot rate (e.g., CIR), forward-rate (i.e., HJM), and non-parametric models (e.g., multivariate kernel estimation.) This is extended to a broader factor structure. While the best model in terms of mean square error (MSE) is the non parametric (MNWK) model, the 3 factor jump diffusion (3FGJD) model performs best among parametric models. In hedging analysis, while these preferred models still outperform within each grouping, the non parametric model is no longer the best performing model, while the 2FCIR is the best model in hedging options in terms of MSE.

Pricing of path-dependent American options by Monte Carlo simulation

In this paper, we evaluate floating-rate bond options, a variant of path-dependent American options, by Monte Carlo simulation. Assuming that the underlying state variable is Markovian, we show that the price of a floating-rate bond option satisfies a dynamic programming equation. The continuation value in the dynamic programming problem is represented by a conditional expectation. It is shown that the conditional expectation can be transformed to an unconditional expectation, using the Malliavin calculus, which in turn enables us to evaluate the price of the floating-rate bond option by Monte Carlo methods. Some numerical examples are given to demonstrate the usefulness of our method.

A NEW FINITE ELEMENT METHOD FOR PRICING OF BOND OPTIONS UNDER TIME INHOMOGENEOUS AFFINE TERM STRUCTURE MODELS OF INTEREST RATES

In this paper we propose a new finite element method for pricing of bond options under time inhomogeneous one-factor affine models of short interest rates: the Hull–White model and the extended CIR model. The stability and weak convergence are established. Numerical results are presented to examine the method and to compare the calibrated models.

Feynman perturbation expansion for the price of coupon bond options and swaptions in quantum finance. I. Theory

European options on coupon bonds are studied in a quantum field theory model of forward interest rates. Swaptions are briefly reviewed. An approximation scheme for the coupon bond option price is developed based on the fact that the volatility of the forward interest rates is a small quantity. The field theory for the forward interest rates is Gaussian, but when the payoff function for the coupon bond option is included it makes the field theory nonlocal and nonlinear. A perturbation expansion using Feynman diagrams gives a closed form approximation for the price of coupon bond option. A special case of the approximate bond option is shown to yield the industry standard one-factor HJM formula with exponential volatility.