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The underlying models for bond options are often based on some linear drift functions such that the option Greeks depend crucially on the mean reversion parameters. Substantial estimation bias may arise when these parameters are estimated using standard methods such as maximum likelihood estimation, leading to a bias in estimating the Greeks. To address this issue, following Phillips and Yu (2005), a jackknife method is adopted in this article. In particular, we apply the method directly to the estimation of option Greeks, rather than the estimation of parameters. This approach is general and computationally inexpensive; hence, it is convenient in practice. The finite-sample performance is investigated in several Monte Carlo studies. At last, in dynamic Delta hedging, we show that the bias reduction in the estimation of option Greeks using the proposed method can achieve some economic value. TOPICS:Derivatives, options, fixed income and structured finance, quantitative methods, simulations Key Findings ▪ Like option pricing, the bias problem is still serious in estimating the Greeks of the options. ▪ The jackknife method is a good method to reduce the estimation bias in estimating the Greeks. ▪ The bias reduction in the estimation of option Greeks using the proposed method can achieve some economic values in option markets around the world.

I develop and estimate an affine short-rate model that incorporates a nonstationary stochastic mean. In my model, the time-varying stochastic mean is subject to a sequence of permanent shocks that can better capture the source of nonlinearity in the drift than existing models. I find that the proposed model provides a better in-sample and out-of-sample fit to observed interest rates and bond prices relative to extant models. More specifically, my model outperforms constant elasticity of volatility models. It follows that the nonstationary stochastic mean model offers new insights to the implied bond option valuation and accounts for the downward bias in bond option prices generally documented in the literature.

Hybrid equity-rate derivatives are commonly traded between financial institutions, but are challenging to price with traditional methods. Especially challenging are those contracts which involve an explicit interest rate (fixing) dependence in the cashflows, which stretches typical measure-change approaches beyond their practical limit. We introduce a framework for pricing equity swaps, equity cap/floors, and other hybrid derivatives under general stochastic short-rate models with a correlated equity. By utilizing the machinery of Continuous Time Markov Chain (CTMC) approximation, and a decoupled representation of the equity-rate model, we derive closed-form approximations for the hybrid contract prices based on a regime-switching model and prove theoretical convergence. The numerical implementation of the method is fast and very accurate, achieving superquadratic convergence in numerical experiments. The framework also provides a practical alternative to traditional approaches such as trees for pricing bonds and bond options under short-rates models which lack closed-form solutions.

In this paper, we analize a novel approach for calibrating the one-factor and the two-factor Hull–White models using swaptions under a market-consistent framework. The technique is based on the pricing formulas for coupon bond options and swaptions proposed by Russo and Fabozzi (J Fixed Income 25:76–82, 2016b; J Fixed Income 27:30–36, 2017b). Under this approach, the volatility of the coupon bond is derived as a function of the stochastic durations. Consequently, the price of coupon bond options and swaptions can be calculated by simply applying standard no-arbitrage pricing theory given the equivalence between the price of a coupon bond option and the price of the corresponding swaption. This approach can be adopted to calibrate parameters of the one-factor and the two-factor Hull–White models using swaptions quoted in the market. It represents an alternative with respect to the existing approaches proposed in the literature and currently used by practitioners. Numerical analyses are provided in order to highlight the quality of the calibration results in comparison with existing models, addressing some computational issues related to the optimization model. In particular, calibration results and sensitivities are provided for the one- and the two-factor models using market data from 2011 to 2016. Finally, an out-of-sample analysis is performed in order to test the ability of the model in fitting swaption prices different from those used in the calibration process.

We propose two methodologies to price sovereign bond options in emerging markets. The motivation is to provide hedging protection against price fluctuations, departing from the not liquid data provided by the stock exchange. Taking this into account, we first compute prices provided by the Jamshidian formula, when modeling the interest rate through Vasicek model, with parameters estimated with the help of the Kalman filter. The second methodology is the pricing strategy provided by the Black-Derman-Toy tree model. A numerical comparison is carried out. The first equilibrium approach provides parsimonious modeling, is less sensitive to daily changes and more robust, while the second non-arbitrage approach provides more fluctuating but also what can be considered more accurate option prices.

Purpose Recently, Stein et al. (2016) studied theoretical properties and parameter estimation of continuous time processes derived as solutions of a generalized Langevin equation (GLE). In this paper, the authors extend the model to a wider class of memory kernels and then propose a bond and bond option valuation model based on the extension of the generalized Langevin process of Stein et al. (2016). Design/methodology/approach Bond and bond option pricing based on the proposed interest rate models presents new difficulties as the standard partial differential equation method of stochastic calculus for bond pricing cannot be used directly. The authors obtain bond and bond option prices by finding the closed form expression of the conditional characteristic function of the integrated short rate process driven by a general Lévy noise. Findings The authors obtain zero-coupon default-free bond and bond option prices for short rate models driven by a variety of Lévy processes, which include Vasicek model and the short rate model obtained by solving a second-order Langevin stochastic differential equation (SDE) as special cases. Originality/value Bond and bond option pricing plays an important role in capital markets and risk management. In this paper, the authors derive closed form expressions for bond and bond option prices for a wider class of interest rate models including second-order SDE models. Closed form expressions may be especially instrumental in facilitating parameter estimation in these models.

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...

Derivative (i.e. option) pricing is essential for modern financial instrumentations. Despite of the previous efforts, the exact analytical forms of the derivative pricing distributions are still challenging to obtain. In this study, we established a quantitative framework using path integrals to obtain the exact analytical solutions of the statistical distribution for bond and bond option pricing for the Vasicek model. We discuss the importance of statistical fluctuations away from the expected option pricing characterized by the distribution tail and their associations to value at risk (VaR). The framework established here is general and can be applied to other financial derivatives for quantifying the underlying statistical distributions.

Due to the presence of stochastic volatility dynamics, the Fong-Vasicek short rate model is more complex but also more realistic than the classical Vasicek version. To enhance the numerical tractability of the Fong-Vasicek model for the calculation of bond option prices, we suggest the use of the Heath-Platen estimator which performs excellently in the related Heston stochastic volatility model. We show that the Heath-Platen estimator reduces the variance and thus the size of confidence intervals dramatically compared to a crude Monte Carlo estimation, which leads to a drastic speed-up in price calculations across different realistic parameter sets.

The pricing of longevity-linked securities depends not only on the stochastic uncertainty of the underlying risk factors, but also the attitude of investors towards those factors. In this research, we investigate how to estimate the market risk premium of longevity risk using investable retirement indexes, incorporating uncertain real interest rates using an affine dynamic Nelson-Siegel model. A multi-cohort aggregate, or systematic, continuous time affine mortality model is used where each risk factor is assigned a market price of mortality risk. To calibrate the market price of longevity risk, a common practice is to make use of market prices, such as longevity-linked securities and longevity indices. We use the BlackRock CoRI Retirement Indexes, which provides a daily level of estimated cost of lifetime retirement income for 20 cohorts in the U.S. Although investment in the index directly is not possible, individuals can invest in funds that track the index. For these 20 cohorts, we assume risk premiums for the common factors are the same across cohorts, but the risk premium of the factors for a specific cohort is allowed to take different values for different cohorts. The market prices of longevity risk are then calibrated by matching the risk-neutral model prices with BlackRock CoRI index values. Closed-form expressions and prices for European options on longevity zero-coupon bonds are derived using the model and compared to prices for standard options on zero coupon bonds. The impact of uncertain mortality on long term option prices is quantified and discussed.

We analyze certain results on the stochastic string modeling of the term structure of interest rates and we apply them to study the sensitivities and the hedging of options with payoff functions homogeneous of degree one. Under the same framework, we use an exact multi-factor extension of Jamshidian (1989) to find the sensitivities for swaptions and we prove that it cannot be applied to captions. We present a new approximate result for pricing options on coupon bonds based on the Fenton-Wilkinson method and we show that it generalizes the fast coupon bond option pricing proposed in Munk (1999). This result can be easily applied to the approximate valuation of swaptions and captions.

We provide closed-form expressions for bond prices in interest rate models based on compact Lie groups. Our approach uses a Doob transform technique and PDE solutions by the Mathieu periodic functions. As a by-product, we derive formulas for bond option prices as well as new identities for the Laplace transform of periodic functionals of Brownian motion and Brownian diffusion processes.

Bond option transactions from a hedging perspective are currently almost non-existent in the South African bond and bond option market. As a result of comments and suggestions made by academics and independent observers a study was conducted in the South African bond options market amongst former and current bond option traders. The goals of the present study was to establish if bond options can be an effective hedging tool in the South African bond market, to conduct empirical tests on the basic option hedging strategies to ascertain these particular strategies’ suitability as hedges against investment risk by using actual market movements in the South African bond market, and to formulate recommendations that could be implemented to re-establish bond options as a viable hedging instruments in South Africa and also introduce it to Africa.

Due to the presence of stochastic volatility dynamics, the Fong-Vasicek short rate model is more complex but also more realistic than the classical Vasicek version. To enhance the numerical tractability of the Fong-Vasicek model for the calculation of bond option prices, we suggest the use of the Heath-Platen estimator which performs excellently in the related Heston stochastic volatility model. We show that the Heath-Platen estimator reduces the variance and thus the size of confidence intervals dramatically compared to a crude Monte Carlo estimation, which leads to a drastic speed-up in price calculations across different realistic parameter sets.

International audience; Jamshidian developed a model for pricing bond options within a Vasicek one-factor framework, with the very useful property that it allows an option on a coupon bond to be decomposed into a set of options on the individual coupons. In the Vasicek framework, the “Jamshidian trick” produces yields to maturity on the coupons that are linear functions of the underlying state variable. But it has not been clear whether this approach could extend to more complicated interest rate processes. In this article, Leblon and Moraux show how to extend the Jamshidian technique to quadratic interest rate processes and use it to derive analytic formulas for European call options on coupon bonds. Finally, they verify that the required conditions appear to hold in the real world.

We consider the valuation of collateralized derivative contracts such as bond option or Caplet contracts. We allow for posting different collaterals such as securities or cash for the derivatives and its hedges. The pricing is based on modeling the joint evolution of collateral rate and the spread between collaterals. The Hull–White models are applied to collateral rate and spread to generate the closed pricing formula for zero coupon bond option. We also derive the pricing formula for Caplet under the Libor Market model and SABR model framework.

Bond option pricing models come in two forms: equilibrium models and arbitrage-free models. The former start from assumptions about the dynamic process that governs the evolution of the instantaneous short rate, which produces a theoretically consistent model of the whole term structure. But unfortunately, real-world bond prices do not follow the model perfectly, and many will appear to be mispriced. By contrast, the Ho–Lee model, the first arbitrage-free model, starts with the observed term structure and sets up a binomial structure that determines how it can evolve over time so that there are no riskless arbitrage profit opportunities embedded in the current or any possible future yield curves. But the original formulation of the Ho–Lee model has the unfortunate property that it is possible for interest rates to go negative or to grow without bound over time, neither of which is acceptable. In this article, the authors present a modified Ho–Lee specification that eliminates the problem by allowing the binomial probabilities to change over time while still maintaining the no-arbitrage restriction.

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.

This paper studies the possibility of using financial regulation that prohibits the use of money substitutes as a tool for mitigating the adverse effects of deviations from the Friedman rule. When inflation is not too high regulation aimed at eliminating money substitutes improves welfare by economizing on transaction costs. The gains from regulation depend on the distribution of income and the level of direct taxation. The area under the demand for money curve is equal to the welfare cost of inflation only when there are no direct taxes and no proportional intermediation cost: otherwise, the area under the demand curve overstates the welfare cost of inflation when money substitutes are not important and understates the welfare cost when money substitutes are important.

The chemical bond approach to the glassy state is discussed. It is based on two concepts: Ovshinsky's idea of bond option as a necessary condition for amorphization, and Dembovsky's notions about hypervalent bonds (HVB) as alternative bonding state responsible for glass specificity and glass formation at all. The third idea is self-organization of HVBs in the form of a bond wave, which provides an interconnected interpretation of three structural levels: short-range order that is different at around HVB and far outside it, medium-range order in the wavefronts populated with HVB, and long-range order due to the bond wave itself. Experimental evidences and practical consequences of the bond-wave picture are discussed.

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.

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 Caratheodory–Fejer 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.

In this paper, we present a discrete time regime switching binomial-like model of the term structure where the regime switches are governed by a discrete time semi-Markov process. We model the evolution of the prices of zero-coupon when given an initial term structure as in the model by Ho and Lee that we aim to extend. We discuss and derive conditions for the model to be arbitrage free and relate this to the notion of martingale measure. We explicitly show that due to the extra source of uncertainty coming from the underlying semi-Markov process, there are an infinite number of equivalent martingale measures. The notion of path independence is also studied in some detail, especially in the presence of regime switches.Wedeal with the market incompleteness by giving an explicit characterization of the minimal entropy martingale measure. We give an application to the pricing of a European bond option both in a Markov and semi-Markov framework. Finally, we draw some conclusions.

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.

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 the bond prices implied by the simulation-based approach and those delivered by ML. Some consequences for the statistical testing of contingent-claim pricing models are discussed. The Author 2009. Published by Oxford University Press on behalf of The Society for Financial Studies. All rights reserved. For Permissions, please e-mail: journals.permissions@oxfordjournals.org., Oxford University Press.

This paper examines the pricing of interest rate derivatives when the interest rate dynamics experience infrequent jump shocks modelled as a Poisson process. The pricing framework adapted was developed by Chiarella and Nikitopoulos to provide an extension of the Heath, Jarrow and Morton model to jump‐diffusions and achieves Markovian structures under certain volatility specifications. Fourier Transform solutions for the price of a bond option under deterministic volatility specifications are derived and a control variate numerical method is developed under a more general state dependent volatility structure, a case in which closed form solutions are generally not possible. In doing so, a novel perspective is provided on control variate methods by going outside a given complex model to a simpler more tractable setting to provide the control variates.

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.

We present a model in which the interest rate reverts to a nonstationary stochastic mean. In our model, the stochastic mean is subject to a sequence of permanent shocks that can better capture the source of nonlinearity in the drift than extant models. We find that our model provides a better fit to observed interest rates relative to a large number of alternative models. In particular, the Vasicek model with a nonstationary mean outperforms constant elasticity of volatility models. The implied bond option price from our model accounts for the downward bias in existing bond option valuation models.

This paper extends the results on quadratic term structure models in continuous time to the discrete time setting. The continuous time setting can be seen as a special case of the discrete time one. Discrete time quadratic models have advantages over their continuous time counterparts as well as over discrete time affine models. Recursive closed form solutions for zero coupon bonds are provided even in the presence of multiple correlated underlying factors, time-dependent parameters, regime changes and “jumps” in the underlying factors. In particular regime changes and “jumps” cannot so easily be accommodated in continuous time quadratic models. Pricing bond options requires simple integration and model estimation does not require a restrictive choice of the market price of risk.

We describe the Bond Market Model, a multi-factor interest rate term structure model, where it is possible to price with Black-like formulas the three classes of over-the-counter plain vanilla options. We derive the prices of caps/floors, bond options and swaptions. A comparison with Libor Market Model and Swap Market Model is discussed in detail, underlining advantages and limits of the different approaches.

Leveraging the explicit formula for European swaptions and coupon‐bond options in the HJM one‐factor model, a semi‐explicit formula for 2‐Bermudan options (also called Canary options) is developed. The European swaption formula is extended to future times. So equipped, one is able to reduce the valuation of a 2‐Bermudan swaption to a single numerical integration at the first expiry date. In that integration the most complex part of the embedded European swaptions valuation has been simplified to perform it only once and not for every point. In a special but very common in practice case, a semi‐explicit formula is provided. Those results lead to a significantly faster and more precise implementation of swaption valuation. The improvements extend even more favourably to sensitivity calculations.

The standard method for valuing a European option on a bond portfolio is developed by Jamshidian. He shows that under certain circumstances the payoff from a bond option can be expressed as a portfolio of payoffs on discount bond options, allowing the option to be valued as a portfolio of options. A limitation of this approach is that it cannot be applied to non-Markovian interest rate processes. This paper develops a method for the valuation of a European option on a bond portfolio that can be applied to both Markovian and non-Markovian interest rate processes.

In this paper, we generalize the classical binomial bond market model of Ho and Lee [2] to the multinomial model. We establish necessary and sufficient conditions for such a bond market model to be arbitrage-free and path independent. We study the bond option pricing and forward-rate equation in the trinomial case.

We hereby present an explicit formula for European options on coupon bearing bonds in the Heath–Jarrow–Morton one factor model with non-stochastic volatility. The formula extends the Jamshidian formula for zero-coupon bonds for special form of volatility. Moreover we present a formula for zero-coupon bonds without condition on the volatility. We provide also an explicit way to compute the hedging ratio (Δ) in order to hedge the options individually.

In this paper, we explore two new tree lattice methods, the piecewise binomial tree and the piecewise trinomial tree for both the bond prices and European/American bond option prices assuming that the short rate is given by a generalized skew Vasicek model with discontinuous drift coefficient. These methods build nonuniform jump size piecewise binomial/trinomial tree based on a tractable piecewise process, which is derived from the original process according to a transform. Numerical experiments of bonds and European/American bond options show that our approaches are efficient as well as reveal several price features of our model.

Pricing bond options under the Cox, Ingersoll and Ross (CIR) model of the term structure of interest rates requires the computation of the noncentral chi-square distribution function. In this article, we compare the performance in terms of accuracy and computational time of alternative methods for computing such probability distributions against an externally tested benchmark. All methods are generally accurate over a wide range of parameters that are frequently needed for pricing bond options, though they all present relevant differences in terms of running times. The iterative procedure of Benton and Krishnamoorthy (Comput. Stat. Data Anal. 43:249–267, 2003) is the most efficient in terms of accuracy and computational burden for determining bond option prices under the CIR assumption.

We carry out group classification of a general bond-option pricing equation. We show that the equation admits a three-dimensional equivalence Lie algebra. We also show that some of the values of the constants which result from group classification give us well-known models in mathematics of finance such as Black-Scholes, Vasicek, and Cox-Ingersoll-Ross. For all such values of these arbitrary constants we obtain Lie point symmetries. Symmetry reductions are then obtained and group invariant solutions are constructed for some cases.

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.

Forward rate dynamics are modeled as a random field. In contrast to multifactor models, random field models offer a parsimonious description of term structure dynamics, while eliminating the self-inconsistent practice of recalibration. The form of the drift of the instantaneous forward rate process necessary to preclude arbitrage under the risk-neutral measure is obtained. Forward risk-adjusted measures are identified and used to price a bond option when the forward volatility structure depends on the square root of the current spot rate. Several classes of tractable random field models are presented. Article published by Oxford University Press on behalf of the Society for Financial Studies in its journal, The Review of Financial Studies.

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.