Coupon Bond Options Research Articles

Calibration of one-factor and two-factor Hull–White models using swaptions

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.

Pricing Coupon Bond Options and Swaptions under theTwo-Factor Hull-White Model

In this article, we propose an alternative approach for pricing bond options and swaptions under the two-factor Hull-White model that differs from existing models used to evaluate these instruments. Assuming that the forward price of a coupon bond is a martingale under the forward risk-neutral measure, the proposed model involves deriving the volatility of the coupon bond as a function of the stochastic durations calculated with respect to the model’s two factors. Once the volatility function is defined, the price of options on coupon bonds can be derived by simply applying standard no-arbitrage pricing theory. Given the equivalence between the price of a coupon bond and the price of the corresponding swaptions, our approach can be adopted to calibrate the parameters of the two-factor Hull-White model using swaptions quoted in the market. The new solution we propose for the calibration can be considered as an alternative with respect to the existing approaches proposed in the literature and currently used by practitioners. Numerical results are provided in order to analyze the features of the proposed model for calibration purposes.

Asset pricing under general collateralization

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.

Asset Pricing Under General Collateralization

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.

Pricing Coupon Bond Options and Swaptions under the One-Factor Hull–White Model

In this article, the authors propose an alternative approach for pricing bond options and swaptions under the one-factor Hull–White model. Their proposal differs from the existing models used to evaluate these type of instruments when the evolution of the term structure of interest rates is modeled by short-rate models. Assuming that the forward price of a coupon bond is a martingale under the forward risk-neutral measure, the proposed model involves deriving the volatility of the coupon bond as a function of its stochastic duration. Once the volatility function is defined, the price of options on coupon bonds can be derived by simply applying the standard no-arbitrage pricing theory in the context of the one-factor Hull–White model. Given the equivalence between the price of a coupon bond and the price of the corresponding swaptions, this approach can be adopted to calibrate the parameters of the one-factor Hull–White model using swaptions quoted in the market. The new solution proposed for the calibration can be considered as an alternative with respect to the existing approaches proposed in the literature and used currently by practitioners. Numerical results are provided in order to analyze the features of the proposed model for calibration purposes. <b>TOPICS:</b>Fixed income and structured finance, derivatives applications, quantitative methods

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.

Simulation of nonlinear interest rates in quantum finance: Libor Market Model

The simulation of the Libor Market Model (LMM) is extensively studied in the framework of quantum finance. The imperfectly correlated Libor rates are simulated based on a Gaussian quantum field and a recursion equation of nontrivial stochastic drift. The Libor options are studied using both the simulation method and the analytical formula. The caplet price of simulation is compared with Black’s caplet formula which can be exactly derived from the LMM. The invariance of caplet price for different forward bond numeraire is verified by using the simulation. The simulation results for coupon bond options and swaptions are compared with the approximate price, which are limited for the reason that the approximate price is derived using the small volatility expansion. The simulation method is shown to have great potential in the application of pricing interest rate instruments.

Pricing interest rate derivatives under stochastic volatility

PurposeThe purpose of this paper is to derive semi‐closed‐form solutions to a wide variety of interest rate derivatives prices under stochastic volatility in affine‐term structure models.Design/methodology/approachThe paper first derives the Frobenius series solution to the cross‐moment generating function, and then inverts the related characteristic function using the Gauss‐Laguerre quadrature rule for the corresponding cumulative probabilities.FindingsThis paper values options on discount bonds, coupon bond options, swaptions, interest rate caps, floors, and collars, etc. The valuation approach suggested in this paper is found to be both accurate and fast and the approach compares favorably with some alternative methods in the literature.Research limitations/implicationsFuture research could extend the approach adopted in this paper to some non‐affine‐term structure models such as quadratic models.Practical implicationsThe valuation approach in this study can be used to price mortgage‐backed securities, asset‐backed securities and credit default swaps. The approach can also be used to value derivatives on other assets such as commodities. Finally, the approach in this paper is useful for the risk management of fixed‐income portfolios.Originality/valueThis paper utilizes a new approach to value many of the most commonly traded interest rate derivatives in a stochastic volatility framework.

Interest rates in quantum finance: Caps, swaptions and bond options

The prices of the main interest rate options in the financial markets, derived from the Libor (London Interbank Overnight Rate), are studied in the quantum finance model of interest rates. The option prices show new features for the Libor Market Model arising from the fact that, in the quantum finance formulation, all the different Libor payments are coupled and (imperfectly) correlated. Black’s caplet formula for quantum finance is given an exact path integral derivation. The coupon and zero coupon bond options as well as the Libor European and Asian swaptions are derived in the framework of quantum finance. The approximate Libor option prices are derived using the volatility expansion. The BGM-Jamshidian (Gatarek et al. (1996) [1], Jamshidian (1997) [2]) result for the Libor swaption prices is obtained as the limiting case when all the Libors are exactly correlated. A path integral derivation is given of the approximate BGM-Jamshidian approximate price.

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.

Pricing American options for interest rate caps and coupon bonds in quantum finance

American option for interest rate caps and coupon bonds are analyzed in the formalism of quantum finance. Calendar time and future time are discretized to yield a lattice field theory of interest rates that provides an efficient numerical algorithm for evaluating the price of American options. The algorithm is shown to hold over a wide range of strike prices and coupon rates. All the theoretical constraints that American options have to obey are shown to hold for the numerical prices of American interest rate caps and coupon bond options. Non-trivial correlation between the different interest rates are efficiently incorporated in the numerical algorithm. New inequalities are conjectured, based on the results of the numerical study, for American options on interest rate instruments.

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.

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

The quantum finance pricing formulas for coupon bond options and swaptions derived by Baaquie [Phys. Rev. E 75, 016703 (2006)] are reviewed. We empirically study the swaption market and propose an efficient computational procedure for analyzing the data. Empirical results of the swaption price, volatility, and swaption correlation are compared with the predictions of quantum finance. The quantum finance model generates the market swaption price to over 90% accuracy.

Price of coupon bond options in a quantum field theory of forward interest rates

European options on coupon bonds are studied in a quantum field theory model of forward interest rates. A approximation scheme for finding the option price is developed based on the fact that the volatility of the forward interest rate is a small quantity. The field theory for the forward interest rates is in effect Gaussian, and when the payoff function for the coupon bonds option is included it makes the field theory exponentially nonlinear. A Feynman perturbation expansion gives a result for the price of Libor swaption that agrees quite well with the market price.

A Semi-explicit Approach to Canary Swaptions in HJM one-factor Model

Leveraging the explicit formula for European swaptions and coupon-bond options in HJM one-factor model, we develop a semi-explicit formula for 2-Bermudan options (also called Canary options). We first extend the European swaption formula to future times. So equipped, we are 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, we also provide a semi-explicit formula. Those results lead to a significantly more precise implementation of swaption valuation. The improvements extend even more favorable to sensitivity calculations.

Pricing Swaptions and Coupon Bond Options in Affine Term Structure Models

We propose an approach to find an approximate price of a swaption in Affine Term Structure Models. Our approach is based on the derivation of approximate dynamics in which the volatility of the Forward Swap Rate is itself an affine function of the factors. Hence we remain in the affine framework and well known results on transforms and transform inversion can be used to obtain swaption prices in ways similar to bond options (i.e. caplets). We demonstrate that we can also obtain a closed form formula for the approximate price which is based on square-root dynamics for the swap rate. The latter approximation is extremely fast while remaining accurate. The method can be easily generalized to price options on coupon bonds. Computational time compares favorably with other approximation methods. Numerical results on the quality of the approximation are excellent. Our results show that in Affine models, analogously to the LIBOR Market Model, LIBOR and Swap rates are driven by approximately the same type of (in this case affine) dynamics.

Pricing Coupon-Bond Options and Swaptions in Affine Term Structure Models

This paper provides a numerically accurate and computationally fast approximation to the prices of European options on coupon-bearing instruments that is applicable to the entire family of affine term structure models. Exploiting the typical shapes of the conditional distributions of the risk factors in affine diffusions, we show that one can reliably compute the relevant probabilities needed for pricing options on coupon-bearing instruments by the same Fourier inversion methods used in the pricing of options on zero-coupon bonds. We apply our theoretical results to the pricing of options on coupon bonds and swaptions, and the calculation of on swap books. As an empirical illustration, we compute the expected exposures implied by several affine term structure models fit to historical swap yields.

Stochastic Duration and Fast Coupon Bond Option Pricing in Multi-Factor Models

This paper generalizes the stochastic duration concept of Cox, Ingersoll, and Ross (1979) to multi-factor diffusion models of the term structure of interest rates. The stochastic duration has time as its unit and measures the sensitivity of the price of a bond (or portfolio of bonds) with respect to any change of the term structure consistent with the model. Some important general properties of the stochastic duration measure are given. For example, it is proven that, under conditions satisfied by most popular term structure models, the familiar Fisher-Weil duration overestimates the interest rate risk of coupon bonds. The stochastic duration is studied in detail in various well-known models. It is also shown that the price of a European option on a coupon bond can be approximated very accurately by a multiple of the price of a European option on a zero-coupon bond with a time to maturity equal to the stochastic duration of the coupon bond. This provides a fast method for pricing European swaptions in multi-factor models.

Valorisation de produits obligataires dans un modèle d'équilibre général en temps discret

A general equilibrium model is defined from the works of Lucas [1978] and Campbell [1986]. Within the framework of this model, we establish various explicit formulas for bond pricing: zero-coupon options, interest rate options, coupon bond options, floating rate coupon bonds. The formulas so obtained are related to the previous literature on bond pricing.