Bond valuation for generalized Langevin processes with integrated Lévy noise

In this paper, we propose a new approach for the pricing of bond options using the relation between the Heath-Jarrow-Morton (HJM) model and the Brace-Gatarek-Musiela (BGM) model. To derive a closed-form solution (CFS) of bond options on the HJM model with the BGM model, we first consider about basic concepts of the HJM model in which is hard to achieve the CFS of bond options. The second obtains the bond pricing equation through the fact that the spot rate is equal to the instantaneous forward rate. Furthermore, we derive the formula of the discount bond price using restrictive condition of Ritchken and Sankarasubramanian (RS). Finally, we get a CFS of bond options using the relation between the HJM volatility function *** f (t ,T ) and the BGM volatility function *** (t ,T ) and give the analytic proof of bond pricing. In particular, we can confirm the humps in the pricing of bond call option occur while the graph of bond put option are decreasing functions of the maturity as the value of *** (tenor) and *** r (volaitility of interest rate) are increasing with two scenarios. This result means a simple and reasonable estimate for the pricing of bond options under the proposed conditions.

Three contents for the pricing of bond options on the arbitrage-free model with jump are included in this paper. The first uses a new technique to derive a Closed-Form Solution (CFS) for bond options on Hull and White (HW) model with jump. The second deals with the pricing of bond option for Heath-Jarrow-Morton (HJM) model based on jump, and the third simulates the proposed models by the Monte Carlo Simulation (MCS). We also analyze the values obtained by the CFS and MCS. There is a substantial difference between bond option prices which are obtained by the HW model with jump and the HJM model based on jump. For this, we use the well-known Mean Standard Error (MSE) and show that lower value of Precision (PCS) in the proposed models corresponds to sharper estimates. In particular, we confirm that the PCS for the HJM based on jump is lower than that for the HW model with jump. Through the empirical simulation of our method suggested, we obtain a better accurate estimation for the pricing of bond options.

A new multi-factor short rate model is presented which is bounded from below by a real-valued function of time. The mean-reverting short rate process is modeled by a sum of pure-jump Ornstein–Uhlenbeck processes such that the related bond prices possess affine representations. Also the dynamics of the associated instantaneous forward rate is provided and a condition is derived under which the model can be market-consistently calibrated. The analytical tractability of this model is illustrated by the derivation of an explicit plain vanilla option price formula. With view on practical applications, suitable probability distributions are proposed for the driving jump processes. The paper is concluded by presenting a post-crisis extension of the proposed short and forward rate model.

Fast approximations of bond option prices under CKLS models

ABSTRACTThis paper evaluates the forecasting and finite sample performance of short‐term interest rate models in a number of countries. Specifically, we run a series of in‐sample and out‐of‐sample tests for both the conditional mean and volatility of one‐factor short rate models, and compare the results to the random walk model. Overall, we find that the out‐of‐sample forecasting performance of one‐factor short rate models is poor, stemming from the inability of the models to accommodate jumps and discontinuities in the time series data. In addition, we perform a series of Monte Carlo analyses similar to Chapman and Pearson to document the finite sample performance of the short rate models when β3 is not restricted to be equal to one. Our results indicate the potential dangers of over‐parameterization and highlight the limitations of short‐term interest rate models.

Implementation of Short-Rate Models - A Case Study of the Black-Derman-Toy Model of Interest Rate

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

This paper presents an analysis of models for Japanese short-term inter- est rate. The models are constructed based on mean reverting model using Bayesian method to capture the dynamics of short-term interest rate. The parameters of our models are estimated by marginal likelihood and posterior expectation and we shall make model selection using the information criterion EIC(extended information cri- terion). An application of the models will be implemented using weekly Japanese average interest rates on certificates of deposit (new issues) less than 30 days in the period from January 2001 to December 2008. 1 Introduction Correct modeling of the short-term interest rate is essential in finace, as it is this rate that is fundamental to the pricing of securities and important for risk man- agement. In the study of the short-term interest rate dynamics, various models have been suggested. There are examples of these models, such as Vasicek model by Vasicek(1977), CIR model by Cox, Ingersoll and Ross(1985) and so on. An empirical comparison of these models was made by Chan, Karolyi, Longstaff and Sanders(1992). In the paper, the param- eters are estimated by generalized method of moments(GMM) and they implemented the hypothesis testing methods developed by Newey and West for evaluation of the models. In the parameter estimation of mean reverting model using GMM, it is known that the results are easy to be influenced by an initial values, and it becomes often unstable. Recently, Ahangarani(2005) employed maximum likelihood method for parameter esti- mation and log likelihood ratio test for model selection, and Kawada(2007) implemented GMM and made model selection using the information criterion EIC(extended information criterion) for various short-term interest rate models. Another articles of the interest rate model, using Bayesian framework, Jones(2003) published study about nonlinear drift of the interest rate model in detail, and Gray(2005) studied continuous time short rate models and the parameters estimated by Markov chain Monte Carlo(MCMC) method. Sanford and Martin(2006) made estimation using MCMC algorithm and model selection is made by Bayes factors for each model calculated using Savage-Dickey density ratio. In addition, a recent study by Hong and Lin(2006) examined a variety of short-term interest rate mod- els including the single-factor diffusion models, GARCH models, Markov regime-switching models and jump-diffusion models for Chinese rates. However, more useful models which capture the dynamics of the short-term interest rate are needed. In this paper, we suggest two hierarchical Bayes models based on mean reverting model. The parameters of the models are treated as random variables so that we have to consider appropriate prior distributions for them. By regarding parameters as random variables, we can consider that model construction and estimation involve necessarily some uncertainty and utilize the prior information of interest rate. In the first hierarchical Bayes model, we assume that volatility is constant, and in the second one, it depends on the

Recombining Trees for One-Dimensional Forward Rate Models

An alternative valuation model for contingent claims

This paper introduces a short rate model in continuous time that adds one or more memory (delay) components to the Merton model [R.C. Merton, A dynamic general equilibrium model of the asset market and its application to the pricing of the capital structure of the firm, Working Paper 497, Sloan School of Management, MIT, Cambridge, 1970; R.C. Merton, Theory of rational option pricing, Bell J. Econ. Manage. Sci. 4 (1973), pp. 141–183.] or the Vasiček model [Vasiček, An equilibrium characterization of the term structure, J. Financ. Econ. 5 (1977), pp. 177–188.] for the short rate. The distribution of the short rate in this model is normal, with the mean depending on past values of the short rate, and a limiting distribution exists for certain values of the parameters. The zero coupon bond price is an affine function of the short rate, whose coefficients satisfy a system of delay differential equations. This system can be solved analytically, obtaining a closed formula. An analytical expression for the instantaneous forward rate is given: it satisfies the risk neutral dynamics of the Heath-Jarrow-Morton model. Formulae for both forward looking and backward looking caplets on overnight risk free rates are presented. Finally, the proposed model is calibrated against forward looking caplets on SONIA rates and the United States yield curve.

The recent financial crisis has led to so-called multi-curve models for the term structure. Here we study a multi-curve extension of short rate models where, in addition to the short rate itself, we introduce short rate spreads. In particular, we consider a Gaussian factor model where the short rate and the spreads are second order polynomials of Gaussian factor processes. This leads to an exponentially quadratic model class that is less well known than the exponentially affine class. In the latter class the factors enter linearly and for positivity one considers square root factor processes. While the square root factors in the affine class have more involved distributions, in the quadratic class the factors remain Gaussian and this leads to various advantages, in particular for derivative pricing. After some preliminaries on martingale modeling in the multi-curve setup, we concentrate on pricing of linear and optional derivatives. For linear derivatives, we exhibit an adjustment factor that allows one to pass from pre-crisis single curve values to the corresponding post-crisis multi-curve values.

Derivative Pricing for a Multi-Curve Extension of the Gaussian, Exponentially Quadratic Short Rate Model

We consider a model for interest rates where the short rate is given under the risk-neutral measure by a time-homogeneous one-dimensional affine process in the sense of Duffie, Filipovic, and Schachermayer. We show that in such a model yield curves can only be normal, inverse, or humped (i.e., endowed with a single local maximum). Each case can be characterized by simple conditions on the present short rate rt. We give conditions under which the short rate process converges to a limit distribution and describe the risk-neutral limit distribution in terms of its cumulant generating function. We apply our results to the Vasicek model, the CIR model, a CIR model with added jumps, and a model of Ornstein–Uhlenbeck type.

In this paper we investigate whether inflation causes the time-varying mean-reverting level in the Balduzzi et al. (Review of Economics and Statistics, Vol. 80, No. 1 (1998), pp. 62–72) short rate model. We find a time-varying mean-reverting level for the UK nominal short rate, but the real short rate mean reverts to a constant. This suggests a monetary source for the time-varying mean-reverting level in nominal short rate models. The time-varying mean factor is closely related to market expectations about future inflation. This suggests that expected future inflation determines the mean-reverting level of the nominal short rate.