Short Rate Models Research Articles

A New S.D.E. and Instantaneous Mean Reversion Rate Formula (Presented via a Numerical Empirical Model Comparison)

The theory in this research is to ascertain that, not all models are good all the time. A comparison, of short-rate one factor models is calibrated and analyzed numerically via a Monte Carlo simulation, using two variance reduction techniques. The first technique is stratified sampling, the second the Sobol Algorithm. An empirical comparison is constructed using criterions of the goodness of fit of the models. The aim is to amend the discussion to ex-ante predictability of stochastic models. Five European exchange rates are examined. A new explicit instantaneous mean reversion formula and a new short rate model are introduced.

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

A Short Note on the Exact Stochastic Simulation Scheme of the Hull-White Model and Its Implementation

In this short note we derive an exact simulation scheme for the joint distribution of (r(t),N(t)), where r denotes the short rate following a Hull-White model and $N$ denotes the numeraire.To sample the correct joint distribution of (r(t),N(t)) our scheme requires a two-factor Brownian driver. In other words: the time-discrete Hull-White model is a two factor model.We investigate the performance of this scheme compared to the classical schemes for correlation dependent products like the LIBOR in arrears and Bermudan swaptions. We show that the exact sampling of the joint distribution is important in some applications, e.g., the valuation of Bermudan options using American Monte-Carlo method. Traditional approximation of the numeraire may generate biases if the model uses larger simulation time-steps.Large time-step simulation of short rate models and the American Monte-Carlo method is of interest in the calculation of portfolio effects (xVAs), where usually many risk factors have to be simulated and computational resources need to be saved.

Implementation of a One-Factor Markov-Functional Interest Rate Model

The interest rate market has been expanding immensely for thirty years, both in term of volumes and diversity of traded contracts. The growing complexity of derivatives has implied a need for sophisticated models in order to price and hedge these products. Three main approaches can be distinguished in interest rates modeling. Short-rate models model the dynamics of the term structure of interest rates by specifying the dynamics of a single rate (the spot rate or the instantaneous spot rate) from which the whole term structure at any point in time can be calculated. The prices of derivatives in these models are quite involved functions of the underlying process which is being modeled. This fact makes these models difficult to calibrate. However the short rate process is easy to follow and hence implementation is straightforward.Unlike short rate models the class of Market Models is formulated in terms of market rates which are directly related to tradable assets. Thus they exhibit better calibration properties than short rate models. However they are high dimensional by construction and tedious to implement.In 1999, Hunt, Kennedy and Pelsser introduced the class of Markov-Functional Models (MFM) aiming at developing models which could match as many market prices as Market Models while maintaining the efficiency of short rate models in calculating derivative prices.After a general overview of the two dominant paradigms in section III, this report will focus on the class of Markov-functional models. Section IV presents the general framework. Then several issues related to the implementation of a one-factor MFM model are analyzed in section V. Finally we will display in section VI some numerical results of the simulations of this one-factor model.

ON THE FOUR-PARAMETER BOND PRICING MODEL

A four-parameter random walk model for the short rate of interest is described in Wilmott et al. (15). For pricing zero-coupon bonds from the resulting partial differential equation based on this short rate model, a certain form of solution requires the solution of two first-order nonlinear ordinary dif- ferential equations. In the present paper we show the interesting result that, for obtaining solutions of the bond pricing equation, neither of these two equa- tions requires any differential equation solving techniques; in fact, both these first-order nonlinear differential equations can be solved simply by elementary integration. We include the corresponding yield curve and its asymptotic be- havior. We identify our results obtained here for the general four-parameter model in the two special cases of Vasicek (14) and Cox, Ingersoll and Ross (4) with those given by these authors.

CONSISTENT YIELD CURVE PREDICTION

AbstractWe present an arbitrage-free non-parametric yield curve prediction model which takes the full discretized yield curve data as input state variable. Absence of arbitrage is a particularly important model feature for prediction models in case of highly correlated data as, for instance, interest rates. Furthermore, the model structure allows to separate constructing the daily yield curve from estimating its volatility structure and from calibrating the market prices of risk. The empirical part includes tests on modeling assumptions, out-of-sample back-testing and a comparison with the Vasiček (1977) short-rate model.

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

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.

Long-Term Growth Rate of Expected Utility for Leveraged ETFs: Martingale Extraction Approach

This paper studies the long-term growth rate of expected utility from holding a leveraged exchanged-traded fund (LETF), which is a constant proportion portfolio of the reference asset. Working with the power utility function, we develop an analytical approach that employs martingale extraction and involves finding the eigenpair associated with the infinitesimal generator of a Markovian time-homogeneous diffusion. We derive explicitly the long-term growth rates under a number of models for the reference asset, including the geometric Brownian motion model, GARCH model, inverse GARCH model, extended CIR model, 3/2 model, quadratic model, as well as the Heston and 3/2 stochastic volatility models. We also investigate the impact of stochastic interest rate such as the Vasicek model and the inverse GARCH short rate model. We determine the optimal leverage ratio for the long-term investor and examine the effects of model parameters.

An Infinite Hidden Markov Model for Short-term Interest Rates

The time-series dynamics of short-term interest rates are important as they are a key input into pricing models of the term structure of interest rates. In this paper we extend popular discrete time short-rate models to include Markov switching of infinite dimension. This is a Bayesian nonparametric model that allows for changes in the unknown conditional distribution over time. Applied to weekly U.S. data we find significant parameter change over time and strong evidence of non-Gaussian conditional distributions. Our new model with an hierarchical prior provides significant improvements in density forecasts as well as point forecasts. We find evidence of recurring regimes as well as structural breaks in the empirical application.

APPLICATION OF MAXIMUM LIKELIHOOD ESTIMATION TO STOCHASTIC SHORT RATE MODELS

The application of maximum likelihood estimation is not well studied for stochastic short rate models because of the cumbersome detail of this approach. We investigate the applicability of maximum likelihood estimation to stochastic short rate models. We restrict our consideration to three important short rate models, namely the Vasicek, Cox–Ingersoll–Ross (CIR) and 3/2 short rate models, each having a closed-form formula for the transition density function. The parameters of the three interest rate models are fitted to US cash rates and are found to be consistent with market assessments.

JOINING THE HESTON AND A THREE-FACTOR SHORT RATE MODEL: A CLOSED-FORM APPROACH

In this paper, we present an innovative hybrid model for the valuation of equity options. Our approach includes stochastic volatility according to Heston (1993) [Review of Financial Studies 6 (2), 327–343] and features a stochastic interest rate that follows a three-factor short rate model based on Hull and White (1994) [Journal of Derivatives 2 (2), 37–48]. Our model is of affine structure, allows for correlations between the stock, the short rate and the volatility processes and can be fitted perfectly to the initial term structure. We determine the zero bond price formula and derive the analytic solution for European type options in terms of characteristic functions needed for fast calibration. We highlight the flexibility of our approach, by comparing the price and implied volatility surfaces of our model with the Heston model, where we in particular focus on the correlation structure. Our approach forms a comprehensive market model with an intuitive correlation structure that connects both the equity and interest market to the market volatility.

Generalised Additive-Gaussian Models of the Term Structure

This draft paper outlines a term structure model that models the short rate as a sum of n Gaussian processes (a 'Gn' model). It is shown that the given specification involves no redundant parameters and gives rise to a non-trivial multifactor term structure model. A closed-form expression, involving only the Gaussian cumulative distribution function, is derived. This appears unique in the literature for short rate models with more than two factors. We then delineate the class of term structure models that can be transformed to the Gn model and can therefore use the closed-form option expression.

A Vasicek-Type Short Rate Model With Memory Effect

We introduce a Vasicek-type short rate model which has two additional parameters representing memory effect. This model presents better results in yield curve fitting than the classical Vasicek model. We derive closed-form expressions for the prices of bonds and bond options. Although the model is non-Markov, there exists an associated Markov process that allows one to apply usual numerical methods to the model. We derive analogs of an affine term structure and term structure equations for the model, and, using them, we present a numerical method to evaluate contingent claims.

Path integral and asset pricing

We give a pragmatic/pedagogical discussion of using Euclidean path integral in asset pricing. We then illustrate the path integral approach on short-rate models. By understanding the change of path integral measure in the Vasicek/Hull-White model, we can apply the same techniques to “lesstractable” models such as the Black-Karasinski model. We give explicit formulas for computing the bond pricing function in such models in the analog of quantum mechanical “semiclassical” approximation. We also outline how to apply perturbative quantum mechanical techniques beyond the “semiclassical” approximation, which are facilitated by Feynman diagrams.

Optimal investment in multidimensional Markov-modulated affine models

In a multidimensional affine framework we consider a portfolio optimization problem with finite horizon, where an investor aims to maximize the expected utility of her terminal wealth. We state a very flexible asset price model that incorporates several risk factors modeled both by diffusion processes and by a Markov chain. Exploiting the affine structure of the model we solve the corresponding Hamilton–Jacobi–Bellman equations explicitly up to an expectation only over the Markov chain or equivalently up to a system of simple ODEs. The relevance of the presented model is illustrated on two examples including a stochastic short rate model with trading in the bond and the stock market, and a multidimensional stochastic volatility and stochastic correlation model. Precise verification results for both examples are provided. Economic interpretations of the models and results complement the theoretical analysis.

Explicit Density Approximations for Local Volatility Models Using Heat Kernel Expansions

Heat kernel perturbation theory is a tool for constructing explicit approximation formulas for the solutions of linear parabolic equations. We review the crux of this perturbative formalism and then apply it to differential equations which govern the transition densities of several local volatility processes. In particular, we compute all the heat kernel coefficients for the CEV and quadratic local volatility models; in the later case, we are able to use these to construct an exact explicit formula for the processes' transition density. We then derive low order approximation formulas for the cubic local volatility model, an affine-affine short rate model, and a generalized mean reverting CEV model. We finally demonstrate that the approximation formulas are accurate in certain model parameter regimes via comparison to Monte Carlo simulations.

A Dual-Curve Short Rate Model with Multi-Factor Stochastic Volatility: I. Asymptotic Analysis

We present a stochastic-volatility, short rate term structure model, which extends the classic multi-factor Hull-White model. This model is designed to fit the swaption implied volatility cube and to incorporate the two-curve modeling paradigm. The model exhibits non-Gaussian forward swap rates whose distributions are parameterized across the dimensions of the volatility cube: underlying tenor, option strike and option expiration. To facilitate rapid model calibration, we establish suitable asymptotic expressions for the bond prices. Furthermore, we derive an effective SABR dynamics for each forward swap rate. Finally, we use the mean field approximation to match the effective SABR parameters corresponding to each swaption to the market levels.

Real-Time Risk Management: An AAD-PDE Approach

We apply adjoint algorithmic differentiation (AAD) to the risk management of derivative securities in the situation where the dynamics of securities prices are given in terms of partial differential equations (PDE). With simple examples, we show how AAD can be applied to both forward and backward PDEs in a straightforward manner. In particular, in the context of one-factor short-rate models for interest rates or default intensity processes, we show how one can compute price sensitivities reliably and orders of magnitude faster than with a standard finite-difference approach. This significantly increased efficiency is obtained by combining (i) the adjoint of a forward PDE for calibrating the parameters of the model, (ii) the adjoint of a backward PDE for pricing the derivative security, and (iii) the implicit function theorem to avoid iterating the calibration procedure.

Pricing credit spread option with Longstaff-Schwartz and GARCH models in Chinese bond market

This paper investigates the mean-reversion and volatile of credit spread time series by using regression and time series analysis in Chinese bond market. Then the Longstaff-Schwartz model and GARCH model are applied to price credit spread put option. The authors compare the features of these two models by employing daily bond prices of government bonds and corporate bonds for the period 2010–2012 in Chinese bond market. The proposed results show that the higher the credit ratings of the corporate bonds are, the lower the prices of the credit spread options are.

A novel partial integrodifferential equation-based framework for pricing interest rate derivatives under jump-extended short-rate models

Interest rate derivatives under jump-extended short-rate models have commonly been valued using lattice methods. Unfortunately, lattice methods have pitfalls, mainly in terms of accuracy, efficiency and ease of programming. This paper proposes a much faster and more accurate valuation method based on partial integrodifferential equations (PIDEs). Spatial differential and integral terms are discretized by fourthorder finite differences and Simpson's rule, respectively. For the time-stepping, we investigate a Crank-Nicolson scheme, an implicit-explicit (IMEX) scheme and an exponential time integration scheme. We demonstrate our method by pricing bonds as well as European and American options on both zero-coupon and coupon bonds under jump-extended constant-elasticity-of-variance processes. Through a variety of test problems, we demonstrate that our PIDE-based approach is remarkably superior to lattice methods in terms of accuracy, speed and ease of implementation. By way of example, pricing a one-year zero-coupon bond under the Vasicek model extended with exponential jumps, our method attains an accuracy of 104 in 0:09 seconds with the IMEX scheme, whereas a truncated tree takes 78 seconds to reach the same accuracy.