Short Rate Models Research Articles

Bond valuation for generalized Langevin processes with integrated Lévy noise

PurposeRecently, 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/approachBond 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.FindingsThe 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/valueBond 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.

LONG-TERM GROWTH RATE OF EXPECTED UTILITY FOR LEVERAGED ETFs: MARTINGALE EXTRACTION APPROACH

This paper studies the long-term growth rate of expected utility or expected return from holding a leveraged exchanged-traded fund (LETF), which is a constant proportion portfolio of the reference asset. We develop a martingale extraction approach to tackle the path-dependence in the expectation and determine the long-term growth rate through the eigenpair associated with the infinitesimal generator of a time-homogeneous Markovian diffusion. The long-term growth rates are derived explicitly 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 [Formula: see text] stochastic volatility models. We also investigate the impact of stochastic interest rate such as the Vasicek model and the inverse GARCH short rate model. Additionally, we determine the optimal leverage ratio that maximizes the long-term growth rate, and examine the effects of model parameters.

Модель временной структуры доходности Лонгстаффа-Швартца и ее расширение

Модель временной структуры доходности Лонгстаффа-Швартца и ее расширение

Scenario generation for long run interest rate risk assessment

We propose a statistical model of the term structure of U.S. treasury yields tailored for long-term probability-based scenario generation and forecasts. Our model is easy to estimate and is able to simultaneously reproduce the positivity, persistence, and factor structure of the yield curve. Moreover, we incorporate heteroskedasticity and time-varying correlations across yields, both prevalent features of the data. The model also features a regime-switching short-rate model. We evaluate the out-of-sample performance of our model in terms of forecasting ability and coverage properties, and find that it improves on the standard Diebold and Li model.

Consistent recalibration of yield curve models

AbstractThe analytical tractability of affine (short rate) models, such as the Vasiček and the Cox–Ingersoll–Ross (CIR) models, has made them a popular choice for modeling the dynamics of interest rates. However, in order to properly account for the dynamics of real data, these models must exhibit time‐dependent or even stochastic parameters. This breaks their tractability, and modeling and simulating become an arduous task. We introduce a new class of Heath–Jarrow–Morton (HJM) models that both fit the dynamics of real market data and remain tractable. We call these models consistent recalibration (CRC) models. CRC models appear as limits of concatenations of forward rate increments, each belonging to a Hull–White extended affine factor model with possibly different parameters. That is, we construct HJM models from “tangent” affine models. We develop a theory for continuous path versions of such models and discuss their numerical implementations within the Vasiček and CIR frameworks.

A new S.D.E. and instantaneous mean reversion rate formula (presented via a numerical empirical model comparison)

A new short-rate model and a new explicit instantaneous mean reversion formula are introduced. The introduction is presented via a comparison of various short-rate one factor models, which are calibrated and analyzed numerically via a Monte Carlo simulation. Two variance reduction techniques, Stratified Sampling and the Sobol Algorithm, are compared. An empirical comparison is constructed using criteria of goodness-of-fit, in five exchange rates. The data is ex-ante ultimately measuring the predictability of the stochastic models and variance reduction.

Calibrating Short Interest Rate Models in Negative Rate Environments

Monetary policy based on “quantitative easing” has been followed actively in the United States and Europe in recent years. It has led to short-term interest rates that have become negative in a number of countries. Most of the standard theoretical interest rate models have the restriction that rates are not allowed to go negative within the model, which makes it impossible to calibrate them to market prices in the current environment. This problem is particularly acute in the market for caps, floors, and swaptions, but it is more pervasive because these markets are often used to calibrate models for other securities. Practitioners either refuse to quote short maturity swaptions when the short rate is negative, or they use one of several alternative approaches for valuation: retaining the Black model but only using positive forward rates; modeling rate shocks as normal rather than lognormal, which allows rates to become arbitrarily negative in the model; or adding a shift to a lognormal density to push it into the region of positive rates. In this article, the authors document the problem of quoting swaptions when rates are negative and compare the three alternatives for calibrating standard interest rate models empirically.

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

In order to value a bond with one or more embedded options, one must consider the volatility of interest rates, as changes in volatility will affect the possibility of the option being exercised. The valuation of bonds with embedded options can be achieved by constructing a binomial or trinomial interest rate tree that models the random evolution of future interest rates and then by using backward-induction to generate the value of the bond at each node. No-arbitrage models such as the Black-Derman-Toy (BDT) model are more commonly used in practice to price financial instruments with embedded options or contingent claims. In this paper therefore, we explore how one of the no-arbitrage short rate models, the BDT model, can be implemented in a practical and reproducible manner. This is the first in our series of discussions on short-rate models. In future essays, we will discuss the implementation of other short-rate models namely the Hull-White, Ho-Lee, Kalotay-Fabozzi-Williams and the Black-Karansinski models of interest rates.

ASYMPTOTICS OF BOND YIELDS AND VOLATILITIES FOR EXTENDED VASICEK MODELS UNDER THE REAL-WORLD MEASURE

Vasicek's short rate model is a mean reverting model of the short rate which permits closed-form pricing formulae of zero coupon bonds and options on zero coupon bonds. This paper supplies proofs which are valid for any single factor mean reverting Gaussian short rate model having time-inhomogeneous parameters. The formulae are for the expected present value of payoffs under the real-world probability measure, known as actuarial pricing. Importantly, we give formulae for asymptotic levels of bond yields and volatilities for extended Vasicek models when suitable conditions are imposed on the model parameters.

Sensitivities under G2++ model of the yield curve

The two-additive-factor Gaussian model G2[Formula: see text] is a famous stochastic model for the instantaneous short rate. It has functional qualities required in various practical purposes, as in Asset Liability Management and in Trading of interest rate derivatives. Though closed formulas for the prices of various main interest-rate instruments are known and used under the G2[Formula: see text] model, it seems that references for the corresponding sensitivities are not clearly presented over the financial literature. To fill this gap is one of our purposes in the present work. We derive here analytic expressions for the sensitivities of zero-coupon bond, coupon-bearing bonds, portfolio of coupon bearing bonds. The sensitivities under consideration here are those with respect to the shocks linked to the unobservable two-uncertainty shock risk/opportunity factors underlying the G2[Formula: see text] model. As a such, the hedging of a position sensitive to the interest rate by means of a portfolio (in accordance with the market participants practice) becomes easily transparent as just resulting from the balance between the various involved sensitivities.

Valuación de la viabilidad de un cambio tecnológico en México utilizando opciones reales

This paper deals with the economic valuation of the electric energy generation system’s technological change in Mexico. Such change is aimed at reducing the pollution resulting from the electric energy generation process, and/or to optimize its operating and production costs, through the substitution of the carbo-electric energy generation system, for a nucleo-electric one, using different types of real options, namely, a technological change real option and a project expansion one. In order to calculate the term structure of risk free interest rate, the short rate model proposed by Cox, Toss and Rubinstein (1979) is used.

Consistency of Extended Nelson-Siegel Curve Families with the Ho-Lee and Hull and White Short Rate Models

Nelson and Siegel curves are widely used to fit the observed term structure of interest rates in a particular date. By the other hand, several interest rate models have been developed such their initial forward rate curve can be adjusted to any observed data, as the Ho-Lee and the Hull and White one factor models. In this work we study the evolution of the forward curve process for each of these models assuming that the initial curve is of Nelson-Siegel type. We conclude that the forward curve process produces curves belonging to a parametric family of curves that can be seen as extended Nelson and Siegel curves. We show that the forward rate curve evolution has a linear or an exponential growth, depending on the particular short rate interest model. We applied the results to Argentinian short and forward rates obtained from the Lebac’s bills yields using the Hull and White short rate model, showing a good estimation of the observed forward rate curve for near dates when the initial forward curve is adjusted with a Nelson and Siegel one.

Dynamic Refinement of the Term Structure - Time Homogenous Term Structure Modeling

In this note we consider a classical term structure model framework, that is, a HJM framework on a time-discrete tenor, like the LIBOR market model, using a sequence of tenor discretization, where the tenors are valid for a specific simulation time interval.The setup then allows to model dynamic refinements of the tenor structure and, as a special case, a quasi time-homogenous tenor structure.A time-homogenous tenor structure has some relevance in exposure simulations, where two requirements come together: it is desirable to model a finer tenor structure on the short end of the rate curve compared to the long end of the rate curve and, this property should persist in the simulation at a future time.The property is easily fulfilled by model, which uses an equally fine discretization at all times, e.g., a short rate model, where simulation time discretization and tenor time discretization coincide.As we will demonstrate, a refinement of the tenor structure via a simple interpolation of forward rates would either introduce a strong restriction on the model's volatility structure (as it is the case for a classical fixed tenor LIBOR market model) or introduce an arbitrage violation. The challenge in the refinement is to simulate the right stochastic drifts. Under a (milder) condition on the model's volatility structure, the drifts can be reconstructed using a single additional state variable. The additional state variable is only needed on the coarse discretization tenors, limiting the computational resources needed to implement the model.In a limit case, the approach can be used to glue together a short rate model for the short end of the rate curve and a term-structure model for the long end of the rate curve in a time-homogenous way.

Bayesian testing for short term interest rate models

In empirical finance, interest rate models have been widely used for modeling short-term interest rate. Under the framework of the hypothesis testing, this paper provides a Bayesian approach for comparing a range of alternative models. These compared models are nested in a general single-factor diffusion process for the short-term interest rate, with each alternative model indexed by the level effect parameter for the volatility. The performance of the developed procedure is illustrated by an empirical example of Eurodollar deposit rates.

Scenario Generation for Long-Run Interest Rate Risk Assessment

We propose a statistical model of the term structure of sovereign yields tailored for long-term probability-based scenario generation and forecasts. While being simple to estimate, our model is able to reproduce simultaneously the positivity of the yield curve, high persistence, factor structure and time varying volatilities and correlations. It features a regime switching short rate model. A complete benchmark of the model following Diebold and Li is performed in terms of forecasting ability and coverage properties. We show that the proposed model improves performance relative to a standard model from the literature.

Option pricing under the Merton model of the short rate in subdiffusive Brownian motion regime

ABSTRACTIn this paper, we propose an extension of the Merton short rate model, which reflects the subdiffusive nature of the short rate dynamics. The subdiffusive property is manifested by the random periods of time, during which the asset price does not change. We derive explicit formulas for European call and put options and present some simulation results for the case of α stable. Moreover, we discuss the implied volatility of this model.

An infinite hidden Markov model for short-term interest rates

Abstract 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 a 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.

Counterparty Credit Exposures for Interest Rate Derivatives using the Stochastic Grid Bundling Method

ABSTRACTThe regulatory credit value adjustment (CVA) for an outstanding over-the-counter (OTC) derivative portfolio is computed based on the portfolio exposure over its lifetime. Usually, the future portfolio exposure is approximated using the Monte Carlo simulation, as the portfolio value can be driven by several market risk-factors. For derivatives, such as Bermudan swaptions, that do not have an analytical approximation for their Mark-to-Market (MtM) value, the standard market practice is to use the regression functions from the least squares Monte Carlo method to approximate their MtM along simulated scenarios. However, such approximations have significant bias and noise, resulting in inaccurate CVA charge. In this paper, we extend the Stochastic Grid Bundling Method (SGBM) for the one-factor Gaussian short rate model, to efficiently and accurately compute Expected Exposure, Potential Future exposure and CVA for Bermudan swaptions. A novel contribution of the paper is that it demonstrates how different measures, for instance spot and terminal measure, can simultaneously be employed in the SGBM framework, to significantly reduce the variance and bias of the solution.

THE PRICING OF QUANTO OPTIONS UNDER THE VASICEK'S SHORT RATE MODEL

We derive a closed-form expression for the price of a Euro- pean quanto call option when both foreign and domestic interest rates follow the Vasicek's short rate model. A quanto is a type of financial derivative whose pay-out currency differs from the natural denomination of its underlying financial variable. A quanto option is a cash-settled, cross-currency derivative whose underlying asset has a payoff in one currency, but the payoff is converted to another currency when the option is settled. For that reason, the correlation between underlying asset and currency exchange rate plays an important role in pricing quanto option. Quanto options in this paper have both the strike price and the underlying asset price denominated in foreign currency. At exercise, the value of the option is calculated as the options intrinsic value in foreign currency, which is then converted to the domestic currency at the fixed exchange rate. This allows investors to obtain exposure to foreign assets without the corresponding foreign exchange risk. Pricing quanto options based on the classical Black-Scholes (1) model, on which most of the research on quanto options has focused, has a weakness of assuming a constant volatility and constant interest rates. To overcome such weakness, in valuing quanto option, it is natural to consider a stochastic volatility or stochastic interest rate models. Despite its importance, very few researches have been done on finding analytic solutions of quanto option prices under a stochastic volatility model primarily due to the sophisticated stochastic processes and inability to obtain the general closed form. However, by assuming constant interest rates, Giese (4) provided a closed-form expression for the price of a quanto option in the Stein-Stein stochastic volatility model, and then Y. Lee et al. (5) got a closed-form expression for the price of a European quanto

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.