Short Rate Volatility Research Articles

One‐Factor Hull—White Model Calibration for CVA Part I: Instrument Selection with a Kink

This paper is the first of a multi-part series on the calibration of the one-factor Hull-White short rate model for the purpose of computing CVAs (and xVAs) with an xVA system. It introduces an atypical bootstrapping scheme for the calibration of the short rate volatility. The second part focuses on the selection of the mean reversion parameter. In both expositions we present long-term time series results for EUR, JPY, and USD, covering the period from the beginning of 2009 (at the earliest) to spring 2020.

One‐Factor Hull—White Model Calibration for CVA Part II: Optimizing the Mean Reversion Parameter

This paper is the second of a multi-part series on the calibration of the one-factor Hull—White short rate model for the purpose of computing CVAs (and XVas) with an XVa system. The first part introduces an atypical bootstrapping scheme for the calibration of the short rate volatility. The present second part focuses on the selection of the mean reversion parameter. In both expositions we present long-term time series results for EUR, JPY, and USD, covering the period from the beginning of 2009 (at the earliest) to spring 2020.

Bermudan option in Singapore Savings Bonds

The Singapore Savings Bonds (SSB) is a unique investment program offered by the Singapore government whereby retail investors can earn risk-free tax-free step-up interest closely matched to Treasury bond rates for up to 10 years and can redeem on any business day prior to maturity without any early redemption penalty. This study analyses the unique design of the SSB and provides a valuation of the Bermudan option for early redemption that is embedded in the SSB. The Black–Derman–Toy model is used to build the interest rate tree, and an iterative method is employed to avoid arbitrary specification of the pre-determined short rate volatility function. This bespoke Bermudan option can have changing strike prices over time. It also has a novel characteristic whereby the value of exercise to a buyer need not equal to the cost of being exercised to a seller. Better understanding of embedded options within government savings bonds leads to innovative designs that may encourage effective citizens’ savings.

One-factor Hull-White Model Calibration for CVA - Part II: Optimizing the Mean Reversion Parameter

This paper is the second of a multi-part series on the calibration of the one-factor Hull-White short rate model for the purpose of computing CVAs (and xVAs) with an xVA system. The first part introduces an atypical bootstrapping scheme for the calibration of the short rate volatility. The present second part focuses on the selection of the mean reversion parameter. In both expositions we present long-term time series results for EUR, JPY, and USD, covering the period from the beginning of 2009 (at the earliest) to spring 2020.

One-factor Hull-White Model Calibration for CVA - Part I: Instrument Selection With a Kink

This paper is the first of a multi-part series on the calibration of the one-factor Hull-White short rate model for the purpose of computing CVAs (and xVAs) with an xVA system. It introduces an atypical bootstrapping scheme for the calibration of the short rate volatility. The second part focuses on the selection of the mean reversion parameter. In both expositions we present long-term time series results for EUR, JPY, and USD, covering the period from the beginning of 2009 (at the earliest) to spring 2020.

American Put Under Stochastic Rates

We prove that in calibrated diffusive models with non-negative spot-rate correlation the calibrated American put values decrease with increasing short rate volatility and attain their maximum in the pure local volatility model. The result holds for Bermudan put options.

How to Extract Short-Rate Expectations from the Extended Vasicek Model

This paper deals with how reasonable it is to accept the pure expectations hypothesis as a guide to the expected future development of the economy as indicated by the expected future short rate. At this aim it tries to extract short rate expectations from the observable term structure of interest rates, short rate volatility and market prices of bund-futures options within the framework of an extended version of Vasicek’s prominent 1977 one-factor-no-arbitrage model. Doing so has its merits in teaching too because it allows to precisely disentangle the interwined concepts of the market price of zerobond price risk on the one hand and term or liquidity premium on the other.

Modelling and Forecasting Short-Term Interest Rate Volatility: A Semiparametric Approach

This paper employs a semiparametric procedure to estimate the diffusion process of short-term interest rate. This method is compared in its ability to capture the dynamics of short rate volatility to a class of one-factor diffusion models where the conditional variance is serially correlated and levels dependent. The Monte Carlo study shows that the semiparametric approach produces more accurate volatility estimates than models that accommodate asymmetry, levels effect and serial dependence in the conditional variance. The empirical evidence based on U.S. three-month Treasury bill rates further indicates that the semiparametric procedure is better than the widely used single-factor diffusion models in forecasting the future volatility of interest rate changes. The improvement in modelling short rate volatility using the new procedure has implications for pricing interest rate derivatives.

Can interest rate volatility be extracted from the cross section of bond yields?

Most affine models of the term structure with stochastic volatility predict that the variance of the short rate should play a ‘dual role’ in that it should also equal a linear combination of yields. However, we find that estimation of a standard affine three-factor model results in a variance state variable that, while instrumental in explaining the shape of the yield curve, is essentially unrelated to GARCH estimates of the quadratic variation of the spot rate process or to implied variances from options. We then investigate four-factor affine models. Of the models tested, only the model that exhibits ‘unspanned stochastic volatility’ (USV) generates both realistic short rate volatility estimates and a good cross-sectional fit. Our findings suggest that short rate volatility cannot be extracted from the cross-section of bond prices. In particular, short rate volatility and convexity are only weakly correlated.

Level-ARCH short rate models with regime switching: Bivariate modeling of US and European short rates

This paper introduces regime switching into level-ARCH models for the short rates of the US, the UK, and Germany. Once regime switching and level effects are included there are no gains from including ARCH effects. It is of secondary importance how the regime switching is specified. The estimated level parameters differ across countries. The corresponding new bivariate models show that the states of the US and UK short rate volatilities are not independent nor identical. There is Granger causality from the US to the UK short rate volatility state but not vice versa. There is no contagion between the US and UK volatility states. Equivalent results apply to the relation between the US and German volatility states.

Level-ARCH Short Rate Models with Regime Switching: Bivariate Modeling of US and European Short Rates

This paper introduces regime switching into level-ARCH models for the short rates of the US, the UK, and Germany. Once regime switching and level effects are included there are no gains from including ARCH effects. It is of secondary importance how the regime switching is specified. The estimated level parameters differ across countries. The corresponding new bivariate models show that the states of the US and UK short rate volatilities are not independent nor identical. There is Granger causality from the US to the UK short rate volatility state but not vice versa. There is no contagion between the US and UK volatility states. Equivalent results apply to the relation between the US and German volatility states.

Calibrating the Black-Derman-Toy model: some theoretical results

The Black-Derman-Toy (BDT) model is a popular one-factor interest rate model that is widely used by practitioners. One of its advantages is that the model can be calibrated to both the current market term structure of interest rate and the current term structure of volatilities. The input term structure of volatility can be either the short term volatility or the yield volatility. Sandmann and Sondermann derived conditions for the calibration to be feasible when the conditional short rate volatility is used. In this paper conditions are investigated under which calibration to the yield volatility is feasible. Mathematical conditions for this to happen are derived. The restrictions in this case are more complicated than when the short rate volatilities are used since the calibration at each time step now involves the solution of two non-linear equations. The theoretical results are illustrated by showing numerically that in certain situations the calibration based on the yield volatility breaks down for apparently plausible inputs. In implementing the calibration from period n to period n + 1, the corresponding yield volatility has to lie within certain bounds. Under certain circumstances these bounds become very tight. For yield volatilities that violate these bounds, the computed short rates for the period (n, n + 1) either become negative or else explode and this feature corresponds to the economic intuition behind the breakdown.

Volatility of the short rate in the rational lognormal model

A recent article of Flesaker and Hughston introduces a one factor interest rate model called the rational lognormal model. This model has a lot to recommend it including guaranteed finite positive interest rates and analytic tractability. Consequently, it has received a lot of attention among practioners and academics alike. However, it turns out to have the undesirable feature of predicting that the asymptotic value of the short rate volatility is zero. This theoretical result is proved rigorously in this article. The outcome of an empirical study complementing the theoretical result is discussed at the end of the article. European call options are valued with the rational lognormal model and a comparably calibrated mean reverting Gaussian model. unsurprisingly, rational lognormal option values are considerably lower than the analogous mean reverting Gaussian option values. In other words, the volatility in the rational lognormal model declines so quickly that options are severely undervalued.

The term structure of real interest rates and the Cox, Ingersoll, and Ross model

This paper estimates real term structures from cross-sections of British government index-linked (‘realrd) bond prices. The Cox, Ingersoll, and Ross (1985) model is then fitted to the same data; the model closely approximates the shapes of the directly-estimated term structures. In contrast to similar studies of the nominal term structure, the long-term zero-coupon yield is quite stable, as the CIR model predicts, and in common with previous studies, the level of implied short rate volatility corresponds well with time series estimates. The other parameters, however, are often highly correlated and intertemporal parameter stability is rejected.