Short Rate Dynamics Research Articles

TERM STRUCTURE MODELING OF SOFR: EVALUATING THE IMPORTANCE OF SCHEDULED JUMPS

As interest rate benchmarks move from LIBOR to overnight risk-free rates (RFR), it has become increasingly important for models to accurately capture the interest rate dynamics at the overnight tenor. Overnight rates closely track central bank policy rate decisions resulting, in highly discontinuous dynamics around scheduled meeting dates. In this paper, we construct a dynamic term structure model, which accounts for the discontinuous short-rate dynamics. We show that the model is able to jointly fit the overnight US policy rate, secured overnight financing rate (SOFR) and SOFR futures rates through the recent Fed hiking cycle. Comparing our model with a standard continuous time-homogeneous short-rate model, we find several indications that our model avoids the clear misspecification of the continuous model, in particular with regard to the short-rate dynamics around meeting dates of the Federal Open Market Committee (FOMC). Thiseffect begins to disappear as the term of the rates under consideration is increased, suggesting that diffusive dynamics are a reasonably accurate reflection of the evolution of market expectations embodied in longer-term interest rates.

Options on bonds: implied volatilities from affine short-rate dynamics

We derive an explicit asymptotic approximation for the implied volatilities of Call options written on bonds assuming the short-rate is described by an affine short-rate model. For specific affine short-rate models, we perform numerical experiments in order to gauge the accuracy of our approximation.

On a convergent power series method to price defaultable bonds in a Vasicek-CIR model

In this paper, we prove that the price of a defaultable bond, under a Vasicek short rate dynamics coupled with a Cox-Ingersoll-Ross default intensity model, is a real analytic function, in a neighborhood of the origin, of the correlation parameter between the Brownian motions driving the processes, used to express the dependence between the short rate and the default intensity of the bond issuer. Employing conditioning and a change of numéraire technique, we obtain a manageable representation of the bond price in this non-affine model which allows us to control its derivatives and assess the convergence of the series. By truncating the expansion at the second order, a quadratic approximation formula for the price is then provided. Finally, practical applications of the result are highlighted by performing a numerical comparison with alternative pricing methodologies.

Short Rate Dynamics: A Fed Funds and SOFR perspective

The Secured Overnight Funding Rate (SOFR) is becoming the main Risk–Free Rate benchmark in US dollars, thus interest rate term structure models need to be updated to reflect the key features exhibited by the dynamics of SOFR and the forward rates implied by SOFR futures. Historically, interest rate term structure modelling has been based on rates of substantially longer time to maturity than overnight, but with SOFR the overnight rate now is the primary market observable. This means that the empirical idiosyncrasies of the overnight rate cannot be ignored when constructing interest rate models in a SOFR–based world. As a rate reflecting transactions in the Treasury overnight repurchase market, the dynamics of SOFR are closely linked to the dynamics Effective Federal Funds Rate (EFFR), which is the interest rate most directly impacted by US monetary policy target rate decisions. Therefore, these rates feature jumps at known times (Federal Open Market Committee meeting dates), and market expectations of these jumps are reflected in prices for futures written on these rates. On the other hand, forward rates implied by Fed Funds and SOFR futures continue to evolve diffusively. The model presented in this paper reflects the key empirical features of SOFR dynamics and is calibrated to futures prices. In particular, the model reconciles diffusive forward rate dynamics with piecewise constant paths of the target short rate.

Critical analysis of the most widespread methodologies for the simulation of the short rate dynamics under extreme market conditions

This study proposes an analysis of the main drawbacks emerged when adopting the traditional short rate dynamics under extreme market conditions such as under negative interest rates. In fact, this condition has led to invalidate the use of the majority of the most popular stochastic differential equations (SDE) reported in the scientific literature. The first part of the paper makes an overall introduction to the problem, analyzing it from different perspectives. Given that the author dealt with these topics in previous research items, the objective of the paper is to focus on only one of these aspects which was not scrutinized before and to examine it in full detail. As a result, the most popular stochastic dynamics are shown in the second part and the problems raised by their numerical integration are then discussed. Starting from the equations for which it is feasible to implement a numerical scheme for their solution, the problem thus becomes how to find a reasonable estimation for the SDE parameters. The third section deals with the problems occurred in the application of Maximum Likelihood Estimation (MLE) caused by the negative interest rates during the implementation of the well-established approaches. For every analyzed dynamics a real market case is provided. The paper highlights the appropriateness of the Hull-White model which can be considered a feasible and reliable solution for simulating short rates also under extreme market conditions.

Estimating Stochastic Volatility under the Assumption of Stochastic Volatility of Volatility

We propose novel nonparametric estimators for stochastic volatility and the volatility of volatility. In doing so, we relax the assumption of a constant volatility of volatility and therefore, we allow the volatility of volatility to vary over time. Our methods are exceedingly simple and far simpler than the existing ones. Using intraday prices for the Standard & Poor’s 500 equity index, the estimates revealed strong evidence that both volatility and the volatility of volatility are stochastic. We also proceeded in a Monte Carlo simulation analysis and found that the estimates were reasonably accurate. Such evidence implies that the stochastic volatility models proposed in the literature with constant volatility of volatility may fail to approximate the discrete-time short rate dynamics.

Oracle Properties, Bias Correction, and Bootstrap Inference for Adaptive Lasso for Time Series M‐Estimators

We derive new theoretical results on the properties of the adaptive least absolute shrinkage and selection operator (adaptive lasso) for possibly nonlinear time series models. In particular, we investigate the question of how to conduct inference on the parameters given an adaptive lasso model. Central to this study is the test of the hypothesis that a given adaptive lasso parameter equals zero, which therefore tests for a false positive. To this end, we introduce a recentered bootstrap procedure and show, theoretically and empirically through extensive Monte Carlo simulations, that the adaptive lasso can combine efficient parameter estimation, variable selection, and inference in one step. Moreover, we analytically derive a bias correction factor that is able to significantly improve the empirical coverage of the test on the active variables. Finally, we apply the adaptive lasso and the recentered bootstrap procedure to investigate the relation between the short rate dynamics and the economy, thereby providing a statistical foundation (from a model choice perspective) for the classic Taylor rule monetary policy model.

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.

Expected Returns in Treasury Bonds

We study risk premium in U.S. Treasury bonds. We decompose Treasury yields into inflation expectations and maturity-specific interest-rate cycles, which we define as variation in yields orthogonal to expected inflation. The short-maturity cycle captures the real short-rate dynamics. Jointly with expected inflation, it comprises the expectations hypothesis (EH) term in the yield curve. Controlling for the EH term, we extract a measure of risk-premium variation from yields. The risk-premium factor forecasts excess bond returns in and out of sample and subsumes the common bond return predictor obtained as a linear combination of forward rates.

Analytic bond pricing for short rate dynamics evolving on matrix Lie groups

We provide closed-form expressions for bond prices in interest rate models based on compact Lie groups. Our approach uses a Doob transform technique and PDE solutions by the Mathieu periodic functions. As a by-product, we derive formulas for bond option prices as well as new identities for the Laplace transform of periodic functionals of Brownian motion and Brownian diffusion processes.

CSA Caps Convexity Impact on Hull & White Calibration

Prices of CSA instruments are observed in a world which is not natural for LIBOR based payoffs in the sense that LIBOR forward is not martingale under CSA-forward measure. From a theoretical perspective, pricing models for collateralized LIBOR based payoffs should account for CSA-convexity, implied when changing from CSA-forward measure to LIBOR-forward measure. In practice CSA-convexity effect is neglected. In the present paper we show how modeling jointly OIS and LIBOR using one factor guassian short rate dynamics allows to capture CSA-convexity on caps and we measure its impact on LIBOR volatility calibration in the Hull & White case.

CSA Caps Convexity Impact on Hull & White Calibration

Prices of CSA instruments are observed in a world which is not natural for LIBOR based payoffs in the sense that LIBOR forward is not martingale under CSA-forward measure. From a theoretical perspective, pricing models for collateralized LIBOR based payoffs should account for CSA-convexity, implied when changing from CSA-forward measure to LIBOR-forward measure. In practice CSA-convexity effect is neglected. In the present paper we show how modeling jointly OIS and LIBOR using one factor guassian short rate dynamics allows to capture CSA-convexity on caps and we measure its impact on LIBOR volatility calibration in the Hull & White case.

Does the Short Rate Revert to its Mean in the Risk-Neutral World?

We propose a methodology to calibrate the non-linear and non-affine short rate models under physical and risk-neutral probability measures, based on the U.S treasury yield curve data. For this purpose, we price the default-free bond using the Markov chain approximation method built on the well-established idea of local consistency. Integrating the standard recursive filtering algorithm and the Markov chain approximation bond price formula, we propose a Markov chain filter in which the transition equation is a general stochastic differential equation for the short rate dynamics, and the measurement equation can be based on observations of any interest rate derivatives. Our simulation experiments show that popular parameter estimators can generate spurious non-linear drift estimates. However, the proposed Markov chain filter does not. In contrast to the existing literature, through our empirical findings we argue that the short rate does not revert to its mean under the risk-neutral probability measure. Our findings corroborate the observations that yields on the short end term structure have remained near 0% for more than 30 months since the implementation of Quantitative Easing monetary policy.

THE CARMA INTEREST RATE MODEL

In this paper, we present a multi-factor continuous-time autoregressive moving-average (CARMA) model for the short and forward interest rates. This model is able to present an adequate statistical description of the short and forward rate dynamics. We show that this is a tractable term structure model and provides closed-form solutions to bond prices, yields, bond option prices, and the term structure of forward rate volatility. We demonstrate the capabilities of our model by calibrating it to a panel of spot rates and the empirical volatility of forward rates simultaneously, making the model consistent with both the spot rate dynamics and forward rate volatility structure.

Inflation Securities Valuation with Macroeconomic-Based No-Arbitrage Dynamics

We develop a model to price inflation and interest rates derivatives using continuous-time dynamics that have some links with macroeconomic monetary DSGE models equipped with a Taylor rule: in particular, the reaction function of the central bank, the bond market liquidity, inflation and growth expectations play an important role. The model can explain the effects of non-standard monetary policies (like quantitative easing or its tapering) and shed light on how central bank policy can affect the value of inflation and interest rates derivatives.The model is built under standard no-arbitrage assumptions. Interestingly, the model yields short rate dynamics that are consistent with a time-varying Hull-White model, therefore making the calibration to the nominal interest curve and options straightforward. Further, we obtain closed forms for both zero-coupon and year-on-year inflation swap and options. The calibration strategy we propose is fully separable, which means that the calibration can be carried out in subsequent simple steps that do not require heavy computation. A market calibration example is provided.The advantages of such structural inflation modelling become apparent when one starts doing risk analysis on an inflation derivatives book: because the model explicitly takes into account economic variables, a trader can easily assess the impact of a change in central bank policy on a complex book of fixed income instruments, which is normally not straightforward if one is using standard inflation pricing models.

Oracle Properties and Finite Sample Inference of the Adaptive Lasso for Time Series Regression Models

We derive new theoretical results on the properties of the adaptive least absolute shrinkage and selection operator (adaptive lasso) for time series regression models. In particular we investigate the question of how to conduct finite sample inference on the parameters given an adaptive lasso model for some fixed value of the shrinkage parameter. Central in this study is the test of the hypothesis that a given adaptive lasso parameter equals zero, which therefore tests for a false positive. To this end we construct a simple (conservative) testing procedure and show, theoretically and empirically through extensive Monte Carlo simulations, that the adaptive lasso combines efficient parameter estimation, variable selection, and valid finite sample inference in one step. Moreover, we analytically derive a bias correction factor that is able to significantly improve the empirical coverage of the test on the active variables. Finally, we apply the introduced testing procedure to investigate the relation between the short rate dynamics and the economy, thereby providing a statistical foundation (from a model choice perspective) to the classic Taylor rule monetary policy model.

Optimal bond portfolios with fixed time to maturity

We study interest rate models where the term structure is given by an affine relation and in particular where the driving stochastic processes are so-called generalized Ornstein–Uhlenbeck processes.For many institutional investors it is natural to consider investment in bonds where the time to maturity of the bonds in the portfolio is kept fixed over time. We show that the return and variance of such a portfolio of bonds which are continuously rolled over, also called rolling horizon bonds, can be expressed using the cumulant generating functions of the background driving Lévy processes associated with the OU processes. This allows us to calculate the efficient mean–variance portfolio. We exemplify the results by a case study on euro swap rates.We also show that if the short rate, in a risk-neutral setting, is given by a linear combination of generalized OU processes, the implied term structure can be expressed in terms of the cumulant generating functions. This makes it possible to quite easily see what kind of term structures can be generated with a particular short rate dynamics.

Understanding Bond Risk Premia

We study risk premium in US Treasury bonds. We decompose Treasury yields into inflation expectations and maturity-specific interest rate cycles, which we define as variation in yields orthogonal to expected inflation. The short-maturity cycle captures the real short-rate dynamics. Jointly with expected inflation, it comprises the expectations hypothesis (EH) term in the yield curve. Controlling for the EH term, we extract a measure of risk premium variation from yields. The risk premium factor forecasts excess bond returns in and out of sample and subsumes the common bond return predictor obtained as a linear combination of forward rates.

PRICING AND CALIBRATION OF A CHOOSER FLEXIBLE CAP

In this paper, we deal with no-arbitrage pricing problems of a chooser flexible cap written on an underlying LIBOR. The chooser flexible cap allows a right for a buyer to exercise a limited and pre-determined number of the interim period caplets in a multiple-period cap agreement. Assuming a common diffusion short rate dynamics, e.g., Hull–White model, we propose a dynamic programming approach for their risk neutral evaluation. This framework is suited to a calibration from an observed initial yield curve and market price data of discount bonds, caplets, and floorlets.

Short Rate Dynamics and Regime Shifts*

ABSTRACTWe characterize the dynamics of the US short‐term interest rate using a Markov regime‐switching model. Using a test developed by Garcia, we show that there are two regimes in the data: In one regime, the short rate behaves like a random walk with low volatility; in another regime, it exhibits strong mean reversion and high volatility. In our model, the sensitivity of interest rate volatility to the level of interest rate is much lower than what is commonly found in the literature. We also show that the findings of nonlinear drift in Aït‐Sahalia and Stanton, using nonparametric methods, are consistent with our regime‐switching model.