Asymmetric Mean Reversion in European Interest Rates: A Two-factor Model

Abstract

This paper tests for asymmetric mean reversion in European short-term interest rates using a combination of the interest rate models introduced by Longstaff and Schwartz (Longstaff, F.A., Schwarts, E.S. (1992) Interest rate volatility and the ferm structure: A two factor general equilibrium model, Journal of Finance, 48, pp. 1259–1282.) and Bali (Bali, T. (2000) Testing the empirical performance of stochastic volatility models of the short-term interest rates, Journal of Financial and Quantitative Analysis, 35, pp. 191–215.). Using weekly rates for France, Germany and the United Kingdom, it is found that short-term rates follow in all instances asymmetric mean reverting processes. Specifically, interest rates exhibit non-stationary behavior following rate increases, but they are strongly mean reverting following rate decreases. The mean reverting component is statistically and economically stronger thus offsetting non-stationarity. Volatility depends on past innovations past volatility and the level of interest rates. With respect to past innovations volatility is asymmetric rising more in response to positive innovations. This is exactly opposite to the asymmetry found in stock returns.     

Pricing of guaranteed minimum withdrawal benefits in variable annuities under stochastic volatility,stochastic interest rates and stochastic mortality via the componentwise splitting method

This paper values guaranteed minimum withdrawal benefit (GMWB) riders embedded in variable annuities assuming that the underlying fund dynamics evolve under the influence of stochastic interest rates, stochastic volatility, stochastic mortality and equity risk. The valuation problem is formulated as a partial differential equation (PDE) which is solved numerically by employing the operator splitting method. Sensitivity analysis of the fair guarantee fee is performed with respect to various model parameters. We find that (i) the fair insurance fee charged by the product provider is an increasing function of the withdrawal rate; (ii) the GMWB price is higher when stochastic interest rates and volatility are incorporated in the model, compared to the case of static interest rates and volatility; (iii) the GMWB price behaves non-monotonically with changing volatility of variance parameter; (iv) the fair fee increases with increasing volatility of interest rates parameter, and increasing correlation between the underlying fund and the interest rates; (v) the fair fee increases when the speed of mean-reversion of stochastic volatility or the average long-term volatility increases; (vi) the GMWB fee decreases when the speed of mean-reversion of stochastic interest rates or the average long-term interest rates increase. We investigate both static and dynamic (optimal) policyholder's withdrawal behaviours; we present the optimal withdrawal schedule as a function of the withdrawal account and the investment account for varying volatility and interest rates. When incorporating stochastic mortality, we find that its impact on the fair guarantee fee is rather small. Our results demonstrate the importance of correct quantification of risks embedded in GMWBs and provide guidance to product providers on optimal hedging of various risks associated with the contract.     

News Arrival, Jump Dynamics, and Volatility Components for Individual Stock Returns

This paper models components of the return distribution, which are assumed to be directed by a latent news process. The conditional variance of returns is a combination of jumps and smoothly changing components. A heterogeneous Poisson process with a time‐varying conditional intensity parameter governs the likelihood of jumps. Unlike typical jump models with stochastic volatility, previous realizations of both jump and normal innovations can feed back asymmetrically into expected volatility. This model improves forecasts of volatility, particularly after large changes in stock returns. We provide empirical evidence of the impact and feedback effects of jump versus normal return innovations, leverage effects, and the time‐series dynamics of jump clustering.     

Regime-switching stochastic volatility and short-term interest rates

In this paper, we introduce regime switching in a two-factor stochastic volatility (SV) model to explain the behavior of short-term interest rates. We model the volatility of short-term interest rates as a stochastic volatility process whose mean is subject to shifts in regime. We estimate the regime-switching stochastic volatility (RSV) model using a Gibbs Sampling-based Markov Chain Monte Carlo algorithm. In-sample results strongly favor the RSV model in comparison to the single-state SV model and Generalized Autoregressive Conditional Heteroscedasticity (GARCH) family of models. Out-of-sample results are mixed and, overall, provide weak support for the RSV model.     

Can negative interest rates really affect option pricing? Empirical evidence from an explicitly solvable stochastic volatility model

The profound financial crisis generated by the collapse of Lehman Brothers and the European sovereign debt crisis in 2011 have caused negative values of government bond yields both in the USA and in the EURO area. This paper investigates whether the use of models which allow for negative interest rates can improve option pricing and implied volatility forecasting. This is done with special attention to foreign exchange and index options. To this end, we carried out an empirical analysis on the prices of call and put options on the US S&P 500 index and Eurodollar futures using a generalization of the Heston model in the stochastic interest rate framework. Specifically, the dynamics of the option’s underlying asset is described by two factors: a stochastic variance and a stochastic interest rate. The volatility is not allowed to be negative, but the interest rate is. Explicit formulas for the transition probability density function and moments are derived. These formulas are used to estimate the model parameters efficiently. Three empirical analyses are illustrated. The first two show that the use of models which allow for negative interest rates can efficiently reproduce implied volatility and forecast option prices (i.e. S&P index and foreign exchange options). The last studies how the US three-month government bond yield affects the US S&P 500 index.     

Carry Trades and Global Foreign Exchange Volatility

We investigate the relation between global foreign exchange (FX) volatility risk and the cross section of excess returns arising from popular strategies that borrow in low interest rate currencies and invest in high interest rate currencies, so‐called “carry trades.” We find that high interest rate currencies are negatively related to innovations in global FX volatility, and thus deliver low returns in times of unexpected high volatility, when low interest rate currencies provide a hedge by yielding positive returns. Furthermore, we show that volatility risk dominates liquidity risk and our volatility risk proxy also performs well for pricing returns of other portfolios.     

Nonlinear Drift And Stochastic Volatility: An Empirical Investigation Of Short-Term Interest Rate Models

In this article I provide new evidence on the role of nonlinear drift and stochastic volatility in interest rate modeling. I compare various model specifications for the short‐term interest rate using the data from five countries. I find that modeling the stochastic volatility in the short rate is far more important than specifying the shape of the drift function. The empirical support for nonlinear drift is weak with or without the stochastic volatility factor. Although a linear drift stochastic volatility model fits the international data well, I find that the level effect differs across countries.     

Smile and default: the role of stochastic volatility and interest rates in counterparty credit risk

In this research, we investigate the impact of stochastic volatility and interest rates on counterparty credit risk (CCR) for FX derivatives. To achieve this we analyse two real-life cases in which the market conditions are different, namely during the 2008 credit crisis where risks are high and a period after the crisis in 2014, where volatility levels are low. The Heston model is extended by adding two Hull–White components which are calibrated to fit the EURUSD volatility surfaces. We then present future exposure profiles and credit value adjustments (CVAs) for plain vanilla cross-currency swaps (CCYS), barrier and American options and compare the different results when Heston-Hull–White-Hull–White or Black–Scholes dynamics are assumed. It is observed that the stochastic volatility has a significant impact on all the derivatives. For CCYS, some of the impact can be reduced by allowing for time-dependent variance. We further confirmed that Barrier options exposure and CVA is highly sensitive to volatility dynamics and that American options’ risk dynamics are significantly affected by the uncertainty in the interest rates.     

Linear-quadratic term structure models – Toward the understanding of jumps in interest rates

We study linear-quadratic term structure models with random jumps in the short rate process where the jump arrival rate follows a stochastic process. Empirical results based on the US data show that incorporating stochastic jump intensity significantly improves model fit to the dynamics of both interest rate and volatility term structure. Our results also show that jump intensity is negatively correlated with interest rate changes and the average size is larger on the downside than upside. Examining the relation between jump intensity and macroeconomic shocks, we find that at monthly frequency, jumps are neither triggered by nor predictive of changes in macroeconomic variables. At daily frequency, however, we document interesting patterns for jumps associated with information shocks.     

Real-world jump-diffusion term structure models

This paper considers interest rate term structure models in a market attracting both continuous and discrete types of uncertainty. The event-driven noise is modelled by a Poisson random measure. Using as numeraire the growth optimal portfolio, interest rate derivatives are priced under the real-world probability measure. In particular, the real-world dynamics of the forward rates are derived and, for specific volatility structures, finite-dimensional Markovian representations are obtained. Furthermore, allowing for a stochastic short rate in a non-Markovian setting, a class of tractable affine term structures is derived where an equivalent risk-neutral probability measure may not exist.     

Testing the local volatility assumption: a statistical approach

In practice, the choice of using a local volatility model or a stochastic volatility model is made according to their respective ability to fit implied volatility surfaces. In this paper, we adopt a different point of view. Indeed, using a purely statistical methodology, we design new procedures aiming at testing the assumption of a local volatility model for the price dynamics, against the alternative of a stochastic volatility model. These test procedures are based only on historical data and do not require any calibration procedures via option prices. We also provide a convincing simulation study and an empirical analysis on future contracts on interest rates.     

Interest Rate Caps “Smile” Too! But Can the LIBOR Market Models Capture the Smile?

Using 3 years of interest rate caps price data, we provide a comprehensive documentation of volatility smiles in the caps market. To capture the volatility smiles, we develop a multifactor term structure model with stochastic volatility and jumps that yields a closed‐form formula for cap prices. We show that although a three‐factor stochastic volatility model can price at‐the‐money caps well, significant negative jumps in interest rates are needed to capture the smile. The volatility smile contains information that is not available using only at‐the‐money caps, and this information is important for understanding term structure models.     

The time-varying asymmetry of exchange rate returns: A stochastic volatility – stochastic skewness model

While the time-varying volatility of financial returns has been extensively modelled, most existing stochastic volatility models either assume a constant degree of return shock asymmetry or impose symmetric model innovations. However, accounting for time-varying asymmetry as a measure of crash risk is important for both investors and policy makers. This paper extends a standard stochastic volatility model to allow for time-varying skewness of the return innovations. We estimate the model by extensions of traditional Markov Chain Monte Carlo (MCMC) methods for stochastic volatility models. When applying this model to the returns of four major exchange rates, skewness is found to vary substantially over time. In addition, stochastic skewness can help to improve forecasts of risk measures. Finally, the results support a potential link between carry trading and crash risk.     

The volatility target effect in structured investment products with capital protection

Designing a structured investment product with capital protection which would be characterized by high capital protection level as well as high equity participation rate is a challenging task in the current market environment. Low interest rates and high volatility levels negatively affect the above key parameters of such investment products. One way to increase the participation rate of a structured investment product with a fixed capital protection level is to use a volatility target (VolTarget) strategy as an underlying asset for a financial option embedded in such a product. We introduce an extended VolTarget mechanism with interest rate dependent volatility target levels and provide a detailed comparative numerical study of European options linked to VolTarget strategies within a hybrid Heston–Vasi?ec model with stochastic volatility and stochastic interest rate.     

Regime shifts in interest rate volatility

I find evidence of regime shifts in interest rate volatility using short-rate data from the U.S., the U.K., Japan, and Canada. The regime shifts, if unaccounted for, could lead to spurious volatility persistence when the volatility processes are estimated with the stochastic volatility (SVOL) model. In contrast, the apparent persistence in volatility drops sharply in three out of the four countries when I estimate the volatility processes with the regime-switching stochastic volatility (RSSV) model. I also contribute to the literature by showing how to account for correlation in the regime-switching stochastic volatility model, which is important for modeling asymmetric volatility.     

Double-jump diffusion model for VIX: evidence from VVIX

This paper studies the continuous-time dynamics of VIX with stochastic volatility and jumps in VIX and volatility. Built on the general parametric affine model with stochastic volatility and jumps in the logarithm of VIX, we derive a linear relationship between the stochastic volatility factor and the VVIX index. We detect the existence of a co-jump of VIX and VVIX and put forward a double-jump stochastic volatility model for VIX through its joint property with VVIX. Using the VVIX index as a proxy for stochastic volatility, we use the MCMC method to estimate the dynamics of VIX. Comparing nested models of VIX, we show that the jump in VIX and the volatility factor are statistically significant. The jump intensity is also stochastic. We analyse the impact of the jump factor on VIX dynamics.     

Further evidence on alternative continuous time models of the short-term interest rate

This paper examines the stochastic behavior of the 1-month interbank rate in ten countries. Various one-factor models are estimated using an exact maximum likelihood estimator, which is based on the recently introduced Gaussian methodology. Interest rate volatility is found to be less sensitive to interest rate levels than stated in the literature. In addition, the constant elasticity variance (CEV) model is superior to other formulations in terms of data fit.     

Unspanned stochastic volatility and fixed income derivatives pricing

We propose a parsimonious ‘unspanned stochastic volatility’ model of the term structure and study its implications for fixed-income option prices. The drift and quadratic variation of the short rate are affine in three state variables (the short rate, its long-term mean and variance) which follow a joint Markov (vector) process. Yet, bond prices are exponential affine functions of only two state variables, independent of the current interest rate volatility level. Because this result holds for an arbitrary volatility process, such a process can be calibrated to match fixed income derivative prices. Furthermore, this model can be ‘extended’ (by relaxing the time-homogeneity) to fit any arbitrary term structure. In its ‘HJM’ form, this model nests the analogous stochastic equity volatility model of Heston (1993) [Heston, S.L., 1993. A closed form solution for options with stochastic volatility. Review of Financial Studies 6, 327–343]. In particular, if the volatility process is specified to be affine, closed-form solutions for interest rate options obtain. We propose an efficient algorithm to compute these prices. An application using data on caps and floors shows that the model can capture very well the implied Black spot volatility surface, while simultaneously fitting the observed term structure.     

Common stochastic volatility trends in international stock returns

The aim of this work is to capture common stochastic trends in weekly volatilities of the Dow Jones, Nikkei, Hang Seng and Strait Times index using a multivariate stochastic volatility (SV) model. The results suggest a very high correlation among the volatility innovations, so that it is examined whether the four series share any common stochastic trends. A Principal Component Analysis and a Factor Analysis in the state space setting reveal that two common stochastic trends can be found to underlie the volatility series. The resulting linear combinations of the volatility series no more exhibit any stochastic trend but are stationary in the state space framework. Thus, it can be concluded that volatilities of the four stock indexes are in essence co-persistent.     

A Complete Markovian Stochastic Volatility Model in the HJM Framework

This paper considers a stochastic volatility version of the Heath, Jarrow and and Morton (1992) term structure model. Market completeness is obtained by adapting the Hobson and Rogers (1998) complete stochastic volatility stock market model to the interest rate setting. Numerical simulation for a special case is used to compare the stochastic volatility model against the traditional Vasicek (1977) model.