Discrete-time Stochastic Volatility Models Research Articles

A note on calculating expected shortfall for discrete time stochastic volatility models

In this paper we consider the problem of estimating expected shortfall (ES) for discrete time stochastic volatility (SV) models. Specifically, we develop Monte Carlo methods to evaluate ES for a variety of commonly used SV models. This includes both models where the innovations are independent of the volatility and where there is dependence. This dependence aims to capture the well-known leverage effect. The performance of our Monte Carlo methods is analyzed through simulations and empirical analyses of four major US indices.

The leverage effect puzzle revisited: Identification in discrete time

The term “leverage effect,” as coined by Black (1976), refers to the tendency of an asset’s volatility to be negatively correlated with the asset’s return. Aït-Sahalia et al. (2013) refer to the “leverage effect puzzle” as the fact that, in spite of a broad agreement that the effect should be present, it is hard to identify empirically. For this purpose, we propose an extension with leverage effect of the discrete time stochastic volatility model of Darolles et al. (2006). This extension is shown to be the natural discrete time analog of the Heston (1993) option pricing model. It shares with Heston (1993) the advantage of structure preserving change of measure: with an exponentially affine stochastic discount factor, the historical and the risk neutral models belong to the same family of joint probability distributions for return and volatility processes. The discrete time approach allows us to make the role of various parameters more transparent: leverage versus volatility feedback effect, connection with daily realized volatility, impact of leverage on the volatility smile, etc. Even more importantly it sheds some new light on the identification of leverage effect and of the various risk premium parameters through link functions in closed form. The price of volatility risk is identified from underlying asset return data, even without option price data, if and only if leverage effect is present. However, the link functions are almost flat if the leverage effect is close to zero, making estimation of the volatility risk price difficult and paving the way for identification robust inference.

A Stochastic Volatility Model With Realized Measures for Option Pricing

Based on the fact that realized measures of volatility are affected by measurement errors, we introduce a new family of discrete-time stochastic volatility models having two measurement equations relating both observed returns and realized measures to the latent conditional variance. A semi-analytical option pricing framework is developed for this class of models. In addition, we provide analytical filtering and smoothing recursions for the basic specification of the model, and an effective MCMC algorithm for its richer variants. The empirical analysis shows the effectiveness of filtering and smoothing realized measures in inflating the latent volatility persistence—the crucial parameter in pricing Standard and Poor’s 500 Index options.

A new class of discrete-time stochastic volatility model with correlated errors

ABSTRACTReturns of risky assets and their time-dependent volatility are often jointly modelled by stochastic volatility models (SVMs). Over the last few decades, several SVMs have been proposed to adequately capture the features of the relationship between the return and its volatility. The earliest SVM considers a hierarchical model, with current return as a function of the current latent volatility, which is further modelled as an auto-regressive process. In an attempt to make SVM more appropriate for complex realistic market behaviour, leverage parameter was introduced, which however led to violation of martingale difference property of the risky part of return (a necessary mean-zero condition that prevents arbitrage opportunities under mild regularity conditions). Subsequently, alternative SVMs had been developed and are currently in use. In this article, we propose mean-corrections for several generalizations of SVM with leverage that capture the complex market behaviour as well as satisfy martingale difference property of returns. We also establish a few theoretical results to characterize the key desirable features of these models, and present comparison with other popular competitors. Furthermore, three real-life examples (CITI bank stock price, euro–USD rate and S&P 500 index returns) have been used to demonstrate the performance of this new class of SVMs.

Double Gamma Stochastic Volatility Model in Discrete Time

In an affine term structure framework we propose a discrete time stochastic volatility model. We derive the characteristic function of the log swap rate under swap measure. Having the characteristic function, we employ the Fourier cosine (COS) technique to price swaptions. Using data on tweleve years of swap rates and swaption premiums, model parameters are estimated using an unscented Kalman filter algorithm.

A New Class of Discrete-Time Stochastic Volatility Model with Correlated Errors

In an efficient stock market, the returns and their time-dependent volatility are often jointly modeled by stochastic volatility models (SVMs). Over the last few decades several SVMs have been proposed to adequately capture the defining features of the relationship between the return and its volatility. Among one of the earliest SVM, Taylor (1982) proposed a hierarchical model, where the current return is a function of the current latent volatility, which is further modeled as an auto-regressive process. In an attempt to make the SVMs more appropriate for complex realistic market behavior, a leverage parameter was introduced in the Taylor's SVM, which however led to the violation of the efficient market hypothesis (EMH, a necessary mean-zero condition for the return distribution that prevents arbitrage possibilities). Subsequently, a host of alternative SVMs had been developed and are currently in use. In this paper, we propose mean-corrections for several generalizations of Taylor's SVM that capture the complex market behavior as well as satisfy EMH. We also establish a few theoretical results to characterize the key desirable features of these models, and present comparison with other popular competitors. Furthermore, four real-life examples (Oil price, CITI bank stock price, Euro-USD rate, and S&P 500 index returns) have been used to demonstrate the performance of this new class of SVMs.

Smile at Errors: A Discrete-Time Stochastic Volatility Framework for Pricing Options with Realized Measures

Motivated by the fact that realized measures of volatility are affected by measurement errors, we introduce a new family of discrete-time stochastic volatility models having two measurement equations relating both the observed returns and realized measures to the latent conditional variance. The first contribution of the paper is to provide an analytical filtering and smoothing recursions for the basic version of the model and an effective MCMC algorithm for richer versions featuring heterogeneous structure in both volatility and leverage. The second contribution is to present the first fully analytical option pricing framework for discrete-time stochastic volatility models.

A Note on the Asymptotic Normality of the Kernel Deconvolution Density Estimator with Logarithmic Chi-Square Noise

This paper studies the asymptotic normality for the kernel deconvolution estimator when the noise distribution is logarithmic chi-square; both identical and independently distributed observations and strong mixing observations are considered. The dependent case of the result is applied to obtain the pointwise asymptotic distribution of the deconvolution volatility density estimator in discrete-time stochastic volatility models.

Assessing the quality of volatility estimators via option pricing

The aim of this paper is to measure and assess the accuracy of different volatility estimators based on high frequency data in an option pricing context. For this, we use a discrete-time stochastic volatility model based on Auto-Regressive-Gamma (ARG) dynamics for the volatility. First, ARG processes are presented both under historical and risk-neutral measure, in an affine stochastic discount factor framework. The model parameters are estimated exploiting the informative content of historical high frequency data. Secondly, option pricing is performed via Monte Carlo techniques. This framework allows us to measure the quality of different volatility estimators in terms of mispricing with respect to real option data, leaving to the ARG volatility model the role of a tool. Our analysis points out that using high frequency intra-day returns allows to obtain more accurate ex post estimation of the true (unobservable) return variation than do the more traditional sample variances based on daily returns, and this is reflected in the quality of pricing. Moreover, estimators robust to microstructure effects show an improvement over the realized volatility estimator. The empirical analysis is conducted on European options written on S&P500 index.

Constructing a Class of Stochastic Volatility Models: Empirical Investigation with Vix Data

We propose a class of discrete-time stochastic volatility models that, in a parsimonious way, captures the time-varying higher moments observed in financial series. Three desirable results are obtained. First, we have a recursive procedure for the log-price characteristic function which allows a semi-analytical formula for option prices as in Heston and Nandi [2000]. Second, we reproduce some features of the VIX Index. Finally, we derive a simple formula for the VIX index and use it for option pricing.

Assessing the Quality of Volatility Estimators via Option Pricing

The aim of this paper is to measure and assess the accuracy of different volatility estimators based on high frequency data in an option pricing context. For this, we use a discrete-time stochastic volatility model based on Auto-Regressive-Gamma (ARG) dynamics for the volatility.First, ARG processes are presented both under historical and risk-neutral measure, in an affine stochastic discount factor framework. The model parameters are estimated exploiting the informative content of historical high frequency data. Secondly, option pricing is performed via Monte Carlo techniques. This framework allows us to measure the quality of different volatility estimators in terms of mispricing with respect to real option data, leaving to the ARG volatility model the role of a tool. Our analysis points out that using high frequency intra-day returns allows to obtain more accurate ex post estimation of the true (unobservable) return variation than do the more traditional sample variances based on daily returns, and this is reflected in the quality of pricing. Moreover, estimators robust to microstructure effects show an improvement over the realized volatility estimator. The empirical analysis is conducted on European options written on S&P500 index.

Stochastic Volatility Models with Application in Option Pricing

Abstract In this paper, we develop a new method for pricing derivatives under stochastic volatility models by viewing the call price as an expected value of a truncated log normal distribution under the risk neutral measure. We also obtain the formula for the estimate of the variance of the call price with the stochastic volatility. Using return data, we estimate the mean and variance of the stochastic volatility of the Black-Scholes model. An extensive empirical analysis of the European call option valuation for S & P 100 Index shows: (i) our method outperforms other compelling stochastic volatility pricing models, (ii) the pricing errors when using our method are quite small even though our estimation procedure is based only on historical return data. Formulas for option pricing and variances derivation for Heston's (1993) continuous time stochastic volatility model and for Taylor's (1973) discrete time stochastic volatility model are also discussed in some detail.

Adaptive estimation of the dynamics of a discrete time stochastic volatility model

This paper is concerned with the discrete time stochastic volatility model Y i = exp ( X i / 2 ) η i , X i + 1 = b ( X i ) + σ ( X i ) ξ i + 1 , where only ( Y i ) is observed. The model is rewritten as a particular hidden model: Z i = X i + ε i , X i + 1 = b ( X i ) + σ ( X i ) ξ i + 1 , where ( ξ i ) and ( ε i ) are independent sequences of i.i.d. noise. Moreover, the sequences ( X i ) and ( ε i ) are independent and the distribution of ε is known. Then, our aim is to estimate the functions b and σ 2 when only observations Z 1 , … , Z n are available. We propose to estimate b f and ( b 2 + σ 2 ) f and study the integrated mean square error of projection estimators of these functions on automatically selected projection spaces. By ratio strategy, estimators of b and σ 2 are then deduced. The mean square risk of the resulting estimators are studied and their rates are discussed. Lastly, simulation experiments are provided: constants in the penalty functions defining the estimators are calibrated and the quality of the estimators is checked on several examples.

Recursive estimation for continuous time stochastic volatility models

Volatility plays an important role in portfolio management and option pricing. Recently, there has been a growing interest in modeling volatility of the observed process by nonlinear stochastic process [S.J. Taylor, Asset Price Dynamics, Volatility, and Prediction, Princeton University Press, 2005; H. Kawakatsu, Specification and estimation of discrete time quadratic stochastic volatility models, Journal of Empirical Finance 14 (2007) 424–442]. In [H. Gong, A. Thavaneswaran, J. Singh, Filtering for some time series models by using transformation, Math Scientist 33 (2008) 141–147], we have studied the recursive estimates for discrete time stochastic volatility models driven by normal errors. In this paper, we study the recursive estimates for various classes of continuous time nonlinear non-Gaussian stochastic volatility models used for option pricing in finance.

A nonparametric approach tothe noise density in stochastic volatility models

We propose a nonparametric method to determine the functional form of the noise density in discrete-time stochastic volatility models of financial returns. Our approach suggests that the assumption of Gaussian noise is often adequate, but we do observe deviations from Gaussian noise for some assets, for instance gold.

Option Valuation with Conditional Heteroskedasticity and Non-Normality

We provide results for the valuation of European style contingent claims for a large class of specifications of the underlying asset returns. Our valuation results obtain in a discrete time, infinite state-space setup using the no-arbitrage principle and an equivalent martingale measure. Our approach allows for general forms of heteroskedasticity in returns, and valuation results for homoskedastic processes can be obtained as a special case. It also allows for conditional non-normal return innovations, which is critically important because heteroskedasticity alone does not suffice to capture the option smirk. We analyze a class of equivalent martingale measures for which the resulting risk-neutral return dynamics are from the same family of distributions as the physical return dynamics. In this case, our framework nests the valuation results obtained by Duan (1995) and Heston and Nandi (2000) by allowing for a time-varying price of risk and non-normal innovations. We provide extensions of these results to more general equivalent martingale measures and to discrete time stochastic volatility models, and we analyze the relation between our results and those obtained for continuous time models.

How to Forecast Long-Run Volatility: Regime Switching and the Estimation of Multifractal Processes

We propose a discrete-time stochastic volatility model in which regime switching serves three purposes. First, changes in regimes capture low-frequency variations. Second, they specify intermediate-frequency dynamics usually assigned to smooth autoregressive transitions. Finally, high-frequency switches generate substantial outliers. Thus a single mechanism captures three features that are typically viewed as distinct in the literature. Maximum-likelihood estimation is developed and performs well in finite samples. Using exchange rates, we estimate a version of the process with four parameters and more than a thousand states. The multifractal outperforms GARCH, MS-GARCH, and FIGARCH in- and out-of-sample. Considerable gains in forecasting accuracy are obtained at horizons of 10 to 50 days.

Testing Models of the Spot Interest Rate Volatility

This paper estimates discrete-time stochastic volatility models of the short-term spot interest rate for Germany and France, 1981-1997. Stochastic volatility interest rate models with both news and square root level effects seem to dominate alternative specifications. We extend previous results by using an exponential functional form for volatility, which is able to mitigate the impact of very large realizations and yield a better fit. Interest rate volatility shows a less persistent behavior than in stock returns and US Treasury bill yields volatility. Positive innovations in interest rates have a greater impact on conditional volatility than negative innovations.

Validity of discrete-time stochastic volatility models in non-synchronous equity markets

The paper investigates the validity of versions of discrete-time stochastic volatility models for index series known to contain component stocks exhibiting non-synchronous trading. The efficient method of moments (EMM) is used to fit versions of the discrete-time stochastic volatility (SV) model. The EMM methodology confronts moment conditions generated by a score generator (SNP) that are valid by construction. The moment generator suggests non-linearity in the index series. The EMM construction shows that a classical discrete time stochastic volatility model is rejected. An extended model incorporating an asymmetric volatility specification validates all the moment scores. Option values from Black and Scholes (BS) and Monte Carlo simulations (MC) seem significantly different. The results suggest that BS does not price asymmetry adequately. Asymmetry suggests increased market risk inducing higher BS call prices and lower (higher) BS put pricing for ATM and OTM options (ITM) relative to MC.

Regime-Switching and the Estimation of Multifractal Processes

We propose a discrete-time stochastic volatility model in which regimeswitching serves three purposes. First, changes in regimes capture low frequency variations, which is their traditional role. Second, they specify intermediate frequency dynamics that are usually assigned to smooth autoregressive processes. Finally, high frequency switches generate substantial outliers. Thus, a single mechanism captures three important features of the data that are typically addressed as distinct phenomena in the literature. Maximum likelihood estimation is developed and shown to perform well in finite sample. We estimate on exchange rate data a version of the process with four parameters and more than a thousand states. The estimated model compares favorably to earlier specifications both in- and out-of-sample. Multifractal forecasts slightly improve on GARCH(1,1) at daily and weekly intervals, and provide considerable gains in accuracy at horizons of 10 to 50 days.