New solvable stochastic volatility models for pricing volatility derivatives

In this paper we discuss a new approach to extend a class of solvable stochastic volatility models (SVM). Usually, classical SVM adopt a CEV process for instantaneous variance where the CEV parameter γ takes just few values: 0—the Ornstein–Uhlenbeck process, 1/2—the Heston (or square root) process, 1—GARCH, and 3/2—the 3/2 model. Some other models, e.g. with γ = 2 were discovered in Henry-Labordére (Analysis, geometry, and modeling in finance: advanced methods in option pricing. Chapman & Hall/CRC Financial Mathematics Series, London, 2009) by making connection between stochastic volatility and solvable diffusion processes in quantum mechanics. In particular, he used to build a bridge between solvable superpotentials (the Natanzon superpotentials, which allow reduction of a Schrödinger equation to a Gauss confluent hypergeometric equation) and existing SVM. Here we propose some new models with ${\gamma \in \mathbb{R}}$ and demonstrate that using Lie’s symmetries they could be priced in closed form in terms of hypergeometric functions. Thus obtained new models could be useful for pricing volatility derivatives (variance and volatility swaps, moment swaps).     

Modeling the leverage effect with copulas and realized volatility

In this paper, we propose the use of static and dynamic copulas to study the leverage effect in the S&P 500 index. Copula models can conveniently separate the leverage effect from the marginal distributions of the return and its volatility. Daily volatility is proxied by a measure of realized volatility, which is constructed from high-frequency data. We uncover a significant leverage effect in the S&P 500 index, and this leverage effect is found to be changing over time in a highly persistent manner. Moreover the dynamic copula models are shown to outperform the static counterparts.     

Specification and estimation of discrete time quadratic stochastic volatility models

I propose a new class of stochastic volatility models that nests the commonly used log normal autoregressive specification. As with the eigenfunction specification of Meddahi (Meddahi, Nour, 2001. An eigenfunction approach for volatility modeling. Unpublished.), the log-quadratic model can generate high kurtosis, a key feature of asset returns, even with Gaussian innovations. I discuss maximum likelihood estimation based on numerical integration of the log-quadratic specification that allows for leverage effects. A small Monte Carlo simulation experiment demonstrates the feasibility of maximum likelihood estimation and the importance of allowing for leverage effects. I fit the log-quadratic specification to the daily S&P 500 index return series and find that it provides a better fit than the commonly used log autoregressive specification with Gaussian and Student-t mean equation innovations.     

Forecasting volatility in GARCH models with additive outliers

We model the volatility of a single risky asset using a multifactor (matrix) Wishart affine process, recently introduced in finance by Gourieroux and Sufana. As in standard Duffie and Kan affine models the pricing problem can be solved through the Fast Fourier Transform of Carr and Madan. A numerical illustration shows that this specification provides a separate fit of the long-term and short-term implied volatility surface and, differently from previous diffusive stochastic volatility models, it is possible to identify a specific factor accounting for the stochastic leverage effect, a well-known stylized fact of the FX option markets analysed by Carr and Wu.     

How Persistent is Stock Return Volatility? An Answer with Markov Regime Switching Stochastic Volatility Models

Abstract:  We propose generalised stochastic volatility models with Markov regime changing state equations (SVMRS) to investigate the important properties of volatility in stock returns, specifically high persistence and smoothness. The model suggests that volatility is far less persistent and smooth than the conventional GARCH or stochastic volatility. Persistent short regimes are more likely to occur when volatility is low, while far less persistence is likely to be observed in high volatility regimes. Comparison with different classes of volatility supports the SVMRS as an appropriate proxy volatility measure. Our results indicate that volatility could be far more difficult to estimate and forecast than is generally believed.     

Multiple time scales and the exponential Ornstein–Uhlenbeck stochastic volatility model

We study the exponential Ornstein–Uhlenbeck stochastic volatility model and observe that the model shows a multiscale behaviour in the volatility autocorrelation. It also exhibits a leverage correlation and a probability profile for the stationary volatility which are consistent with market observations. All these features make the model quite appealing since it appears to be more complete than other stochastic volatility models also based on a two-dimensional diffusion. We finally present an approximate solution for the return probability density designed to capture the kurtosis and skewness effects.     

The microstructural foundations of leverage effect and rough volatility

We show that typical behaviors of market participants at the high frequency scale generate leverage effect and rough volatility. To do so, we build a simple microscopic model for the price of an asset based on Hawkes processes. We encode in this model some of the main features of market microstructure in the context of high frequency trading: high degree of endogeneity of market, no-arbitrage property, buying/selling asymmetry and presence of metaorders. We prove that when the first three of these stylized facts are considered within the framework of our microscopic model, it behaves in the long run as a Heston stochastic volatility model, where a leverage effect is generated. Adding the last property enables us to obtain a rough Heston model in the limit, exhibiting both leverage effect and rough volatility. Hence we show that at least part of the foundations of leverage effect and rough volatility can be found in the microstructure of the asset.     

Statistical properties of demand fluctuation in the financial market

Numerical integration methods for stochastic volatility models in financial markets are discussed. We concentrate on two classes of stochastic volatility models where the volatility is either directly given by a mean-reverting CEV process or as a transformed Ornstein–Uhlenbeck process. For the latter, we introduce a new model based on a simple hyperbolic transformation. Various numerical methods for integrating mean-reverting CEV processes are analysed and compared with respect to positivity preservation and efficiency. Moreover, we develop a simple and robust integration scheme for the two-dimensional system using the strong convergence behaviour as an indicator for the approximation quality. This method, which we refer to as the IJK (137) scheme, is applicable to all types of stochastic volatility models and can be employed as a drop-in replacement for the standard log-Euler procedure.     

Overnight information and stochastic volatility: A study of European and US stock exchanges

This paper provides a comprehensive evaluation of the predictive ability of information accumulated during nontrading hours for a set of European and US stock indexes. We introduce a stochastic volatility model, which conditions on lagged overnight information, distinguishes between the nontrading periods of weeknights, weekends, holidays and long weekends, and allows for an asymmetric leverage effect on the impact of overnight news. We implement Bayesian methods for estimation and ranking of the empirical models, and find two key results: (i) there is substantial predictive ability in financial information accumulated during nontrading hours; and (ii) the performance of stochastic volatility models improves considerably by separating the asymmetric impact of positive and negative news made available over weeknights, weekends, holidays and long weekends.     

The leverage effect puzzle: the case of European sovereign credit default swap market

One of the stylized facts about the behaviour of time series is that their volatility exhibits asymmetrical responses to good and bad news. In the case of stock markets, volatility seems to rise when the stock price decreases and fall when the stock price increases. This so-called “leverage effect” was first described by Black (Proceedings of the 1976 meeting of the business and economic statistics section, pp 177–181, 1976). The concept is not new and has already been comprehensively studied and implemented in many volatility models (GARCH and SV) in the form of an additional parameter in the volatility equation. However, there is no study or a theoretical explanation of the leverage effect in sovereign credit default swap spreads (hereinafter: sCDS). In this article, we discuss the possible behaviour of sCDS volatility and explain it by way of reference to the Prospect Theory by Kahneman and Tversky (Econometrica 47(2):263–292, 1979). We estimate a series of stochastic volatility models with the leverage effect, proposed by Yu (J Econom 127(2):165–178, 2005). In this model, the “leverage effect” is, in fact, the same as a coefficient of the correlation between the current return of an asset and its expected future volatility. We show that the effect does exist and differs across markets. As far as the safe European markets are concerned, the parameter is negative; in the case of extremely risky economies—it is positive. In markets of medium risk the effect varies depending on the relationship between the perceived risk and the value of the sCDS premium.     

Forecasting daily variability of the S&P 100 stock index using historical,realised and implied volatility measurements

The increasing availability of financial market data at intraday frequencies has not only led to the development of improved volatility measurements but has also inspired research into their potential value as an information source for volatility forecasting. In this paper, we explore the forecasting value of historical volatility (extracted from daily return series), of implied volatility (extracted from option pricing data) and of realised volatility (computed as the sum of squared high frequency returns within a day). First, we consider unobserved components (UC-RV) and long memory models for realised volatility which is regarded as an accurate estimator of volatility. The predictive abilities of realised volatility models are compared with those of stochastic volatility (SV) models and generalised autoregressive conditional heteroskedasticity (GARCH) models for daily return series. These historical volatility models are extended to include realised and implied volatility measures as explanatory variables for volatility. The main focus is on forecasting the daily variability of the Standard & Poor's 100 (S&P 100) stock index series for which trading data (tick by tick) of almost 7 years is analysed. The forecast assessment is based on the hypothesis of whether a forecast model is outperformed by alternative models. In particular, we will use superior predictive ability tests to investigate the relative forecast performances of some models. Since volatilities are not observed, realised volatility is taken as a proxy for actual volatility and is used for computing the forecast error. A stationary bootstrap procedure is required for computing the test statistic and its p-value. The empirical results show convincingly that realised volatility models produce far more accurate volatility forecasts compared to models based on daily returns. Long memory models seem to provide the most accurate forecasts.     

SUBORDINATED BINOMIAL OPTION PRICING

We extend the binomial option pricing model to allow for more accurate price dynamics while retaining computational simplicity. The asset price in each binomial period evolves according to two independent and successive Bernoulli trials on trade occurrence/nonoccurrence and up/down price movement. Subordination leads to a trinomial tree with stochastic volatility in calendar time. We derive utility‐dependent valuation results incorporating the leverage effect and test the model empirically.     

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.     

The role of jumps and leverage in forecasting volatility in international equity markets

We analyze the importance of jumps and the leverage effect on forecasts of realized volatility in a large cross-section of 18 international equity markets, using daily realized measures data from the Oxford-Man Realized Library, and two widely employed empirical models for realized volatility that allow for jumps and leverage. Our out-of-sample forecast evaluation results show that the separation of realized volatility into a continuous and a discontinuous (jump) component is important for the S&P 500, but of rather limited value for the remaining 17 international equity markets that we analyze. Only for 6 equity markets are significant and sizable forecast improvements realized at the one-step-ahead horizon, which, nevertheless, deteriorate quickly and abruptly as the prediction horizon increases. The inclusion of the leverage effect, on the other hand, has a much larger impact on all 18 international equity markets. Forecast gains are not only highly significant, but also sizeable, with gains remaining significant for forecast horizons of up to one month ahead.     

On Valuing Participating Life Insurance Contracts with Conditional Heteroscedasticity

In this paper, we consider a novel approach for the fair valuation of a participating life insurance policy when the dynamics of the reference portfolio underlying the policy are governed by an Asymmetric Power GARCH (APGARCH) model with innovations having a general parametric distribution. The APGARCH model provides a flexible way to incorporate the effect of conditional heteroscedasticity or time-varying conditional volatility and nests a number of important symmetric or asymmetric ARCH-type models in the literature. It also provides a flexible way to capture both the memory effect of the conditional volatility and the asymmetric effects of past positive and negative returns on the current conditional volatility, called the leverage effect. The key valuation tool here is the conditional Esscher transform of Bühlmann et al. (1996, 1998). The conditional Esscher transform provides a convenient and flexible way for the fair valuation under different specifications of the conditional heteroscedastic models. We illustrate the practical implementation of the model using the S&P 500 index as a proxy for the reference portfolio. We also conduct sensitivity analysis of the fair value of the policy with respect to the parameters in the APGARCH model to document the impacts of different conditional volatility models nested in the APGARCH model and the leverage effect on the fair value. The results of the analysis reveal that the memory effect of the conditional volatility has more significant impact on the fair value of the policy than the leverage effect.     

Estimation and pricing under long-memory stochastic volatility

We treat the problem of option pricing under a stochastic volatility model that exhibits long-range dependence. We model the price process as a Geometric Brownian Motion with volatility evolving as a fractional Ornstein–Uhlenbeck process. We assume that the model has long-memory, thus the memory parameter H in the volatility is greater than 0.5. Although the price process evolves in continuous time, the reality is that observations can only be collected in discrete time. Using historical stock price information we adapt an interacting particle stochastic filtering algorithm to estimate the stochastic volatility empirical distribution. In order to deal with the pricing problem we construct a multinomial recombining tree using sampled values of the volatility from the stochastic volatility empirical measure. Moreover, we describe how to estimate the parameters of our model, including the long-memory parameter of the fractional Brownian motion that drives the volatility process using an implied method. Finally, we compute option prices on the S&P 500 index and we compare our estimated prices with the market option prices.     

Valuation of VIX derivatives

We conduct an extensive empirical analysis of VIX derivative valuation models before, during, and after the 2008–2009 financial crisis. Since the restrictive mean-reversion and heteroskedasticity features of existing models yield large distortions during the crisis, we propose generalisations with a time-varying central tendency, jumps, and stochastic volatility, analyse their pricing performance, and implications for term structures of VIX futures and volatility “skews.” We find that a process for the log of the observed VIX combining central tendency and stochastic volatility reliably prices VIX derivatives. We also uncover a significant risk premium that shifts the long-run volatility level.     

Box-Cox stochastic volatility models with heavy-tails and correlated errors

This paper presents a Markov chain Monte Carlo (MCMC) algorithm to estimate parameters and latent stochastic processes in the asymmetric stochastic volatility (SV) model, in which the Box-Cox transformation of the squared volatility follows an autoregressive Gaussian distribution and the marginal density of asset returns has heavy-tails. We employed the Bayes factor and the Bayesian information criterion (BIC) to examine whether the Box-Cox transformation of squared volatility is favored against the log-transformation. When applying the heavy-tailed asymmetric Box-Cox transformed SV model, three competing SV models and the t-GARCH(1,1) model to continuously compounded daily returns of the Australian stock index, we find that the Box-Cox transformation of squared volatility is strongly favored by Bayes factors and BIC against the log-transformation. While both criteria strongly favor the t-GARCH(1,1) model against the heavy-tailed asymmetric Box-Cox transformed SV model and the other three competing SV models, we find that SV models fit the data better than the t-GARCH(1,1) model based on a measure of closeness between the distribution of the fitted residuals and the distribution of the model disturbance. When our model and its competing models are applied to daily returns of another five stock indices, we find that in terms of SV models, the Box-Cox transformation of squared volatility is strongly favored against the log-transformation for the five data sets.     

Realizing smiles: Options pricing with realized volatility

We develop a discrete-time stochastic volatility option pricing model exploiting the information contained in the Realized Volatility (RV), which is used as a proxy of the unobservable log-return volatility. We model the RV dynamics by a simple and effective long-memory process, whose parameters can be easily estimated using historical data. Assuming an exponentially affine stochastic discount factor, we obtain a fully analytic change of measure. An empirical analysis of Standard and Poor's 500 index options illustrates that our model outperforms competing time-varying and stochastic volatility option pricing models.     

Short-term asymptotics for the implied volatility skew under a stochastic volatility model with Lévy jumps

The implied volatility skew has received relatively little attention in the literature on short-term asymptotics for financial models with jumps, despite its importance in model selection and calibration. We rectify this by providing high order asymptotic expansions for the at-the-money implied volatility skew, under a rich class of stochastic volatility models with independent stable-like jumps of infinite variation. The case of a pure-jump stable-like Lévy model is also considered under the minimal possible conditions for the resulting expansion to be well defined. Unlike recent results for “near-the-money” option prices and implied volatility, the results herein aid in understanding how the implied volatility smile near expiry is affected by important features of the continuous component, such as the leverage and vol-of-vol parameters. As intermediary results, we obtain high order expansions for at-the-money digital call option prices, which furthermore allow us to infer analogous results for the delta of at-the-money options. Simulation results indicate that our asymptotic expansions give good fits for options with maturities up to one month, underpinning their relevance in practical applications, and an analysis of the implied volatility skew in recent S&P 500 options data shows it to be consistent with the infinite variation jump component of our models.