Efficient estimation of drift parameters in stochastic volatility models

We study the parametric problem of estimating the drift coefficient in a stochastic volatility model , where Y is a log price process and V the volatility process. Assuming that one can recover the volatility, precisely enough, from the observation of the price process, we construct an efficient estimator for the drift parameter of the diffusion V. As an application we present the efficient estimation based on the discrete sampling with δ _n →0 and n δ _n →∞. We show that our setup is general enough to cover the case of ‘microstructure noise’ for the price process as well.      

Bayesian testing volatility persistence in stochastic volatility models with jumps

Whether or not there is a unit root persistence in volatility of financial assets has been a long-standing topic of interest to financial econometricians and empirical economists. The purpose of this article is to provide a Bayesian approach for testing the volatility persistence in the context of stochastic volatility with Merton jump and correlated Merton jump. The Shanghai Composite Index daily return data is used for empirical illustration. The result of Bayesian hypothesis testing strongly indicates that the volatility process doesn’t have unit root volatility persistence in this stock market.     

Gradient-based simulated maximum likelihood estimation for stochastic volatility models using characteristic functions

Parameter estimation and statistical inference are challenging problems for stochastic volatility (SV) models, especially those driven by pure jump Lévy processes. Maximum likelihood estimation (MLE) is usually preferred when a parametric statistical model is correctly specified, but traditional MLE implementation for SV models is computationally infeasible due to high dimensionality of the integral involved. To overcome this difficulty, we propose a gradient-based simulated MLE method under the hidden Markov structure for SV models, which covers those driven by pure jump Lévy processes. Gradient estimation using characteristic functions and sequential Monte Carlo in the simulation of the hidden states are implemented. Numerical experiments illustrate the efficiency of the proposed method.     

Estimating stochastic volatility models using integrated nested Laplace approximations

Volatility in financial time series is mainly analysed through two classes of models; the generalized autoregressive conditional heteroscedasticity (GARCH) models and the stochastic volatility (SV) ones. GARCH models are straightforward to estimate using maximum-likelihood techniques, while SV models require more complex inferential and computational tools, such as Markov Chain Monte Carlo (MCMC). Hence, although provided with a series of theoretical advantages, SV models are in practice much less popular than GARCH ones. In this paper, we solve the problem of inference for some SV models by applying a new inferential tool, integrated nested Laplace approximations (INLAs). INLA substitutes MCMC simulations with accurate deterministic approximations, making a full Bayesian analysis of many kinds of SV models extremely fast and accurate. Our hope is that the use of INLA will help SV models to become more appealing to the financial industry, where, due to their complexity, they are rarely used in practice.     

Joint analysis and estimation of stock prices and trading volume in Barndorff-Nielsen and Shephard stochastic volatility models

We introduce a variant of the Barndorff-Nielsen and Shephard stochastic volatility model where the non-Gaussian Ornstein–Uhlenbeck process describes some measure of trading intensity like trading volume or number of trades instead of unobservable instantaneous variance. We develop an explicit estimator based on martingale estimating functions in a bivariate model that is not a diffusion, but admits jumps. It is assumed that both the quantities are observed on a discrete grid of fixed width, and the observation horizon tends to infinity. We show that the estimator is consistent and asymptotically normal and give explicit expressions of the asymptotic covariance matrix. Our method is illustrated by a finite sample experiment and a statistical analysis of IBM? stock from the New York Stock Exchange and Microsoft Corporation? stock from Nasdaq during a history of five years.     

Geometric Asian option pricing in general affine stochastic volatility models with jumps

In this paper, we present some results on Geometric Asian option valuation for affine stochastic volatility models with jumps. We shall provide a general framework into which several different valuation problems based on some average process can be cast, and we shall obtain closed form solutions for some relevant affine model classes.     

Quadratic hedging in affine stochastic volatility models

We determine the variance-optimal hedge for a subset of affine processes including a number of popular stochastic volatility models. This framework does not require the asset to be a martingale. We obtain semiexplicit formulas for the optimal hedging strategy and the minimal hedging error by applying general structural results and Laplace transform techniques. The approach is illustrated numerically for a Lévy-driven stochastic volatility model with jumps as in Carr et al. (Math Finance 13:345–382, 2003).      

Pricing via recursive quantization in stochastic volatility models

We provide the first recursive quantization-based approach for pricing options in the presence of stochastic volatility. This method can be applied to any model for which an Euler scheme is available for the underlying price process and it allows one to price vanillas, as well as exotics, thanks to the knowledge of the transition probabilities for the discretized stock process. We apply the methodology to some celebrated stochastic volatility models, including the Stein and Stein [Rev. Financ. Stud. 1991, (4), 727–752] model and the SABR model introduced in Hagan et al. [Wilmott Mag., 2002, 84–108]. A numerical exercise shows that the pricing of vanillas turns out to be accurate; in addition, when applied to some exotics like equity-volatility options, the quantization-based method overperforms by far the Monte Carlo simulation.     

General approximation schemes for option prices in stochastic volatility models

In this paper we develop a general method for deriving closed-form approximations of European option prices and equivalent implied volatilities in stochastic volatility models. Our method relies on perturbations of the model dynamics and we show how the expansion terms can be calculated using purely probabilistic methods. A flexible way of approximating the equivalent implied volatility from the basic price expansion is also introduced. As an application of our method we derive closed-form approximations for call prices and implied volatilities in the Heston [Rev. Financial Stud., 1993, 6, 327–343] model. The accuracy of these approximations is studied and compared with numerically obtained values.     

A class of stochastic volatility models and the q-optimal martingale measure

This paper proposes an approach under which the q-optimal martingale measure, for the case where continuous processes describe the evolution of the asset price and its stochastic volatility, exists for all finite time horizons. More precisely, it is assumed that while the ‘mean–variance trade-off process’ is uniformly bounded, the volatility and asset are imperfectly correlated. As a result, under some regularity conditions for the parameters of the corresponding Cauchy problem, one obtains that the qth moment of the corresponding Radon–Nikodym derivative does not explode in finite time.     

Sequential Monte Carlo for fractional stochastic volatility models

In this paper, we consider a fractional stochastic volatility model, that is a model in which the volatility may exhibit a long-range dependent or a rough/antipersistent behaviour. We propose a dynamic sequential Monte Carlo methodology that is applicable to both long memory and antipersistent processes in order to estimate the volatility as well as the unknown parameters of the model. We establish a central limit theorem for the state and parameter filters and we study asymptotic properties (consistency and asymptotic normality) for the filter. We illustrate our results with a simulation study and we apply our method to estimate the volatility and the parameters of a long-range dependent model for S& P 500 data.     

Moment explosions in stochastic volatility models

In this paper, we demonstrate that many stochastic volatility models have the undesirable property that moments of order higher than 1 can become infinite in finite time. As arbitrage-free price computation for a number of important fixed income products involves forming expectations of functions with super-linear growth, such lack of moment stability is of significant practical importance. For instance, we demonstrate that reasonably parametrized models can produce infinite prices for Eurodollar futures and for swaps with floating legs paying either Libor-in-arrears or a constant maturity swap rate. We systematically examine the moment explosion property across a spectrum of stochastic volatility models. We show that lognormal and displaced-diffusion type models are easily prone to moment explosions, whereas CEV-type models (including the so-called SABR model) are not. Related properties such as the failure of the martingale property are also considered.

---

Electronic Supplementary Material Supplementary material is available for this article at and is accessible for authorized users.      

A new technique for calibrating stochastic volatility models: the Malliavin gradient method

We discuss the application of gradient methods to calibrate mean reverting stochastic volatility models. For this we use formulas based on Girsanov transformations as well as a modification of the Bismut–Elworthy formula to compute the derivatives of certain option prices with respect to the parameters of the model by applying Monte Carlo methods. The article presents an extension of the ideas to apply Malliavin calculus methods in the computation of Greek's.     

Correlations among cryptocurrencies: Evidence from multivariate factor stochastic volatility model

This paper is the first study to apply the multivariate factor stochastic volatility model (MFSVM) for analyzing the correlations among six cryptocurrencies. We use MFSVM with the Bayesian estimation procedure for the period from August 8, 2015, to January 1, 2020. According to the findings, there is a significant positive correlation between price volatility values of Bitcoin and Litecoin. Besides, the volatility values of Ethereum have a positive correlation with both Ripple and Stellar. There is also a positive correlation between the volatility values of Ripple and Dash. These findings are robust to consider different correlation networks. The evidence implies that Bitcoin is mainly related to Litecoin, but Ethereum is associated with other cryptocurrencies.     

Online Kernel estimation of stationary stochastic diffusion models

Nonparametric regression has recently become important in quantitative finance due to its distribution-free property. However, this advantage does not come without any cost. As large sample sizes are always required to adequately estimate local structures, nonparametric regression is computationally intensive in real applications. This paper proposes an online method to decrease the computational cost of nonparametric regression for estimating stationary stochastic diffusion models. We establish asymptotic behaviours of the proposed estimators under appropriate conditions. Numerical examples and an empirical study of US 3-month treasury bill rates are illustrated. The application to financial risk management is also taken into consideration.     

A new approach for option pricing under stochastic volatility

We develop a new approach for pricing European-style contingent claims written on the time T spot price of an underlying asset whose volatility is stochastic. Like most of the stochastic volatility literature, we assume continuous dynamics for the price of the underlying asset. In contrast to most of the stochastic volatility literature, we do not directly model the dynamics of the instantaneous volatility. Instead, taking advantage of the recent rise of the variance swap market, we directly assume continuous dynamics for the time T variance swap rate. The initial value of this variance swap rate can either be directly observed, or inferred from option prices. We make no assumption concerning the real world drift of this process. We assume that the ratio of the volatility of the variance swap rate to the instantaneous volatility of the underlying asset just depends on the variance swap rate and on the variance swap maturity. Since this ratio is assumed to be independent of calendar time, we term this key assumption the stationary volatility ratio hypothesis (SVRH). The instantaneous volatility of the futures follows an unspecified stochastic process, so both the underlying futures price and the variance swap rate have unspecified stochastic volatility. Despite this, we show that the payoff to a path-independent contingent claim can be perfectly replicated by dynamic trading in futures contracts and variance swaps of the same maturity. As a result, the contingent claim is uniquely valued relative to its underlying’s futures price and the assumed observable variance swap rate. In contrast to standard models of stochastic volatility, our approach does not require specifying the market price of volatility risk or observing the initial level of instantaneous volatility. As a consequence of our SVRH, the partial differential equation (PDE) governing the arbitrage-free value of the contingent claim just depends on two state variables rather than the usual three. We then focus on the consistency of our SVRH with the standard assumption that the risk-neutral process for the instantaneous variance is a diffusion whose coefficients are independent of the variance swap maturity. We show that the combination of this maturity independent diffusion hypothesis (MIDH) and our SVRH implies a very special form of the risk-neutral diffusion process for the instantaneous variance. Fortunately, this process is tractable, well-behaved, and enjoys empirical support. Finally, we show that our model can also be used to robustly price and hedge volatility derivatives.     

A link between complete models with stochastic volatility and ARCH models

In this paper, we propose a heteroskedastic model in discrete time which converges, when the sampling interval goes to zero, towards the complete model with stochastic volatility in continuous time described in Hobson and Rogers (1998). Then, we study its stationarity and moment properties. In particular, we exhibit a specific model which shares many properties with the GARCH(1,1) model, establishing a clear link between the two approaches. We also prove the consistency of the pseudo conditional likelihood maximum estimates for this specific model.Received: December 2002Mathematics Subject Classification: 90A09, 60J60, 62M05JEL Classification: C32This work was supported in part by Dynstoch European network. Thanks to David Hobson for introducing me to these models, and to Valentine Genon-Catalot for numerous and very fruitful discussion on this work. The author is also grateful to Uwe Kuchler for various helpful suggestions, and to two referees and an associate editor for their comments and suggestions.     

Multiperiod hedging in the presence of stochastic volatility

In this paper, we characterize the multiperiod minimum-risk hedge strategy within the stochastic volatility (SV) framework and compare it to other hedge strategies on the basis of hedging performance. Using crude oil markets as an example, we demonstrate that the SV model is appropriate in depicting price behaviour. However, ex ante and ex post comparisons indicate that the SV strategy is inferior to conventional hedging strategies. There is also evidence that the Generalized Autoregressive Conditional Heteroskedasticity (GARCH) strategy may be better than the SV strategy, at least in terms of variance reduction.     

Risk minimization in stochastic volatility models: model risk and empirical performance

In this paper the performance of locally risk-minimizing delta hedge strategies for European options in stochastic volatility models is studied from an experimental as well as from an empirical perspective. These hedge strategies are derived for a large class of diffusion-type stochastic volatility models, and they are as easy to implement as usual delta hedges. Our simulation results on model risk show that these risk-minimizing hedges are robust with respect to uncertainty and misconceptions about the underlying data generating process. The empirical study, which includes the US sub-prime crisis period, documents that in equity markets risk-minimizing delta hedges consistently outperform usual delta hedges by approximately halving the standard deviation of the profit-and-loss ratio.