Maximum likelihood estimation of stochastic volatility models

We develop and implement a method for maximum likelihood estimation in closed-form of stochastic volatility models. Using Monte Carlo simulations, we compare a full likelihood procedure, where an option price is inverted into the unobservable volatility state, to an approximate likelihood procedure where the volatility state is replaced by proxies based on the implied volatility of a short-dated at-the-money option. The approximation results in a small loss of accuracy relative to the standard errors due to sampling noise. We apply this method to market prices of index options for several stochastic volatility models, and compare the characteristics of the estimated models. The evidence for a general CEV model, which nests both the affine Heston model and a GARCH model, suggests that the elasticity of variance of volatility lies between that assumed by the two nested models.     

Bayesian range-based estimation of stochastic volatility models

Alizadeh, Brandt, and Diebold [2002. Journal of Finance 57, 1047–1091] propose estimating stochastic volatility models by quasi-maximum likelihood using data on the daily range of the log asset price process. We suggest a related Bayesian procedure that delivers exact likelihood based inferences. Our approach also incorporates data on the daily return and accommodates a nonzero drift. We illustrate through a Monte Carlo experiment that quasi-maximum likelihood using range data alone is remarkably close to exact likelihood based inferences using both range and return data.     

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.      

Some nonstandard stochastic volatility models and their estimation using structured hidden Markov models

We introduce a number of nonstandard stochastic volatility (SV) models and examine their performance when applied to the series of daily returns on several stocks listed on the New York Stock Exchange. The nonstandard models under investigation extend both the observation process and the volatility-generating process of basic SV models. In particular, we consider dependent as well as independent mixtures of autoregressive components as the log-volatility process, and include in the observation equation a lower bound on the volatility. We also consider an experimental SV model that is based on conditionally gamma-distributed volatilities.Our estimation method is based on the fact that an SV model can be approximated arbitrarily accurately by a hidden Markov model (HMM), whose likelihood is easy to compute and to maximize. The method is close, but not identical, to those of Fridman and Harris (1998), Bartolucci and De Luca (2001, 2003) and Clements et al. (2006), and makes explicit the useful link between HMMs and the methods of those authors. Likelihood-based estimation of the parameters of SV models is usually regarded as challenging because the likelihood is a high-dimensional multiple integral. The HMM approximation is easy to implement and particularly convenient for fitting experimental extensions and variants of SV models such as those we introduce here. In addition, and in contrast to the case of SV models themselves, simple formulae are available for the forecast distributions of HMMs, for computing appropriately defined residuals, and for decoding, i.e. estimating the volatility of the process.     

Moment-based estimation of stochastic volatility

This paper makes use of the distributional information contained in high-frequency data to test for the specification of the functional form of the volatility process within the class of stochastic volatility models.     

Portfolio optimization in discrete time with proportional transaction costs under stochastic volatility

This paper is devoted to evaluating the optimal self-financing strategy and the optimal trading frequency for a portfolio with a risky asset and a risk-free asset. The objective is to maximize the expected future utility of the terminal wealth in a stochastic volatility setting, when transaction costs are incurred at each discrete trading time. A HARA utility function is used, allowing a simple approximation of the optimization problem, which is implementable forward in time. For each of various transaction cost rates, we find the optimal trading frequency, i.e. the one that attains the maximum of the expected utility at time zero. We study the relation between transaction cost rate and optimal trading frequency. The numerical method used is based on a stochastic volatility particle filtering algorithm, combined with a Monte-Carlo method. The filtering algorithm updates the estimate of the volatility distribution forward in time, as new stock observations arrive; these updates are used at each of these discrete times to compute the new portfolio allocation.     

Affine fractional stochastic volatility models

By fractional integration of a square root volatility process, we propose in this paper a long memory extension of the Heston (Rev Financ Stud 6:327–343, 1993) option pricing model. Long memory in the volatility process allows us to explain some option pricing puzzles as steep volatility smiles in long term options and co-movements between implied and realized volatility. Moreover, we take advantage of the analytical tractability of affine diffusion models to clearly disentangle long term components and short term variations in the term structure of volatility smiles. In addition, we provide a recursive algorithm of discretization of fractional integrals in order to be able to implement a method of moments based estimation procedure from the high frequency observation of realized volatilities.     

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.     

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).     

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.

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Nonparametric estimation for a stochastic volatility model

Consider discrete-time observations (X _{? δ} )_1≤?≤n+1 of the process X satisfying $dX_{t}=\sqrt{V_{t}}dB_{t}Consider discrete-time observations (X _{ℓ δ} )_{1≤ℓ≤n+1} of the process X satisfying dX_t=?{V_t}dB_tdX_{t}=\sqrt{V_{t}}dB_{t} , with V a one-dimensional positive diffusion process independent of the Brownian motion B. For both the drift and the diffusion coefficient of the unobserved diffusion V, we propose nonparametric least square estimators, and provide bounds for their risk. Estimators are chosen among a collection of functions belonging to a finite-dimensional space whose dimension is selected by a data driven procedure. Implementation on simulated data illustrates how the method works.     

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.     

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).      

Autoregressive stochastic volatility models with heavy-tailed distributions: A comparison with multifactor volatility models

This paper examines two asymmetric stochastic volatility models used to describe the heavy tails and volatility dependencies found in most financial returns. The first is the autoregressive stochastic volatility model with Student's t-distribution (ARSV-t), and the second is the multifactor stochastic volatility (MFSV) model. In order to estimate these models, the analysis employs the Monte Carlo likelihood (MCL) method proposed by Sandmann and Koopman [Sandmann, G., Koopman, S.J., 1998. Estimation of stochastic volatility models via Monte Carlo maximum likelihood. Journal of Econometrics 87, 271–301.]. To guarantee the positive definiteness of the sampling distribution of the MCL, the nearest covariance matrix in the Frobenius norm is used. The empirical results using returns on the S&P 500 Composite and Tokyo stock price indexes and the Japan–US exchange rate indicate that the ARSV-t model provides a better fit than the MFSV model on the basis of Akaike information criterion (AIC) and the Bayes information criterion (BIC).     

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.     

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.     

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.     

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.