Stochastic Volatility Research Papers - Academia.edu (original) (raw)

Stochastic volatility modelling of financial processes has become increasingly popular. The proposed models usually contain a stationary volatility process. We will motivate and review several nonparametric methods for estimation of the density of the volatility process. Both models based on discretely sampled continuous time processes and discrete time models will be discussed.

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- Stochastic Volatility, Density Estimation, Discrete time, Kernel density estimate

In this article, we consider Bayesian inference procedures to test for a unit root in Stochastic Volatility (SV) models. Unit-root tests for the persistence parameter of the SV models, based on the Bayes Factor (BF), have been recently introduced in the literature. In contrast, we propose a flexible class of priors that is noninformative over the entire support of the persistence parameter (including the non-stationarity region). In addition, we show that our model fitting procedure is computationally efficient (using the software WinBUGS). Finally, we show that our proposed test procedures have good frequentist properties in terms of achieving high statistical power, while maintaining low total error rates. We illustrate the above features of our method by extensive simulation studies, followed by an application to a real data set on exchange rates.

- by Sujit Ghosh
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- Statistics, Bayesian Inference, Markov Chain Monte Carlo, Stochastic Volatility

This article details a Bayesian analysis of the Nile river flow data, using a simple state space model. This allows the article to concentrate on implementation issues surrounding this model. For this data set, Metropolis-Hastings and Gibbs sampling algorithms are implemented in the programming language Ox. The Markov chain Monte Carlo methods only provide output conditioned upon the full data set. For filtered output, conditioning only on past observations, the particle filter is introduced. The sampling methods are flexible, and this advantage is used to extend the model to incorporate a stochastic volatility process. The volatility changes both in the Nile data, and for comparison also in daily S&P 500 return data, are investigated. The posterior density of parameters and states is found to provide information on which elements of the model are easily identifiable, and which elements are estimated with less precision.

- by Charles Bos
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- Time Series, Bayesian Analysis, Sampling methods, Stochastic Volatility

We extend to the Heston stochastic volatility framework the parity result of McDonald and Schroder (1998) for American call and put options.

- by Anna Battauz
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- Stochastic Volatility

In this note, we prove that in asset price models with lognormal stochastic volatility, when the correlation coecien t between the Brownian motion driving the volatility and the one driving the actualized asset price is positive, this... more

- by Benjamin Jourdain
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- Stochastic Volatility, Brownian Motion, Asset Prices, Asset pricing model

There is strong empirical evidence that long-term interest rates contain a time-varying risk premium. Interest rate options may contain information about this risk premium because their prices are sensitive to the volatility and market prices of the risk factors that drive interest rates. We use the joint time series of swap rates and interest rate cap prices to estimate dynamic term structure models. The risk premiums that we estimate using option prices are better able to predict excess returns for long-term swaps over short-term swaps. Moreover, in contrast to previous literature, the most succesful models for predicting excess returns have risk factors with stochastic volatility. We also show that the models we estimate using option prices match the failure of the expectations hypothesis. * We thank

- by Caio Almeida
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- Econometrics, Time Series, Forecasting, Stochastic Volatility

- by Rene Garcia
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- Latent variable, Stochastic Volatility, Latent Variables, Option pricing

This paper investigates the consequences of stochastic volatility for pricing spot foreign currency options. A diffusion model for exchange rates with stochastic volatility is proposed and estimated. The parameter estimates are then used to price foreign currency options and the predictions are compared to observed market prices. We find that allowing volatility to be stochastic results in a much better fit to the empirical distribution of the Canada-U.S. exchange rate, and that this improvement in fit results in more accurate predictions of observed option prices.

- by Stuart Turnbull
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- Econometrics, Prediction, Money, Parameter estimation

A Maximum Likelihood (ML) approach based upon an Efficient Importance Sampling (EIS) procedure is used to estimate several extensions of the standard Stochastic Volatility (SV) model for daily financial return series. EIS provides a highly generic procedure for a very accurate Monte Carlo (MC) evaluation of the marginal likelihood which depends upon high-dimensional interdependent integrals. Extensions of the standard SV model being analyzed only require minor modifications in the ML-EIS procedure. Furthermore, EIS can also be applied for filtering which provides the basis for several diagnostic tests. Our empirical analysis indicates that extensions such as a seminonparametric specification of the error term distribution in the return equation dominate the standard SV model. Finally, we also apply the ML-EIS approach to a multivariate factor model with stochastic volatility. D Journal of Empirical Finance 10 (2003) 505 -531

- by Jean-françois Richard
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- Econometrics, Empirical Finance, Monte Carlo, Marginal Likelihood

A price process is scale-invariant if and only if the returns distribution is independent of the price measurement scale. We show that most stochastic processes used for pricing options on financial assets have this property and that many models not previously recognised as scale-invariant are indeed so. We also prove that price hedge ratios for a wide class of contingent claims under a wide class of pricing models are model-free. In particular, previous results on model-free price hedge ratios of vanilla options based on scale-invariant models are extended to any contingent claim with homogeneous pay-off, including complex, path-dependent options. However, model-free hedge ratios only have the minimum variance property in scale-invariant stochastic volatility models when price-volatility correlation is zero. In other stochastic volatility models and in scale-invariant local volatility models, model-free hedge ratios are not minimum variance ratios and our empirical results demonstrate that they are less efficient than minimum variance hedge ratios.

- by Carol Alexander
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- Applied Mathematics, Stochastic Process, Path Dependence, Stochastic Volatility

In this paper, we present a highly efficient approach to price variance swaps with discrete sampling times. We have found a closed-form exact solution for the partial differential equation (PDE) system based on the Heston (1993) two-factor stochastic volatility model embedded in the framework proposed by Little and Pant (2001). In comparison with all the previous approximation models based on the assumption of continuous sampling time, the current research of working out a closed-form exact solution for variance swaps with discrete sampling times at least serves for two major purposes: (i) to verify the degree of validity of using a continuous-sampling-time approximation for variance swaps of relatively short sampling period; (ii) to demonstrate that significant errors can result from still adopting such an assumption for a variance swap with small sampling frequencies or long tenor. Other key features of our new solution approach include: (a) with the newly found analytic solution, all the hedging ratios of a variance swap can also be analytically derived; (b) numerical values can be very efficiently computed from the newly found analytic formula.

- by Guanghua Lian
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- Applied Mathematics, Economics, Mathematical Finance, Stochastic Volatility

In this paper we compare the predictive abilility of Stochastic Volatility (SV) models to that of volatility forecasts implied by option prices. We develop an SV model with implied volatility as an exogeneous var able in the variance equation which facilitates the use of statistical tests for nested models; we refer to this model as the SVX model. The SVX

- by Eugenie Hol and +1
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- Monte Carlo, Volatility Forecasting, Maximum Likelihood, Stochastic Volatility

This paper surveys recent developments in the theory of option pricing. The emphasis is on the interplay between option prices and investors' impatience and their aversion to risk. The traditional view, steeped in the risk-neutral approach to derivative pricing, has been that these preferences play no role in the determination of option prices. However, the usual lognormality assumption required to obtain preference-free option pricing formulas is at odds with the empirical properties of financial assets. The lognormality assumption is easily reconcilable with those properties by the introduction of a latent state variable whose values can be interpreted as the states of the economy. The presence of a covariance risk with the state variable makes option prices depend explicitly on preferences. Generalized option pricing formulas, in which preferences matter, can explain several well-known empirical biases associated with preference-free models such as that of Black and Scholes (...

- by Richard Luger
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- Stochastic Volatility, Option pricing

This study develops a GARCH-type model, i.e., the variance-gamma GARCH (VG GARCH) model, based on the two major strands of option pricing literature. The first strand of the literature uses the variance-gamma process, a time-changed Brownian motion, to model the underlying asset price process such that the possible skewness and excess kurtosis on the distributions of asset returns are considered. The second strand of the literature considers the propagation of the previously arrived news by including the feedback and leverage effects on price movement volatility in a GARCH framework. The proposed VG GARCH model is shown to obey a locally risk-neutral valuation relationship (LRNVR) under the sufficient conditions postulated by . This new model provides a unified framework for estimating the historical and risk-neutral distributions, and thus facilitates option pricing calibration using historical underlying asset prices. An empirical study is performed comparing the proposed VG GARCH model with four competing pricing models: benchmark Black-Scholes, ad hoc Black-Scholes, normal NGARCH, and stochastic volatility VG. The performance of the VG GARCH model versus these four competing models is then demonstrated.

- by Lie-Jane Kao
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- Mathematical Sciences, Stochastic Volatility, Brownian Motion, Empirical Study

This paper examines option pricing in a universe in which it is assumed that markets are incomplete. It derives multiperiod discrete time option bounds based on stochastic dominance considerations for a risk-averse investor holding only the underlying asset, the riskless asset and (possibly) the option for any type of underlying asset distribution, discrete or continuous. It then considers the limit behavior of these bounds for special categories of such distributions as trading becomes progressively more dense, tending to continuous time. It is shown that these bounds nest as special cases most, if not all, existing arbitrage-and equilibrium-based option pricing models. Thus, when the underlying asset follows a generalized diffusion both bounds converge to a single value. For jumpdiffusion processes, stochastic volatility models, and GARCH processes the bounds remain distinct and define several new option pricing results containing as special cases the arbitrage-based results.

- by Ioan Oancea and +1
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- Stochastic dominance, Risk Aversion, Stochastic Volatility, Option pricing

The delta hedging performance of deterministic local volatility models is poor, with most studies showing that even the simple constant volatility Black-Scholes model performs better. But when the local volatility model is extended to capture stochastic dynamics for the spot volatility process the hedge ratios change. Here we derive the local volatility hedge ratios that are consistent with a stochastic spot volatility and show that the stochastic local volatility model is equivalent to the market model for implied volatilities. We also quantify the hedging error that arises from residual hedging uncertainty and provide an empirical example based on a stochastic normal mixture diffusion model for asset returns.

- by Carol Alexander
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- Stochastic Volatility, Hedging, Implied Volatility

The Heston model stands out from the class of stochastic volatility (SV) models mainly for two reasons. Firstly, the process for the volatility is non-negative and mean-reverting, which is what we observe in the markets. Secondly, there exists a fast and easily implemented semi-analytical solution for European options. In this article we adapt the original work of Heston (1993) to a foreign exchange (FX) setting. We discuss the computational aspects of using the semi-analytical formulas, performing Monte Carlo simulations, checking the Feller condition, and option pricing with FFT. In an empirical study we show that the smile of vanilla options can be reproduced by suitably calibrating three out of five model parameters.

- by Rafał Weron and +1
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- Computational Finance, Monte Carlo Simulation, Stochastic Volatility, Empirical Study

This paper considers the problem of pricing American options when the dynamics of the underlying are driven by both stochastic volatility following a square root process as used by , and by a Poisson jump process as introduced by Merton (1976). Probability arguments are invoked to find a representation of the solution in terms of expectations over the joint distribution of the underlying process.

The Pearson diffusions is a flexible class of diffusions defined by having linear drift and quadratic squared diffusion coefficient. It is demonstrated that for this class explicit statistical inference is feasible. Explicit optimal martingale estimating functions are found, and the corresponding estimators are shown to be consistent and asymptotically normal. The discussion covers GMM, quasi-likelihood, and nonlinear weighted least squares estimation too, and it is discussed how explicit likelihood or approximate likelihood inference is possible for the Pearson diffusions. A complete model classification is presented for the ergodic Pearson diffusions. The class of stationary distributions equals the full Pearson system of distributions. Well-known instances are the Ornstein-Uhlenbeck processes and the square root (CIR) processes. Also diffusions with heavy-tailed and skew marginals are included. Special attention is given to a skew t-type distribution. Explicit formulae for the conditional moments and the polynomial eigenfunctions are derived. The analytical tractability is inherited by transformed Pearson diffusions, integrated Pearson diffusions, sums of Pearson diffusions, and stochastic volatility models with Pearson volatility process. For the non-Markov models explicit optimal prediction based estimating functions are found and shown to yield consistent and asymptotically normal estimators.

- by Julie Forman
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- Statistics, Statistical Inference, Stochastic Volatility, Spectral method

This report covers the important topic of stochastic volatility modelling with an emphasis on linear state models. The approach taken focuses on comparing models based on their ability to flt the data and their forecasting performance. To this end several parsimonious stochastic volatility models are estimated using realised volatility, a volatility proxy from high frequency stock price data. The

- by paresh date
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- High Frequency, Linear Model, Kalman Filtering, Stochastic Volatility

Le CIRANO est un organisme sans but lucratif constitué en vertu de la Loi des compagnies du Québec. Le financement de son infrastructure et de ses activités de recherche provient des cotisations de ses organisations-membres, d'une subvention d'infrastructure du ministère de la Recherche, de la Science et de la Technologie, de même que des subventions et mandats obtenus par ses équipes de recherche.

- by Eric Renault
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- Neural Network, Long Memory, Statistical Inference, Latent variable

We consider the pricing of options when delta hedging only takes place at discrete intervals. We show how to include transaction costs, jumps and stochastic volatility while optimally, but discretely, dynamically hedging. Copyright © 2009... more

- by Hyungsok Ahn
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- Transaction Costs, Stochastic Volatility, Poisson Process, Option pricing

- by Gerald Cheang
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- Economics, Density-functional theory, Quantitative Finance, Mathematical Sciences

This essay is aimed to provide a straightforward and sufficiently accessible demonstration of some known procedures for stochastic volatility model. It reviews the important related concepts, gives informal derivations of the methods and can be useful as a cookbook for a novice. The exposition is confined to classical (non-Bayesian) framework and discrete-time formulations.

- by Alexander Tsyplakov
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- Stochastic Volatility, Importance Sampling, Bayesian framework

Building on realized variance and bipower variation measures constructed from high-frequency financial prices, we propose a simple reduced form framework for effectively incorporating intraday data into the modeling of daily return volatility. We decompose the total daily return variability into the continuous sample path variance, the variation arising from discontinuous jumps that occur during the trading day, as well as the overnight return variance. Our empirical results, based on long samples of highfrequency equity and bond futures returns, suggest that the dynamic dependencies in the daily continuous sample path variability are well described by an approximate long-memory HAR-GARCH model, while the overnight returns may be modeled by an augmented GARCH type structure. The dynamic dependencies in the non-parametrically identified significant jumps appear to be well described by the combination of an ACH model for the time-varying jump intensities coupled with a relatively simple log-linear structure for the jump sizes. Finally, we discuss how the resulting reduced form model structure for each of the three components may be used in the construction of out-of-sample forecasts for the total return volatility.

- by Tim Bollerslev
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- Finance, Econometrics, Long Memory, High Frequency
- by Donald Lien
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- Applied Economics, Option Valuation, Stochastic Volatility, Economics Finance

A characteristic function-based method is proposed to estimate the time-changed Lévy models, which take into account both stochastic volatility and infinite-activity jumps. The method facilitates computation and overcomes problems related to the discretization error and to the non-tractable probability density. Estimation results and option pricing performance indicate that the infiniteactivity model performs better than the finite-activity one. By introducing a jump component in the volatility process, a double-jump model is also investigated.

- by Fulvio Ortu
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- Econometrics, Statistics, Stochastic Volatility, Performance Model

This paper analyzes the volatility structure of commodity derivatives markets. The model encompasses stochastic volatility that may be unspanned by futures contracts. A generalized hump-shaped volatility specification is assumed that entails a finite-dimensional affine model for the commodity futures curve and quasi-analytical prices for options on commodity futures. An empirical study of the crude oil futures volatility structure is carried out using an extensive database of futures prices as well as futures option prices spanning 21 years. The study supports a hump-shaped, partially spanned stochastic volatility specification. Factor hedging, which takes into account shocks to both the volatility processes and the futures curve, depicts the presence of unspanned components in the volatility of commodity futures and the outperformance of the hump-shaped volatility in comparison to the more popular exponential decaying volatility. This hump shaped feature is more pronounced when the market is volatile.

- by Boda Kang and +2
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- Mechanical Engineering, Energy Economics, Applied Economics, Stochastic Volatility

It is known that Heston's stochastic volatility model exhibits moment explosion, and that the critical moment s+ can be obtained by solving (numerically) a simple equation. This yields a leading order expansion for the implied volatility at large strikes: σBS(k, T ) 2 T ∼ Ψ(s+ − 1) × k (Roger Lee's moment formula). Motivated by recent "tail-wing" refinements of this moment formula, we first derive a novel tail expansion for the Heston density, sharpening previous work of Drȃgulescu and Yakovenko [Quant. Finance 2, 6 (2002), 443-453], and then show the validity of a refined expansion of the type σBS(k, T ) 2 T = (β1k 1/2 + β2 + . . . ) 2 , where all constants are explicitly known as functions of s+, the Heston model parameters, spot vol and maturity T . In the case of the "zero-correlation" Heston model such an expansion was derived by Gulisashvili and Stein [Appl. Math. Optim. 61, 3 (2010), 287-315]. Our methods and results may prove useful beyond the Heston model: the entire quantitative analysis is based on affine principles: at no point do we need knowledge of the (explicit, but cumbersome) closed form expression of the Fourier transform of log ST (equivalently: Mellin transform of ST ); what matters is that these transforms satisfy ordinary differential equations of Riccati type. Secondly, our analysis reveals a new parameter ("critical slope"), defined in a model free manner, which drives the second and higher order terms in tail-and implied volatility expansions.

- by Archil Gulisashvili
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- Economics, Quantitative analysis, Derivative Pricing, Quantitative Finance

We consider a class of asset pricing models, where the risk-neutral joint process of log-price and its stochastic variance is an affine process in the sense of . First we obtain conditions for the price process to be conservative and a martingale. Then we present some results on the long-term behavior of the model, including an expression for the invariant distribution of the stochastic variance process. We study moment explosions of the price process, and provide explicit expressions for the time at which a moment of given order becomes infinite. We discuss applications of these results, in particular to the asymptotics of the implied volatility smile, and conclude with some calculations for the Heston model, a model of Bates and the Barndorff-Nielsen-Shephard model.

We consider the model Zi = Xi + εi, for i.i.d. Xi's and εi's and independent sequences (Xi) i∈N and (εi) i∈N . The density fε of ε1 is assumed to be known, whereas the one of X1, denoted by g, is unknown. Our aim is to estimate linear functionals of g, ψ, g for a known function ψ. We propose a general estimator of ψ, g and study the rate of convergence of its quadratic risk as a function of the smoothness of g, fε and ψ. Different contexts with dependent data, such as stochastic volatility and AutoRegressive Conditionally Heteroskedastic models, are also considered. An estimator which is adaptive to the smoothness of unknown g is then proposed, following a method studied by Laurent et al. (Preprint (2006)) in the Gaussian white noise model. We give upper bounds and asymptotic lower bounds of the quadratic risk of this estimator. The results are applied to adaptive pointwise deconvolution, in which context losses in the adaptive rates are shown to be optimal in the minimax sense. They are also applied in the context of the stochastic volatility model.

- by Fabienne Comte
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- Model Selection, Stochastic Volatility, Laplace Transform, Density Estimation

This paper proposes two types of stochastic correlation structures for Multivariate Stochastic Volatility (MSV) models, namely the constant correlation (CC) MSV and dynamic correlation (DC) MSV models, from which the stochastic covariance structures can easily be obtained. Both structures can be used for purposes of determining optimal portfolio and risk management strategies through the use of correlation matrices, and for calculating Value-at-Risk (VaR) forecasts and optimal capital charges under the Basel Accord through the use of covariance matrices. A technique is developed to estimate the DC MSV model using the Markov Chain Monte Carlo (MCMC) procedure, and simulated data show that the estimation method works well. Various multivariate conditional volatility and MSV models are compared via simulation, including an evaluation of alternative VaR estimators. The DC MSV model is also estimated using three sets of empirical data, namely Nikkei 225 Index, Hang Seng Index and Strait Times Index returns, and significant dynamic correlations are found. The Dynamic

- by Michael McAleer
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- Approximation Theory, Econometrics, Data Analysis, Portfolio Management

There are two unique volatility surfaces associated with any arbitrage-free set of standard European option prices, the implied volatility surface and the local volatility surface. Several papers have discussed the stochastic differential equations for implied volatilities that are consistent with these option prices but the static and dynamic no-arbitrage conditions are complex, mainly due to the large (or even infinite) dimensions of the state probability space. These no-arbitrage conditions are also instrument-specific and have been specified for some simple classes of options. However, the problem is easier to resolve when we specify stochastic differential equations for local volatilities instead. And the option prices and hedge ratios that are obtained by making local volatility stochastic are identical to those obtained by making instantaneous volatility or implied volatility stochastic. After proving that there is a one-to-one correspondence between the stochastic implied volatility and stochastic local volatility approaches, we derive a simple dynamic no-arbitrage condition for the stochastic local volatility model that is model-specific. The condition is very easy to check in local volatility models having only a few stochastic parameters.

- by Carol Alexander
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- Stochastic Volatility

We examine a general multi-factor model for commodity spot prices and futures valuation. We extend the multi-factor long-short model in [1] and [2] in two important aspects: firstly we allow for both the long and short term dynamic factors to be mean reverting incorporating stochastic volatility factors and secondly we develop an additive structural seasonality model. Then a Milstein discretized non-linear stochastic volatility state space representation for the model is developed which allows for futures and options contracts in the observation equation. We then develop numerical methodology based on an advanced Sequential Monte Carlo algorithm utilising Particle Markov chain Monte Carlo to perform calibration of the model jointly with the filtering of the latent processes for the long-short dynamics and volatility factors. In this regard we explore and develop a novel methodology based on an adaptive Rao-Blackwellised version of the Particle Markov chain Monte Carlo methodology. In doing this we deal accurately with the non-linearities in the state-space model which are therefore introduced into the filtering framework. We perform analysis on synthetic and real data for oil commodities.

- by Gareth Peters and +1
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- Applied Mathematics, Statistics, Statistical Computing, Computational Finance

In this paper, we propose a multivariate model for financial assets which incorporates jumps, skewness, kurtosis and stochastic volatility, and discuss its applications in the context of equity and credit risk. In the former case we describe the stochastic behavior of a series of stocks or indexes, in the latter we apply the model in a multi-firm, value-based default model.

- by Elisa Luciano
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- Economics, Derivative Pricing, Quantitative Finance, Mathematical Sciences

In this paper we consider the pricing of an American call option whose underlying asset dynamics evolve under the influence of two independent stochastic volatility processes of the Heston (1993) type. We derive the associated partial differential equation (PDE) of the option price using hedging arguments and Ito's lemma. An integral expression for the general solution of the PDE is presented by using Duhamel's principle and this is expressed in terms of the joint transition density function for the driving stochastic processes. We solve the Kolmogorov PDE for the joint transition density function by first transforming it to a corresponding system of characteristic PDEs using a combination of Fourier and Laplace transforms. The characteristic PDE system is solved by using the method of characteristics. With the full price representation in place, numerical results are presented by first approximating the early exercise surface with a bivariate log linear function. We perform numerical comparisons with results generated by the method of lines algorithm and note that our approach is very competitive in terms of accuracy.

- by Jonathan Ziveyi
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- Stochastic Process, Stochastic Volatility, Method of Lines, Laplace Transform
- by Andrea Pascucci
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- Path Dependence, Mathematical Sciences, Stochastic Volatility, Minimum variance

In this paper we analyze asset returns models with diffusion part and jumps in returns with stochastic volatility either from diffusion or pure jump part. We consider different specifications for the pure jump part including compound Poisson, Variance Gamma and Levy α-stable jumps. Monte Carlo Markov chain algorithm is constructed to estimate models with latent Variance Gamma and Levy α−stable jumps. Our construction corrects for separability problems in the state space of the MCMC for Levy α−stable jumps. We apply our model specifications for analysis of S&P500 daily returns. We find, that models with infinite activity jumps and stochastic volatility from diffusion perform well in capturing S&P500 returns characteristics. Models with stochastic volatility from jumps cannot represent excess kurtosis and tails of returns distributions. One-day and one-week ahead prediction and VaR performance characterizing conditional returns distribution rejects Variance Gamma jumps in favor of Levy α−stable jumps in returns.

- by Harald Uhlig and +1
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- Econometrics, Stochastic Volatility, Monte Carlo Markov Chain, State Space

We propose a new and intuitive risk-neutral valuation model for real estate derivatives which are linked to autocorrelated indices. We model the observed index with an autoregressive model which can be estimated using standard econometric techniques. The resulting index behavior can easily be analyzed and closed-form pricing solutions are derived for forwards, swaps and European put and call options. We demonstrate the application of the model by valuing a put option on a house price index. Autocorrelation in the index returns appears to have a large impact on the option value. We also study the effect of an over-or undervalued real estate market. The observed effects are significant and as expected.

- by A. Pelsser
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- Real Estate, Seasonality, Price Discovery, Stochastic Volatility

The stochastic volatility model usually incorporates asymmetric effects by introducing the negative correlation between the innovations in returns and volatility. In this paper, we propose a new asymmetric stochastic volatility model, based on the leverage and size effects. The model is a generalization of the exponential GARCH (EGARCH) model of . We consider categories for asymmetric effects, which describes the difference among the asymmetric effect of the EGARCH model, the threshold effects indicator function of Glosten, Jagannathan and Runkle (1992), and the negative correlation between the innovations in returns and volatility. The new model is estimated by the efficient importance sampling method of Liesenfeld and Richard (2003), and the finite sample properties of the estimator are investigated using numerical simulations. Four financial time series are used to estimate the alternative asymmetric SV models, with empirical asymmetric effects found to be statistically significant in each case. The empirical results for S&P 500 and Yen/USD returns indicate that the leverage and size effects are significant, supporting the general model.

- by Michael McAleer
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- Econometrics, Numerical Simulation, Financial time series, Stochastic Volatility

In this paper we compare the forecast performance of continuous and discrete-time volatility models. In discrete time, we consider more than ten GARCH-type models and an asymmetric autoregressive stochastic volatility model. In... more

- by Helena Veiga
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- Volatility Forecasting, Maximum Likelihood, Stochastic Volatility, Particle Filter

Guaranteed annuity options are options providing the right to convert a policyholder's accumulated funds to a life annuity at a fixed rate when the policy matures. These options were a common feature in UK retirement savings contracts issued in the 1970's and 1980's when interest rates were high, but caused problems for insurers as the interest rates began to fall in the 1990's. Currently, these options are frequently sold in the US and Japan as part of variable annuity products. The last decade the literature on pricing and risk management of these options evolved. Until now, for pricing these options generally a geometric Brownian motion for equity prices is assumed. However, given the long maturities of the insurance contracts a stochastic volatility model for equity prices would be more suitable. In this paper explicit expressions are derived for prices of guaranteed annuity options assuming stochastic volatility for equity prices and either a 1-factor or 2-factor Gaussian interest rate model. The results indicate that the impact of ignoring stochastic volatility can be significant.

- by A. Pelsser and +1
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- Economics, Risk Management, Mathematical Sciences, Stochastic Volatility

We give a survey of the methods involved in portfolio selection with partial observation. We describe both the theoretical and numerical aspects related to these optimization problems. The presentation is divided in two parts. In the first one, we focus on continuous-time problem : here, the mean rates of return of the asset prices are not directly observable. Investors observe only asset prices. By the method of change of probability and innovation process in filtering theory, the partial observation portfolio selection problem is transformed into a full observation one with the additional filter state variable, for which one may apply the martingale or PDE approach. We investigate different cases for the modelling of the unobervable mean rate of return : Bayesian, linear-Gaussian and finite state Markov chain. In the second part, we consider discretetime optimization problems : this context includes the case of unobservable volatility. We are then concerned with the numerical approximation of the optimization problem under partial observation. This is achieved by performing a quantization of the pair process filter-observation, and using backward dynamic programming formulas. Several numerical experiments illustrate the resuts for hedging problems in the context of partially observed stochastic volatility models.

- by Huyen Pham
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- Optimization Problem, Portfolio Optimization, Stochastic Volatility, Rate of return

Consider an option on a stock whose volatility is unknown and stochastic. An agent assumes this volatility to be a specific function of time and the stock price, knowing that this assumption may result in a misspecification of the volatility. However, if the misspecified volatility dominates the true volatility, then the misspecified price of the option dominates its true price. Moreover, the option hedging strategy computed under the assumption of the misspecified volatility provides an almost sure one-sided hedge for the option under the true volatility. Analogous results hold if the true volatility dominates the misspecified volatility. These comparisons can fail, however, if the misspecified volatility is not assumed to be a function of time and the stock price. The positive results, which apply to both European and American options, are used to obtain a bound and hedge for Asian options.

- by Monique Jeanblanc
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- Applied Mathematics, Mathematical Finance, Stochastic Volatility, Mathematical

We provide a new framework for estimating the systematic and idiosyncratic jump tail risks in financial asset prices. The theory underlying our estimates are based on in-fill asymptotic arguments for directly identifying the systematic and idiosyncratic jumps, together with conventional long-span asymptotics and Extreme Value Theory (EVT) approximations for consistently estimating the tail decay parameters and tail dependencies. On implementing the estimation procedures with a panel of high-frequency intraday prices for a large cross-section of individual stocks and the aggregate S&P 500 market portfolio, we find that the distributions of the systematic and idiosyncratic jumps are both generally heavy-tailed and not necessarily symmetric. Our estimates also point to the existence of strong dependencies between the market-wide jumps and the corresponding systematic jump tails for all of the stocks in the sample. Moreover, we show how the jump tail dependencies deduced from the high-frequency data together with the day-today temporal variation in the volatility are able to "explain" most of the "extreme" joint dependencies observed for the individual stocks and the market portfolio at the daily level.

- by Tim Bollerslev
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- Econometrics, Extreme Value Theory, Stochastic Volatility, Systematic Risk

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... more

- by Maxwell King
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- Stochastic Process, Econometrics, Empirical Finance, Markov Chain Monte Carlo

This study compares the performance of several methods to calculate the Value-at-Risk of the six main ASEAN stock markets. We use filtered historical simulations, GARCH models, and stochastic volatility models. The out-of-sample performance is analyzed by various backtesting procedures. We find that simpler models fail to produce sufficient Value-at-Risk forecasts, which appears to stem from several econometric properties of the return distributions. With stochastic volatility models, we obtain better Value-at-Risk forecasts compared to GARCH. The quality varies over forecasting horizons and across markets. This indicates that, despite a regional proximity and homogeneity of the markets, index volatilities are driven by different factors.

- by Thomas Walther and +1
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- ASEAN, Stochastic Volatility, GARCH, Value at Risk

§ GARCH option pricing models have the advantage of a well-established econometric foundation.

- by Emese Lazar and +1
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- Stochastic Volatility, GARCH, Markov switching, Levy Process