Inference for a Class of Stochastic Volatility Models Using Option and Spot Prices: Application of a Bivariate Kalman Filter

In this paper Bayesian methods are applied to a stochastic volatility model using both the prices of the asset and the prices of options written on the asset. Posterior densities for all model parameters, latent volatilities and the market price of volatility risk are produced via a Markov Chain Monte Carlo (MCMC) sampling algorithm. Candidate draws for the unobserved volatilities are obtained in blocks by applying the Kalman filter and simulation smoother to a linearization of a nonlinear state space representation of the model. Crucially, information from both the spot and option prices affects the draws via the specification of a bivariate measurement equation, with implied Black–Scholes volatilities used to proxy observed option prices in the candidate model. Alternative models nested within the Heston (1993) framework are ranked via posterior odds ratios, as well as via fit, predictive and hedging performance. The method is illustrated using Australian News Corporation spot and option price data.     

Efficient Monte Carlo option pricing under CEV model

One of the financial model with nonconstant volatiltiy is the constant elasticity of varinace model, or CEV model for short. The CEV model is an altrnative to the Black–Scholes model of stock price movements. In this diffusion process, unlike the Black–Scholes model, the volatility is a function of the stock price and involves two parameters. In this article, we propose an efficient Monte-Carlo algorithm for pricing arithmetic Asian option under CEV model. In an earlier work by Mehrdoust, an efficient Monte Carlo simulation algorithm for pricing arithmetic Asian options under Black–Scholes model is proposed. The proposed algorithm has proved extremely successful in decreasing the standard deviation and the error of simulation in pricing of the arithmetic Asian options. In this article, we find that the proposed algorithm under the geometric Brownian motion assumption in the Black–Scholes model can effectively apply for pricing arithmetic Asian options when the stock price process follows the CEV model. Numerical experiments show that our algorithm gives very accurate results.     

Derivation of Kurtosis and Option Pricing Formulas for Popular Volatility Models with Applications in Finance

This article discusses some topics relevant to financial modeling. The kurtosis of a distribution plays an important role in controlling tail-behavior and is used in edgeworth expansion of the call prices. We present derivations of the kurtosis for a number of popular volatility models useful in financial applications, including the class of random coefficient GARCH models. Option pricing formulas for various classes of volatility models are also derived and a simple proof of the option pricing formula under the Black–Scholes model is given.     

Examining the interrelation dynamics between option and stock markets using the Markov-switching vector error correction model

This study examines the dynamics of the interrelation between option and stock markets using the Markov-switching vector error correction model. Specifically, we calculate the implied stock prices from the Black–Scholes 6 Black, F. and Scholes, M. 1973. The pricing of options and corporate liabilities. J. Polit. Econ., 81: 637–659. [Crossref], [Web of Science ®] [Google Scholar] model and establish a statistic framework in which the parameter of the price discrepancy between the observed and implied prices switches according to the phase of the volatility regime. The model is tested in the US S&P 500 stock market. The empirical findings of this work are consistent with the following notions. First, while option markets react more quickly to the newest stock–option disequilibrium shocks than spot markets, as found by earlier studies, we further indicate that the price adjustment process occurring in option markets is pronounced when the high variance condition is concerned, but less so during the stable period. Second, the degree of the co-movement between the observed and implied prices is significantly reduced during the high variance state. Last, the lagged price deviation between the observed and implied prices functions as an indicator of the variance-turning process.     

Analytical and numerical studies on the second-order asymptotic expansion method for European option pricing under two-factor stochastic volatilities

The celebrated Black–Scholes model made the assumption of constant volatility but empirical studies on implied volatility and asset dynamics motivated the use of stochastic volatilities. Christoffersen in 2009 showed that multi-factor stochastic volatilities models capture the asset dynamics more realistically. Fouque in 2012 used it to price European options. In 2013, Chiarella and Ziveyi considered Christoffersen’s ideas and introduced an asset dynamics where the two volatilities of the Heston type act separately and independently on the asset price, and using Fourier transform for the asset price process and double Laplace transform for the two volatilities processes, solved a pricing problem for American options. This paper considers the Chiarella and Ziveyi model and parameterizes it so that the volatilities revert to the long-run-mean with reversion rates that mimic fast (for example daily) and slow (for example seasonal) random effects. Applying asymptotic expansion method presented by Fouque in 2012, we make an extensive and detailed derivation of the approximation prices for European options. We also present numerical studies on the behavior and accuracy of our first- and second-order asymptotic expansion formulas.     

Inference for a Class of Stochastic Volatility Models Using Option and Spot Prices: Application of a Bivariate Kalman Filter

In this paper Bayesian methods are applied to a stochastic volatility model using both the prices of the asset and the prices of options written on the asset. Posterior densities for all model parameters, latent volatilities and the market price of volatility risk are produced via a Markov Chain Monte Carlo (MCMC) sampling algorithm. Candidate draws for the unobserved volatilities are obtained in blocks by applying the Kalman filter and simulation smoother to a linearization of a nonlinear state space representation of the model. Crucially, information from both the spot and option prices affects the draws via the specification of a bivariate measurement equation, with implied Black-Scholes volatilities used to proxy observed option prices in the candidate model. Alternative models nested within the Heston (1993) framework are ranked via posterior odds ratios, as well as via fit, predictive and hedging performance. The method is illustrated using Australian News Corporation spot and option price data.     

Conditional expectation determination based on the J-process using Malliavin calculus applied to pricing American options

The aim of our paper is to elaborate a theoretical methodology based on the Malliavin calculus to calculate the following conditional expectation (P_t(X_t)|(X_s)) for s≤t where the only state variable follows a J-process [Jerbi Y. A new closed-form solution as an extension of the Black—Scholes formula allowing smile curve plotting. Quant Finance. 2013; Online First Article. doi:10.1080/14697688.2012.762458]. The theoretical results are applied to the American option pricing, consisting of an extension of the work of Bally et al. [Pricing and hedging American options by Monte Carlo methods using a Malliavin calculus approach. Monte Carlo Methods Appl. 2005;11-2:97–133], as well as the J-process (with additional parameters λ and θ) is an extension of the Wiener process. The introduction of the aforesaid parameters induces skewness and kurtosis effects, i.e. smile curve allowing to fit with the reality of financial market. In his work Jerbi [Jerbi Y. A new closed-form solution as an extension of the Black–-Scholes formula allowing smile curve plotting. Quant Finance. 2013; Online First Article. doi:10.1080/14697688.2012.762458] showed that the use of the J-process is equivalent to the use of a stochastic volatility model based on the Wiener process as in Heston's. The present work consists on extending this result to the American options. We studied the influence of the parameters λ and θ on the American option price and we find empirical results fitting with the options theory.     

HEAVY-TAILED-DISTRIBUTED THRESHOLD STOCHASTIC VOLATILITY MODELS IN FINANCIAL TIME SERIES

To capture mean and variance asymmetries and time‐varying volatility in financial time series, we generalize the threshold stochastic volatility (THSV) model and incorporate a heavy‐tailed error distribution. Unlike existing stochastic volatility models, this model simultaneously accounts for uncertainty in the unobserved threshold value and in the time‐delay parameter. Self‐exciting and exogenous threshold variables are considered to investigate the impact of a number of market news variables on volatility changes. Adopting a Bayesian approach, we use Markov chain Monte Carlo methods to estimate all unknown parameters and latent variables. A simulation experiment demonstrates good estimation performance for reasonable sample sizes. In a study of two international financial market indices, we consider two variants of the generalized THSV model, with US market news as the threshold variable. Finally, we compare models using Bayesian forecasting in a value‐at‐risk (VaR) study. The results show that our proposed model can generate more accurate VaR forecasts than can standard models.     

Volatility,Momentum, and Time-Varying Skewness in Foreign Exchange Returns

This article tests a stochastic volatility model of exchange rates that links both the level of volatility and its instantaneous covariance with returns to pathwise properties of the currency. In particular, the model implies that the return–volatility covariance behaves like a weighted average of recent returns and hence switches signs according to the direction of trends in the data. This implies that the skewness of the finite-horizon return distribution likewise switches sign, leading to time-varying implied volatility “smiles” in options prices. The model is fit and assessed using Bayesian techniques. Some previously reported volatility results are accounted for by the fitted models. The predicted pattern of skewness dynamics accords well with that found in historical options prices.     

Common-factor stochastic volatility modelling with observable proxy

Multi-asset modelling is of fundamental importance to financial applications such as risk management and portfolio selection. In this article, we propose a multivariate stochastic volatility modelling framework with a parsimonious and interpretable correlation structure. Building on well-established evidence of common volatility factors among individual assets, we consider a multivariate diffusion process with a common-factor structure in the volatility innovations. Upon substituting an observable market proxy for the common volatility factor, we markedly improve the estimation of several model parameters and latent volatilities. The model is applied to a portfolio of several important constituents of the S&P500 in the financial sector, with the VIX index as the common-factor proxy. We find that the prediction intervals for asset forecasts are comparable to those of more complex dependence models, but that option-pricing uncertainty can be greatly reduced by adopting a common-volatility structure. The Canadian Journal of Statistics 48: 36–61; 2020 © 2020 Statistical Society of Canada     

A Stochastic Volatility Model With Realized Measures for Option Pricing

Abstract

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

Empirical Performance and Asset Pricing in Hidden Markov Models

Abstract

To improve the empirical performance of the Black-Scholes model, many alternative models have been proposed to address leptokurtic feature, volatility smile, and volatility clustering effects of the asset return distributions. However, analytical tractability remains a problem for most alternative models. In this article, we study a class of hidden Markov models including Markov switching models and stochastic volatility models, that can incorporate leptokurtic feature, volatility clustering effects, as well as provide analytical solutions to option pricing. We show that these models can generate long memory phenomena when the transition probabilities depend on the time scale. We also provide an explicit analytic formula for the arbitrage-free price of the European options under these models. The issues of statistical estimation and errors in option pricing are also discussed in the Markov switching models.     

Option pricing for generalized distributions

The Black Scholes formula has been widely used to price financial instruments. The derivation of this formula is based on the assumption of lognormally distributed returns which is often in poor agreement with actual data. An option pricing formula based on the generalized beta of the second kind (GB2) is presented. This formula includes the Black Scholes formula as a special case and accommodates a wide variety of nonlognormally distributed returns. The sensitivity of option values to departures from the skewness and kurtosis associated with the lognormal distribution is investigated.     

Valuing equity-indexed annuities with icicled barrier options

Inspired by the recent popularity of autocallable structured products, this paper intends to enhance equity-indexed annuities (EIAs) by introducing a new class of barrier options, termed icicled barrier options. The new class of options has a vertical (icicled) barrier along with the horizontal one of the ordinary barrier options, which may act as an additional knock-in or knock-out trigger. To improve the crediting method of EIAs, we propose a new EIA design, termed autocallable EIA, with payoff structure similar to the autocallable products except for the minimum guarantee, and further investigate the possibility of embedding various icicled barrier options into the plain point-to-point or the ratchet EIAs. Explicit pricing formulas for the proposed EIAs and the icicled barrier options are obtained under the Black–Scholes model. To the purpose, we derive the joint distribution of the logarithmic returns at the icicled time and the maturity, and their running maximum. As an application of the well-known reflection principle, the derivation itself is an interesting probability problem and the joint distribution plays a key role in the subsequent pricing stage. Our option pricing result can be easily transferred to EIAs or other equity-linked products. The pricing formulas for the EIAs and the options are illustrated through numerical examples.     

INDEX TO VOLUME 4 (1986)

This article provides an empirical investigation of the risk-neutral variance process and the market price of variance risk implied in the foreign-currency options market. There are three principal contributions. First, the parameters of Heston's mean-reverting square-root stochastic volatility model are estimated using dollar/mark option prices from 1987 to 1992. Second, it is shown that these implied parameters can be combined with historical moments of the dollar/mark exchange rate to deduce an estimate of the market price of variance risk. These estimates are found to be nonzero, time varying, and of sufficient magnitude to imply that the compensation for variance risk is a significant component of the risk premia in the currency market. Finally, the out-of-sample test suggests that the historical variance and the Hull and White implied variance contain no more information than that imbedded in the Heston implied variance.     

基于BP神经网络的S＆P500指数期权定价

期权定价理论源于影响期权价格的变量和期权价格之间的非线性关系,传统的Black-Scholes期权定价公式过于严格的假设削弱了该公式在现实中的适用性,使其在理论与应用上均存在缺陷。因此,能够以任意精度近似复杂非线性系统的神经网络运用于期权定价。分别利用BP神经网络和Black-Scholes期权定价公式对S＆P 500指数看跌期权进行定价,实证结果表明BP神经网络的定价结果要优于Black-Scholes定价公式。     

ASSESSING AND TESTING FOR THRESHOLD NONLINEARITY IN STOCK RETURNS

This paper proposes a test for threshold nonlinearity in a time series with generalized autore‐gressive conditional heteroscedasticity (GARCH) volatility dynamics. This test is used to examine whether financial returns on market indices exhibit asymmetric mean and volatility around a threshold value, using a double‐threshold GARCH model. The test adopts the reversible‐jump Markov chain Monte Carlo idea of Green, proposed in 1995, to calculate the posterior probabilities for a conventional GARCH model and a double‐threshold GARCH model. Posterior evidence favouring the threshold GARCH model indicates threshold nonlinearity with asymmetric behaviour of the mean and volatility. Simulation experiments demonstrate that the test works very well in distinguishing between the conventional GARCH and the double‐threshold GARCH models. In an application to eight international financial market indices, including the G‐7 countries, clear evidence supporting the hypothesis of threshold nonlinearity is discovered, simultaneously indicating an uneven mean‐reverting pattern and volatility asymmetry around a threshold return value.     

A Standardized Normal-Laplace Mixture Distribution Fitted to Symmetric Implied Volatility Smiles

This article proposes to use a standardized version of the normal-Laplace mixture distribution for the modeling of tail-fatness in an asset return distribution and for the fitting of volatility smiles implied by option prices. Despite the fact that only two free parameters are used, the proposed distribution allows arbitrarily high kurtosis and uses one shape parameter to adjust the density function within three standard deviations for any specified kurtosis. For an asset price model based on this distribution, the closed-form formulas for European option prices are derived, and subsequently the volatility smiles can be easily obtained. A regression analysis is conducted to show that the kurtosis, which is commonly used as an index of tail-fatness, is unable to explain the smiles satisfactorily under the proposed model, because the additional shape parameter also significantly accounts for the deviations revealed in smiles. The effectiveness of the proposed parsimonious model is demonstrated in the practical examples where the model is fitted to the volatility smiles implied by the NASDAQ market traded foreign exchange options.     

Pricing arithmetic Asian option under a two-factor stochastic volatility model with jumps

This paper presents an efficient Monte Carlo simulation scheme based on the variance reduction methods to evaluate arithmetic average Asian options in the context of the double Heston's stochastic volatility model with jumps. This paper consists of two essential parts. The first part presents a new flexible stochastic volatility model, namely, the double Heston model with jumps. In the second part, by combining two variance reduction procedures via Monte Carlo simulation, we propose an efficient Monte Carlo simulation scheme for pricing arithmetic average Asian options under the double Heston model with jumps. Numerical results illustrate the efficiency of our method.     

Option pricing under model and parameter uncertainty using predictive densities

The theoretical price of a financial option is given by the expectation of its discounted expiry time payoff. The computation of this expectation depends on the density of the value of the underlying instrument at expiry time. This density depends on both the parametric model assumed for the behaviour of the underlying, and the values of parameters within the model, such as volatility. However neither the model, nor the parameter values are known. Common practice when pricing options is to assume a specific model, such as geometric Brownian Motion, and to use point estimates of the model parameters, thereby precisely defining a density function.We explicitly acknowledge the uncertainty of model and parameters by constructing the predictive density of the underlying as an average of model predictive densities, weighted by each model's posterior probability. A model's predictive density is constructed by integrating its transition density function by the posterior distribution of its parameters. This is an extension to Bayesian model averaging. Sampling importance-resampling and Monte Carlo algorithms implement the computation. The advantage of this method is that rather than falsely assuming the model and parameter values are known, inherent ignorance is acknowledged and dealt with in a mathematically logical manner, which utilises all information from past and current observations to generate and update option prices. Moreover point estimates for parameters are unnecessary. We use this method to price a European Call option on a share index.