Fourier-cosine method for pricing forward starting options with stochastic volatility and jumps

This article provides an efficient method for pricing forward starting options under stochastic volatility model with double exponential jumps. The forward characteristic function of the log asset price is derived and thereby forward starting options are well evaluated by Fourier-cosine technique. Based on adaptive simulated annealing algorithm, the model is calibrated to obtain the estimated parameters. Numerical results show that the pricing method is accurate and fast. Double exponential jumps have pronounced impacts on long-term forward starting options prices. Stochastic volatility model with double exponential jumps fits forward implied volatility smile pretty well in contrast to stochastic volatility model.     

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.     

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.     

Methodology for stochastic volatility process calibration application to the CAC 40 index

In considering volatility as a stochastic, the aim of this paper is to estimate the four parameters related to a particular stochastic process named P1 and based on a Wiener–Levy process. We present the methodology to estimate its four parameters. We calibrate this theoretical model P1 to the CAC 40 index real data. In the same time, we test the normality of the random variables related to the two Wiener–Levy processes. The calibration is performed using the implemented aforesaid algorithm. We compare the stochastic process P1 with another process named P2 and to the Heston [Closed form solution for options with stochastic volatility with application to bonds and currency options, Rev. Financ. Stud. 6(2) (1993), pp. 327–343] process named H0 and to two other improved Heston processes named H1 and H2. For the empirical study, the same algorithm is used to calibrate the five processes. The calibration is based on a database including the CAC 40 index daily ‘closing fixing’ values for the time period from 3rd January 2005 to 22nd January 2007. The data are divided into 18 classes relative to 18 different contracts of European calls on the CAC 40 index. As a result, we find that, the normality test of the CAC 40 index is rejected which is in accordance with the previous original works dealing with this problem. For the five volatility processes, the normality test is verified almost for the same contracts. We also find that according to the used data, the process P1 and its equivalent H1 are the best for calibration.     

Kalman filter-based modelling and forecasting of stochastic volatility with threshold

We propose a parametric nonlinear time-series model, namely the Autoregressive-Stochastic volatility with threshold (AR-SVT) model with mean equation for forecasting level and volatility. Methodology for estimation of parameters of this model is developed by first obtaining recursive Kalman filter time-update equation and then employing the unrestricted quasi-maximum likelihood method. Furthermore, optimal one-step and two-step-ahead out-of-sample forecasts formulae along with forecast error variances are derived analytically by recursive use of conditional expectation and variance. As an illustration, volatile all-India monthly spices export during the period January 2006 to January 2012 is considered. Entire data analysis is carried out using EViews and matrix laboratory (MATLAB) software packages. The AR-SVT model is fitted and interval forecasts for 10 hold-out data points are obtained. Superiority of this model for describing and forecasting over other competing models for volatility, namely AR-Generalized autoregressive conditional heteroscedastic, AR-Exponential GARCH, AR-Threshold GARCH, and AR-Stochastic volatility models is shown for the data under consideration. Finally, for the AR-SVT model, optimal out-of-sample forecasts along with forecasts of one-step-ahead variances are obtained.     

Bayesian estimation and the application of long memory stochastic volatility models

A new sampling-based Bayesian approach to the long memory stochastic volatility (LMSV) process is presented; the method is motivated by the GPH-estimator in fractionally integrated autoregressive moving average (ARFIMA) processes, which was originally proposed by J. Geweke and S. Porter-Hudak [The estimation and application of long memory time series models, Journal of Time Series Analysis, 4 (1983) 221–238]. In this work, we perform an estimation of the memory parameter in the Bayesian framework; an estimator is obtained by maximizing the posterior density of the memory parameter. Finally, we compare the GPH-estimator and the Bayes-estimator by means of a simulation study and our new approach is illustrated using several stock market indices; the new estimator is proved to be relatively stable for the various choices of frequencies used in the regression.     

An empirical Bayesian forecast in the threshold stochastic volatility models

In the area of finance, the stochastic volatility (SV) model is a useful tool for modelling stock market returns. However, there is evidence that asymmetric behaviour of stock returns exists. A threshold SV (THSV) model is provided to capture this behaviour. In this study, we introduce a robust model created through empirical Bayesian analysis to deal with the uncertainty between the SV and THSV models. A Markov chain Monte Carlo algorithm is applied to empirically select the hyperparameters of the prior distribution. Furthermore, the value at risk from the resulting predictive distribution is also given. Simulation studies show that the proposed empirical Bayes model not only clarifies the acceptability of prediction but also reduces the risk of model uncertainty.     

A fractionally integrated Wishart stochastic volatility model

There has recently been growing interest in modeling and estimating alternative continuous time multivariate stochastic volatility models. We propose a continuous time fractionally integrated Wishart stochastic volatility (FIWSV) process, and derive the conditional Laplace transform of the FIWSV model in order to obtain a closed form expression of moments. A two-step procedure is used, namely estimating the parameter of fractional integration via the local Whittle estimator in the first step, and estimating the remaining parameters via the generalized method of moments in the second step. Monte Carlo results for the procedure show a reasonable performance in finite samples. The empirical results for the S&P 500 and FTSE 100 indexes show that the data favor the new FIWSV process rather than the one-factor and two-factor models of the Wishart autoregressive process for the covariance structure.     

Modelling stochastic volatility using generalized t distribution

In modelling financial return time series and time-varying volatility, the Gaussian and the Student-t distributions are widely used in stochastic volatility (SV) models. However, other distributions such as the Laplace distribution and generalized error distribution (GED) are also common in SV modelling. Therefore, this paper proposes the use of the generalized t (GT) distribution whose special cases are the Gaussian distribution, Student-t distribution, Laplace distribution and GED. Since the GT distribution is a member of the scale mixture of uniform (SMU) family of distribution, we handle the GT distribution via its SMU representation. We show this SMU form can substantially simplify the Gibbs sampler for Bayesian simulation-based computation and can provide a mean of identifying outliers. In an empirical study, we adopt a GT–SV model to fit the daily return of the exchange rate of Australian dollar to three other currencies and use the exchange rate to US dollar as a covariate. Model implementation relies on Bayesian Markov chain Monte Carlo algorithms using the WinBUGS package.     

An analytical approximation method for pricing barrier options under the double Heston model

Abstract

The purpose of the paper is to provide an efficient pricing method for single barrier options under the double Heston model. By rewriting the model as a singular and regular perturbed BS model, the double Heston model can separately mimic a fast time-scale and a slow time-scale. With the singular and regular perturbation techniques, we analytically derive the first-order asymptotic expansion of the solution to a barrier option pricing partial differential equation. The convergence and efficiency of the approximate method is verified by Monte Carlo simulation. Numerical results show that the presented asymptotic pricing method is fast and accurate.     

Bivariate degradation modelling with marginal heterogeneous stochastic processes

In this paper, we consider that the degradation of two performance characteristics of a product can be modelled by stochastic processes and jointly by copula functions, but different stochastic processes govern the behaviour of each performance characteristic (PC) degradation. Different heterogeneous and homogeneous models are presented considering copula functions and different combinations of the most used stochastic processes in degradation analysis as marginal distributions. This is an important aspect to consider because the behaviour of the degradation of each PC may be different in its nature. As the joint distributions of the proposed models result in complex distributions, the estimation of the parameters of interest is performed via MCMC. A simulation study is performed to compare heterogeneous and homogeneous models. In addition, the proposed models are implemented to crack propagation data of two terminals of an electronic device, and some insights are provided about the product reliability under heterogeneous models.     

Bayesian analysis of multivariate stochastic volatility with skew return distribution

Multivariate stochastic volatility models with skew distributions are proposed. Exploiting Cholesky stochastic volatility modeling, univariate stochastic volatility processes with leverage effect and generalized hyperbolic skew t-distributions are embedded to multivariate analysis with time-varying correlations. Bayesian modeling allows this approach to provide parsimonious skew structure and to easily scale up for high-dimensional problem. Analyses of daily stock returns are illustrated. Empirical results show that the time-varying correlations and the sparse skew structure contribute to improved prediction performance and Value-at-Risk forecasts.     

Variable dimension via stochastic volatility model using FX rates

In this paper, changepoint analysis is applied to stochastic volatility (SV) models which aim to understand the locations and movements of high frequency FX financial time series. Bayesian inference using the Markov Chain Monte Carlo method is performed using a process called variable dimension for SV parameters. Interesting results are that FX series have locations where one or more positions of the sequence correspond to systemic changes, and overall non-stationarity, in the returns process. Furthermore, we found that the changepoint locations provide an informative estimate for all FX series. Importantly in most cases, the detected changepoints can be identified with economic factors relevant to the country concerned. This helps support the fact that macroeconomics news and the movement in financial price are positively related.     

Forecasting interest rates volatilities by GARCH (1,1) and stochastic volatility models

In this paper, we compare the forecast ability of GARCH(1,1) and stochastic volatility models for interest rates. The stochastic volatility is estimated using Markov chain Monte Carlo methods. The comparison is based on daily data from 1994 to 1996 for the ten year swap rates for Deutsch Mark, Japanese Yen, and Pound Sterling. Various forecast horizons are considered. It turns out that forecasts based on stochastic volatility models are in most cases superiour to those obtained by GARCH(1,1) models.     

Computational aspects of pricing foreign exchange options with stochastic volatility and stochastic interest rates

This paper presents an analytic result for the price of a European call option on a foreign exchange currency rate. Market volatility is assumed correlated with the exchange rate and interest rates, domestic and foreign, are assumed to be stochastic. Integrals involving interest rates are derived, characteristic functions are produced, and, with evaluation, the nature of the integrals involved in Fourier inversion is examined. By comparison with FX market data, some of the effects of the nature of stochastic interest rates upon option prices are examined.     

On periodic autoregressive stochastic volatility models: structure and estimation

To capture both the volatility evolution and the periodicity feature in the autocorrelation structure exhibited by many nonlinear time series, a Periodic AutoRegressive Stochastic Volatility (PAR-SV ) model is proposed. Some probabilistic properties, namely the strict and second-order periodic stationarity, are provided. Furthermore, conditions for the existence of higher-order moments are established. The autocovariance structure of the squares and higher order powers of the PAR-SV process is studied. Its dynamic properties are shown to be consistent with financial time series empirical findings. Ways in which the model may be estimated are discussed. Finally, a simulation study of the performance of the proposed estimation methods is provided and the PAR-SV is applied to model the spot rates of the euro and US dollar both against the Algerian dinar. The empirical analysis shows that the proposed PAR-SV model can be considered as a viable alternative to the periodic generalized autoregressive conditionally heteroscedastic (PGARCH) model.     

GMM estimation of a realized stochastic volatility model: A Monte Carlo study

This article investigates alternative generalized method of moments (GMM) estimation procedures of a stochastic volatility model with realized volatility measures. The extended model can accommodate a more general correlation structure. General closed form moment conditions are derived to examine the model properties and to evaluate the performance of various GMM estimation procedures under Monte Carlo environment, including standard GMM, principal component GMM, robust GMM and regularized GMM. An application to five company stocks and one stock index is also provided for an empirical demonstration.     

Deciding between GARCH and stochastic volatility via strong decision rules

The GARCH and stochastic volatility (SV) models are two competing, well-known and often used models to explain the volatility of financial series. In this paper, we consider a closed form estimator for a stochastic volatility model and derive its asymptotic properties. We confirm our theoretical results by a simulation study. In addition, we propose a set of simple, strongly consistent decision rules to compare the ability of the GARCH and the SV model to fit the characteristic features observed in high frequency financial data such as high kurtosis and slowly decaying autocorrelation function of the squared observations. These rules are based on a number of moment conditions that is allowed to increase with sample size. We show that our selection procedure leads to choosing the model that fits best, or the simplest model under equivalence, with probability one as the sample size increases. The finite sample size behavior of our procedure is analyzed via simulations. Finally, we provide an application to stocks in the Dow Jones industrial average index.