Parameters estimations for continuous-time stochastic volatility models

This study performs a comparative analysis of various GARCH and stochastic volatility (SV) models using the BIST 100 index. The models we examine include traditional GARCH (1,1) models and SV models with an AR (1) log-volatility process. Additionally, we consider more flexible models that incorporate jump components, volatility-in-mean, leverage effects, and innovations following a t-distribution or a moving average. The empirical findings reveal the following findings: (1) Stochastic Volatility (SV) models generally demonstrate better performance when compared to their GARCH counterparts. (2) The inclusion of a jump component and innovations following a tdistribution notably enhances the performance of the standard GARCH model while having less impact on the SV model. (3) The presence of a

This thesis presents our study on using the hybrid stochastic-local volatility model for option pricing. Many researchers have demonstrated that stochastic volatility models cannot capture the whole volatility surface accurately, although the model parameters have been calibrated to replicate the market implied volatility data for near at-the-money strikes. On the other hand, the local volatility model can reproduce the implied volatility surface, whereas it does not consider the stochastic behaviour of the volatility. To combine the advantages of stochastic volatility (SV) and local volatility (LV) models, a class of stochastic-local volatility (SLV) models has been developed. The SLV model contains a stochastic volatility component represented by a volatility process and a local volatility component represented by a so-called leverage function. The leverage function can be roughly seen as a ratio between local volatility and conditional expectation of stochastic volatility. The difficulty of implementing the SLV model lies in the calibration of the leverage function. In the thesis, we first review the fundamental theories of stochastic differential equations and the classic option pricing models, and study the behaviour of the volatility in the context of FX market. We then introduce the SLV model and illustrate our implementation of the calibration and pricing procedure. We apply the SLV model to exotic option pricing in the FX market and compare pricing results of the SLV model with pure local volatility and pure stochastic volatility models. Numerical results show that the SLV model can match the implied volatility surface very well as well as improve the pricing performance for barrier options. In addition, we further discuss some extensions of the SLV project, such as parallelization potential for accelerating option pricing and pricing techniques for window barrier options. Although the SLV model we use in the thesis is not entirely new, we contribute to the research in the following aspects: 1) we investigate the hybrid volatility modeling thoroughly from theoretical backgrounds to practical implementations; 2) we resolve some critical issues in implementing the SLV model such as developing a fast and stable numerical method to derive the leverage function; and 3) we build a robust calibration and pricing platform under the SLV model, which can be extended for practical uses.

This thesis presents our study on using the hybrid stochastic-local volatility model for option pricing. Many researchers have demonstrated that stochastic volatility models cannot capture the whole volatility surface accurately, although the model parameters have been calibrated to replicate the market implied volatility data for near at-the-money strikes. On the other hand, the local volatility model can reproduce the implied volatility surface, whereas it does not consider the stochastic behaviour of the volatility. To combine the advantages of stochastic volatility (SV) and local volatility (LV) models, a class of stochastic-local volatility (SLV) models has been developed. The SLV model contains a stochastic volatility component represented by a volatility process and a local volatility component represented by a so-called leverage function. The leverage function can be roughly seen as a ratio between local volatility and conditional expectation of stochastic volatility. The difficulty of implementing the SLV model lies in the calibration of the leverage function. In the thesis, we first review the fundamental theories of stochastic differential equations and the classic option pricing models, and study the behaviour of the volatility in the context of FX market. We then introduce the SLV model and illustrate our implementation of the calibration and pricing procedure. We apply the SLV model to exotic option pricing in the FX market and compare pricing results of the SLV model with pure local volatility and pure stochastic volatility models. Numerical results show that the SLV model can match the implied volatility surface very well as well as improve the pricing performance for barrier options. In addition, we further discuss some extensions of the SLV project, such as parallelization potential for accelerating option pricing and pricing techniques for window barrier options. Although the SLV model we use in the thesis is not entirely new, we contribute to the research in the following aspects: 1) we investigate the hybrid volatility modeling thoroughly from theoretical backgrounds to practical implementations; 2) we resolve some critical issues in implementing the SLV model such as developing a fast and stable numerical method to derive the leverage function; and 3) we build a robust calibration and pricing platform under the SLV model, which can be extended for practical uses.

Modeling Energy Price Dynamics: GARCH Versus Stochastic Volatility

Modeling energy price dynamics: GARCH versus stochastic volatility

Term Structure and Volatility: Lessons from the Eurodollar Markets

This paper introduces a general class of stochastic volatility models that can serve as a basis for modeling and estimating simultaneous equation systems involving multivariate volatility processes. The class consists of processes obtained as monotone polynomial transformations of so-called stochastic autoregressive volatility (SARV) models. The class permits a flexible modeling of volatility and avoids strong distributional assumptions. Most of the standard stochastic volatility models are incorporated in the framework, including the lognormal autoregressive model and stochastic volatility generalizations of GARCH and EGARCH. General conditions for strict statinarity, ergodicity, and the existence of finite lower-order unconditional moments for SARV are established. Closed-form expressions for the lowerorder moments constitute the basis for the proposed generalized method of moments estimation procedure. As an illustration we consider an informationdriven model of the return volatility–trading volume system for financial assets. Parameters with structural interpretation are estimated and an extended Kalmanfilter procedure allows for the construction of volatility forecasts based on the bivariate return-volume series. Empirical results are based on daily NYSE data for IBM over 1973–1991.

In this work we study the effects by including threshold, constant and time-dependent correlation in stochastic volatility (SV) models to capture the asymmetry relationship between stock returns and volatility. We develop SV models which include only time-dependent correlated innovations and both threshold and time-dependent correlation, respectively. It has been shown in literature that the SV model with only constant correlation does a better job of capturing asymmetry than threshold stochastic volatility (TSV) model. We show here that the SV model with time-dependent correlation performs better than the model with constant correlation on capturing asymmetry, and the comprehensive model with both threshold and time-dependently correlated innovations dominates models with pure threshold, constant and time-dependent correlation, and both threshold and constant correlation as well. In our comprehensive model, volatility and returns are time-dependently correlated, where the time-varying correlation is negative, and the volatility is more persistent, less volatile and higher following negative returns as expected. An empirical study is provided to illustrate our findings.

L 1-estimation of a location parameter is studied for the “product type” stochastic volatility models. The asymptotic distribution of the L 1-estimator is established under general conditions on the behavior of the distribution function of the errors near zero.

Estimating volatility and model parameters of stochastic volatility models with jumps using particle filter

This paper proposes a new approach to analyze stock return asymmetry and quantiles. We also present a new scale mixture of uniform (SMU) representation for the asymmetric Laplace distribution (ALD). The use of the SMU for a probability distribution is a data augmentation technique that simplifies the Gibbs sampler of the Bayesian Markov chain Monte Carlo algorithms. We consider a stochastic volatility (SV) model with an ALD error distribution. With the SMU representation, the full conditional distribution for some parameters is shown to have closed form. It is also known that the ALD can be used to obtain the coefficients of quantile regression models. This paper also considers a quantile SV model by fixing the skew parameter of the ALD at specific quantile level. Simulation study shows that the proposed methodology works well in both SV and quantile SV models using Bayesian approach. In the empirical study, we analyze index returns of the stock markets in Australia, Japan, Hong Kong, Thailand, and the UK and study the effect of S&P 500 on these returns. The results show the significant return asymmetry in some markets and the influence by S&P 500 in all markets at all quantile levels. Copyright © 2014 John Wiley & Sons, Ltd.

Ordinal stochastic volatility (OSV) models were recently developed and fitted by Müller & Czado (2009) to account for the discreteness of financial price changes, while allowing for stochastic volatility (SV). The model allows for exogenous factors both on the mean and volatility level. A Bayesian approach using Markov Chain Monte Carlo (MCMC) is followed to facilitate estimation in these parameter driven models. In this paper the applicability of the OSV model to financial stocks with different levels of trading activity is investigated and the influence of time between trades, volume, day time and the number of quotes between trades is determined. In a second focus we compare the performance of OSV models and SV models. The analysis shows that the OSV models which account for the discreteness of the price changes perform quite well when applied to such data sets.

The calendar spread options (CSOs) on agricultural commodities, most notably corn, soybeans and wheat, allow market participants to hedge the roll-over risk of futures contracts. Despite the interest from agricultural businesses, there is lack of both theoretical and empirical research on pricing and hedging performances of CSOs. We propose to price and hedge CSOs under geometric Brownian motion (GBM) and stochastic volatility (SV) models. We estimate the model parameters by using implied state-generalized method of moments (IS-GMM) and evaluate the in-sample and out- of-sample pricing and hedging performances. We find that the average pricing errors of the SV model are 0.79% for corn, 0.75% for soybeans and 1.2% for wheat; the pricing and hedging performance of the SV model are mostly superior to the benchmark GBM model, both in and out of sample, with only one exception where the out-of-sample hedging error for the GBM model for market makers is slightly better than the SV model.

Efficient, accurate, and fast Markov Chain Monte Carlo estimation methods based on the Implicit approach are proposed. In this article, we introduced the notion of Implicit method for the estimation of parameters in Stochastic Volatility models. Implicit estimation offers a substantial computational advantage for learning from observations without prior knowledge and thus provides a good alternative to classical inference in Bayesian method when priors are missing. Both Implicit and Bayesian approach are illustrated using simulated data and are applied to analyze daily stock returns data on CAC40 index.

On leverage in a stochastic volatility model