Stochastic Evolution Equations for Large Portfolios of Stochastic Volatility Models

This paper studies the question of filtering and maximizing terminal wealth from expected utility in a partially information stochastic volatility models. The special features is that the only information available to the investor is the one generated by the asset prices, and the unobservable processes will be modeled by a stochastic differential Using the change of measure techniques, the partial observation context can be transformed into a full information context such that coefficients depend only on past history of observed prices (filters processes). Adapting the stochastic non-linear filtering, we show that under some assumptions on the model coefficients, the estimation of the filters depend on a priori models for the trend and the stochastic volatility. Moreover, these filters satisfy a stochastic partial differential equations named Kushner-Stratonovich equations. Using the martingale duality approach in this partially observed incomplete model, we can characterize the value function and the optimal portfolio. The main result here is that the dual value function associated to the martingale approach can be expressed, via the dynamic programming approach, in terms of the solution to a semilinear partial differential equation. We illustrate our results with some examples of stochastic volatility models popular in the financial literature.

In this study we present a closed‐form, exact solution for the pricing of VIX futures in a stochastic volatility model with simultaneous jumps in both the asset price and volatility processes. The newly derived formula is then used to show that the well‐known convexity correction approximations can sometimes lead to large errors. Utilizing the newly derived formula, we also conduct an empirical study, the results of which demonstrate that the Heston stochastic volatility model is a good candidate for the pricing of VIX futures. While incorporating jumps into the underlying price can further improve the pricing of VIX futures, adding jumps to the volatility process appears to contribute little improvement for pricing VIX futures. © 2011 Wiley Periodicals, Inc. Jrl Fut Mark

We introduce a hybrid stochastic volatility model where the asset price process follows the Heston model and interest rates are governed by a two-factor stochastic model. Two cases are considered. First, it is assumed that interest rates and asset prices are uncorrelated. The characteristic function method is used to derive semi-analytical pricing formulae for plain vanilla options. In the second case we introduce a correlation between the asset price process and the short rate process and use Monte Carlo simulations for pricing options. To reduce the stochastic error, we implement the control variate method where an estimator of the option value for the uncorrelated case is used as a control variate. The options are priced with a varying correlation coefficient. We observe that the control variate method allows us to speed up Monte Carlo computations by a factor with the magnitude of several hundreds. The efficiency of the method is higher for smaller values of the correlation coefficient. We then study the impact a correlation between the two processes has on option prices. It has been noticed that the call option price is an increasing function of the correlation coefficient.

Stochastic volatility (SV) models take a very important role in financial market, while there still exist some difficulties in estimating parameters in SV models. In this paper, a unified parameter estimation algorithm is proposed to estimate continuous-time SV model. Parameters in equity prices and volatilities stochastic processes are estimated separately after orthogonalization of Brownian motion in SV models. A closed-form of Maximum Likelihood Estimation (MLE) and moment estimation results for four parameters in SV models are deducted. Five typical SV models are simulated to reveal the characteristics in empirical volatilities data. Parameter estimation results are presented to prove the convergence of estimation algorithm. We also compare our algorithm with Least Square (LS) estimations and numerical solutions of all the parameters in with MLE method, respectively. LS can only work in a specific form of SV models and the convergence speed is limited. Numerical solutions of the likelihood function with four parameters are unstable and can not converge to true values of all the parameters. Thus our algorithm works more efficiently than other methods.

Fractional Brownian Motions, Fractional Noises and Applications

In this paper, we propose a fractional stochastic volatility jump-diffusion model which extends the Bates(1996) model. Where we model the volatility as a fractional process. Extensive empirical studies show that the distributions of the logarithmic returns of financial asset usually exhibit properties of self-similarity and long-range dependence and since the fractional Brownian motion has these two important properties, it has the ability to capture the behavior of underlying asset price. Further incorporating jumps into the stochastic volatility framework gives further freedom to financial mathematicians to fit both the short and long end of the implied volatility surface. We propose a stochastic model which contains both fractional and jump process. Then we price options using Monte Carlo simulations along with a variance reduction technique(antithetic variates). We use market data from the S&P 500 index and we compare our results with the Heston and Bates model using error measures. The results show our model greatly outperforms previous models in terms of estimation accuracy.

In this paper we prove the existence of a unique solution for a class of stochastic parabolic partial differential equations in bounded domains, with Dirichlet boundary conditions. The main tool is an equivalence result, provided by the stochastic characteristics method, between the stochastic equations under investigation and a class of deterministic parabolic equations with moving boundaries, depending on random coefficients. We show the existence of the solution to this last problem, thus providing a solution to the former.

Term Structure and Volatility: Lessons from the Eurodollar Markets

Estimation of Jump-Diffusion Processes via Empirical Characteristic Functions

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.

This article introduces a discrete‐time fractional stochastic volatility model (FSV) based on fractional Gaussian noise. The new model includes the standard stochastic volatility model as a special case and has the same limit as the fractional integrated stochastic volatility (FISV) model, which is the continuous‐time fractional Ornstein–Uhlenbeck process. A simulated maximum likelihood method, which maximizes the time‐domain log‐likelihood function calculated by the importance sampling technique, and a frequency‐domain quasi maximum likelihood method (or quasi Whittle) are employed to estimate the model parameters. Simulation studies suggest that, while both estimation methods can accurately estimate the model, the simulated maximum likelihood method outperforms the quasi Whittle method. As an illustration, we fit the FSV and FISV models with the proposed estimation techniques to the S&P 500 composite index over a sample period spanning 45 years.

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.

Based on an extension of the process of investors' expectations to stochastic volatility we derive asset price processes in a general continuous time pricing kernel framework. Our analysis suggests that stochastic volatility of asset price processes results from the fact that investors do not know the risk of an asset and therefore the volatility of the process of their expectations is stochastic, too. Furthermore, our model is consistent with empirical studies reporting negative correlation between asset prices and their volatility as well as significant variations in the Sharpe ratio.

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.