High‐Frequency Instruments and Identification‐Robust Inference for Stochastic Volatility Models

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.

A New Class of Discrete-Time Stochastic Volatility Model with Correlated Errors

Returns of risky assets and their time-dependent volatility are often jointly modelled by stochastic volatility models (SVMs). Over the last few decades, several SVMs have been proposed to adequately capture the features of the relationship between the return and its volatility. The earliest SVM considers a hierarchical model, with current return as a function of the current latent volatility, which is further modelled as an auto-regressive process. In an attempt to make SVM more appropriate for complex realistic market behaviour, leverage parameter was introduced, which however led to violation of martingale difference property of the risky part of return (a necessary mean-zero condition that prevents arbitrage opportunities under mild regularity conditions). Subsequently, alternative SVMs had been developed and are currently in use. In this article, we propose mean-corrections for several generalizations of SVM with leverage that capture the complex market behaviour as well as satisfy martingale difference property of returns. We also establish a few theoretical results to characterize the key desirable features of these models, and present comparison with other popular competitors. Furthermore, three real-life examples (CITI bank stock price, euro–USD rate and S&P 500 index returns) have been used to demonstrate the performance of this new class of SVMs.

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.

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.

Bayesian Inference of Long-Memory Stochastic Volatility via Wavelets

In this study, we undertake a systematic study of historic market volatility spanning roughly five preceding decades. We focus specifically on the time series of the realized volatility (RV) of the S&P500 index and its distribution function. As expected, the largest values of RV coincide with the largest economic upheavals of the period: Savings and Loan Crisis, Tech Bubble, Financial Crisis and Covid Pandemic. We address the question of whether these values belong to one of the three categories: Black Swans (BS), that is, they lie on scale-free, power-law tails of the distribution; Dragon Kings (DK), defined as statistically significant upward deviations from BS; or Negative Dragons Kings (nDK), defined as statistically significant downward deviations from BS. In analyzing the tails of the distribution with RV>40, we observe the appearance of “potential” DK, which eventually terminate in an abrupt plunge to nDK. This phenomenon becomes more pronounced with the increase in the number of days over which the average RV is calculated—here from daily, n=1, to “monthly”, n=21. We fit the entire distribution with a modified Generalized Beta (mGB) distribution function, which terminates at a finite value of the variable but exhibits a long power-law stretch prior to that, as well as a Generalized Beta Prime (GB2) distribution function, which has a power-law tail. We also fit the tails directly with a straight line on a log-log scale. In order to ascertain BS, DK or nDK behavior, all fits include their confidence intervals and p-values are evaluated for the data points to check whether they can come from the respective distributions.

Equity-linked annuity pricing with cliquet-style guarantees in regime-switching and stochastic volatility models with jumps

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

Modeling Energy Price Dynamics: GARCH Versus Stochastic Volatility

Modeling energy price dynamics: GARCH versus stochastic volatility

In the past years, there has been an extensive investigation of the class of stochastic volatility models for the evaluation of options and complex derivatives. These models have proven to be extremely useful in generalizing the classic Black–Scholes economy and accounting for discrepancies between observation and predictions in the simple log-normal, constant-volatility model. In this paper, we study the structure of an options market with a stochastic volatility that will eventually vanish (i.e., reaches zero) for very short periods of time with probability of one. We investigate the form of pricing measures in this situation, first in a simple binomial case, and then for a diffusion model, by constructing a weak approximation in discrete space and continuous time. The market described allows fleeting arbitrage opportunities, since a vanishing volatility prevents the construction of an equivalent measure, so that pricing contingent claims are, a priori, not obvious. Nevertheless, we can still produce a fair pricing equation. Let us note that this issue is not only of theoretical relevance, as the phenomenon of very low volatility has indeed been observed in the financial markets and the economy for quite a long time in the recent past.

Term Structure and Volatility: Lessons from the Eurodollar Markets

This paper introduces a new class of stochastic volatility models for asset prices, the generalized Heston Nandi GARCH (GHN‐GARCH), with the primary objective of optimal dynamic asset allocation under expected utility theory for constant relative risk aversion investors. We study some of its theoretical properties, and demonstrate that the GHN‐GARCH class of models permits closed‐form solutions for optimal allocation and value function. We introduce and study in more detail an example of this class, the 4/2‐HN‐GARCH model, inspired by the continuous‐time 4/2 stochastic volatility model of Grasselli. A robust parameter estimation procedure is developed, and a numerical analysis is performed.