Stochastic Volatility Jump-diffusion Research Articles

Bayesian estimation of stochastic volatility jump diffusion model parameters using S&P 500 and VIX data

The Stochastic Volatility Jump Diffusion (SVJD) model is a common tool used for option pricing. Existing literature has employed many techniques in order to estimate the parameters of this model, including both Bayesian and frequentist approaches. In 2010, Duan and Yeh [Jump and volatility risk premiums implied by VIX. J Econ Dyn Control. 2010;34(11):2232–2244] used a frequentist approach with maximum likelihood estimation that included the VIX, the Chicago Board Options Exchange (CBOE) Volatility Index, which carries real time S&P 500 option price information in addition to the S&P 500 data. In this paper, we combine the previous Bayesian approach in Cape et al. [Estimating Heston's and Bates' models parameters using Markov Chain Monte Carlo simulation. J Stat Comput Simul. 2015;85(11):2295–2314] with the inclusion of VIX by Duan and Yeh. We provide the derivations of the conditional posterior distributions of the SVJD model's parameters and a guide to a Markov Chain Monte Carlo (MCMC) algorithm used to estimate the parameters. We apply the algorithm to the historical S&P 500 and VIX data and provide the subsequent parameter estimates. We offer these tools for others performing similar research with our R code available upon request.

Analytical solution of the Dupire-like equation in calibration to the generalized stochastic volatility jump-diffusion model for option pricing

Analytical solution of the Dupire-like equation in calibration to the generalized stochastic volatility jump-diffusion model for option pricing

A Simulation and Empirical Study of the Maximum Likelihood Estimator for Stochastic Volatility Jump-Diffusion Models

Abstract We investigate the behaviour of the maximum likelihood estimator (MLE) for stochastic volatility jump-diffusion models commonly used in financial risk management. A simulation study shows the practical conditions under which the MLE behaves according to theory. In an extensive empirical study based on nine indices and more than 6000 individual stocks, we nonetheless find that the MLE is unable to replicate key higher moments. We then introduce a moment-targeted MLE – robust to model misspecification – and revisit both simulation and empirical studies. We find it performs better than the MLE, improving the management of financial risk.

Interaction between Sovereign Quanto Credit Default Swap Spreads and Currency Options

This study analyzes the term structures of sovereign quanto credit default swap (CDS) spreads and currency options, which are driven by anticipated currency depreciation risk following sovereign credit default (Twin Ds). We develop consistent pricing models for these instruments using a jump-diffusion stochastic volatility model, which allows us to decompose the term structure into the risk components. We find a common risk factor between the intensity process of sovereign credit risk and the stochastic volatility of the exchange rate, and the depreciation risk mainly captures the dependence structure between these markets during periods of high market stress in the Eurozone countries. Depreciation risk is an important component of sovereign quanto CDS spreads and is evident in the negative slope of the volatility smile in the currency option market.

Option Pricing with Fractional Stochastic Volatilities and Jumps

Empirical studies suggest that asset price fluctuations exhibit “long memory”, “volatility smile”, “volatility clustering” and asset prices present “jump”. To fit the above empirical characteristics of the market, this paper proposes a fractional stochastic volatility jump-diffusion model by combining two fractional stochastic volatilities with mixed-exponential jumps. The characteristic function of the log-return is expressed in terms of the solution of two-dimensional fractional Riccati equations of which closed-form solution does not exist. To obtain the explicit characteristic function, we approximate the pricing model by a semimartingale and convert fractional Riccati equations into a classic PDE. By the multi-dimensional Feynman-Kac theorem and the affine structure of the approximate model, we obtain the solution of the PDE with which the explicit characteristic function and its cumulants are derived. Based on the derived characteristic function and Fourier cosine series expansion, we obtain approximate European options prices. By differential evolution algorithm, we calibrate our approximate model and its two nested models to S&P 500 index options and obtain optimal parameter estimates of these models. Numerical results demonstrate the pricing method is fast and accurate. Empirical results demonstrate our approximate model fits the market best among the three models.

Optimal portfolio problem for an insurer under mean-variance criteria with jump-diffusion stochastic volatility model

This paper studies an insurer's optimal investment portfolio under the mean-variance criterion. The financial market consists of a riskless bond and a risky asset, and the latter's volatility is random. We are extending the Cox–Ingersoll–Ross (CIR) model to the case with jumps, where it is modeled by a jump-diffusion stochastic differential equation (SDE). We use a Lévy SDE to describe the risk process we have, in which we extend the classic Cramér-Lundberg model to the Lévy process, and additionally introduce the stochastic volatility into this model. We assume that the insurer in question is a mean-variance optimizer. In other words, the decision that this insurer faces is to simultaneously maximize and minimize the mean and variance of his/her terminal wealth by selecting an optimal portfolio. We have uncovered closed-form solutions to the mean-variance problem with respect to the efficient strategy and efficient frontier by solving for expected utility maximization of a quadratic function through the martingale method. Finally, we give a numerical example that analyzes the economic behavior of the efficient frontier.

Robust consumption and portfolio choice with derivatives trading

This paper analyzes the optimal consumption and portfolio choice problem for an ambiguity-averse investor with recursive preferences. The investor has access to both the stock and derivatives markets. The stock price process follows a stochastic volatility jump-diffusion model and the investor can have different levels of ambiguity about diffusion and jump risks, respectively. We obtain an exact analytical solution for investors with unit elasticity of intertemporal substitution of consumption and an approximate solution otherwise. We find that the optimal consumption policy is more sensitive to ambiguity aversion with respect to diffusion risks than the jump risk. The optimal exposures to diffusion and jump risks are significantly affected by the corresponding ambiguity aversions in the complete market; however, the optimal stock investment is relatively insensitive to jump misspecification in the incomplete market. We also show that taking into consideration ambiguity aversion to diffusion risks and participating in the derivatives market are essential to reduce the potential welfare loss, while the impact of jump misspecification is marginal.

Investigating the effects of illiquidity on credit risks via new liquidity augmented stochastic volatility jump diffusion model

Liquidity is extremely important not only within the context of financial markets but also in every scale of economic transactions. In this study, within the realm of financial markets, we configure liquidity as an independent stochastic process moderating the fluidity of all transactions and hence dynamically changing asset values. This study’s asset value process ignoring liquidity is modelled with a stochastic volatility jump-diffusion () model and that model is augmented with the incorporation of a liquidity process. The new model is called liquidity augmented stochastic volatility jump-diffusion () model. The simulation results suggest that model outperforms the models without liquidity. The application of model on the estimation of probabilities of default and credit risk spreads, using actual financial data of the selected companies listed in Dow Jones 30, reveals that neglecting the liquidity dimension in asset valuation might lead to inefficient assessments of risks. The models without an illiquidity process underestimate the probabilities of default and credit spread risks in comparison to the liquidity augmented model, model, and we believe this might have accounted for the ignored risks that caused the 2007–2008 financial crisis in a great degree.

JDOI variance reduction method and the pricing of American-style options

This article revisits the Diffusion Operator Integral (DOI) variance reduction technique originally proposed in Heath and Platen (2002) and extends its theoretical concept to the pricing of American-style options under (time-homogeneous) Lévy stochastic differential equations. The resulting Jump Diffusion Operator Integral (JDOI) method can be combined with numerous Monte Carlo-based stopping-time algorithms, including the ubiquitous least-squares Monte Carlo (LSMC) algorithm of Longstaff and Schwartz (cf. Carriere (1996) and Longstaff and Schwartz (2001)). We exemplify the usefulness of our theoretical derivations under a concrete, though very general jump-diffusion stochastic volatility dynamics and test the resulting LSMC-based version of the JDOI method. The results provide evidence of a strong variance reduction when compared with a simple application of the LSMC algorithm and proves that applying our technique on top of Monte Carlo-based pricing schemes provides a powerful way to speed-up these methods.

The Theory of Uncertaintism

The stock jumps of the underlying assets underpinning the Margrabe options have been studied by Cheang and Chiarella [Cheang, GH and Chiarella C (2011). Exchange options under jump-diffusion dynamics. Applied Mathematical Finance, 18(3), 245–276], Cheang and Garces [Cheang, GHL and Garces LPDM (2020). Representation of exchange option prices under stochastic volatility jump-diffusion dynamics. Quantitative Finance, 20(2), 291–310], Cufaro Petroni and Sabino [Cufaro Petroni, N and Sabino P (2020). Pricing exchange options with correlated jump diffusion processes. Quantitate Finance, 20(11), 1811–1823], and Ma et al. [Ma, Y, Pan D and Wang T (2020). Exchange options under clustered jump dynamics. Quantitative Finance, 20(6), 949–967]. Although the authors argue that they explored stock jumps under Hawkes processes, those processes are the Poisson process in their applications. Thus, they studied Hawkes processes in-between two assets while this study explores Hawkes process within any asset. Furthermore, the Poisson process can be flipped into Hawkes process and vice versa. In terms of hedging, this study uses specific Greeks (rho and phi) while some of the mentioned studies used other Greeks (Delta, Theta, Vega, and Gamma). Moreover, hedging is carried out under static and dynamic environments. The results illustrate that the jumpy Margrabe option can be extended to complex barrier option and waiting to invest option. In addition, hedging strategies are robust both under static and dynamic environments.

Hedging of options for jump-diffusion stochastic volatility models by Malliavin calculus

Hedging of options for jump-diffusion stochastic volatility models by Malliavin calculus

Electricity price modelling with stochastic volatility and jumps: An empirical investigation

Over the past few years, the electricity derivatives market has experienced a substantial growth in the volume of trade and the diversity of available products. This has led to a rich data environment that requires more sophisticated and accurate modelling approaches for electricity spot prices. This paper deals with an analysis of continuous-time stochastic volatility jump-diffusion processes in the context of pricing of futures contracts written on electricity spots. We formulate a variety of models which aim to capture the most prominent characteristics and stylised facts of the electricity spot market including mean reversion, seasonality, extreme volatility, and spikes. The proposed modelling framework extends the existing models by incorporating mean reversion, stochastic volatility, and jumps in both the underlying spot price process and its volatility. The modelling parameters are estimated using the Markov Chain Monte Carlo (MCMC) technique for the Australian electricity market. We find that incorporating stochastic volatility and jumps in both the underlying electricity spot price and its volatility is absolutely essential to accurately fit the observed electricity spot prices. We derive futures prices in a semi-closed form and confirm flexibility of the proposed models by their ability to fit the observed spot and futures prices in the Australian electricity market.

Investigating the Effects of Illiquidity on Credit Risks via New Liquidity Augmented Stochastic Volatility Jump Diffusion Model

Investigating the Effects of Illiquidity on Credit Risks via New Liquidity Augmented Stochastic Volatility Jump Diffusion Model

JDOI Variance Reduction Method and the Pricing of American-Style Options

The present article revisits the Diffusion Operator Integral (DOI) variance reduction technique originally proposed in [HP02] and extends its theoretical concept to the pricing of American-style options under (time-homogeneous) Levy stochastic differential equations. The resulting Jump Diffusion Operator Integral (JDOI) method can be combined with numerous Monte Carlo based stopping-time algorithms, including the ubiquitous least-squares Monte Carlo (LSMC) algorithm of Longstaff and Schwartz (cf. [Car96], [LS01]). We exemplify the usefulness of our theoretical derivations under a concrete, though very general jump-diffusion stochastic volatility dynamics and test the resulting LSMC based version of the JDOI method. The results provide evidence of a strong variance reduction when compared with a simple application of the LSMC algorithm and proves that applying our technique on top of Monte Carlo based pricing schemes provides a powerful way to speed-up these methods.

Option Pricing under Two-Factor Stochastic Volatility Jump-Diffusion Model

Empirical evidence shows that single-factor stochastic volatility models are not flexible enough to account for the stochastic behavior of the skew, and certain financial assets may exhibit jumps in returns and volatility. This paper introduces a two-factor stochastic volatility jump-diffusion model in which two variance processes with jumps drive the underlying stock price and then considers the valuation on European style option. We derive a semianalytical formula for European vanilla option and develop a fast and accurate numerical algorithm for the computation of the option prices using the fast Fourier transform (FFT) technique. We compare the volatility smile and probability density of the proposed model with those of alternative models, including the normal jump diffusion model and single-factor stochastic volatility model with jumps, respectively. Finally, we provide some sensitivity analysis of the model parameters to the options and several calibration tests using option market data. Numerical examples show that the proposed model has more flexibility to capture the implied volatility term structure and is suitable for empirical work in practice.

An Intelligent Learning and Ensembling Framework for Predicting Option Prices

ABSTRACT Estimating option prices and implied volatilities are critical for option risk management and trading. Common strategies in previous studies have relied on parametric models, including the stochastic volatility model (SV), jump-diffusion model (JD), and Black-Scholes model (BS). However, these models are built on several strict and idealistic assumptions, including lognormality and sample-path continuity. In addition, previous studies on option pricing mainly relied on its own market-level indicators without considering the effect of other concurrent options. To address these challenges, we propose an intelligent learning and ensembling framework based on convolutional neural network (CNN). Specifically, the customized nonparametric learning approach is first utilized to estimate option prices. Second, several traditional parametric models are also applied to estimate these prices. The estimated prices are combined by a CNN to obtain the final estimations. Our experiments based on Chinese SSE 50 ETF options demonstrate that the proposed intelligent framework outperforms the traditional SV model, JD model, and BS model with at least 41.52% performance enhancement in terms of RMSE.

European option pricing under stochastic volatility jump-diffusion models with transaction cost

In this paper, we consider an underlying general stochastic volatility jump-diffusion model. Option pricing under this general model with transaction costs will lead to handling with nonlinear partial integro-differential equations (here after PIDE). In this case, option replication in a discrete-time framework with transaction costs and the non-uniqueness option pricing in such incomplete market will be studied. Observing and introducing a traded proxy for the volatility in the modern market, we acquire a nonlinear PIDE in the advent of transaction costs. Under appropriate regularity conditions, the existence of the strong solution to this pricing problem has been proved. The corresponding self-financing and positions readjustment for pricing a portfolio will also be discussed in the end.

Representation of exchange option prices under stochastic volatility jump-diffusion dynamics

In this article, we provide representations of European and American exchange option prices under stochastic volatility jump-diffusion (SVJD) dynamics following models by Merton [Option pricing when underlying stock returns are discontinuous. J. Financ. Econ., 1976, 3(1-2), 125–144], Heston [A closed-form solution for options with stochastic volatility with applications to bond and currency options. Rev. Financ. Stud., 1993, 6(2), 327–343], and Bates [Jumps and stochastic volatility: Exchange rate processes implicit in Deutsche Mark options. Rev. Financ. Stud., 1996, 9(1), 69–107]. A Radon–Nikodým derivative process is also introduced to facilitate the shift from the objective market measure to other equivalent probability measures, including the equivalent martingale measure. Under the equivalent martingale measure, we derive the integro-partial differential equation that characterizes the exchange option prices. We also derive representations of the European exchange option price using the change-of-numéraire technique proposed by Geman et al. [Changes of numéraire, changes of probability measure and option pricing. J. Appl. Probab., 1995, 32(2), 443–458] and the Fourier inversion formula derived by Caldana and Fusai [A general closed-form spread option pricing formula. J. Bank. Finance, 2013, 37, 4893–4906], and show that these two representations are comparable. Lastly, we show that the American exchange option price can be decomposed into the price of the European exchange option and an early exercise premium.

A Theory of Scenario Generation

We show how to reduce financial asset prices to low-dimensional scenario trees. Applied to time series, the scenarios and their probabilities become time-varying factors. From S&P 500 options, two or three time-varying scenarios suffice to forecast returns, implied variance or skewness on par, or better, than extant multivariate stochastic volatility jump-diffusion models, while reducing the computational effort to fractions of a second.

Use of Adapted Particle Filters in SVJD Models

Particle Filter algorithms for filtering latent states (volatility and jumps) of Stochastic-Volatility Jump-Diffusion (SVJD) models are being explained. Three versions of the SIR particle filter with adapted proposal distributions to the jump occurrences, jump sizes, and both are derived and their performance is compared in a simulation study to the un-adapted particle filter. The filter adapted to both the jump occurrences and jump sizes achieves the best performance, followed in their respective order by the filter adapted only to the jump occurrences and the filter adapted only to the jump sizes. All adapted particle filters outperformed the unadapted particle filter.