An Efficient Numerical Method for Pricing Options Under Stochastic Volatility with Jump Model

Recent studies have shown that stochastic volatility in a continuous-time framework provides an excellent fit for financial asset returns when combined with finite-activity Merton's type compound Poisson Jump-diffusion models. However, we demonstrate that stochastic volatility does not play a central role when incorporated with infinite-activity Levy type pure jump models such as variance-gamma and normal inverse Gaussian processes to model high and low frequency historical time-series SP500 index returns. In addition, whether sources of stochastic volatility are diffusions or jumps are not relevant to improve the overall empirical fits of returns. Nevertheless, stochastic diffusion volatility with infinite-activity Levy jumps processes considerably reduces SP500 index call option in-sample and out-of-sample pricing errors of long-term ATM and OTM options, which contributed to a substantial improvement of pricing performances of SVJ and EVGSV models, compared to constant volatility Levy-type pure jumps models and/or stochastic volatility model without jumps. Interestingly, unlike asset returns, whether pure Levy jumps specifications are finite or infinite activity is not an important factor to enhance option pricing model performances once stochastic volatility is incorporated. Option prices are computed via improved Fast Fourier Transform algorithm using characteristic functions to match arbitrary log-strike grids with equal intervals with each moneyness and maturity of actual market option prices considered in this paper.

We extend the scheme developed in B. During, A. Pitkin, ”High-order compact finite difference scheme for option pricing in stochastic volatility jump models”, 2017, to the so-called stochastic volatility with contemporaneous jumps (SVCJ) model, derived by Duffie, Pan and Singleton. The performance of the scheme is assessed through a number of numerical experiments, using comparisons against a standard second-order central difference scheme. We observe that the new high-order compact scheme achieves third order convergence alongside improvements in efficiency and computation time.

We extend the scheme developed in B. During, A. We extend the scheme developed in B. During, A. Pitkin, High-order compact finite difference scheme for option pricing in stochastic volatility jump models, 2019, to the so-called stochastic volatility with contemporaneous jumps (SVCJ) model, derived by Duffie, Pan and Singleton. The performance of the scheme is assessed through a number of numerical experiments, using comparisons against a standard second-order central difference scheme. We observe that the new high-order compact scheme achieves fourth order convergence and discuss the effects on efficiency and computation time.

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.

Consistent pricing of VIX and equity derivatives with the 4/2 stochastic volatility plus jumps model

Using intraday data from the KOSPI 200 Index options, we examine the pricing performance of alternative option pricing models. For comparison, we consider the Black and Scholes (Journal of Political Economy, 81, 1973, 637) model, a simple traders’ rule known as the ad hoc Black‐Scholes model, the deterministic volatility model, the stochastic volatility model, and the stochastic volatility with jumps model. Contrary to the findings of Jackwerth and Rubinstein (Recovering stochastic processes from option prices), Li and Pearson (A “horse race” among competing option pricing models using S&P 500 Index options), and Kim (Journal of Futures Markets, 29, 2009, 999) using daily data, we find that the most complicated model, namely the stochastic volatility with jumps model, shows the best performance for pricing the KOSPI 200 Index options. Overall, our evidence from intraday data indicates that the traders’ rules do not dominate mathematically more sophisticated models.

Option pricing using the fast Fourier transform under the double exponential jump model with stochastic volatility and stochastic intensity

Stochastic volatility of the futures prices of emission allowances: A Bayesian approach

This paper offers a new approach for pricing options on assets with stochastic volatility. We start by constructing the surface of Black-Scholes implied volatilities for (readily observable) liquid, European call options with varying strike prices and maturities. Then, we show that the implied volatility of an at-the-money call option with time-to-maturity going to zero is equal to the underlying asset's instantaneous (stochastic) volatility. We then model the stochastic processes followed by the implied volatilities of options of all maturities and strike prices jointly with the stock price, and find a no-arbitrage condition that their drift must satisfy. Finally, we use the resulting arbitrage-free joint process for the stock price and its volatility to price other derivatives, such as standard but illiquid options as well as exotic options using numerical methods. The great advantage of our approach is that, when pricing these other derivatives, we are secure in the knowledge that the model values the hedging instruments - namely the stock and the simple, liquid options - consistently with the market. Our approach can easily be extended to allow for stochastic interest rates and a stochastic dividend yield, which may be particularly relevant to the pricing of currency and commodity options. We can also extend our model to price bond options when the term structure of interest rates has stochastic volatility.

Graphical User Interface for pricing Cryptocurrency Options under the Stochastic Volatility with Correlated Jumps model

A particle filter based method to price American option under partial observation framework is introduced. Assuming the underlying price process is driven by unobservable latent factors, the pricing methodology should contain inference on latent factors in addition to the original least-squares Monte Carlo approach of Longstaff and Schwartz. Sequential Monte Carlo is a widely applied technique to provide such inference. Applications on stochastic volatility models has been introduced by Rambharat, who assume that volatility is a latent stochastic process, and capture information about it using particle filter based “summary vectors”. This paper investigates this particle filter based pricing methodology, with an extension to a stochastic volatility jump model, stochastic volatility correlated jump model (SVCJ), and auxiliary particle filter (APF) introduced first by Pitt and Shephard. In the APF algorithm of SVCJ model, it also provides a modification version to enhance the performance in the resampling step. A detailed implementation and numerical examples of the algorithm are provided. The algorithm is also applied to empirical data.

This study, attempts to estimate and compare four different models of jump-diffusion class combined with stochastic volatility that are based on stochastic differential equations, and their parameters latent variables are estimated by Markov chain Monte Carlo (MCMC) methods. In the Stochastic Volatility with Correlated Jumps (SVCJ) model, volatilities are scholastic, and the term jump is added to both scholastic prices and volatilities. The results of this study showed that this model is more efficient than the others are, as it provides a significantly better fit to the data, and therefore, corrects the shortcomings of the previous models and that it is closer to the actual market prices. Therefore, our estimating model under the Monte Carlo simulation allows an analysis on oil prices during certain times in the periods of tension and shock in the oil market

This paper presents a new numerical method for pricing American call options when the volatility of the price of the underlying stock is stochastic. By exploiting a log-linear relationship of the optimal exercise boundary with respect to volatility changes, we derive an integral representation of an American call price and the early exercise premium which holds under stochastic volatility. This representation is used to develop a numerical method for pricing the American options based on an approximation of the optimal exercise boundary by Chebyshev polynomials. Numerical results show that our numerical approach can quickly and accurately price American call options both under stochastic and/or constant volatility.

Partial integro-differential equation (PIDE) formulations are often preferable for pricing options under models with stochastic volatility and jumps, especially for American-style option contracts. We consider the pricing of options under such models, namely the Bates model and the so-called stochastic volatility with contemporaneous jumps (SVCJ) model. The nonlocality of the jump terms in these models leads to matrices with full matrix blocks. Standard discretization methods are not viable directly since they would require the inversion of such a matrix. Instead, we adopt a two-step implicit-explicit (IMEX) time discretization scheme, the IMEX-CNAB scheme, where the jump term is treated explicitly using the second-order Adams--Bashforth (AB) method, while the rest is treated implicitly using the Crank--Nicolson (CN) method. The resulting linear systems can then be solved directly by employing LU decomposition. Alternatively, the systems can be iterated under a scalable algebraic multigrid (AMG) method. For pricing American options, LU decomposition is employed with an operator splitting method for the early exercise constraint or, alternatively, a projected AMG method can be used to solve the resulting linear complementarity problems. We price European and American options in numerical experiments, which demonstrate the good efficiency of the proposed methods.

Price jumps, stochastic volatility, seasonality and stochastic cost of carry have been considered separately, but not collectively, for pricing agricultural commodity futures and options. Focusing on corn, soybeans and wheat, we propose a comprehensive model that incorporates all four features. We employ a special Markov Chain Monte Carlo (MCMC) algorithm, which is new in the agricultural commodity derivatives pricing literature, to estimate the proposed stochastic volatility (SV) and stochastic volatility with jumps (SVJ) models. Along with various model diagnostic tests, the in-sample and out-of-sample pricing and hedging results generally, with few exceptions, lend support for the SVJ model.