Volatility Derivatives in Market Models with Jumps

Statistics & Probability Letters, 2012

In this paper, we deal with the pricing of European style options when the dynamics of the risky underlying asset are driven by a Markov-modulated jump diffusion with stochastic volatility. We investigate the Radon-Nikodym derivative for the minimal martingale measure and a partial differential equation approach for pricing European options. An optimal hedging strategy in terms of local risk minimization is obtained.

2018

The Bates model provides a parsimonious fit to implied volatility surfaces, and its usefulness in developed markets is well documented. However, there is a lack of research assessing its applicability to developing markets. Additionally, research surrounding its usefulness for hedging long term liabilities is limited, despite its frequent use for this purpose. This dissertation dissects the dynamics of the Bates model into the Heston and Merton models in order to separately examine the effects of stochastic volatility and jumps. Challenges surrounding application of this model are investigated through an evaluation of risk-neutral calibration and simulation methods. The model's ability to fit the implied volatility surfaces from the JSE Top 40 equity index is analysed. Lastly, an evaluation of the model's delta and vega hedging performance is presented by comparing it to the hedge performance of other commonly used models. Firstly, I would like to thank my internal supervisors, Obeid Mahomed and Professor David Taylor for their guidance over the course of this dissertation. Thank you to Neil Kennedy for the initial contributions to this report, as well as for the dissertation topic. I thank my family, especially my parents, Jill and Steve Gorven for their endless love and support. I would not have survived my university career without it. More thanks goes to the 2017 M.Phil class for making this tough degree an enjoyable and fulfilling experience. Lastly, thank you to BANKSETA for the funding they provided for this year of study.

Journal of Applied Mathematics and Stochastic Analysis, 2008

In this paper, generalizing results in Alòs, , we see that the dependence of jumps in the volatility under a jump-diffusion stochastic volatility model, has no effect on the short-time behaviour of the at-the-money implied volatility skew, although the corresponding Hull and White formula depends on the jumps. Towards this end, we use Malliavin calculus techniques for Lévy processes based on Løkka (2004), .

1996

We consider the following stochastic di erential equation (S.D.E.) for describing nancial data evolution: dX t = b(t; X t) dt + (t)h(X t) dW t ; X(0) = x with a stochastic volatility (t) (e.g. the combination of a di usion and a jump process). We prove the existence and positivity of the solution of a Cox-Ingersoll-Ross type S.D.E. with time varying coe cients which is a special case of our model. From observation on X t at times t i (with non regular sampling scheme), we propose a non-parametric estimator for the volatility that is optimal in a certain way. We show its pointwise convergence and its asymptotic normality. We propose an estimator for the volatility jump times and prove a Central Limit Theorem. The application of these estimators to the BTP futures (Italian ten year bond futures) and Lira 1 month deposit Eurorates seems to con rm the adequacy of the proposed model.

Finance and Stochastics, 2007

In this paper we use the Malliavin calculus techniques to obtain an expression for the short-time behavior of the at-the-money implied volatility skew for a generalization of the Bates model, where the volatility does not need to be neither a difussion, nor a Markov process, as the examples in section 7 show. This expression depends on the derivative of the volatility in the sense of Malliavin calculus.

Quantitative Finance, 2013

This paper considers the problem of pricing American options when the dynamics of the underlying are driven by both stochastic volatility following a square root process as used by , and by a Poisson jump process as introduced by Merton (1976). Probability arguments are invoked to find a representation of the solution in terms of expectations over the joint distribution of the underlying process.

To improve the empirical performance of the Black-Scholes model, many alternative models have been proposed to address the leptokurtic feature of the asset return distribution, volatility smile and the effects of volatility clustering phenomenon. However, analytical tractability remains a problem for most of the alternative models. In this paper, we propose a Markov jump diffusion model, that can not only incorporate both the leptokurtic feature and volatility smile, but also present the economic features of volatility clustering and long memory. To evaluate derivatives prices, we apply Lucas’s general equilibrium framework to provide closed form formulas for option and futures prices. When the jump size follows a specific distribution, for instance a lognormal distribution and a default probability, we write explicit analytic formulas for the equilibrium prices. Through these formulas, we illustrate the effect of jumps, via stochastic intensity, on implied volatility and volatility...

In this paper we find numerical solutions for the pricing problem in jump diffusion markets. We utilize a model in which the underlying asset price is generated by a process that consists of a Brownian motion and an independent compensated Poisson process. By risk neutral pricing the option price can be expressed as an expectation. We simulate the option price numerically using the Monte Carlo method.

SSRN Electronic Journal, 2001

This article reviews a pricing model, suitable for variance-gamma jump processes, based on the method of lines. The method accuracy is studied using European style calls as a benchmark. Implementation details for continuously and discretely monitored barrier options, and American and Bermudan options are given. function for the variance-gamma pure jump process. Consequently, the model of lines provides a method for pricing derivative claims in the variance gamma model when the model parameters reduce the arrival distribution to an Erlang distribution. As will be demonstrate later on, the general case can often be recovered with high precision by interpolation or extrapolation methods. The differential equations that arise in the model of lines are similar to the Black-Scholes equations, with the following important distinction: time derivatives are replaced by finite differences, while derivatives with respect to stock price remain intact. The pricing functions along each line are found to satisfy a system of inhomogeneous ordinary differential-difference equations, which admit simple analytic solutions for most options. Although the situation is similar to Carr's equations for American style options with randomized maturity [6], there are important differences. The key difference being that calendar time in Carr's solution must be reinterpreted as financial time in the model of lines. Because trading occurs in calendar time, not in financial time, the re-interpretation breaks risk neutrality. To restore risk neutrality, the stock price must be scaled and the option price discounted from one time-step to the next. Under this adjustment, the model of lines reproduces the exactup to negligible roundoff errors -prices of European style options in which the underlying follows a variance-gamma process. The solution scheme for European style puts and calls can easily be modified to exactly price barrier and Bermudan options contingent on information on the lines only. Path-dependent options requiring continuous monitoring, such as American options and barrier options can also be priced efficiently. However, in these cases, the model of lines produces approximate prices, as the exercise boundaries are assumed to be piecewise constant between lines. The model of lines enjoys the same calibration efficiencies and empirical explanatory power of the variance-gamma model. Nonetheless, it is still interesting to compare it with the better known stochastic volatility models. Diffusion models where volatility is stochastic have been considered by a number of authors, including Hull and White [17], Wiggins [29], Scott [27], Melino and Turnbull [22], Heston [15], [16]. Stochastic volatility models based on GARCH, such as in Duan [13], have the added advantage that the postulated process for the underlying asset is well justified by historical time series. As in jump models, the effect of stochastic volatility can formally be interpreted as inducing a random time change. Furthermore, these

arXiv (Cornell University), 2018

This paper presents the solution to a European option pricing problem by considering a regime-switching jump diffusion model for the underlying financial asset price dynamics. The regimes are assumed to be the results of an observed pure jump process, driving the values of interest rate and volatility coefficient. The pure jump process is assumed to be a semi-Markov process on finite state space. This consideration helps to incorporate a specific type of memory influence in the asset price. Under this model assumption, the locally risk minimizing prices of the European type path-independent options are found. The Föllmer-Schweizer decomposition is adopted to show that the option price satisfies an evolution problem, as a function of time, stock price, market regime, and the stagnancy period. To be more precise, the evolution problem involves a linear, parabolic, degenerate and non-local system of integro-partial differential equations. We have established existence and uniqueness of the classical solution to the evolution problem in an appropriate class. This has helped us in obtaining the optimal hedging.

Journal of Mathematics and Statistics, 2013

In this study, we present the application of Time Changed Levy method to model a jump-diffusion process with stochastic volatility and stochastic interest rate. We apply the Lewis Fourier transform method as well as the risk neutral expectation pricing method to derive a formula for a European option pricing. These combining methods give quite a short route to derive the formula and make it efficient to compute option prices. We also show the calibration of our model to the real market with global and local optimization algorithms.

Applied Stochastic Models in Business and Industry, 2017

In this paper, we introduce a unifying approach to option pricing under continuous‐time stochastic volatility models with jumps. For European style options, a new semi‐closed pricing formula is derived using the generalized complex Fourier transform of the corresponding partial integro‐differential equation. This approach is successfully applied to models with different volatility diffusion and jump processes. We also discuss how to price options with different payoff functions in a similar way.In particular, we focus on a log‐normal and a log‐uniform jump diffusion stochastic volatility model, originally introduced by Bates and Yan and Hanson, respectively. The comparison of existing and newly proposed option pricing formulas with respect to time efficiency and precision is discussed. We also derive a representation of an option price under a new approximative fractional jump diffusion model that differs from the aforementioned models, especially for the out‐of‐the money contracts....

SSRN Electronic Journal, 2003

arXiv (Cornell University), 2020

We consider option pricing using a discrete-time Markov switching stochastic volatility with co-jump model, which can model volatility clustering and varying mean-reversion speeds of volatility. For pricing European options, we develop a computationally efficient method for obtaining the probability distribution of average integrated variance (AIV), which is key to option pricing under stochastic-volatility-type models. Building upon the efficiency of the European option pricing approach, we are able to price an American-style option, by converting its pricing into the pricing of a portfolio of European options. Our work also provides constructive guidance for analyzing derivatives based on variance, e.g., the variance swap. Numerical results indicate our methods can be implemented very efficiently and accurately.