Option Pricing Under a Double Exponential Jump Diffusion Model (original) (raw)
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Journal of Applied Mathematics
Our work is aimed at modeling the American option price by combining the dynamic programming and the optimal stopping time under two asset price models. In doing so, we attempt to control the theoretical error and illustrate the asymptotic characteristics of each model; thus, using a numerical illustration of the convergence of the option price to an equilibrium price, we can notice its behavior when the number of paths tends to be a large number; therefore, we construct a simple estimator on each slice of the number of paths according to an upper and lower bound to control our error. Finally, to highlight our approach, we test it on different asset pricing models, in particular, the exponential Lévy model compared to the simple Black and Scholes model, and we will show how the latter outperforms the former in the real market (Microsoft “MSFT” put option as an example).
Pricing American Options in a Jump Diffusion Model
2011 14th IEEE International Conference on Computational Science and Engineering, 2011
In this study, we use the McKean's integral equation to evaluate the American option price for constant jump diffusion model . The early exercise boundary is approximated by a multipiece exponential function. Numerical results show that the proposed method improves Chesney and Jeanblance results for larger dividend rate American options.
Pricing discrete path-dependent options under a double exponential jump–diffusion model
Journal of Banking & Finance, 2013
We provide methodologies to price discretely monitored exotic options when the underlying evolves according to a double exponential jump diffusion process. We show that discrete barrier or lookback options can be approximately priced by their continuous counterparts' pricing formulae with a simple continuity correction. The correction is justified theoretically via extending the corrected diffusion method of Siegmund (1985). We also discuss the jump effects on the performance of this continuity correction method. Numerical results show that this continuity correction performs very well especially when the proportion of jump volatility to total volatility is small. Therefore, our method is sufficiently of use for most of time.
Numerical Valuation of European and American Options under Kou's Jump-diffusion Model
Numerical methods are developed for pricing European and American options under Kou's jump-diffusion model, which assumes the price of the underlying asset to behave like a geometrical Brownian motion with a drift and jumps whose size is log-double-exponentially distributed. The price of a European option is given by a partial integro-differential equation (PIDE), while American options lead to a linear complementarity problem (LCP) with the same operator. Spatial differential operators are discretized using finite differences on nonuniform grids, and time stepping is performed using the implicit Rannacher scheme. For the evaluation of the integral term easy-to-implement re-cursion formulas are derived which have optimal computational cost. When pricing European options the resulting dense linear systems are solved using a stationary iteration. For American options two ways to solve the LCPs are described: an operator slitting method and a penalty method. Numerical experiments confirm that the developed methods are very efficient, as fairly accurate option prices can be computed in a few milliseconds on a PC.
Pricing Asian Options Under a Hyper-Exponential Jump Diffusion Model
Operations Research, 2012
We obtain a closed-form solution for the double-Laplace transform of Asian options under the hyper-exponential jump diffusion model (HEM). Similar results are only available previously in the special case of the Black-Scholes model (BSM). Even in the case of the BSM, our approach is simpler as we essentially use only Itô's formula and do not need more advanced results such as those of Bessel processes and Lamperti's representation. As a by-product we also show that a well-known recursion relating to Asian options has a unique solution in a probabilistic sense. The double-Laplace transform can be inverted numerically via a two-sided Euler inversion algorithm. Numerical results indicate that our pricing method is fast, stable, and accurate, and performs well even in the case of low volatilities.
Numerical simulations for the pricing of options in jump diffusion markets
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.
A Closed-Form Option Valuation Formula in Markov Jump Diffusion Models
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...
Pricing American exchange options in a jump-diffusion model
Journal of Futures Markets, 2007
A way to estimate the value of an American exchange option when the underlying assets follow jump-diffusion processes is presented. The estimate is based on combining a European exchange option and a Bermudan exchange option with two exercise dates by using Richardson extrapolation as proposed by R. Geske and H. Johnson (1984). Closed-form solutions for the values of European and Bermudan exchange options are derived. Several numerical examples are presented, illustrating that the early exercise feature may have a significant economic value. The results presented should have potential for pricing over-the-counter options and in particular for pricing real options.
Pricing double-barrier options under a flexible jump diffusion model
Operations Research Letters, 2009
In this paper we present a Laplace transform-based analytical solution for pricing double-barrier options under a flexible hyper-exponential jump diffusion model (HEM). The major theoretical contribution is that we prove non-singularity of a related high-dimensional matrix, which guarantees the existence and uniqueness of the solution.