Numerical Schemes For Pricing Asian Options Under State Dependent Regime Switching Jump Diffusion Models
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| Author | : Duy-Minh Dang |
| Publisher | : |
| Total Pages | : 30 |
| Release | : 2018 |
| Genre | : |
| ISBN | : |
We propose numerical schemes for pricing Asian options when the underlying asset price follows a very general state-dependent regime-switching jump-diffusion process. Under this model, the price of the option can be obtained by solving a highly complex system of coupled two-dimensional parabolic partial integro-differential equations (PIDEs) via iterative techniques. One of the proposed schemes is provably convergent to the solution of the system of PIDEs. In addition, by treating the coupling and integral terms explicitly, over each iteration of the scheme, the pricing problem under this scheme can be partitioned into independent pricing sub-problem, with communication at the end of the iteration. Hence, this method allows for a very natural and easy-to-implement, yet efficient, parallelization of the solution process on multi-core architectures. We illustrate the accuracy and efficiency of the proposed methods by several numerical examples.
| Author | : Justin Kirkby |
| Publisher | : |
| Total Pages | : 39 |
| Release | : 2020 |
| Genre | : |
| ISBN | : |
Utilizing frame duality and a FFT-based implementation of density projection we develop a novel and efficient transform method to price Asian options for very general asset dynamics, including regime switching Levy processes and other jump diffusions as well as stochastic volatility models with jumps. The method combines Continuous-Time Markov Chain (CTMC) approximation, with Fourier pricing techniques. In particular, our method encompasses Heston, Hull-White, Stein-Stein, 3/2 model as well as recently proposed Jacobi, alpha-Hypergeometric, and 4/2 models, for virtually any type of jump amplitude distribution in the return process. This framework thus provides a unified approach to pricing Asian options in stochastic jump diffusion models and is readily extended to alternative exotic contracts. We also derive a characteristic function recursion by generalizing the Carverhill-Clewlow factorization which enables the application of transform methods in general. Numerical results are provided to illustrate the effectiveness of the method. Various extensions of this method have since been developed, including the pricing of barrier, American, and realized variance derivatives.
| Author | : Granville Sewell |
| Publisher | : John Wiley & Sons |
| Total Pages | : 224 |
| Release | : 2018-10-09 |
| Genre | : Mathematics |
| ISBN | : 1119507936 |
Solve engineering and scientific partial differential equation applications using the PDE2D software developed by the author Solving Partial Differential Equation Applications with PDE2D derives and solves a range of ordinary and partial differential equation (PDE) applications. This book describes an easy-to-use, general purpose, and time-tested PDE solver developed by the author that can be applied to a wide variety of science and engineering problems. The equations studied include many time-dependent, steady-state and eigenvalue applications such as diffusion, heat conduction and convection, image processing, math finance, fluid flow, and elasticity and quantum mechanics, in one, two, and three space dimensions. The author begins with some simple "0D" problems that give the reader an opportunity to become familiar with PDE2D before proceeding to more difficult problems. The book ends with the solution of a very difficult nonlinear problem, which requires a moving adaptive grid because the solution has sharp, moving peaks. This important book: Describes a finite-element program, PDE2D, developed by the author over the course of 40 years Derives the ordinary and partial differential equations, with appropriate initial and boundary conditions, for a wide variety of applications Offers free access to the Windows version of the PDE2D software through the author’s website at www.pde2d.com Offers free access to the Linux and MacOSX versions of the PDE2D software also, for instructors who adopt the book for their course and contact the author at www.pde2d.com Written for graduate applied mathematics or computational science classes, Solving Partial Differential Equation Applications with PDE2D offers students the opportunity to actually solve interesting engineering and scientific applications using the accessible PDE2D.
| Author | : Pengfei Luo |
| Publisher | : |
| Total Pages | : 34 |
| Release | : 2015 |
| Genre | : |
| ISBN | : |
We consider an irreversible investment in a project, which generates cash flow following a double exponential jump-diffusion process and its expected return is governed by a continuous-time two-state Markov chain. If the expected return is observable, we present explicit expressions for the pricing and timing of the option to invest. With partial information, i.e. if the expected return is unobservable, we provide an explicit project value and an integral-differential equation for the pricing and timing of the option. We show a method to measure the information value, i.e. the difference between the values of the option to invest under the two cases. We present numerical solutions by finite difference methods if jumps are absent. By numerical analysis, we find that: (i) The value of the option to invest increases with the belief on economic boom; (ii) If investors are more uncertain about the state of the economy, information is more valuable; (iii) The more likely the transition from boom to recession, the less the value of the option; (iv) The bigger the dispersion of the expected return, the higher the information value; (v) A higher cash flow volatility induces a less information value.
| Author | : George Yin |
| Publisher | : Springer |
| Total Pages | : 593 |
| Release | : 2019-07-16 |
| Genre | : Mathematics |
| ISBN | : 3030254984 |
This volume collects papers, based on invited talks given at the IMA workshop in Modeling, Stochastic Control, Optimization, and Related Applications, held at the Institute for Mathematics and Its Applications, University of Minnesota, during May and June, 2018. There were four week-long workshops during the conference. They are (1) stochastic control, computation methods, and applications, (2) queueing theory and networked systems, (3) ecological and biological applications, and (4) finance and economics applications. For broader impacts, researchers from different fields covering both theoretically oriented and application intensive areas were invited to participate in the conference. It brought together researchers from multi-disciplinary communities in applied mathematics, applied probability, engineering, biology, ecology, and networked science, to review, and substantially update most recent progress. As an archive, this volume presents some of the highlights of the workshops, and collect papers covering a broad range of topics.
| Author | : Tao Wu |
| Publisher | : |
| Total Pages | : 122 |
| Release | : 2010 |
| Genre | : Options (Finance) |
| ISBN | : |
| Author | : Rama Cont |
| Publisher | : |
| Total Pages | : 39 |
| Release | : 2004 |
| Genre | : |
| ISBN | : |
We present a finite difference method for solving parabolic partial integro-differential equations with possibly singular kernels which arise in option pricing theory when the random evolution of the underlying asset is driven by a Levy process or, more generally, a time-inhomogeneous jump-diffusion process. We discuss localization to a finite domain and provide an estimate for the localization error under an integrability condition on the Levy measure. We propose an explicit-implicit time-stepping scheme to solve the equation and study stability and convergence of the schemes proposed, using the notion of viscosity solution. Numerical tests are performed for the Merton jump-diffusion model and for the Variance Gamma model with smooth and non-smooth payoff functions. Our scheme can be used for European and barrier options, applies in the case of pure-jump models or degenerate diffusion coefficients, and extends to time-dependent coefficients.
| Author | : Saize Stefane Draiva |
| Publisher | : LAP Lambert Academic Publishing |
| Total Pages | : 56 |
| Release | : 2015-12-16 |
| Genre | : |
| ISBN | : 9783659779503 |
In this book we derive the analytical solutions to the American-style Asian Options under jump-diffusion processes. The similar problem was studied by Hansen and Jorgensen (2000), but they considered the diffusion case. First of all we transform the problem into one-state variable problem (the dual problem). To this new problem, we find its general analytical solution by using theories from Hansen and Jorgensen (2000), Merton (1976) and H. Pham. Also we derive the analytical solutions to the particular cases, when the average is geometric and arithmetic. In the arithmetic average case, the one-state variable is not a geometric Brownian motion, so we approximate it to a geometric Brownian motion by using the Wilkisson aproximation. At the end of this book we have some numerical results comparing the earlier exercise boundaries in diffusion and jump-diffusion cases.
| Author | : Santtu Salmi |
| Publisher | : |
| Total Pages | : 18 |
| Release | : 2013 |
| Genre | : |
| ISBN | : |
We propose families of IMEX time discretization schemes for the partial integro-differential equation derived for the pricing of options under a jump diffusion process. The schemes include the families of IMEX-midpoint, IMEXCNAB and IMEX-BDF2 schemes. Each family is defined by a convex parameter c ∈ [0, 1], which divides the zeroth-order term due to the jumps between the implicit and explicit part in the time discretization. These IMEX schemes lead to tridiagonal systems, which can be solved extremely efficiently. The schemes are studied through Fourier stability analysis and numerical experiments. It is found that, under suitable assumptions and time step restrictions, the IMEX-midpoint family is absolutely stable only for c = 0, while the IMEX-CNAB and the IMEX-BDF2 families are absolutely stable for all c ∈ [0, 1]. The IMEX-CNAB c = 0 scheme produced the smallest error in our numerical experiments.
| Author | : Svetlana Boyarchenko |
| Publisher | : |
| Total Pages | : 0 |
| Release | : 2008 |
| Genre | : |
| ISBN | : |
The pricing problem for American options in Markov-modulated Lévy models is solved. The early exercise boundaries and prices are calculated using a generalization of Carr's randomization procedure for regime-switching models. The pricing procedure is efficient even if the number of states is large provided the transition rates are not large w.r.t. the riskless rates. The payoffs and riskless rates may depend on a state. Special cases are stochastic volatility models and models with stochastic interest rate; both must be modeled as finite-state Markov chains. In contrast with the earlier version of the method, an explicit algorithm is formulated for wide classes of Lévy processes, and FFT and iFFT are used.
