10,459 research outputs found
Reduced Order Models for Pricing European and American Options under Stochastic Volatility and Jump-Diffusion Models
European options can be priced by solving parabolic partial(-integro)
differential equations under stochastic volatility and jump-diffusion models
like Heston, Merton, and Bates models. American option prices can be obtained
by solving linear complementary problems (LCPs) with the same operators. A
finite difference discretization leads to a so-called full order model (FOM).
Reduced order models (ROMs) are derived employing proper orthogonal
decomposition (POD). The early exercise constraint of American options is
enforced by a penalty on subset of grid points. The presented numerical
experiments demonstrate that pricing with ROMs can be orders of magnitude
faster within a given model parameter variation range
Application of Operator Splitting Methods in Finance
Financial derivatives pricing aims to find the fair value of a financial
contract on an underlying asset. Here we consider option pricing in the partial
differential equations framework. The contemporary models lead to
one-dimensional or multidimensional parabolic problems of the
convection-diffusion type and generalizations thereof. An overview of various
operator splitting methods is presented for the efficient numerical solution of
these problems.
Splitting schemes of the Alternating Direction Implicit (ADI) type are
discussed for multidimensional problems, e.g. given by stochastic volatility
(SV) models. For jump models Implicit-Explicit (IMEX) methods are considered
which efficiently treat the nonlocal jump operator. For American options an
easy-to-implement operator splitting method is described for the resulting
linear complementarity problems.
Numerical experiments are presented to illustrate the actual stability and
convergence of the splitting schemes. Here European and American put options
are considered under four asset price models: the classical Black-Scholes
model, the Merton jump-diffusion model, the Heston SV model, and the Bates SV
model with jumps
Pricing American Options on Jump-Diffusion Processes using Fourier Hermite Series Expansions
This paper presents a numerical method for pricing American call options where the underlying asset price follows a jump-diffusion process. The method is based on the Fourier-Hermite series expansions of Chiarella, El-Hassan & Kucera (1999), which we extend to allow for Poisson jumps, in the case where the jump sizes are log-normally distributed. The series approximation is applied to both European and American call options, and algorithms are presented for calculating the option price in each case. Since the series expansions only require discretisation in time to be implemented, the resulting price approximations require no asset price interpolation, and for certain maturities are demonstrated to produce both accurate and efficient solutions when compared with alternative methods, such as numerical integration, the method of lines and finite difference schemes.American options; jump-diusion; Fourier-Hermite series expansions; free boundary problem
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