3 research outputs found
Evaluating Callable and Putable Bonds: An Eigenfunction Expansion Approach
We propose an efficient method to evaluate callable and putable bonds under a
wide class of interest rate models, including the popular short rate diffusion
models, as well as their time changed versions with jumps. The method is based
on the eigenfunction expansion of the pricing operator. Given the set of call
and put dates, the callable and putable bond pricing function is the value
function of a stochastic game with stopping times. Under some technical
conditions, it is shown to have an eigenfunction expansion in eigenfunctions of
the pricing operator with the expansion coefficients determined through a
backward recursion. For popular short rate diffusion models, such as CIR,
Vasicek, 3/2, the method is orders of magnitude faster than the alternative
approaches in the literature. In contrast to the alternative approaches in the
literature that have so far been limited to diffusions, the method is equally
applicable to short rate jump-diffusion and pure jump models constructed from
diffusion models by Bochner's subordination with a L\'{e}vy subordinator
Implicit-explicit predictor-corrector methods combined with improved spectral methods for pricing European style vanilla and exotic options
In this paper we present a robust numerical method to solve several types of European style option pricing problems. The governing equations are described by variants of Black-Scholes partial differential equations (BS-PDEs) of the reaction-diffusion-advection type. To discretise these BS-PDEs numerically, we use the spectral methods in the asset (spatial) direction and couple them with a third-order implicit-explicit predictor-corrector (IMEX-PC) method for the discretisation in the time direction. The use of this high-order time integration scheme sustains the better accuracy of the spectral methods for which they are well-known. Our spectral method consists of a pseudospectral formulation of the BS-PDEs by means of an improved Lagrange formula. On the other hand, in the IMEX-PC methods, we integrate the diffusion terms implicitly whereas the reaction and advection terms are integrated explicitly. Using this combined approach, we first solve the equations for standard European options and then extend this approach to digital options, butterfly spread options, and European calls in the Heston model. Numerical experiments illustrate that our approach is highly accurate and very efficient for pricing financial options such as those described above
Robust Spectral Methods for Solving Option Pricing Problems
Doctor Scientiae - DScRobust Spectral Methods for Solving Option Pricing Problems
by
Edson Pindza
PhD thesis, Department of Mathematics and Applied Mathematics, Faculty of
Natural Sciences, University of the Western Cape
Ever since the invention of the classical Black-Scholes formula to price the financial
derivatives, a number of mathematical models have been proposed by numerous researchers
in this direction. Many of these models are in general very complex, thus
closed form analytical solutions are rarely obtainable. In view of this, we present a
class of efficient spectral methods to numerically solve several mathematical models of
pricing options. We begin with solving European options. Then we move to solve their
American counterparts which involve a free boundary and therefore normally difficult
to price by other conventional numerical methods. We obtain very promising results
for the above two types of options and therefore we extend this approach to solve
some more difficult problems for pricing options, viz., jump-diffusion models and local
volatility models. The numerical methods involve solving partial differential equations,
partial integro-differential equations and associated complementary problems which are
used to model the financial derivatives. In order to retain their exponential accuracy,
we discuss the necessary modification of the spectral methods. Finally, we present
several comparative numerical results showing the superiority of our spectral methods