5 research outputs found
BENCHOPâSLV: the BENCHmarking project in Option PricingâStochastic and Local Volatility problems
In the recent project BENCHOPâthe BENCHmarking project in Option Pricing we found that Stochastic and Local Volatility problems were particularly challenging. Here we continue the effort by introducing a set of benchmark problems for this type of problems. Eight different methods targeted for the Stochastic Differential Equation (SDE) formulation and the Partial Differential Equation (PDE) formulation of the problem, as well as Fourier methods making use of the characteristic function, were implemented to solve these problems. Comparisons are made with respect to time to reach a certain error level in the computed solution for the different methods. The implemented Fourier method was superior to all others for the two problems where it was implemented. Generally, methods targeting the PDE formulation of the problem outperformed the methods for the SDE formulation. Among the methods for the PDE formulation the ADI method stood out as the best performing one
Calibration of Local Volatility Model with Stochastic Interest Rates by Efficient Numerical PDE Method
Long maturity options or a wide class of hybrid products are evaluated using
a local volatility type modelling for the asset price S(t) with a stochastic
interest rate r(t). The calibration of the local volatility function is usually
time-consuming because of the multi-dimensional nature of the problem. In this
paper, we develop a calibration technique based on a partial differential
equation (PDE) approach which allows an efficient implementation. The essential
idea is based on solving the derived forward equation satisfied by P(t; S;
r)Z(t; S; r), where P(t; S; r) represents the risk neutral probability density
of (S(t); r(t)) and Z(t; S; r) the projection of the stochastic discounting
factor in the state variables (S(t); r(t)). The solution provides effective and
sufficient information for the calibration and pricing. The PDE solver is
constructed by using ADI (Alternative Direction Implicit) method based on an
extension of the Peaceman-Rachford scheme. Furthermore, an efficient algorithm
to compute all the corrective terms in the local volatility function due to the
stochastic interest rates is proposed by using the PDE solutions and grid
points. Different numerical experiments are examined and compared to
demonstrate the results of our theoretical analysis
An adjoint method for the exact calibration of stochastic local volatility models
Abstract: This paper deals with the exact calibration of semidiscretized stochastic local volatility (SLV) models to their underlying semidiscretized local volatility (LV) models. Under an SLV model, it is common to approximate the fair value of European-style options by semidiscretizing the backward Kolmogorov equation using finite differences. In the present paper we introduce an adjoint semidiscretization of the corresponding forward Kolmogorov equation. This adjoint semidiscretization is used to obtain an expression for the leverage function in the pertinent SLV model such that the approximated fair values defined by the LV and SLV models are identical for non-path-dependent European-style options. In order to employ this expression, a large non-linear system of ODEs needs to be solved. The actual numerical calibration is performed by combining ADI time stepping with an inner iteration to handle the non-linearity. Ample numerical experiments are presented that illustrate the effectiveness of the calibration procedure