Calibration of Local Volatility Model with Stochastic Interest Rates by Efficient Numerical PDE Method

TL;DR

This paper introduces an efficient PDE-based calibration method for local volatility models with stochastic interest rates, enabling faster pricing of long maturity options and hybrid products.

Contribution

The paper develops a PDE approach using ADI scheme for quick calibration of local volatility with stochastic interest rates, improving computational efficiency.

Findings

PDE method significantly reduces calibration time.

The approach accurately captures the impact of stochastic interest rates.

Numerical experiments validate the effectiveness of the proposed method.

Abstract

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.