Calibrating local volatility in inverse option pricing using the Levenberg–Marquardt method

Abstract

We derive an iterative algorithm for an Inverse Problem of Option Pricing. The aim is to determine the local volatility such that the corresponding solutions of the Black–Scholes equation match the quoted market prices. Market data are given as option prices with different strikes and maturities. This leads to a parameter estimation problem for parabolic differential equations. This inverse problem is nonlinear and ill-posed. To overcome these difficulties, we apply an inexact Newton method. Thus, we derive a regularizing iterative algorithm of Levenberg–Marquardt type. As an underlying operator, we introduce the propagation operator. It maps, for given initial data, the unknown volatility to the final data. Then, we determine its Gâteaux-differential operator via a parabolic equation of second order. Moreover, its corresponding adjoint, called backpropagation operator, is obtained via a backward parabolic equation. Finally, to illustrate the efficiency of the method, we present some numerical results for simulated data and for observations extracted from the market. Besides, we discuss different strategies for choosing the regularizing parameters.