Recovery of time dependent volatility coefficient by linearization

TL;DR

This paper presents a method to reconstruct time-dependent local volatility from option prices at two expiration times using linearization, Laplace transforms, and a numerical algorithm, ensuring uniqueness and stability.

Contribution

It introduces a linearization approach for inverse volatility problems, simplifies the integral operators, and provides a numerical method with proven uniqueness and stability results.

Findings

Unique solvability of the inverse problem in basic functional spaces.

Stability results under smallness conditions of the spatial interval.

A practical numerical algorithm based on the linearized analysis.

Abstract

We study the problem of reconstruction of special special time dependent local volatility from market prices of options with different strikes at two expiration times. For a general diffusion process we apply the linearization technique and we conclude that the option price can be obtained as the sum of the Black-Scholes formula and of an operator $W$ which is linear in perturbation of volatility. We further simplify the linearized inverse problem and obtain unique solvability result in basic functional spaces. By using the Laplace transform in time we simplify the kernels of integral operators for $W$ and we obtain uniqueness and stability results for for volatility under natural condition of smallness of the spacial interval where one prescribes the (market) data. We propose a numerical algorithm based on our analysis of the linearized problem.