Filling the gaps smoothly

TL;DR

This paper extends a local volatility calibration method by using a piecewise linear variance and incorporating interest rates and dividends, maintaining analytical tractability with hypergeometric functions.

Contribution

It introduces a novel extension of existing calibration techniques, enabling more flexible modeling with analytical solutions under more realistic market conditions.

Findings

The method successfully calibrates local volatility models with piecewise linear variance.

Inclusion of interest rates and dividends enhances model realism.

Analytical solutions are derived using hypergeometric functions.

Abstract

The calibration of a local volatility models to a given set of option prices is a classical problem of mathematical finance. It was considered in multiple papers where various solutions were proposed. In this paper an extension of the approach proposed in LiptonSepp2011 is developed by i) replacing a piecewise constant local variance construction with a piecewise linear one, and ii) allowing non-zero interest rates and dividend yields. Our approach remains analytically tractable; it combines the Laplace transform in time with an analytical solution of the resulting spatial equations in terms of Kummer's degenerate hypergeometric functions.