Positive Stochastic Collocation for the Collocated Local Volatility Model

TL;DR

This paper explores applying stochastic collocation to boundary-constrained assets, identifies limitations of polynomial collocation, examines issues in the collocated local volatility model, and derives an analytical expression for Dupire local volatility.

Contribution

It introduces a novel application of stochastic collocation to boundary-constrained assets and provides an analytical formula for local volatility derived from option prices.

Findings

Polynomial collocation struggles with lognormal distributions.

Potential issues in the collocated local volatility model are identified.

An analytical expression for Dupire local volatility is derived.

Abstract

This paper presents how to apply the stochastic collocation technique to assets that can not move below a boundary. It shows that the polynomial collocation towards a lognormal distribution does not work well. Then, the potentials issues of the related collocated local volatility model (CLV) are explored. Finally, a simple analytical expression for the Dupire local volatility derived from the option prices modelled by stochastic collocation is given.