Semi-analytical pricing of barrier options in the time-dependent $\lambda$-SABR model

TL;DR

This paper develops a semi-analytical method for pricing barrier options within the time-dependent $\lambda$-SABR model, extending previous approaches to handle stochastic volatility with time-dependent features.

Contribution

It introduces a modified integral transform method and derives a Fourier-Bessel series solution for barrier options in the $\lambda$-SABR model, including the linear mixed Volterra-Fredholm equation.

Findings

Method achieves comparable speed and accuracy to finite-difference methods at small maturities.

Outperforms finite-difference methods at high maturities.

Simplistic RBF implementation effectively solves the LMVF.

Abstract

We extend the approach of Carr, Itkin and Muravey, 2021 for getting semi-analytical prices of barrier options for the time-dependent Heston model with time-dependent barriers by applying it to the so-called $\lambda$-SABR stochastic volatility model. In doing so we modify the general integral transform method (see Itkin, Lipton, Muravey, Generalized integral transforms in mathematical finance, World Scientific, 2021) and deliver solution of this problem in the form of Fourier-Bessel series. The weights of this series solve a linear mixed Volterra-Fredholm equation (LMVF) of the second kind also derived in the paper. Numerical examples illustrate speed and accuracy of our method which are comparable with those of the finite-difference approach at small maturities and outperform them at high maturities even by using a simplistic implementation of the RBF method for solving the LMVF.