Semi-analytical pricing of barrier options in the time-dependent Heston model

TL;DR

This paper introduces a semi-analytical method using general integral transforms for pricing barrier options within the time-dependent Heston stochastic volatility model, extending previous 1D techniques to 2D for improved speed and accuracy.

Contribution

It generalizes the GIT method to two-dimensional stochastic volatility models, enabling efficient and accurate pricing of barrier options with time-dependent features.

Findings

Method achieves high speed and accuracy compared to finite-difference methods.

Extension to 2D models allows for applications in finance and physics.

Numerical examples demonstrate practical effectiveness of the approach.

Abstract

We develop the general integral transforms (GIT) method for pricing barrier options in the time-dependent Heston model (also with a time-dependent barrier) where the option price is represented in a semi-analytical form as a two-dimensional integral. This integral depends on yet unknown function $\Phi(t,v)$ which is the gradient of the solution at the moving boundary $S = L(t)$ and solves a linear mixed Volterra-Fredholm equation of the second kind also derived in the paper. Thus, we generalize the one-dimensional GIT method, developed in (Itkin, Lipton, Muravey, Generalized integral transforms in mathematical finance, WS, 2021) and the corresponding papers, to the two-dimensional case. In other words, we show that the GIT method can be extended to stochastic volatility models (two drivers with inhomogeneous correlation). As such, this 2D approach naturally inherits all advantages of the corresponding 1D methods, in particular, their speed and accuracy. This result is new and has various applications not just in finance but also in physics. Numerical examples illustrate high speed and accuracy of the method as compared with the finite-difference approach.