Solving the partial differential equation for pricing barrier options via trigonometric expansions

Abstract

Barrier options, although highly liquid financial derivatives, present notable pricing challenges. In this thesis, we present a novel pricing approach for valuing continuously-monitored knock-out barrier options within the framework of stochastic volatility models.

The underlying process is firstly modelled under geometric Brownian motion and, subsequently, under Heston's stochastic volatility model. A key insight is that the value of a barrier option can be expressed as a single-dimensional integral, whereby the integrand involves the so-called survival density function, which captures the barrier-brea-ching information. Therefore, the option can be valued using the one-dimensional COS method for European options, once the Fourier series coefficients of the survival density are obtained.

The coefficients of the sine series expansion of the survival density function are, in fact, a continuous function, which is closely related to the characteristic function of the density. This motivates us to directly recover that function, which we refer to as the target function herewith, by selecting an appropriate series expansion for it. Thereafter, we insert this series expansion into the partial differential equation (PDE) that the target function should satisfy, which can be derived from the pricing PDE. This results in a linear system, solving which we obtain the coefficients needed to reconstruct the target function. Notably, this approach is particularly advantageous when the reference values of option prices are limited or unavailable, as it relies solely on the PDE to calibrate the series coefficients.

Our choice of series expansion is driven by the need for precise global and local approximations. Our research shows that a proper expansion for the target function can be built up from integrating a two-dimensional Fourier series of its first derivative with respect to time, marking our second pivotal insight. That results in a trigonometric expansion that significantly enhances the accuracy compared to a direct Fourier series expansion on the target function. Finally, our third pivotal insight: applying a change of variables further improves error convergence.

Extensive testing results suggest that, for similar accuracy levels, our approach greatly outperforms Monte Carlo simulations in terms of computation time. It also demonstrates superior computational efficiency and accuracy compared to other advanced numerical methods in the existing literature.
The benefits of this method become even more prominent when pricing a large number of options simultaneously.

Numerical tests reveal algebraic convergence for the series expansion reconstruction. For the option price, theoretical error analysis aligns with our findings, also predicting algebraic convergence.