Diffusion processes for density of the integral of the quadratic brownian bridge and Asian options - Rohra Sonakshi Mahesh (Nanyang Technological University, 2021)
The first objective of this project is to study the density of the integral of the quadratic brownian motion/bridge. Densities for both time integrals are approximated by gamma and log-normal distributions using a conditional moment matching approach. The conditional density is multiplied by the normal density of brownian motion to obtain a joint density. In addition, a Monte Carlo simulation is implemented to derive the true density of both integrals. A two dimensional Monte Carlo simulation is used to find the true joint density. Lastly, the structure of the planar quadratic langevin diffusion density from Franchi’s paper is graphically studied and comparisons are made between all four densities. It is found by graphical observation that for not too small but not too large values of time, the planar quadratic langevin diffusion density is the best approximation to the true density. The second objective is to compare pricing methods for Asian options. An introduction to option pricing and Asian options is provided, followed by two different approaches to price Asian options: the density approximations and partial differential equations. The first section discusses how the density approximations found using moment matching in Chapter 2 can be applied to the Cox-Ingersoll-Ross process to price Asian options, while the second section focuses largely on PDE methods using the Geometric Brownian Motion model. We refer to Brown’s framework of using a diffusion process to obtain an arbitrary PDE for Asian options, which can be used to recover any Asian option PDE. Our project extends this to two more PDEs. This is followed by Numerical Analysis of pricing methods and a summary of numerical results. Parallels are drawn between the two areas of study by the umbrella theme of diffusion processes: the planar quadratic langevin diffusion in the first case which is used to match the true joint density of the time integral of quadratic brownian motion, and the generalised diffusion process used in Brown’s framework in the second case. The density approximations by moment matching are also applied in both cases, which showcases the versatility of the method.