The Design and Analysis of Lattice Algorithms for Multivariate Option Pricing Models

Financial derivatives are financial instruments whose payoff depends on some fundamental financial assets. They are essential tools for speculation and risk-management. For pricing derivatives, analytical formulas are rare for most ones. When this happens, derivatives must be priced by numerical methods such as lattices. This thesis focuses on the lattices for the multivariate pricing models including GARCH (generalized autoregressive conditional heteroskedastic)-type models, the Hilliard-Schwartz (HS) model, which nests the Hull-White (HW) and SABR models, the Chambers-Lu (CL) model, and the multi-asset options. Let n denote the number of time steps and Δt denote the duration of a time step. This thesis proves that the Ritchken-Trevor lattice for GARCH-type models explodes if n exceeds some threshold and the mean-tracking lattice for GARCH-type models grows only quadratically if n does not exceed some threshold. Hilliard and Schwartz give a cubic-time lattice for their model. But this thesis proves that the HS lattice inherently has negative transition probabilities and any valid lattices for the HW, SABR, and HS models must have at least a subexponential complexity. A lattice for multi-asset options is said to be (correlation) optimal if it guarantees validity as long as the correlations between all pairs of assets fall within −1+O(Δt^0.5) and 1−O(Δt^0.5). Many lattices in the literature are analyzed in the thesis according as their optimality and ability to handle barrier options. This thesis proposes a new multi-asset lattice called the hexanomial lattice that can align a layer of lattice nodes with a barrier for each asset for excellent convergence and is provably optimal for the bivariate case. Finally, to exploit the massive computing power of GPUs (graphics processing units), this thesis takes the CL lattice as an example and implements it on the GPU and CPU (central processing unit). The numerical results show that up to a hundred-fold speedup can be achieved by the GPU over the CPU.