On accurate and provably efficient GARCH option pricing algorithms

Abstract

The trinomial-tree GARCH option pricing algorithms of Ritchken and Trevor (1999) and Cakici and Topyan (2000) are claimed to be efficient and accurate. However, this thesis proves that both algorithms generate trees that explode exponentially when the number of partitions per day, n, exceeds a typically small number determined by the model parameters. Worse, when explosion happens, the tree cannot grow beyond a certain maturity, making it useless for pricing derivatives with a longer maturity. Meanwhile, a small n has accuracy problems and does not prevent explosion. This thesis then presents a trinomial-tree GARCH option pricing algorithm that solves the above problems.The algorithm provably does not have the short-maturity problem. Furthermore, the tree size is guaranteed to be quadratic if n is less than a threshold easily determined by the model parameters. This result for the first time puts a tree-based GARCH option pricing algorithm in the same complexity class as binomial and trinomial trees under the Black-Scholes model. Extensive numerical evaluation is conducted to confirm the analytical results and the accuracy of the algorithm.

Option pricing under Markov‐switching GARCH processes

“…Let be a fixed constant that determines the space of the gap. To strike an overall balance between accuracy and convergence speed, this study follows the suggestions of Lyuu and Wu (2005) to set the space parameter as (8) After determining the gap between adjacent logarithmic prices on the grid, the lattice algorithm is constructed in the following. …”

“…Because the number of distinct paths to any particular node and any particular regime may increase exponentially as the number of time periods increases, it is not feasible to track all distinct volatilities. We thus follow the idea of the RT tree to keep track of only the maximum and minimum variances at node for each regime, which are denoted as and , respectively, and span the range between and at node to K values for each regime by using Lyuu and Wu's (2005) log-linear interpolation scheme. They are calculated as…”

An efficient and accurate lattice for pricing derivatives under a jump-diffusion process

“…Despite the ability to better model the real world stock price, these alternative processes give rise to new problems for lattice methods. For example, a lattice for the GARCH process may still give rise to an exponential-time pricing algorithm [22]. The major contribution of this paper is an efficient and flexible lattice for the jump-diffusion process.…”

Option Pricing Under GARCH Processes Using PDE Methods

“…For options with early-exercise features, Ritchken and Trevor (1999) propose a lattice algorithm based on a recombining multinomial tree, where each node of the discretized asset-price tree stores the maximum and the minimum variance of all paths to that node; this algorithm suffers from exponential explosion and is improved upon by Cakici and Topyan (2000) and by Lyuu and Wu (2005), who use mean tracking to reduce the size of the tree. Duan and Simonato (2001) propose an approximation of the GARCH process by a finite-state Markov chain with two state variables; and Duan et al (2003) propose using analytical approximations to reduce the dimensionality of a binomial tree from 2 to 1.…”