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Title: On Quantiles of Brownian Motion and Quantile Options
Keywords: Option Pricing, alpha-quantile Options, Discretization Error, Tree Method, Euler Scheme, Richardson Extrapolation
Issue Date: 18-Aug-2010
Citation: ZHU YONGTING (2010-08-18). On Quantiles of Brownian Motion and Quantile Options. ScholarBank@NUS Repository.
Abstract: Option is one of the most popular derivatives traded in the market. In the most important model, the Black-Scholes model, for option pricing, the price of the underlying asset is assumed to satisfy the geometric Brownian motion, which is a very interesting object itself. In this thesis, we provide a quick review about the theory of option pricing in the first part of this thesis. It is well known that the Black-Scholes formula solves the pricing problem for a European option. However, for other types of options, usually there is no such closed-form formula. Numerical methods are developed to solve these problems. In this thesis, we consider the alpha-quantile options. Partially because the alpha-quantiles of a Brownian motion are highly path-dependent, many fundamental problems are still open. One problem is the discretization error between the alpha-quantile of a Brownian motion and that of the Gaussian random walk. This is the first step to connect the price of continuously and discretely monitored alpha-quantile options. We have found a difference between the strong order of convergence of the discretization error for genuine alpha-quantiles (0 < alpha < 1) and that for the maximum (alpha=1) by simulation. Another problem is with the pricing of alpha-quantile options. Although the risk-neutral pricing formula for European-style alpha-quantile options is given in Dassios (1995), it still needs numerical method such as the forward shooting method proposed by Kwok and Lau (2001) and a Monte Carlo method proposed by Ballotta and Kyprianou (2001). However, these existing methods cannot be extended to price the American-style alpha-quantile options. In this thesis, we propose a tree method which, to our knowledge, is the first solution to price American-style alpha-quantile options. We show how Richardson extrapolation can be applied to improve the accuracy of our lattice method.
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