Volume Weighted Average Price Options

Stace, Antony William (2007). Volume Weighted Average Price Options PhD Thesis, School of Physical Sciences, University of Queensland.

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Author Stace, Antony William
Thesis Title Volume Weighted Average Price Options
School, Centre or Institute School of Physical Sciences
Institution University of Queensland
Publication date 2007
Thesis type PhD Thesis
Supervisor Dr Graeme Chandler
Abstract/Summary In this work, we developed methods to price both fixed and floating strike Volume Weighted Average Price (VWAP) options. VWAP options have a payoff that is dependent on both the stock price and volume of stock traded over the lifetime of the option. We concentrate on the valuation of European style, fixed and floating strike call options. Little additional effort is required to adapt the results of this work to value puts. First, introductory mathematical material is presented, followed by a general overview of options and how they are priced. The VWAP is then formally introduced, and some examples of VWAP options in practice are given. Expectations and a partial differential equation that describes the price of the options are then constructed. Throughout, we assume the stock price evolves as a geometric Brownian motion process, and the volume is a fast mean reverting process. Upper and lower bounds are established for the price of both fixed and floating strike VWAP options. These bounds are given as analytic formulae and are independent of the volume model. The bounds are obtained from a simple put-call type parity. We then interpret these bounds further. The upper bounds represent the minimum cost of hedging all of the risk. A Monte Carlo investigation is then performed; this gives a vivid picture of how the VWAP options behave. It also allows later results to be benchmarked. In addition to vanilla Monte Carlo, some effective controls variates are found. We conclude the Monte Carlo investigation by finding the Greeks by the finite difference, pathwise, and likelihood methods. Next, approximations to the prices of the VWAP options are obtained by matching the first two moments of the VWAP to a lognormal distribution. For the fixed strike, a partial differential equation similar to the Black-Scholes-Merton one is obtained. When the market price of risk is constant, an analytic expression is found. For a non-constant market price of risk, a numerical method must be used. The floating strike case gives a two dimensional partial differential equation that needs to be solved numerically. In addition to pricing call options, we also demonstrate a method for the valuation of a VWAP digital option. The partial differential equation which describes the prices of the VWAP options is then solved using finite differences. We use explicit, Crank-Nicolson, and Alternating Direction Implicit schemes. Solving by finite differences is a challenging problem as there are four state variables as well as time, and an incomplete set of boundary conditions are known. Additionally, we report several similarity reductions which are effective in simplifying the partial differential equation for the floating strike option. In the final approach to pricing, a series solution to the price is assumed and the first two terms are found. The analysis assumes that the volume process is a fast mean reverting process. The first term of the series is described by a partial differential equation, which is very similar to the one that arises when pricing the Asian option. Numerical experiments established that the first term of the expansion approximates the true solution well when the mean reversion rate is high. Chapter 10 concludes this work with an overview of the results presented in this thesis, along with possible directions for further work.

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Created: Fri, 21 Nov 2008, 16:12:13 EST