Research being undertaken by our current postgraduates:

Development of Novel Numerical Algorithms for The Estimation of Implied Volatility and Other Financial Market Parameters

During the last two centuries, financial markets have been affected by many crises as well as the fast developments in financial theories and models which have been introduced as a consequence of the crises. In recent years, mathematical methods and financial theories are combined in practice to show and understand financial market’s behaviour as well as address its issues. One financial market’s principle which has been intensively investigated is option pricing. An option is a financial agreement which grants to its holder the right to buy or to sell an amount of the stock at a specific time. An option was theoretically commenced by Louis Bachelier in 1900. In fact, his formula depends on the assumption that stock prices follows a Brownian motion with zero drift. Many models have been subsequently deduced to obtain the valuation of options. For instance, the Black-Scholes equation is one of the simplest formulas to value efficiently options without transaction costs on buying and selling the options and underlying assets in a complete market.

The Black-Scholes model was introduced by Black and Scholes in 1973 to value a European option. It has been considered as a seminal model for many subsequent publications, which have modified it to comprise other financial parameters. Black and Scholes assumed that the drift (μ) and volatility (σ) are constant, the stock price follows a Geometric Brownian Motion and there are no costs involved in transactions. Consequently, they proved that the valuation of an European option satisfies the following partial differential equation:

Vt + 1/2 σ2S2VSS + rSVS − rV = 0,

where V denotes the option price, S > 0 is the underlying stock price, t is the time, r > 0 is a constant riskless interest rate, and σ is the volatility. These parameters can be observed in the financial market, except the volatility, which has a significant impact on option pricing. As a result, many researchers have concentrated on dealing with an implied volatility phenomenon in their projects. It is clear that it is important to estimate the implied volatility accurately in order to measure the uncertainty of the future price of an option in financial markets.

These studies are new and practically useful because they address problems which have been never solved before. Novel efficient, robust and accurate numerical methods/algorithms will be developed from this research to deal with the proposed problems and provide stepping stones for dealing with more complex problems in the future. These algorithms will be helpful for investors as well financial engineers to valuate options accurately in financial markets.

Assistance in statistics is available for Postgraduates students by research at the UWA Centre for Applied Statistics.