A holy grail of mathematics appears solved!

Apr 27, 2006

A University of Wollongong mathematician, Associate Professor Song-Ping Zhu, has cracked a formula that will have important implications for researchers, traders and investors working on the valuation of American-style stock options.

One of Professor Zhuís areas of long-term interest has been developing analytical as well as numerical techniques suitable for the options and futures price modelling. Options are financial derivatives popularly used by companies and large financial institutions in their risk management.

Their valuation is, however, a quite complicated process as the value of an option depends on, among many other factors, the underlying asset (such as stock) value, the time left for the option contract to expire, the strike price at which the option can be exercised, the interest rate that banks pay for the fund deposited with them and the volatility that measures the market fluctuations.

The valuation of options was never a simple job until two mathematicians/economists, Fischer Black and Myron Scholes, derived a mathematical equation (widely referred to as the Black-Scholes equation) in 1973 to guide the pricing of options and many similar financial derivatives.

The formula derived by Fischer Black and Myron Scholes as an exact solution of the Black-Scholes equation has been widely accepted by the financial market as a guide for pricing the so-called European options. Over time the significance of their discovery was fully recognised and in 1997 the Nobel Prize for Economics was awarded to Myron Scholes and Robert Merton. (Merton worked in a similar area at about the same time. Black died in 1995 and Nobel Prizes are not awarded posthumously).However, in todayís financial markets worldwide, popularly traded options are of American style. Unlike European options, American options can be exercised at anytime prior to expiry.

This has changed the problem mathematically into a so-called moving boundary problem and the solution process consequently becomes much more complicated. Mathematicians worldwide have been working for years to find an exact solution of the Black-Scholes equation for the valuation of American options -- many concluded that such a solution did not exist. This conclusion would now appear to be incorrect with Professor Zhuís newly-found exact solution of the Black-Scholes equation for American options.

Professor Zhu presented his work at the ANZIAM2005 Conference (the 41st annual Australia and New Zealand Industrial and Applied Mathematics meeting) and the 2005 Quantitative Methods in Finance conference. He was also an invited speaker to present his new solution at the 3rd SPIE International Symposium (Noise and Fluctuations in Econophysics and Finance) held at Austin, Texas, USA.

His findings have triggered widespread excitement among his mathematical colleagues who are confident that this long-standing problem has finally been solved. Professor Zhu has now had his journal paper, ďAn Explicit and Exact Solution of the Value of American Put and its Optimal Exercise BoundaryĒ accepted for publication in the journal, Quantitative Finance.

He has already presented his findings to financial experts in the Macquarie Bank. Dr Ahmed El-Feki, an Associate Director of Macquarie Bankís Quantitative Applications Division, said the solution could potentially be used for index and foreign exchange options.

Most importantly, according to Dr El-Feki this newly-found exact solution would be more useful as a benchmark to validate other numerical models rather than a pricing tool used by traders. The solution technique can potentially be applied to find exact solutions of other American style exotic options.

Academically, having an exact solution even for the most basic American put option would represent a milestone in quantitative finance, said financial mathematics expert, Professor Ken Kortanek of the University of Pittsburgh.

For further information contact Associate Professor Song-Ping Zhu on (02) 4221 3807 or via spz@uow.edu.au



UOW mathematician, Associate Professor Song-Ping Zhu

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