Analytically pricing American options

Impact: International recognition for solving a long-standing problem in financial mathematics

Finding an analytical solution to a long-standing problem in financial mathematics – the pricing of American options under the Black-Scholes model – was for many years thought to be almost impossible.

Yet the problem was solved, and announced in 2006 in the international journal Quantitative Finance, after a long and rigorous review process, by UOW mathematician, Professor Song-Ping Zhu.

This landmark discovery of a solution in infinite series form with each term of the solution being ‘constructed’ recursively for this notoriously difficult problem brought attention to the quality of mathematical research underway at UOW, which endures today.

Black and Scholes’ celebrated 1973 paper established a pricing framework for financial derivatives and presented an explicit formula for European options. Since then, use of mathematics in social sciences, particularly in finance and economics, has exploded.

A much more difficult problem, however, was to find an analytical solution for the price of American options. Unlike European options, which can only be exercised at the expiry, American options can be exercised at any time prior to expiry.

The fundamental difficulty in pricing American options mathematically lies in the fact that it is a highly nonlinear moving boundary problem, while pricing any European-style options remains a linear problem.

Prior to Professor Zhu’s breakthrough, researchers believed that analytical solutions in any form do not exit; market practitioners always resorted to numerical solutions. Many researchers even openly claimed that finding an analytical answer was an impossible task.

The publication of Professor Zhu’s solution had at least two major impacts in financial mathematics research. Firstly, it demonstrated that it was possible to find an analytic explicit solution for the price of American options, and opened up the possibility of other forms of solution as well.

Secondly, the solution can be used as a benchmark for other numerical solutions to validate numerical accuracy before their adoption by market practitioners.

More significantly, Professor Zhu has continued his high quality research in the field of financial mathematics, resulting in a number of publications in high quality journals, with one of his recent papers published by Mathematical Finance, the top international journal for financial mathematics.