Options: Current perspectives

This article provides an overview of risk-neutral valuation methodology and presents historical milestones in the development of quantitative …nance. It also discusses current challenges and new perspectives in model choice, pricing and hedging. 1 Historical background In 1973, Black, Scholes and Merton ([5], [13]) developed a method for the valuation of a European option based on the idea of perfect replication of its payo¤. Their approach demonstrates how to act in an uncertain environment so that relevant risks are controlled. Around the same time, trading of options on common stocks started in the Chicago Board Options Exchange. Theory met practice and an exciting and fruitful journey started on the crossroads of economics, …nance and mathematics. Its impact was phenomenal in both academia and industry. New areas of research were created, and numerous educational and training activities were established. The derivatives market grew at an unprecedented rate and in‡uenced the development of other markets. Complex mathematical modelling and technical sophistication, predominant elements in theory and applications in engineering and natural sciences, now entered the theory and practice of …nance. This was not the …rst time that stochastic modelling touched …nance. At the beginning of the 20th century, in his pioneering doctoral work, Bachelier ([2]) proposed a stochastic model, based on normality assumptions on their returns, for stock prices. In many aspects, however, his work was ahead of its time and had no impact for years to come. What was the Black, Scholes, and Merton option valuation approach? A European call option is a contract that gives its owner the right to buy the underlying stock at a given price, K and a given maturity, T . Their model, The author acknowledges partial support from the National Science Foundation (NSF Grants DMS-0091946 and DMS-FRG-0456118).