A direct formulation of implied volatility in the Black-Scholes model

Philippe Jacquinot¹* and Nikolay Sukhomlin²

¹Gregor - University of Paris 1, Panthéon-Sorbonne, France.

²Autonomous University of Santo Domingo, Dominican Republic.

*Corresponding author. E-mail: ph.jacquinot@laposte.net.

Accepted 5 April, 2010

 Abstract

The inverse problem of option pricing, also known as market calibration, attracted the attention of a large number of practitioners and academics, from the moment that Black-Scholes formulated their model. The search for an explicit expression of volatility as a function of the observable variables has generated a vast body of literature, forming a specific branch of quantitative finance. But up to now, no exact expression of implied volatility has been obtained. The main result of this paper is such an exact expression. Firstly, a formula was deduced analytically. Secondly, it was shown that this expression is actually an exact inversion, using simulated data. Thirdly, it was shown that the methodology can be used to express implied volatility in more sophisticated models, such as the Blenman and Clark model. In the conclusion, discussion of the results was made.

Key words: Black-Scholes model, inverse problem, implied volatility, conservation law.