The Black-Scholes option pricing problem in mathematical finance: generalization and extensions for a large class of stochastic processes
The ability to price risks and devise optimal investment strategies in the presence of an uncertain "random" market is the cornerstone of modem finance theory. We first consider the simplest such problem of a so-called "European call option" initially solved by Black and Scholes using Ito stochastic calculus for markets modelled by a log-Brownian stochastic process. A simple and powerful formalism is presented which allows us to generalize the analysis to a large class of stochastic processes, such as ARCH, jump or Lévy processes. We also address the case of correlated Gaussian processes, which is shown to be a good description of three different market indices (MATIF, CAC40, FTSEIOO). Our main result is the introduction of the concept of an optimal strategy in the sense of (functional) minimization of the risk with respect to the portfolio. If the risk may be made to vanish for particular continuous uncorrelated 'quasi-Gaussian' stochastic processes (including Black and Scholes model), this is no longer the case for more general stochastic processes. The value of the residual risk is obtained and suggests the concept of risk-corrected option prices. In the presence of very large deviations such as in Lévy processes, new criteria for national fixing of the option prices are discussed. We also apply our method to other types of options, 'Asian', 'American', and discuss new possibilities ('double-decker'...). The inclusion of transaction costs leads to the appearance of a natural characteristic trading time scale.