We see that the path integral approach to solving the problem of pricing options when the volatility is stochastic and correlated with the stock price is a fruitful one. It has given rise to theoretical results such as a full solution for the volatility process in Baaquie. In this thesis, we have found the useful expression (4.65) which enables us to express the option price for the general volatility process as an expectation over certain functionals on the probability space spanned by the volatility paths. The next step would be to derive the distribution functions for these functionals. If this task can be accomplished, we will have an analytic solution for this relatively general volatility process. Considering that path integral techniques in finance are very new, these successes are not insignificant.
Analytic solutions, are, of course, only one side of the coin. If we have efficient numerical algorithms to find the solution, the lack of an analytic solution does not constitute a serious hindrance. As we have shown, extremely efficient Monte Carlo methods can be found using the solution provided by the path integral methods since a large number of degrees of freedom have been integrated out. We can also directly investigate the propagator for the stock prices which is not so easily done using more standard Monte Carlo techniques.
The empirical question of whether these stochastic volatility processes can adequately describe the market is somewhat more difficult. Part of this difficulty is defining what the initial volatility actually means. The shortest time period over which average volatility figures are available is 10 days. This means that the initial volatility is essentially known only in a very imprecise manner. The situation gets worse when we consider that most option trading occurs just a few days before maturity. The time period over which volatility data is collected and distributed by the exchanges and financial information agencies such as Reuters and Bloomberg has been set by convention and is not really binding. Therefore, it is still possible to directly analyze the trades to obtain short term volatility information. Hence, any meaningful empirical study will necessarily be large scale and involve considerable effort. Preliminary tests such as the one done in this thesis and the fact that relatively general volatility processes can be studied using path integral techniques suggest that it is not unlikely that positive results might be achieved.