Posouzení vybraných možností zefektivnění simulace Monte Carlo při opčním oceňování[Examination of selected improvement approaches to Monte Carlo simulation in option pricing]

In general, there exist many ways to detect the fair value of financial derivatives. However, each of them is suitable for different purposes. For example, when the payoff function is not very simple or the underlying process is too complex, the approach of Monte Carlo simulation can be useful. Unfortunately, the plain Monte Carlo simulation needs a very high number of independent paths to get reliable results. It is the reason why an improvement of the plain approach should be applied to decrease the number of paths required in order to get reliable results. In this paper we study more closely several such approaches and examine their potential of increasing the efficiency. To be more exact, we apply the antithetic variates method and stratified sampling approaches, including their combinations in order to get the fair price of a plain vanilla call. We consider three distinct underlying processes: geometric Brownian motion, variance gamma model and normal inverse Gaussian model. We also verify the confidence interval for the option price. We did not find any improvements of examined methods for complex processes considering the definition via two or more independent random numbers. However, if the required accuracy is very high, it might be useful to apply the stratification to the distribution function of the complex process.