Stochastic Volatility Models and Option Pricing

<p style="text-align: justify;">In the modern financial world options are fairly widespread because they allow making significant profits with small initial investments as well as hedging various risks. Consequently both the total number of contracts and the total notional value of all options often reach an enormous size thus increasing the requirements on options&rsquo; evaluation accuracy. Since the traditional models with constant volatility cannot account for some important properties of real prices of underlying assets the models with non-constant volatility have been invented. In the present paper stochastic volatility models are reviewed along with constant volatility models. Option pricing algorithm within the Heston framework together with the calibration problem is considered on the basis of American stock options. We propose a rigorous refutation of one of the assumptions of constant volatility models &ndash; an assumption of log-normal distribution of asset prices. In order to calculate a fair value of selected options a contemporary method of multinomial trees is applied. Furthermore to demonstrate its appropriateness we provide a comparison of this method with the alternative &ndash; the Monte-Carlo method.</p>