The focus of this dissertation is modeling volatility in option pricing by the Black-Scholes formula. A major drawback of the formula is that the returns from assets are assumed to have constant volatility over time. The empirical evidence is overwhelmingly against it. In this dissertation, we allow random volatility for estimating call option prices by Black-Scholes formula and by Monte Carlo simulation. The Black-Scholes formula follows from an assumption that assets evolve according to a Geometric Brownian Motion with constant volatility. This dissertation allows time-varying random volatility in the Geometric Brownian Motion to outline a proof of the formula, thus addressing this drawback. To estimate option prices with the Black-Scholes, the dissertation considers its expectation with respect to two potential probability models of random volatility. Unfortunately, a closed form expression of the expectation of the formula for computing the option prices is intractable. Then the dissertation settles with using an approximation which to its credit incorporates in it the kurtosis of the probability model of random volatility. To our knowledge, option pricing methods in literature do not incorporate kurtosis information. The option pricing with random volatility is pursued for two stochastic volatility models. One model is a member of generalized auto regressive conditional heteroscedasticity (GARCH). The second is a member of Stochastic Volatility models. For each model, estimation of their parameters is outlined. Two real financial series data are then used to illustrate estimation of the option prices, and compared them with those from the Black-Scholes formula with constant volatility. Motivated by a Monte Carlo procedure in the literature for option pricing when the volatility follows a GARCH model, this dissertation lays a foundation for future research to simulate option prices when the random volatility is assumed to follow a Stochastic Volatility model instead of GARCH. / Statistics

This work aims to describe the method of recursive estimation of time series with conditional volatility, used mainly in finance. First, there are described the basic types of models with conditional heteroskedasticity (GARCH) and princi- ples of state-space modeling demonstrated by means of linear models AR and ARMA. Subsequently, there are derived algorithms for recursive estimation of parameters of the GARCH model and its possible modifications including the ones for which recursive estimation formulas have not been yet derived in lit- erature. These algorithms are tested in a simulation study, where their appli- cability in practice is investigated. Finally, we apply these algorithms to real high-frequency data from the stock exchange. The practical part is done us- ing the software Mathematica 11.3. The work also serves as an overview of the current state of online modeling of financial time series. 1