The famous Black-Scholes formula provided the first mathematically sound mechanism to price financial options. It is based on the assumption, that daily random stock returns are identically normally distributed and hence stock prices follow a stochastic process with a constant volatility. Observed prices, at which options trade on the markets, don¡¯t fully support this hypothesis. Options corresponding to different strike prices trade as if they were driven by different volatilities. To capture this so-called volatility smile, we need a more sophisticated option-pricing model assuming that the volatility itself is a random process. The price we have to pay for this stochastic volatility model is that such models are computationally extremely intensive to simulate and hence difficult to fit to observed market prices. This difficulty has severely limited the use of stochastic volatility models in the practice. In this project we propose to overcome the obstacle of computational complexity by executing the simulations in a massively parallel fashion on the graphics processing unit (GPU) of the computer, utilizing its hundreds of parallel processors. We succeed in generating the trillions of random numbers needed to fit a monthly options contract in 3 hours on a desktop computer with a Tesla GPU. This enables us to accurately price any derivative security based on the same underlying stock. In addition, our method also allows extracting quantitative measures of the riskiness of the underlying stock that are implied by the views of the forward-looking traders on the option markets.

The Black-Scholes model is one of the most important concepts in modern financial theory. It was developed in 1973 by Fisher Black, Robert Merton and Myron Scholes and is still widely used today, and regarded as one of the best ways of determining fair prices of options. In the classical Black-Scholes model for the market, it consists of an essentially riskless bond and a single risky asset. So far there is a number of straightforward extensions of the Black-Scholes analysis. Here we consider more complex products where each component in a portfolio entails several variables with constraints. This leads to elegant models based on multivariable stochastic integration, and describing several securities simultaneously. We derive a general asymptotic solution in a short time interval using the heat kernel expansion on a Riemannian metric. We then use our formula to predict the better price of options on multiple underlying assets. Especially, we apply our method to the case known as the one of two-color rainbow ptions, outperformance option, i.e., the special case of the model with two underlying assets. This asymptotic solution is important, as it explains hidden effects in a class of financial models.

In this paper, we study mathematical properties of a generalized bivariate Ornstein-Uhlenbeck model for financial assets. Originally introduced by Lo and Wang, this model possesses a stochastic drift term which influences the statistical properties of the asset in the real (observable) world. Furthermore, we generali- ze the model with respect to a time-dependent (but still non-random) volatility function. Although it is well-known, that drift terms - under weak regularity conditions - do not affect the behaviour of the asset in the risk-neutral world and consequently the Black-Scholes option pricing formula holds true, it makes sense to point out that these regularity conditions are fulfilled in the present model and that option pricing can be treated in analogy to the Black-Scholes case.

In this paper, we study mathematical properties of a generalized bivariate Ornstein-Uhlenbeck model for financial assets. Originally introduced by Lo and Wang, this model possesses a stochastic drift term which influences the statistical properties of the asset in the real (observable) world. Furthermore, we generali- ze the model with respect to a time-dependent (but still non-random) volatility function. Although it is well-known, that drift terms - under weak regularity conditions - do not affect the behaviour of the asset in the risk-neutral world and consequently the Black-Scholes option pricing formula holds true, it makes sense to point out that these regularity conditions are fulfilled in the present model and that option pricing can be treated in analogy to the Black-Scholes case.

This thesis focuses on guaranteed investment funds, which have become very popular among investors in the Czech Republic in recent years. The reason for this popularity is the conservativeness of a typical domestic investor, who appreciates the lower bound for the value of his investment. Guaranteed funds characteristically have a complex structure and valuation of their profitability based solely on intuition is therefore impossible. This analysis tries to provide an answer to the question if investing in these funds is reasonable. A large part of the thesis is dedicated to the option theory and option valuation methods, including the famous Black-Scholes formula, as guaranteed investment funds have the characteristics of an option. Thanks to the complicated structure of these products, the analysis itself is done by Monte Carlo simulation.

Wir bauen das verallgemeinerte diskrete Modell des zu Grunde liegenden Aktienpreisprozesses, der als eine bessere Annäherung an den Aktienpreisprozess dient als der klassische zufällige Spaziergang. Das verallgemeinerte Multinomial-Modell des Option-Preises in Bezug auf das neue Modell des Aktienpreisprozesses wird erhalten. Das entsprechende asymptotische Verfahren erlaubt, die verallgemeinerte Black-Scholes Formel zu erhalten, die die Formel als einen Begrenzungsfall des verallgemeinerten diskreten Option-Preis Modells bewertet.