A closed-form pricing formula of European options is obtained by Fischer Black and Myron Scholes (1973). In such a European option, the payoff depends `linearly' on the underlying asset price at the expiration time. An power option has a payoff which depends nonlinearly on the underlying asset price at the expiration time by raising a certain exponent. In the Black-Scholes model, a closed-form formula of a power option is obtained by Esser (2004). This paper extends Esser's result to the generalized Black- Scholes model. That is, we derive a closed-form pricing formula of a power option in the case when both the interest rate and the stock volatility are time-dependent.

The famous Black-Scholes partial differential equation is one of the most widely used and researched equations in modern financial engineering to address the complex evaluations in the financial markets. This thesis investigates a numerical technique, using a fourth-order discretization in time and space, to solve a generalized version of the classical Black-Scholes partial differential equation. The numerical discretization in space consists of a fourth order centered difference approximation in the interior points of the spatial domain along with a fourth order left and right sided approximation for the points near the boundary. On the other hand, the temporal discretization is made by implementing a Runge-Kutta order four (RK4) method. The designed approximations are analyzed numerically with respect to stability and convergence properties.

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.