Abstract
In the book "Analytic Methods and the Black-Scholes models" (in german), I consider stochastic volatility models and the price determination of European derivatives in these models.
In the first chapter, I derive the stochastic basics which are essential for understanding financial mathematics. Therefore, I construct the Itô-Integral which is elementary for handling stochastic differential equations. The first chapter ends with the proof of the multidimensional Itô-Formula, so that a differentiation of functions of stochastic processes is possible.
In the second chapter, I define basic concepts of financial mathematics and illustrate them in case of the usual Black-Scholes model. After that I introduce the riskless price determination via equivalent martingale measures and demonstrate the procedure again in case of the usual Black-Scholes model. After I have introduced the implied volatility model and proved existence and uniqueness of a pricing function in this model, I consider stochastic volatility models and transfer the ideas mentioned before to this type of model. The second chapter ends with the formulation of the Cauchy initial value problem which is equivalent to determine the pricing function in this model.
In the third chapter, I prove the existence and uniqueness of the solution for a general Cauchy initial value problem in the Hilbert-Space L^2(R^2) using Galerkin-Methods and the abstract theory about semigroups.
In the fourth chapter, I prove that the solution of the Cauchy initial value problem is analytic (in space and time) under certain conditions on the coefficients of the differential operator.
In the fifth chapter, I solve the problem that in practice the initial conditions are not in L^2(R^2) by considering the problem in a weighted L^2(R^2) space.
In the last chapter, I simulate the problem numerically. Here I consider on the one hand, methods for solving a Cauchy initial value problem, especially the finite difference method and Galerkin's Method and on the other hand, methods for solving the problem stochastically, especially Monte-Carlo simulations.