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The main two problems of continuous-time financial mathematics are option pricing and portfolio optimization. In this thesis, various new aspects of these major topics of financial mathematics will be discussed. In all our considerations we will assume the standard diffusion type setting for securitiy prices which is today well-know under the term "Black-Scholes model". This setting and the basic results of option pricing and portfolio optimization are surveyed in the first chapter. The next three chapters deal with generalizations of the standard portfolio problem, also know as "Merton's problem". Here, we will always use the stochastic control approach as introduced in the seminal papers by Merton (1969, 1971, 1990). One such problem is the very realistic setting of an investor who is faced with fixed monetary streams. More precisely, in addition to maximizing the utility from final wealth via choosing an investment strategy, the investor also has to fulfill certain consumption needs. Also the opposite situation, an additional income stream can now be taken into account in our portfolio optimization problem. We consider various examples and solve them on one hand via classical stochastic control methods and on the other hand by our new separation theorem. This together with some numerical examples forms Chapter 2. Chapter 3 is mainly concerned with the portfolio problem if the investor has different lending and borrowing rates. We give explicit solutions (where possible) and numerical methods to calculate the optimal strategy in the cases of log utility and HARA utility for three different modelling approaches of the dependence of the borrowing rate on the fraction of wealth financed by a credit. The further generalization of the standard Merton problem in Chapter 4 consists in considering simultaneously the possibilities for continuous and discrete consumption. In our general approach there is a possibility for assigning the different consumption times different weights which is a generalization of the usual way of making them comparable via discounting. Chapter 5 deals with the special case of pricing basket options. Here, the main problem is not path-dependence but the multi-dimensionality which makes it impossible to give usuefull analytical representations of the option price. We review the literature and compare six different numerical methods in a systematic way. Thereby we also look at the influence of various parameters such as strike, correlation, forwards or volatilities on the erformance of the different numerical methods. The problem of pricing Asian options on average spot with average strike is the topic of Chapter 6. We here apply the bivariate normal distribution to obtain an approximate option price. This method proves to be very reliable and e±cient for the valuation of different variants of Asian options on average spot with average strike.

In nancial mathematics stock prices are usually modelled directly as a result of supply and demand and under the assumption that dividends are paid continuously. In contrast economic theory gives us the dividend discount model assuming that the stock price equals the present value of its future dividends. These two models need not to contradict each other - in their paper Korn and Rogers (2005) introduce a general dividend model preserving the stock price to follow a stochastic process and to be equal to the sum of all its discounted dividends. In this paper we specify the model of Korn and Rogers in a Black-Scholes framework in order to derive a closed-form solution for the pricing of American Call options under the assumption of a known next dividend followed by several stochastic dividend payments during the option's time to maturity.

We present a parsimonious multi-asset Heston model. All single-asset submodels follow the well-known Heston dynamics and their parameters are typically calibrated on implied market volatilities. We focus on the calibration of the correlation structure between the single-asset marginals in the absence of sucient liquid cross-asset option price data. The presented model is parsimonious in the sense that d(d􀀀1)=2 asset-asset cross-correlations are required for a d-asset Heston model. In order to calibrate the model, we present two general setups corresponding to relevant practical situations: (1) when the empirical cross-asset correlations in the risk neutral world are given by the user and we need to calibrate the correlations between the driving Brownian motions or (2) when they have to be estimated from the historical time series. The theoretical background, including the ergodicity of the multidimensional CIR process, for the proposed estimators is also studied.