The aim of this mini-course is to provide a systematic analysis of
valuation formulas for derivatives in finance which are based on
Fourier transforms. In the first part we concentrate on the case
where the underlying security is modeled by an exponential
semimartingale in general. This covers e.g. stock prices, indices
and FX rates. In particular Lévy processes as drivers are studied
in detail. A great variety of payoff functions and specific processes
can be considered within this framework. Formulas for derivatives
which depend on multidimensional underlyers are considered as well.
The Fourier based approach allows also to compute Greeks.
In the second part of this mini-course we will introduce jump processes (or inhomogeneous Levy processes) for modelling the dynamics of energy prices. We analyse multi-factor Ornstein-Uhlenbeck processes with stochatsic volatility as a general class of spot price models, and link these to forward prices. Our models will be motivated by stylized facts of energy prices, like mean-reversion, seasonality and spikes. Finally, we study pricing of options on forwards, based on Fourier methods analysed in detail in the first part of the mini-course by Professor Eberlein. |