Pricing of Financial Derivatives

The efficient and accurate valuation of financial derivatives, for example options, is one of the main tasks of computational finance. Thereby, the fair value of a derivative is determined by the payoff structure of the financial product at hand and a suitable stochastic model for the financial market. The resulting problem is then either an expectation (usually a multivariate integral) or a partial (integro-) differential equation. In most cases there exist not closed-form solutions for these problems and numerical methods have to be used for their computation.

Current challenges are more and more complex financial products, such as compound or multi-asset options, as well as sophisticated market models, such as jump-diffusion models.

In this project, we develop efficient and robust numerical algorithms and realize them on parallel supercomputers. Thereby, modern computational methods, such as multilevel Monte Carlo and Quasi-Monte Carlo simulation, dimension-adaptive sparse grid quadrature, and sparse multinomial trees are employed. These new methods allow the computation of high accuracy solutions and simultaneously a substantial reduction of computing times.

1. T. Gerstner and M. Griebel:
   Sparse grids.
   In Encyclopedia of Quantitative Finance, J. Wiley & Sons, 2009.

2. T. Gerstner and M. Holtz:
   Valuation of performance-dependent options.
   Applied Mathematical Finance, 15(1):1-20, 2008.

3. T. Gerstner and M. Holtz:
   Geometric tools for the valuation of performance-dependent options.
   In Computational Finance and its Application II, pp. 161-170, WIT Press, 2006.

4. T. Gerstner, M. Holtz and R. Korn:
   Valuation of performance-dependent options in a Black-Scholes framework.
   In Numerical Methods for Finance, pp. 203-214. Chapman & Hall/CRC, 2007