Seminar in applied mathematics and statistics
SPEAKER: Lina von Sydow (Univ. Upsala)
TITLE: Numerical option pricing using adaptive finite differences
ABSTRACT:
We consider pricing of options by solving a Black-Scholes-Merton-type partial differential equation. For the spatial discretization we employ adaptive finite differences on structured but non-equidistant grids. By comparing solutions on two grids with different discretization parameters, the local truncation error can be estimated. From this estimate, new spatial grids can be computed that yield a final solution that fulfill a prescribed error tolerance.
Different models for the underlying assets are considered such as the standard Black-Scholes-Merton market model as well as more elaborate models such as the Bates model, which includes stochastic volatility and jumps in the underlying asset. In time we employ a discontinuous Galerkin discretization if there is no jump in the underlying asset. If such jumps are present, we use an implicit-explicit (IMEX) scheme in order to avoid dense blocks in the linear system of equations that has to be solved each time-step.
For multi-asset option, i.e. when the option depends on several underlying assets, we run into the “curse of dimensionality”. For options issued on highly correlated assets we reduce the dimensionality by introducing a principal component analysis and asymptotic expansions. This way we can compute accurate solutions by solving only lower-dimensional problems.
Numerical experiments that illustrate the efficiency of the proposed method will be presented.
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UPCOMING SEMINARS:
November 18, 15:15: Catherine Donnelly (Heriot-Watt University)
December 2, 15:15: Harry Zheng (Imperial College)
December 9, 11:15: Lina von Sydow (Univ. Uppsala)
December 9, 15.15: Anand Vidyashankar (George Mason University)