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Low-dimensional approximations of high-dimensional asset price models
We consider high-dimensional asset price models that are reduced in their dimension in order to reduce the complexity of the problem or the effect of the curse of dimensionality in the context of option pricing. We apply model order reduction (MOR) to obtain a reduced system. MOR has been previously studied for asymptotically stable controlled stochastic systems with zero initial conditions. However, stochastic differential equations modeling price processes are uncontrolled, have non-zero initial states and are often unstable. Therefore, we extend MOR schemes and combine ideas of techniques known for deterministic systems. This leads to a method providing a good pathwise approximation. After explaining the reduction procedure, the error of the approximation is analyzed and the performance of the algorithm is shown conducting several numerical experiments. Within the numerics section, the benefit of the algorithm in the context of option pricing is pointed out.
Path-wise approximation of the Cox-Ingersoll-Ross process
The Doss-Sussmann (DS) approach is used for simulating the Cox-Ingersoll-Ross (CIR) process. The DS formalism allows for expressing trajectories of the CIR process by solutions of some ordinary differential equation (ODE) that depend on realizations of the Wiener process involved. Via simulating the first-passage times of the increments of the Wiener process to the boundary of an interval and solving an ODE, we approximately construct the trajectories of the CIR process. From a conceptual point of view the proposed method may be considered as an exact simulation approach.