Simulation and Estimation of Diffusion Processes: Applications in Finance

Diffusion processes are the most commonly used models in mathematical finance, and are used extensively not only by academics but also practitioners. Nowadays a wide range of models, that can capture many of the effects observed in financial markets, are available. A very important task is to calibrate the models to observed market data and to achieve a good fit, since a slight misspecification can have large monetary consequences. The focus of this thesis is to investigate both theoretical and computational aspects of parameter estimation for diffusion processes.

In the first paper we consider adaptive calibration where the model parameters are considered to be part of a hidden dynamic state. We then use filtering techniques to estimate the parameter paths. An optimal method for tuning the hyperparameters using the expectation maximization algorithm is presented. The method is evaluated on both simulated and real data, where it is shown to be robust.

The second and third paper cover simulation-based methods for density estimation of diffusion processes using multilevel Monte Carlo estimation. This is a technique that uses simulation on a hierarchy of discretization levels in order to reduce computational complexity. In the second paper we provide an improvement to existing multilevel kernel density estimation by proposing a bandwidth choice that takes model-specific information into account. The third paper extends a simulated maximum likelihood algorithm to the multilevel Monte Carlo framework. Both methods are evaluated on simulated data, where they are shown to provide improvements to the compared methods.

The fourth paper introduces a software package for high-performance simulation of diffusion processes in the Julia programming language. Specific features of Julia are utilized in order to create a simulation library that performs significantly better in terms of computational speed compared to other available libraries, while allowing models to be defined using mathematical notation instead of code.