Note: I'm now at Georgia State University


You can contact me at aspeight@gsu.edu.
You can download the final version of my thesis here. Comments are welcome.

Thesis Abstract

To calibrate a model, one adjusts parameters so it gives correct answers to a set of questions with known answers. In this thesis, we consider the problem of calibrating a class of financial models. Specifically, we consider models common in financial economics, macroeconomics, and financial engineering based on continuous-time problems of stochastic control. In particular, the Method of Moments can be regarded as a calibration problem. We ask the question: If a model must be solved numerically, how difficult is it to calibrate? While numerical methods for solving stochastic control problems are well studied in both industry and academy, the inverse problem of parameter calibration has received comparatively little attention.

To solve the calibration problem numerically, we propose a multigrid technique that couples the calibration process with the model solver. For well-behaved problems, we find that the calibration problem can be solved for about three times the cost of solving the control problem with a fixed set of parameters. Computational evidence suggests that this holds independent of the number of parameters to be calibrated. In short: If you can solve a model numerically, it is within your computational budget to calibrate as well, provided it can be calibrated.

We illustrate this technique and its limitations with a series of examples. One example is a recent model from financial economics. This has only one parameter to calibrate but requires careful formulation to realize good numerical results. Another example we consider is a classical optimal stopping problem in two-dimensions. In this problem, the location of the stopping boundary is viewed as an infinite-dimensional parameter that must be chosen to satisfy the smooth pasting and value matching conditions. Fourier analysis of the pseudo-differential operators implicit in this problem show that the natural formulation is ill-posed for numerical purposes. This analysis suggests a reformulation to regularize the problem without introducing distortions and a process for smoothing errors. This is used to construct a fast multigrid solver.