Efficient Pricing of High-Dimensional American-Style Derivatives: A Robust Regression Monte Carlo Method

Abstract

Pricing high-dimensional American-style derivatives is still a challenging task, as the complexity of numerical methods for solving the underlying mathematical problem rapidly grows with the number of uncertain factors. We tackle the problem of developing efficient algorithms for valuing these complex financial products in two ways. In the first part of this thesis we extend the important class of regression-based Monte Carlo methods by our Robust Regression Monte Carlo (RRM) method. The key idea of our proposed approach is to fit the continuation value at every exercise date by robust regression rather than by ordinary least squares; we are able to get a more accurate approximation of the continuation value due to taking outliers in the cross-sectional data into account. In order to guarantee an efficient implementation of our RRM method, we suggest a new Newton-Raphson-based solver for robust regression with very good numerical properties. We use techniques of the statistical learning theory to prove the convergence of our RRM estimator. To test the numerical efficiency of our method, we price Bermudan options on up to thirty assets. It turns out that our RRM approach shows a remarkable convergence behavior; we get speed-up factors of up to over four compared with the state-of-the-art Least Squares Monte Carlo (LSM) method proposed by Longstaff and Schwartz (2001). In the second part of this thesis we focus our attention on variance reduction techniques. At first, we propose a change of drift technique to drive paths in regions which are more important for variance and discuss an efficient implementation of our approach. Regression-based Monte Carlo methods might be combined with the Andersen-Broadie (AB) method (2004) for calculating lower and upper bounds for the true option value; we extend our ideas to the AB approach and our technique leads to speed-up \nfactors of up to over twenty. Secondly, we research the effect of using quasi-Monte Carlo techniques for producing lower and upper bounds by the AB approach combined with the LSM method and our RRM method. In our study, efficiency has high priority and we are able to accelerate the calculation of bounds by factors of up to twenty. Moreover, we suggest some simple but yet powerful acceleration techniques; we research the effect of replacing the double precision procedure for the exponential function and introduce a modified version of the AB approach. We conclude this thesis by combining the most promising approaches proposed in this thesis, and, compared with the state-of-the-art AB method combined with the LSM method, it turns out that our ultimate algorithm shows a remarkable performance; speed-up factors of up to over sixty are quite possible.

Cite this work

No public copy is stored for this record, so only the metadata is cited.

Stored only when both papers are in the corpus — 16 citations exist upstream.

Papers that cite the same references — the relational fallback while embeddings are still backfilling.