Mathematical Finance Seminar
October 18, 2001 , 5:30 PM to 7:00 PM
Alexander Levin, The Dime Bancorp
Pricing path-dependent mortgages on finite-difference grids
It has become a common and trivial perception that most time-efficient pricing structures, trees and grids, become impractical when employed for valuation of mortgages: they call for additional dimensions caused by different path-dependent features.
The purpose of this talk is to demonstrate that some sources of mortgage path-dependence are spurious and can be avoided via a simple problem transformation; others can be "cured" by a proper model selection such as "active-passive" decomposition of mortgage pools that allows for simulating path-dependent burnout effect. Finally, for a "non-curable" path-dependence such as clean-up call or senior/sub structure, we suggest employing a robust ad?hoc approximate finite?difference valuation method (with stable pricing errors not specifically exposed to path?dependence) and complement it with a specially designed control variate. The novelty of this approach is in using the first (out of 3) valuation run (i.e. pricing the original instrument under the approximate method) not only in the final pricing formula, P=P1+P3?P2, but in constructing the control variate itself. We discuss control variate tricks such as "synchronized knock-in" (for the clean-up feature), or matching average lives (for senior/sub and simple CMO structures). In each case the final result is surprisingly practical: we get rather accurate pricing for most non-CMO mortgage instruments and even some CMOs without ever leaving a low-dimensional finite difference grid!