Variance Reduction & Martingale Tests
Document ID: 2007-1236 (previously #81)
Published on: 1st June 2007
Author: Steven Morrison
Market-consistent valuation of insurers’ liabilities usually involves the use of Monte Carlo scenarios produced by an Economic Scenario Generator. The complex, path-dependent, nature of these liabilities usually prohibits the use of other valuation techniques, such as the use of analytic formulae. Unfortunately the Monte Carlo technique is subject to statistical error (commonly known as “sampling error”) meaning that, for any finite number of scenarios, the valuation calculated using these scenarios is just an estimate of the “true” ESG model value. Sampling error is particularly evident when valuing assets for which we know the "true" model value, such as underlying assets (i.e. the "martingale" or "1=1" test). To reduce the size of this sampling error, a number of "variance reduction" techniques have been proposed:
- “Path adjustment”, where we adjust the underlying paths of the equity index in such a way that the 1=1 test is passed exactly.
- “Weighted Monte Carlo”, where we adjust the weights assigned to each path in such a way that the 1=1 test is passed exactly.
- “Control Variates”, in which we modify the option cash flows according to the deviation of the underlying path from it’s mean. In each case we demonstrate how the technique can be implemented in practice using a simple worked example, and highlight its potential benefits and drawbacks.
