Weighted Monte Carlo: Calibrating the Smile and Preserving Martingale Condition

TL;DR

This paper introduces a weighted Monte Carlo calibration method that maintains the martingale condition by adding synthetic options, improving pricing accuracy for exotic options.

Contribution

It proposes a novel calibration algorithm that preserves the martingale condition by incorporating synthetic options, enhancing the accuracy of exotic option pricing.

Findings

The method effectively calibrates to vanilla options and forwards.

Adding synthetic options preserves the martingale condition.

Significant price impacts are demonstrated on a geometric cliquet option.

Abstract

Weighted Monte Carlo prices exotic options calibrating the probabilities of previously generated paths by a regular Monte Carlo to fit a set of option premiums. When only vanilla call and put options and forward prices are considered, the Martingale condition might not be preserved. This paper shows that this is indeed the case and overcomes the problem by adding additional synthetic options. A robust, fast and easy-to-implement calibration algorithm is presented. The results are illustrated with a geometric cliquet option which shows how the price impact can be significant.