Likelihood Ratio Weighting

Likelihood ratio weighting is the mechanism used to correct the bias introduced when sampling from a distribution other than the target distribution. When performing importance sampling, the simulated outcomes are drawn from a proposal distribution, which means they do not represent the true probability of the event in the original distribution.

To adjust for this, each simulated result is multiplied by a weight equal to the ratio of the target probability density to the proposal probability density. This weight ensures that the final estimate remains unbiased, effectively undoing the skew created by the artificial sampling method.

In financial modeling, this is used to accurately value instruments that depend on rare tail events, such as catastrophic market crashes or extreme liquidity dry-ups. Without this weighting, the simulation would produce incorrect results that do not reflect the true risk of the underlying assets.

It is a critical mathematical step that bridges the gap between efficient computation and accurate financial valuation. By properly weighting samples, analysts can gain deep insights into the tail risks of crypto portfolios.