Rare-event pricing and risk management of financial derivatives: A subset simulation framework

Standard Monte Carlo (MC) simulation becomes inefficient for pricing and risk-managing financial derivatives whose values depend on low-probability events: the coefficient of variation (CV) of a rare-event probability estimator scales as the inverse square root of the target probability at fixed sample size. This paper develops a common rare-event simulation framework, based on subset simulation (SubSim), that decomposes a small target probability into a product of moderate conditional probabilities estimated via Markov chain Monte Carlo. We demonstrate the framework in three settings: barrier option pricing under Heston stochastic volatility, likelihood-ratio hedging sensitivities for those options, and a single-name credit default swap (CDS) spread-implied trigger example. We also document, to the best of our knowledge, a likelihood-ratio score for the conditional-volatility sensitivity of a barrier option under Heston stochastic volatility. The barrier experiments show lower relative estimator dispersion in demanding positive-payoff regions, with an MC-to-SubSim price CV ratio reaching approximately 2.52 on the reported grid.