Superhedging And Quantile Hedging

In a complete market, Black-Scholes tells you the one hedge that replicates an option's payoff exactly, so the option has a unique fair price. Most real markets aren't complete — jumps, stochastic volatility, transaction costs, or trading restrictions mean no trading strategy can replicate every payoff exactly. So what's the "right" hedge when perfect replication is impossible? Two answers sit at opposite ends of a spectrum. Superhedging asks for a portfolio that is guaranteed to cover the payoff in every single scenario, no matter how bad — the cheapest such portfolio is the superhedging price, and it's typically the most conservative (and expensive) valuation the option could have. Quantile hedging instead fixes a maximum acceptable probability of shortfall, say 5%, and finds the cheapest hedge that succeeds at least 95% of the time, accepting a real chance of loss in the remaining scenarios in exchange for spending far less capital upfront.

The trade-off is stark: superhedging is safe but often absurdly expensive — for some exotic payoffs the superhedging price is as high as the worst-case payout itself, making it useless as a trading benchmark. Quantile hedging is a deliberate compromise: a trading desk decides it's acceptable to be wrong 1 time in 20, and prices the hedge accordingly, dramatically shrinking the capital tied up. The same idea shows up whenever a risk manager sets a shortfall probability instead of demanding zero-risk coverage — it's the hedging analogue of choosing a VaR confidence level rather than insisting on the maximum possible loss.