You buy a one-year option at 20% implied and you expect 25% realised. You can run the delta hedge using the 20% delta or the 25% delta. Which locks in a known profit, which does not, and what is the trade-off?

You buy a one-year option at 20% implied and you expect 25% realised. You can run the delta hedge using the 20% delta or the 25% delta. Which locks in a known profit, which does not, and what is the trade-off?

Approach: Write the hedged profit and loss under each choice of hedge volatility, one giving a running accrual and the other a difference of two model values fixed at inception.

Hedging at the realised 25% locks in the known profit V(25) - V(20) at inception, while hedging at the 20% implied gives the same expected profit accrued along a random path. Hedging with the 25% delta makes the total profit equal to the difference of the two Black-Scholes values on day one, roughly 0.4*S*sqrt(T)*5 volatility points, and nothing about the path can change it, though the mark to market swings during the year and can be deeply negative before it converges. Hedging with the 20% delta makes the profit an accrual of 0.5*dollar gamma*(sigma_realised^2 - sigma_implied^2) each day, so it is always non-negative when realised beats implied and it is path dependent, paying most when the stock spends its volatile days near the strike. The choice is between a certain total with an uncertain path and an uncertain total with a daily profit that is never negative, and the second is what most desks run because the hedging error is bounded by the gamma they can see rather than by a volatility forecast they cannot verify until expiry.

Follow-up: Realised volatility comes in at 25% but arrives entirely in the last month after the stock has drifted 30% away from the strike. What did each hedge earn?

Key concepts: delta hedging, path dependence, realised volatility, hedging error, dollar gamma.