Delta Hedging in Practice: Discrete Hedges and Where the P&L Comes From

Black–Scholes assumes continuous, costless hedging. Nobody hedges continuously. This lesson is about the gap — where a delta hedger's P&L actually comes from, how noisy it is, and why the whole activity is a bet on realized versus implied volatility.

The P&L identity

Hold an option delta-hedged and expand its price change to second order (the Taylor lesson doing real work). Over a small interval, the hedged position's P&L is approximately

$$\text{P\&L} \approx \tfrac{1}{2}\Gamma S^2\left(\frac{\Delta S}{S}\right)^2 + \Theta\,\Delta t \;=\; \tfrac{1}{2}\Gamma S^2\left[\,r_{\text{realized}}^2 - \sigma_{\text{implied}}^2\,\Delta t\,\right]$$

using the Black–Scholes relation $\Theta \approx -\tfrac12 \Gamma S^2 \sigma_{\text{implied}}^2$ for a hedged book. Read it aloud: each period, a long-gamma position earns gamma on the squared move and pays theta priced at implied vol. The move beats implied → profit; a quiet period → theta bleeds. Summed over the option's life, the hedger's total P&L is (dollar-gamma-weighted) realized variance minus implied variance. Direction cancels; volatility remains. That's the precise sense in which "trading options delta-hedged = trading realized vs implied vol" — the sentence from the Greeks lesson, now with its formula.

Discrete hedging: the error term

Rebalancing $n$ times over the option's life instead of continuously leaves a hedging error — a zero-mean noise on top of the vol bet, with standard deviation scaling like $1/\sqrt{n}$ (double your rebalance frequency, shave the noise by √2 — the CLT's fingerprints again). Practical consequences interviewers fish for:

• Daily hedging on a 3-month option leaves quite visible noise; you can be right about vol and still lose on an unlucky path. Vol traders diversify across names and dates for exactly this reason.
• More frequent hedging costs more (spreads, impact — the microstructure course), so the hedge frequency is a real optimization: noise ↓ vs costs ↑. There is no "correct" answer, only the trade-off, and saying so is the answer.
• Hedging at gaps fails entirely: a jump realizes $\tfrac12\Gamma(\Delta S)^2$ before any rebalance can react — great when long gamma, brutal when short. This is why short-gamma books fear gaps, not drifts, and why jump risk isn't hedgeable with delta at all.

Which delta? The vol-input choice

Subtle and beloved by senior interviewers: hedging with deltas computed at implied vol locks in a smooth accrual of (realized − implied) variance but a path-dependent total; hedging at your forecast vol makes the expected P&L equal your forecast edge but with noisier accrual. Neither is wrong — it's a choice of what your P&L measures. Mentioning that the choice exists puts you in the top decile of candidates on this topic.

The interview version

"You buy a 1-month option at 20 implied and hedge daily; the stock realizes 25. Do you make money for sure?" — In expectation yes; for sure no — discrete-hedging noise and the dollar-gamma weighting of when the vol arrives (variance realized while your gamma was small earns little) can flip an in-expectation winner. "Where does the money show up day to day?" — Rebalancing: long gamma has you buying dips and selling rallies; the scalps are the realized-vol income, netted against theta each night. Concrete, mechanical, and exactly how a vol desk would describe its own P&L.