Discrete Hedging Error

The Black-Scholes derivation hedges continuously: rebalance the stock position every infinitesimal instant, and the randomness cancels perfectly, leaving a riskless portfolio. No real desk can do that — trading has a minimum tick, a bid-ask spread, and a human or server that only checks the book periodically. Rebalance once a day instead, and the stock is free to wander between rebalances without your hedge tracking it. That leftover wander doesn't average away to nothing; it shows up as genuine, random profit and loss, even though the option was theoretically "hedged" the whole time.

A tightrope walker who only checks their balance once a minute

A tightrope walker who continuously senses their balance stays exactly centred. One who can only check once a minute will drift off-centre between checks — sometimes a gust helps, sometimes it pushes them further off, and by the next correction they've accumulated genuine, unpredictable sway. Discrete hedging error is exactly that accumulated sway: the gap between where a continuously-rebalanced hedge would be and where a periodically-rebalanced one ends up.

Where the error comes from

Between two rebalance dates, the stock moves by some amount $\Delta S$ the hedger did not react to until the next check-in. Because the option's value curve is bent (gamma), the true change in its value differs from what the old, stale delta predicted, by a term driven by gamma:

$$\text{hedging error over one interval} \;\approx\; \tfrac{1}{2}\Gamma\big[(\Delta S)^2 - \sigma^2 S^2 \, \Delta t\big].$$

In plain English: the error each period is gamma times the gap between the actual squared move and the expected squared move (the variance the model priced in). If the stock moved more than its "typical" amount, the option holder was underhedged and gains; if less, they were overhedged and lose. Averaged over many periods this term has mean zero — unbiased — but its variance shrinks only as you rebalance more often, roughly as $1/N$ for $N$ rebalances.

Worked example 1: one day's hedging error

A market maker is long one call, gamma $\Gamma = 0.05$, on a stock with $\sigma = 20\%$ annualised, at $100, rebalancing once a day ($\Delta t = 1/252 = 0.00397$). The "typical" expected squared move for the day is $\sigma^2 S^2 \Delta t = (0.04)(10{,}000)(0.00397) = 1.588$.

Suppose the stock actually moves $1.80 (slightly larger-than-typical, so $(\Delta S)^2 = 3.24$).

$$\text{error} = \tfrac{1}{2}(0.05)\big[3.24 - 1.588\big] = 0.025 \times 1.652 = 0.0413.$$

That is a gain of about $0.0413 on this one day, purely because the stock moved more than its priced-in typical amount and the hedger held, until the next rebalance, a position that benefited from exactly that.

Worked example 2: how the error shrinks with more rebalancing

Compare rebalancing once a day (252 times a year) versus four times a day (1,008 times a year). The standard deviation of accumulated hedging error scales roughly as $1/\sqrt{N}$ — going from 252 to 1,008 rebalances should shrink it by roughly $\sqrt{4}=2$.

If daily rebalancing gives a standard deviation of $0.60, four-times-daily brings it to roughly $0.30. Halve it again with 16 rebalances a day, to about $0.15. The error never reaches zero at any finite frequency — only the continuous-time limit removes it, and no real desk can execute that.

Drag the volatility slider and watch how far a path can wander between two points you might only be checking once a day. Every bit of wander that happens between your rebalances, above or below what the model priced in, is exactly the raw material of discrete hedging error.

What this means in practice

Discrete hedging error is a big part of why a delta-hedged book's daily P&L is genuinely noisy even when nothing has "gone wrong" — a desk running the theoretically correct hedge still sees day-to-day swings from the gap between discrete and continuous rebalancing, distinct from actual model or execution mistakes. It is also why rebalancing frequency is a genuine trade-off: rebalance more often and this error shrinks, but trading costs rise, and the optimal frequency sits in between.