Rebalancing Bands For Delta Hedging

In theory, delta-hedging an option means continuously trading the underlying stock to keep your net delta at zero every instant the price moves. In practice, continuous trading is impossible and would bankrupt you in transaction costs even if it weren't — every tiny re-hedge pays the bid-ask spread and commissions. A rebalancing band solves this by only re-hedging when the position's delta drifts outside a chosen tolerance, say ±0.05 per contract, instead of after every tick.

Concretely, suppose a market maker is short a call option with delta −0.40 and holds 40 shares of stock as an offsetting hedge, keeping net delta near zero. If the desk uses a band of ±0.05, it does nothing as the option's delta wanders between −0.35 and −0.45 with small price moves, since the resulting hedge error is judged small enough to tolerate. Only when the delta drifts past −0.45 (or −0.35) does the desk trade stock to bring it back to flat. A tighter band (say ±0.01) tracks the true delta more closely but forces far more trades and racks up far more transaction costs; a wider band saves on costs but leaves the position more exposed to price swings between rebalances.

Choosing the band width is a direct trade-off between hedging error and trading costs, and it widens naturally when transaction costs are high, volatility is low (so delta drifts slowly anyway), or gamma is small (so a given price move barely changes delta in the first place).