What Happens When You Rebalance a Delta Hedge Discretely?

Multiply each option's delta by its signed quantity and contract multiplier, then use the stock to offset the total. Recalculate after spot, time or implied volatility changes. Continuous rebalancing is the ideal used in Black–Scholes. A practical hedge uses discrete trades because stock is bought and sold in finite lots and each rebalance has a cost.

Delta is a slope at today's stock price and model inputs. When you sell a call with positive delta, your position loses value as the stock rises. Buying stock offsets that loss to first order. The slope itself moves as the stock moves. Gamma measures that change, so positions with large absolute gamma can require larger adjustments after the same spot move.

Trading more frequently can reduce exposure to drift in the hedge ratio between trades. It also incurs spread and possibly market impact. A trader may use a schedule, such as at fixed times, or rebalance when net delta leaves a predefined band. A sensible comparison measures both hedge error and trading cost using the same option position and price scenarios.

Measure the exposure that builds up between trades

Let $n$ be the signed number of contracts, $m$ the multiplier, and $h$ the number of shares held. For small changes in spot $\Delta S$ , time $\Delta t$ , and implied volatility $\Delta\sigma$ , a local approximation to portfolio P&L, before carry and costs, is

\Delta\Pi\approx(nm\Delta+h)\Delta S+ nm\left(\tfrac12\Gamma(\Delta S)^2+\Theta\Delta t+\nu\Delta\sigma\right).

Here

\Delta,\Gamma,\Theta

and

\nu

are per-share Greeks. Set

h=-nm\Delta

to remove the first-order spot term. If other inputs are fixed, the new target position changes by approximately

-nm\Gamma\Delta S

. This update becomes less accurate for large moves and near a payoff kink, where the Greeks can vary rapidly.

In the continuous-time replication model, the hedger can reset the stock position at every instant. Between discrete trades the same hedge ratio is held through potentially material changes in spot. That timing difference creates hedge error even with a correct option model and frictionless fills at each rebalance.

Two short calls after a two-dollar rally

You sell two calls with a multiplier of $100$ . Delta is $0.45$ and gamma is $0.03$ per dollar. You buy $90$ shares to neutralise $-200(0.45)=-90$ share equivalents. If the stock rallies $2$ , the new delta is approximately $0.51$ . Your stock position remains $90$ shares against an option delta near $-102$ . You buy about $12$ additional shares to return to neutral.

Over that move the option’s second-order spot contribution is about

-200(0.5)(0.03)(2^2)=-12

dollars. The trade of

12

shares changes future exposure and leaves the prior

12

-dollar estimate as part of P&L already earned. Time decay, volatility changes and execution costs can alter the realised result.

After the rally your short calls have roughly 102 shares of negative delta, while the hedge still holds 90 shares. Buying about 12 more shares restores delta neutrality at the new price. It cannot undo the P&L you accumulated on the way there. For this short-gamma position, the rebalancing trade goes in the direction of the spot move. Rebalancing more frequently reduces the time spent with stale delta and usually increases the number of trades. A desk chooses how to balance those effects using its liquidity, cost and risk limits.

Use the option strategy designer to inspect how a position's net Greeks change as spot moves.

References and next steps

Deriving the Black–Scholes PDE With a Delta Hedge
Dispatch · How option Greeks change

Measure the exposure that builds up between trades

Two short calls after a two-dollar rally

Move volatility and expiry in Black–Scholes

References and next steps