Sizing a long butterfly for early assignment and a long option for gamma

Early assignment on a long butterfly

A long butterfly is a debit structure built from one lower-strike long option, two short options at a middle strike, and one higher-strike long option. It stays contract-neutral while the opening debit caps dollar risk. Early assignment is an obligation to meet a short option that is exercised before expiration, and it can temporarily create a stock inventory inside that otherwise hedged spread.

Early assignment of short contracts in a long butterfly does not raise the original dollar risk, because the book remains hedged and contract-neutral. A one-lot 30/35/40 long call butterfly opened for a 1.50 debit still has that 1.50 debit as maximum dollar risk after assignment on a short 35 call.

The remaining vertical spreads

A vertical spread is a two-strike option spread. A long butterfly can be read as one bull vertical plus one bear vertical that offset each other until one side is closed.

With the underlying at 37 after that assignment, exercising the long 30 call offsets the short stock and completes the 30/35 bull vertical, but five points of risk remain on the open 35/40 bear call spread. Even if the two remaining verticals are closed at a wash, exposure is still limited to the original 1.50 debit.

Margin, forced close, and time premium

The added risk of early assignment is operational: extra margin for short stock, or a forced close if the account cannot carry short stock, and an exercise-driven exit can realize the maximum loss sooner.

If remaining long options can be closed in the open market while they still hold time premium, the net loss can be smaller than an exercise-only unwind, though still below the outcome associated with the underlying finishing at the short strike at expiration.

Gamma, theta, and implied volatility on long options

A long call or long put also carries positive gamma, so a favorable one-point move can change premium by more than a static-delta estimate. Delta is the estimated change in an option's premium for a one-point move in the underlying. Gamma is the amount by which delta itself changes when the underlying moves one point.

An at-the-money long call with a 50 delta and 30 gamma would move to an 80 delta after a one-point rise, implying about 0.65 of premium change if an average 65 delta is used rather than 0.50.

Gamma is largest when the option is at the money and expiration is near, while same-month options already well in or out of the money have almost no gamma.

The long-gamma benefit is paid for with negative theta, and secondarily with long implied-volatility exposure, because at-the-money near-expiry gamma can increase when implied volatility falls. Long options carry negative theta because unused premium can expire worthless. Long options are typically long implied volatility, which can interact with how large near-expiry gamma becomes.