Delta Neutral Is Not Risk Free: Gamma and Hedging Costs

A hedging screen shows zero delta, yet the position loses money the following day. That is not necessarily a broken hedge. Delta neutrality removes the estimated first-order effect of a small move in the underlying at one moment. It does not freeze the option price, remove curvature or stop time passing.

The useful question is therefore not whether the position is labelled neutral. It is which exposure was neutralized, for how long, under which model, and at what cost. A reproducible example makes the distinction between a valid local hedge and a promise of stable profit.

Zero delta is a snapshot, not a permanent position

Delta measures an option's local price sensitivity to its underlying. Gamma measures how that delta changes as the underlying moves. A share position can offset delta, but ordinary shares have no option-like gamma. For a plain long call or put, gamma is positive; reversing the position reverses its sign. A portfolio can combine both, so its net exposure must be added with the correct quantities and signs.

Consider an invented European call with a stock price and strike of 100, 30 calendar days remaining, annualized volatility of 20%, and zero interest and dividends. In a Black–Scholes calculation using 365 days per year, its value is about 2.2872 per underlying share. Delta is 0.511436, gamma 0.069548, daily theta −0.038109, and vega 0.114326 per one percentage-point change in volatility.

Assume ten contracts with a multiplier of 100: 1,000 option units. This multiplier is an assumption, not a rule for every market. The initial delta hedge shorts 511.436 shares. Fractional shares are allowed in this model exercise to isolate the mechanism; real trading increments introduce rounding exposure. Financing, stock borrowing and transaction costs are initially set to zero.

A price move changes the hedge you need

If the stock immediately rises by 2, with time and volatility unchanged, the local estimate of the change in position delta is 1,000 × 0.069548 × 2 = 139.10 shares. The original short position no longer offsets the call's new delta. Returning to neutrality would require approximately 139 additional shares sold short, subject to a fresh model calculation.

For a fall, a positive-gamma position requires buying back some of its short stock. This sell-after-a-rise, buy-after-a-fall pattern explains the appeal of gamma hedging, but it is not evidence of a free gain. The option premium pays for convexity, time decay works against this particular long position, and repeated trades consume the bid–ask spread.

The estimate uses the starting gamma, not an unchanging law over a large move. Near a strike close to expiry, sensitivities can change sharply. A gap can move the market past several intended hedge levels before an order can execute. Neutrality calculated before the gap cannot retroactively remove the loss or financing need it creates.

Separate the curvature benefit from the cost of one day

Keep the original share hedge unchanged until the next calendar day. Hold volatility at 20% and reduce maturity from 30 to 29 days. A local approximation is: hedged profit or loss ≈ 1,000 × [½ × gamma × (stock-price change)² + daily theta]. The first-order delta term has cancelled; the remaining terms have not.

All amounts are hypothetical currency units before costs. The last column reprices the call with the same model, then adds the profit or loss on the original short-share hedge. It is exact within those stated model inputs, not a guaranteed executable market price. Changing gamma and higher-order effects explain why the full up and down results are not symmetrical.

Even a stock-price increase can coexist with a hedged loss: the +1 scenario does not generate enough curvature benefit to offset one day's decay. The local break-even absolute move is about 1.047, obtained from √(−2 × daily theta ÷ gamma). That number belongs only to this one-day, unchanged-volatility, fixed-hedge approximation. It is not a universal trading threshold or a break-even realized-volatility estimate for a rebalanced strategy.

Volatility can overturn the price-only result

Now take the +2 one-day scenario but reduce implied volatility from 20% to 18%. That is a decline of two percentage points, not a 2% relative decline. The starting-vega estimate adds a loss of 1,000 × 0.114326 × 2 = 228.65. Combining local terms gives roughly 100.99 − 228.65 = −127.66.

Full model revaluation gives about −112.18 instead. The difference is informative: a finite price move, the passage of time and a volatility change interact, while the small-change approximation holds the original sensitivities fixed. Do not relabel that approximation as a precise mark-to-market forecast.

A stock hedge does not neutralize vega. Nor does a single volatility input describe every strike and maturity in an actual portfolio. Changing skew, different expiry exposures, dividends or early-exercise features require a model and scenario set suited to the contracts. The cash-flow logic of put–call parity is another reminder that funding and contract terms belong in option comparisons, rather than being dismissed as technical details.

More rebalancing can reduce exposure and still reduce profit

Rebalancing more frequently generally reduces the time spent with a stale delta, but increases turnover. Suppose a separate execution exercise has total absolute share turnover of 800 and an assumed all-in execution cost of 0.03 per share. The bill is 24. A pre-cost result of 20 becomes −4. Count buys and sells once each in turnover; do not double-count a cost already included in the spread assumption. Stock-borrow and funding charges would be additional.

A fixed time schedule, a delta band and an event-triggered hedge are different rules. Compare them on the same price path with the same spread, slippage and financing assumptions. A test using the next bar's known price, continuous frictionless execution or a midpoint unavailable to the trader cannot establish the profitability of a real hedge.

The short-gamma position has the opposite rebalancing pressure: it tends to buy after rises and sell after falls. Time decay may help a plain short-option position, but a large move can more than erase that income. Open interest alone does not establish who holds the long and short sides, so it cannot prove the sign of dealers' aggregate gamma.

What a defensible hedge check should record

Record the instrument, exercise style, settlement, expiry, multiplier, position sign and actual executable quote. Then reconcile the option delta in underlying units with the share or futures hedge, including any basis mismatch and rounding. State whether theta is per calendar day or another convention and whether vega is quoted per volatility percentage point.

Run price-up, price-down and unchanged-price cases, then add time and volatility changes together. Reprice the whole position for larger shocks instead of extending a single gamma indefinitely. Finally compare estimated and realized profit after spread, slippage, funding and borrowing, and define when a stale model, a gap or inadequate liquidity invalidates the planned adjustment.

The conclusion should be narrow and testable: the portfolio had near-zero first-order price exposure at this timestamp under these inputs. It should never be enlarged into “the position cannot lose.” A hedge is a managed set of remaining exposures, not the absence of them.