What is Charm? Definition, Formula, and Example
Charm is the second-order option Greek measuring how an option's delta changes as time passes, holding price and volatility constant — the force that pulls delta toward 0 or 1 as expiration approaches.
What is Charm?
Charm — formally delta decay, written ∂Δ/∂t — is the second-order option Greek that measures the rate of change of delta with respect to the passage of time, holding the underlying price and implied volatility constant. Where theta tells you how much an option's price decays per day, charm tells you how much its directional exposure decays per day. A call with charm of −0.02 loses 0.02 of delta per day from time alone. Charm is why an out-of-the-money option's delta bleeds toward zero and an in-the-money option's delta drifts toward 1.00 as expiration approaches, even if the stock never moves.
How Charm is Calculated
For a European call under Black-Scholes, charm is:
Charm_call = −N′(d₁) · [2(r − q)T − d₂·σ√T] / [2T·σ√T] − q·e^(−qT)·N(d₁)
where N′(d₁) is the standard normal density at d₁, d₂ = d₁ − σ√T, r is the risk-free rate, q the dividend yield, σ implied volatility, and T time to expiration in years. Put charm adds q·e^(−qT)·N(−d₁) instead.
The practical reading matters more than the formula: charm is largest in absolute terms for at-the-money options near expiration, exactly where delta is most unstable. For a zero-dividend stock, call charm at the money is negative (delta decays down toward 0.50's neighborhood from above) when rates are low — but the sign flips depending on whether the option is ITM or OTM. OTM calls have negative charm (delta → 0); ITM calls have positive charm (delta → 1).
Worked Example: SPY Weekly Options
Consider SPY at $550 with 5 days to expiration, 20% implied volatility, and a 5% risk-free rate. The $560 call — roughly 1.8% out of the money — has a delta near 0.22 and a charm of approximately −0.03 per day. If SPY goes nowhere for two days, that call's delta falls to roughly 0.16 purely from time passing. A market maker who is delta hedging a short position in 10,000 of these calls starts short 2,200 deltas of hedge; two days later the book is only short 1,600 deltas, and the desk buys back 600 SPY-share-equivalents without any price move. Aggregate that across the entire options chain and charm-driven rehedging becomes a measurable flow into expiration — one input behind end-of-week pinning near large open-interest strikes.
When Traders Use Charm
Market makers and vol desks monitor charm daily because it changes their hedge without any trade occurring — the book's delta decays overnight and must be rebalanced on the open. 0DTE traders live inside extreme charm: on expiration day, at-the-money delta whips between 0 and 1, and charm is the dominant term in that instability. Sellers of weekly covered calls benefit from negative charm eroding the short call's delta, reducing directional risk as the week progresses.
Limitations and Common Misconceptions
Charm assumes constant implied volatility — in reality, IV drifts into events and weekends, so realized delta decay deviates from the model. Retail platforms rarely display charm, so traders misattribute overnight delta changes to price movement they can't find. And charm is a continuous-time derivative; over weekends, three days of decay hit at once on Monday's open, which is why Monday-morning rehedging flows cluster.