Charm
Charm (also called delta decay or DdeltaDtime) is a second-order option Greek that measures the rate of change of an option's delta with respect to time — specifically, how much the option's delta is expected to change over one day as time passes, holding all other variables constant.
Key takeaways
- Charm = ∂Delta/∂t = ∂²V/∂S∂t; it quantifies the daily decay of an option's directional sensitivity as expiration approaches.
- For at-the-money options, charm is typically negative for calls and positive for puts: delta converges to 0.5 (calls) and −0.5 (puts) slowly far from expiration, then rapidly near expiration.
- Delta hedgers must rebalance their delta positions over weekends and holidays to account for charm, since the position's effective delta changes through time passage alone.
- Charm is particularly significant for options that are near-the-money with imminent expiration, where delta can change dramatically in a single day.
- Institutional options dealers use charm to anticipate and pre-position for the delta decay that will occur over non-trading periods, reducing hedging costs and gamma risk.
Explanation
Charm is one of the 'higher-order Greeks' (sometimes called 'the Greeks of the Greeks' or vanna/charm/vomma group), which become critically important to options market-makers managing large books with positions across many strikes and expirations. While most retail options participants focus on the first-order Greeks (delta, gamma, theta, vega), sophisticated volatility desks must actively manage second-order sensitivities to maintain robust hedging as market conditions evolve.
In the Black-Scholes framework, charm for a European call is: Charm = −e^(−qT) × N'(d1) × [2rT − d2σ√T] / (2T√T), where q is the continuous dividend yield, N'(d1) is the standard normal PDF at d1, r is the risk-free rate, σ is volatility, and T is time to expiration. For a European put, charm has the same formula with a sign adjustment. The key practical takeaway is that charm is largest in magnitude for at-the-money options near expiration, where the binary outcome (expire in-the-money vs. out-of-the-money) creates rapid delta changes.
The practical significance of charm becomes most apparent when considering over-the-weekend delta hedging. An options dealer with a portfolio of near-expiry, at-the-money equity options carries significant charm exposure: over a three-day weekend (Friday close to Monday open), three days of charm accrue without any opportunity to rebalance. If the portfolio is long gamma (long options), the dealer is short charm — their long deltas will decay toward zero (calls) or their short deltas will decay toward zero (puts) over the weekend. This 'carry' from delta decay must be weighed against the gamma profits from market movement.
Charm is closely related to the concept of the 'pin risk' at expiration. As an option approaches expiration near its strike, charm becomes extremely large: a 0.50-delta at-the-money option can swing to delta 1.0 (if ending in-the-money) or delta 0.0 (if ending out-of-the-money) within hours. Market-makers managing net-short gamma positions near expiration must monitor charm carefully to anticipate the rebalancing trades that will be required if the underlying moves near the strike just before expiry.
For portfolio risk management purposes, charm (along with vanna, the change of delta with respect to volatility) is included in the broader sensitivity reporting of institutional options books. A risk system that ignores charm may not accurately represent the P&L trajectory of a delta-hedged options book over a weekend or holiday period, leading to unexpected mark-to-market movements when markets reopen.
Formula
Charm = ∂Δ/∂t = −e^(−qT) × n(d1) × [2rT − d2σ√T] / (2T√T) for European call
Example
An options dealer is long 5,000 contracts (500,000 shares) of a tech stock at-the-money call option expiring in three days, with delta of 0.52 and charm of −0.04 per day. The dealer is delta-neutral after selling 260,000 shares short. Over a three-day weekend, three days of charm accrue: −0.04 × 3 = −0.12 delta change. The call's delta decays from 0.52 to approximately 0.40 through time passage alone. When markets open Monday, the dealer's delta position has shifted: their long 500,000-share call position now has a delta of 200,000 shares (0.40 × 500,000), but their hedge is still 260,000 shares short — leaving them net short 60,000 shares of delta. The dealer must buy back 60,000 shares at Monday's open to restore delta neutrality, generating anticipated demand for the stock.
Related terms
At The Money Call Option Delta Distant Months Dividend Dividend Yield Equity European Option Extrinsic Value Gamma Greeks Hedging