Charm (delta decay)
Charm is the second-order option greek that measures how an option's delta changes as time passes, with price and volatility unchanged. It explains why out-of-the-money deltas drift toward zero and in-the-money deltas toward one as expiry nears, and why a delta-hedged book must trade even on a quiet day.
Senzoukria · Glossary · Updated September 2026
At a glance
- Definition
- Change in delta as calendar time passes (∂Δ/∂t)
- Black-Scholes, r = q = 0
- Charm = φ(d1)·d2 / (2τ) per year, same for calls and puts
- Also called
- Delta decay, DdeltaDtime
Formula
Charm is the derivative of delta with respect to the passage of time. In Black-Scholes with zero rate and dividend it simplifies to φ(d1)·d2/(2τ) per year, identical for a call and a put of the same strike. With a rate r and no dividend, one equivalent form is Γ·S·(σ·d2/(2√τ) − r), which ties charm to gamma. Dividing by 365 gives the delta change per calendar day. Sign conventions differ between texts, since some define charm on time to expiry rather than on time elapsed; the numbers below use time elapsed.
Worked example
Spot 100, implied volatility 20%, zero rate, ten days to expiry. The 105 call has a delta of 0.073 and a charm of −0.010 per day: tomorrow, with nothing else changed, its delta is 0.062. The 95 put has a delta of −0.059 and a charm of +0.009 per day, bringing it to −0.050. Both out-of-the-money deltas shrink toward zero. An in-the-money option drifts the other way, toward a delta of one in absolute value.
- Charm grows as expiry approaches because of the 1/τ factor.
- Close to expiry and close to the strike, charm becomes unstable, since delta is jumping between 0 and 1.
- Over a weekend the calendar-day convention applies two or three days of charm at once.
Why hedgers and GEX models care
A book kept delta-neutral must re-hedge the charm of every position each day, even if spot and implied volatility have not moved. Aggregated over a chain with a positioning assumption, this becomes charm exposure: the dollar delta a hedger would have to buy or sell per day under that assumption. It is often cited to explain drifts in the days before a large expiry. That story rests on the dealer positioning convention, which open interest cannot confirm, and on hedgers actually trading the underlying rather than netting risk elsewhere.
In Senzoukria
No chain provider publishes charm, so the GEX module derives it from the same inputs as gamma and anchors it on the provider's gamma, using charm = Γ·S·(σ·d2/(2√τ) − r) per year with an assumed fixed rate and zero dividend yield. Each leg's charm is scaled as open interest × charm × 100 × spot ÷ 365, in dollars of delta per day, with calls added and puts subtracted as for gamma. The Charm exposure tile, the per-strike values and the Surface page's Charm exposure option all use this figure. Expired legs, legs without implied volatility or gamma are counted and excluded, never set to zero, and the regime panel warns when vanna and charm rest on a low share of the legs.
Related
In the same section
- d1 and d2
- Chart template
- CFTC and NFA
- Circuit breakers
- Central limit order book
- Clearing house
- SKEW Index
- Click trading
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Frequently asked questions
- Is charm the same as theta?
- No. Theta is the change in the option's value as time passes; charm is the change in its delta. Theta tells you what the position earns or loses per day, charm tells you how much the hedge must be adjusted per day.
- Does charm exposure predict a drift into expiry?
- It quantifies a hedge adjustment under a stated positioning assumption. Whether that adjustment reaches the futures tape, and in which direction, depends on who really holds the options and how they hedge, which no public data shows. Treat it as context to test, not as a forecast.