Choose the time variable before reading the sign
Let t be elapsed time and T be time remaining until a fixed expiry, measured in years. Since dT/dt = -1, elapsed-time charm χ = ∂Δ/∂t = -∂Δ/∂T. NAG and OpenGamma use this negative remaining-time derivative. If another source reports ∂Δ/∂T, reverse its sign before comparing it with this page. Always check whether the unit is per year, calendar day or trading day.
- Delta is the option price sensitivity to a small underlying price change. A daily charm of +0.015 is an additive delta change rate, not a 1.5% relative increase in delta.
- Using 252 instead of 365 changes the unit convention. Do not combine a trading-day rate with a calendar-day time shift, especially over weekends. The examples use actual remaining calendar days divided by 365.
| Quantity | Definition | Meaning |
|---|
| Elapsed-time charm | χ = -∂Δ/∂T | Positive means delta increases as the clock advances |
| Remaining-time derivative | ∂Δ/∂T = -χ | Positive means delta increases when more time remains |
| Daily charm here | χday = χ / 365 | Delta units per elapsed calendar day |
| Theta | Θ = -∂V/∂T | Option value change per unit of elapsed time |
Reproduce charm from European option delta
Use the Black-Scholes-Merton model with spot S, strike K, annualized decimal volatility σ, continuous annual risk-free rate r and dividend yield q. Define d1 = [ln(S/K) + (r - q + σ²/2)T] / (σ√T), with N the standard normal cumulative distribution and φ its density. Call delta is exp(-qT)N(d1); put delta is exp(-qT)[N(d1) - 1].
- Define A = [ln(S/K) - (r - q + σ²/2)T] / (2σT√T). Differentiating delta gives annual call charm χcall = exp(-qT)[qN(d1) + φ(d1)A]. Annual put charm χput = χcall - q exp(-qT). Divide either result by 365 for the daily convention here.
- Use φ(x) = exp(-x²/2) / √(2π). Require finite inputs with S > 0, K > 0, σ > 0 and T > 0. Enter 20% volatility as 0.20 in the formula. These equations assume European exercise and a continuous dividend yield.
- All following numerical scenarios use K = $100, σ = 0.20 and r = q = 0. Within each time comparison, S also stays fixed. These are hypothetical calculations, not observed contracts or market quotes.
Compare moneyness at 30, seven and two days
Each row advances the same hypothetical call by one calendar day. The spot price changes only between comparison groups. The last column is the full delta recalculation, Δ(D - 1) - Δ(D), rather than the starting charm multiplied by a day. Figures use unrounded calculations and show eight decimal places.
- The OTM call delta falls toward zero; the ITM call delta rises toward one. At the strike, delta approaches 0.5 from above under these zero-rate, zero-yield assumptions. Exactly at expiry, the payoff has a kink at the strike, so its ordinary spot derivative is undefined there.
- These rows do not establish that charm always grows until expiry. At a fixed spot strictly away from the strike, charm eventually tends toward zero in this model as delta settles near zero or one. Large changes concentrate in a narrowing region near the strike.
- Moneyness alone does not determine the sign for every input set. Rates, dividend yield, volatility and remaining time enter the formula. Even with r = q = 0, call charm changes sign at ln(S/K) = σ²T/2, slightly above the strike.
- For matching puts with q = 0, subtract one from every call delta. The delta changes and charm values stay the same: the $98 put becomes more negative as time passes, while the $102 put approaches zero. With nonzero q, call and put charm differ by q exp(-qT) annually.
| Fixed spot / call moneyness | Days remaining | Starting delta | Next-day delta | Starting charm / day | Full delta change |
|---|
| $98 / OTM | 30 → 29 | 0.37309255 | 0.37063238 | -0.00240407 | -0.00246017 |
| $98 / OTM | 7 → 6 | 0.23712834 | 0.21915766 | -0.01639603 | -0.01797068 |
| $98 / OTM | 2 → 1 | 0.08735639 | 0.02713804 | -0.05447761 | -0.06021835 |
| $100 / ATM | 30 → 29 | 0.51143575 | 0.51124359 | -0.00019054 | -0.00019216 |
| $100 / ATM | 7 → 6 | 0.50552457 | 0.50511478 | -0.00039459 | -0.00040979 |
| $100 / ATM | 2 → 1 | 0.50295308 | 0.50208815 | -0.00073826 | -0.00086493 |
| $102 / ITM | 30 → 29 | 0.64581062 | 0.64782558 | +0.00196346 | +0.00201496 |
| $102 / ITM | 7 → 6 | 0.76694487 | 0.78379707 | +0.01531909 | +0.01685220 |
| $102 / ITM | 2 → 1 | 0.91068676 | 0.97107795 | +0.05369597 | +0.06039119 |
Verify the derivative with a central time difference
Keep spot at $102 and use D = 7 remaining calendar days. Choose a bump h in days with 0 < h < D. Daily elapsed-time charm is approximately [Δ(D - h) - Δ(D + h)] / (2h). The smaller remaining-time value comes first in the numerator. Reversing the order calculates the remaining-time derivative.
- For h = 0.01 day, delta at 6.99 days is 0.7670981953 and delta at 7.01 days is 0.7667918132. Their difference divided by 0.02 is 0.0153191057 per day. The analytic annual value is 5.59146866; dividing it by 365 gives 0.0153190922 per day.
- If the bump is expressed in years instead, divide the resulting derivative by 365 once. Do not divide a difference already calculated per day by 365 again.
- Keep full precision until after subtraction. Rounded four-decimal deltas can erase a small bump or distort it substantially. Compare several bumps for convergence; arbitrarily tiny steps increase cancellation and rounding error.
| Time bump h | Daily charm from delta difference |
|---|
| 1 day | +0.01545523 |
| 0.1 day | +0.01532044 |
| 0.01 day | +0.01531911 |
| Analytic daily charm | +0.01531909 |
Use full delta changes for longer time shifts
At $102 spot and seven days remaining, the one-day linear estimate is 0.76694487 + 0.01531909 = 0.78226396. Full recalculation at six days gives 0.78379707. At two days remaining, the same method estimates 0.96438273 for tomorrow, while the full one-day delta is 0.97107795. The approximation becomes less reliable when the time step is a large part of the remaining life.
- Assume one long equity option contract represents exactly 100 shares. The seven-day starting delta exposure is about 76.694487 shares. Local charm implies about +1.531909 shares of delta drift per day; the full seven-to-six-day change is +1.685220 shares. These units describe exposure, not dollars earned.
- Multiply by signed contract quantity and the verified share multiplier. A short position reverses the exposure and drift signs. Adjusted contracts and non-equity products require their actual deliverables and units; do not assume every multiplier is 100.
- Observed next-day delta also reflects spot moves, IV changes and other inputs. Holding yesterday and today quotes side by side does not isolate charm. For a combined scenario, revalue delta with every changed input rather than attributing the entire difference to elapsed time.
Check the contract, then calculate the time comparison
TradingView documents delta in its options chain and time shifts in Strategy Builder what-if scenarios. Select the exact contract and a single long leg, choose the Delta profile, and compare the same underlying price with IV shift set to zero and time shift set to +1 day. Record the expiry time, quantity and display units. This compares modeled delta profiles; it does not require a displayed Charm field.
- In Pineify AI Stocks & Options Picker, select the ticker, Options Chain tab and expiration. The chain shows available contract delta and IV; expanded rows include other provider Greeks. Its inspected interface does not display Charm. A missing Greek or IV remains unknown rather than becoming zero.
- For the controlled example, open the Option Greeks Calculator and select Call, spot 102, strike 100, seven days, IV 20, rate 0 and dividend yield 0. Read Delta, then change only the remaining days to six. Subtract the first result from the second for the one-day change. The calculator accepts explicit inputs and does not supply a Charm output.
- The calculator displays rounded values and uses an approximate normal CDF, so use the formulas above for the eight-decimal finite-difference check. Its time input has a one-day minimum; use a model that accepts the actual remaining fraction for intraday expiry analysis.
- A futures option needs the correct futures contract and a suitable pricing model. Do not substitute a futures price into this equity spot model while leaving its carry assumptions unchanged. A stock chart alone does not provide a missing option IV, dividend assumption or exercise model.
Stop the smooth calculation at its boundaries
This model assumes constant volatility and rates, continuous dividend yield and European exercise. Discrete dividends, early exercise and assignment, jumps, volatility surface changes and settlement rules can alter actual delta behavior. Agreement between a derivative and its finite differences verifies the calculation, not the market model or an executable hedge.
- At expiry, use the contractual payoff and settlement treatment. Do not enter zero or negative remaining time in the displayed charm formula or take a central difference across expiry. Reject nonpositive spot, strike or volatility in this formula rather than concealing a division by zero.
- Use the exact expiration timestamp and the model day-count convention near expiry. Rounding a few hours to one day changes the scenario. A calendar-day teaching example does not predict how a market will price a weekend or an event.
- Stale or wide quotes can make implied volatility and displayed Greeks unreliable. Missing inputs, unknown provider conventions or a mismatched contract prevent a clean comparison. Extra decimal places cannot resolve those differences.
- Charm alone cannot infer dealer inventory or the direction of market-wide hedging flows. Position ownership, other Greeks, financing and execution matter. A delta-neutral position at one instant can develop exposure as time passes.
