Question it answers
If spot does nothing overnight, how can my hedge ratio still drift?
Black–Scholes & Greeks · BSM-16 · Advanced
Charm
What does Charm measure, in which units, and what does it not predict?
Why it matters
Delta-neutral books can become non-neutral with time alone.
Sign
Depends on option type, moneyness, volatility, rates, and dividends.
Use
Useful for short-dated hedge planning.
What you will learn and prerequisites
- Explain Charm in plain language.
- Reproduce the unit conversion.
- Compare a local estimate with model revaluation.
Charm is calendar-time drift in Delta under fixed other inputs. It explains why the same hedge can become stale even if spot does not move.
The definition here is −∂Delta/∂T. A remaining-time convention reported elsewhere has the opposite sign.
Read the derivative as a sentence
The formula uses A, the derivative of d1 with respect to remaining maturity. Inserting the correct call or put Delta gives the corresponding Charm. Divide annual Charm by 365 for the daily display.
A one-day estimate becomes less reliable when the derivative changes rapidly near expiry. Do not substitute tiny time at the boundary and call it exact.
Use it within its scope
Use Charm for hedge planning, not a forecast of daily price drift. Aggregate signed quantities first. A desk maintaining neutrality changes its hedge oppositely to the book's Delta drift, conditional on actual rebalancing.
The model boundary
The smooth formulas require positive spot, strike, volatility and remaining time. Zero time and zero volatility still admit explicit value limits, but smooth Greek availability needs separate rules. No invalid or zero input is silently replaced with a default. Continuous dividends, constant parameters and European exercise are modeling assumptions, not claims about every real option.
One complete numerical experiment
The shared fictional European call uses spot 100, strike 100, 30 calendar days, sigma 25%, continuous rate 4% and yield 1%. Its BSM value is 2.977574155; d1=0.070239317 and d2=-0.001433455. Values are per option unit before contract multiplication.
Raw Charm is -0.164628389932. The displayed convention gives -0.000451036685 Delta change per calendar day. Isolate one calendar day passing while holding every other input fixed.
Starting Delta is 0.527564616403. The local estimate after the shock is 0.527113579718; a full-model recomputation gives 0.527109602581. Approximation minus recomputation is +0.000003977137. The difference shows the limitation of a local derivative, not a claim that the market must follow either number.
The graph varies only its labeled horizontal input. It is a series of controlled states, not historical observations or a future trajectory. Read its units before interpreting sign or magnitude.
Open the calculation without leaving this lesson
Charm across controlled states
Vertical: Delta change per calendar day
Horizontal: Calendar days remaining
1. Fictional example
Read the graph as a data table
| Calendar days remaining | Fictional example |
|---|---|
| 1 | -0.00254388 |
| 2 | -0.00179448 |
| 3 | -0.00146243 |
| 5 | -0.00112933 |
| 7 | -0.000952023 |
| 10 | -0.000793885 |
| 15 | -0.000645151 |
| 20 | -0.000556415 |
| 30 | -0.000451037 |
| 45 | -0.000364827 |
| 60 | -0.000313313 |
| 90 | -0.000252001 |
A correct number can be misread
The stated unit is Delta change per calendar day. Mixing raw decimal derivatives with point shocks or calendar time with remaining time can magnify or reverse the result.
The controlled comparison also omits live spread, surface dynamics, unsupported exercise effects and execution. A local sensitivity is not a market forecast or a guarantee of realized P/L.
Check your understanding
Try the question before revealing its answer. These are reading checks, not trading certification or a site completion record.
What is the displayed Charm in this example?
Reveal the worked answer
-0.000451036685 Delta change per calendar day.
Always state the unit.
Does model agreement prove the same live-market change?
Reveal the worked answer
No.
Actual inputs, instrument features and executable quotes can differ.
Translate Charm into its specific input, affected quantity and unit before applying it. Revalue when the local approximation no longer fits.
Try it in the working lab
- Record Delta at 30 days and then 29 days.
- Compare the change with Charm/day at the starting state.
- Verify that more time remaining has the opposite derivative sign from time passing.
Your live inputs are independent of the fixed worked example above. Return to the “Lesson example” preset to reproduce it.
My reading record
Personal review marks are not quiz passes, course certification or trading approval. Old plugin progress is not overwritten.
Sources, assumptions and downloadable script
External references support definitions and conventions. Worked scenarios and prose are original. No current quotes or dealer inventory are claimed.
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