# Charm

**BSM-16 · Black–Scholes & Greeks · Advanced**

**Lesson question:** What does Charm measure, in which units, and what does it not predict?

Original fictional examples; educational writing, not live market data, orders or production approval.

## By the end

- Explain Charm in plain language.
- Reproduce the unit conversion.
- Compare a local estimate with model revaluation.

**Prerequisites:** BSM-07, BSM-08, BSM-10

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.

## Calculation popup: Calculate Charm

```text
A=∂d1/∂T=[(r−q+sigma²/2)T−ln(S/K)]/(2 sigma T^(3/2)); Charm=q Delta−e^(−qT)phi(d1)A
```

T is years remaining; calendar time has the opposite derivative sign. N is the standard normal CDF and phi its density.

1. d1=0.070239316937; d2=-0.001433455448.

2. Raw=-0.164628389932; display multiplier=0.0027397260274.

3. Displayed=-0.000451036685.

4. Local affected measure=0.527113579718.

5. Recomputed affected measure=0.527109602581.

**Result:** -0.000451036685 Delta change per calendar day

## Graph: Charm across controlled states

**Horizontal:** Calendar days remaining  
**Vertical:** Delta change per calendar day

Original BSM calculation with only the x-axis input varied. All other inputs are stated in the example; the curve is not a prediction.

Complete numerical data are embedded in `scripts/black-scholes-greeks/BSM-16.js` and displayed as an accessible table in the HTML lesson.

## 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

### What is the displayed Charm in this example?

**Answer:** -0.000451036685 Delta change per calendar day.

**Why:** Always state the unit.

### Does model agreement prove the same live-market change?

**Answer:** No.

**Why:** Actual inputs, instrument features and executable quotes can differ. **Critical risk check.**

## Takeaway

Translate Charm into its specific input, affected quantity and unit before applying it. Revalue when the local approximation no longer fits.

## References and provenance

The cited references support definitions and conventions. The teaching prose, tickets and numerical scenarios are original; they are not attributed as historical market observations.

- [OIC Greeks](https://www.optionseducation.org/advancedconcepts/understanding-options-greeks)

- [OIC Vega](https://www.optionseducation.org/advancedconcepts/vega)


## Plain-English quick read

### Question it answers

If spot does nothing overnight, how can my hedge ratio still drift?

### 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.


## Try it in the working calculator

1. Record Delta at 30 days and then 29 days.
2. Compare the change with Charm/day at the starting state.
3. Verify that more time remaining has the opposite derivative sign from time passing.

The working lab uses fictional shared inputs; the printed worked example above stays fixed. Open it from the lesson's same-page Live Greeks Lab control. The engine prices European calls and puts with continuous yield and calendar days/365.


### Formula helper

For the helper A in this written formula, A = ∂d₁/∂T = [(r−q+σ²/2)T−ln(S/K)]/(2σT√T). The live lab labels the same helper D to avoid confusion with a discounted-price term.