# Epsilon

**BSM-12 · Black–Scholes & Greeks · Core**

**Lesson question:** What does Epsilon 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 Epsilon in plain language.
- Reproduce the unit conversion.
- Compare a local estimate with model revaluation.

**Prerequisites:** BSM-03

Epsilon, sometimes called dividend Rho, measures sensitivity to a continuous yield assumption. Higher modeled yield reduces a call and increases a put at otherwise identical inputs.

The derivative is not a cash dividend paid to an option holder. Stock ownership and option rights have different entitlements.

## Read the derivative as a sentence

The identity −T×S×Delta makes the signs transparent: positive call Delta gives negative Epsilon, while negative put Delta gives positive Epsilon. Divide by 100 for a percentage-point yield change.

Real discrete dividends and American exercise are not completely represented by smooth q. A yield approximation must not be described as a full dividend-and-exercise model.

## Use it within its scope

Check what a platform's carry input actually represents. A cash amount cannot be inserted into a field expecting annual percentage yield. Apply the same explicit yield shock in both the local estimate and full revaluation.

## 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 Epsilon is -4.336147532076. The displayed convention gives -0.043361475321 option-value change per percentage-point yield change. Isolate yield +1 percentage point while holding every other input fixed.

Starting option value is 2.977574155222. The local estimate after the shock is 2.934212679901; a full-model recomputation gives 2.934417833834. Approximation minus recomputation is -0.000205153933. 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 Epsilon

```text
Epsilon=∂V/∂q=−T S Delta; displayed=raw/100
```

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=-4.336147532076; display multiplier=0.01.

3. Displayed=-0.043361475321.

4. Local affected measure=2.934212679901.

5. Recomputed affected measure=2.934417833834.

**Result:** -0.043361475321 option-value change per percentage-point yield change

## Graph: Epsilon across controlled states

**Horizontal:** Calendar days remaining  
**Vertical:** option-value change per percentage-point yield change

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-12.js` and displayed as an accessible table in the HTML lesson.

## A correct number can be misread

The stated unit is option-value change per percentage-point yield change. 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 Epsilon in this example?

**Answer:** -0.043361475321 option-value change per percentage-point yield change.

**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 Epsilon 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

### Plain English

Epsilon is the option’s dividend-yield exposure.

### Typical call sign

Negative: higher dividend yield tends to reduce call value.

### Typical put sign

Positive: higher dividend yield tends to increase put value.

### Practical note

For discrete dividends, a continuous-yield model is only an approximation.


## Try it in the working calculator

1. Change continuous yield by +1 percentage point with other shocks zero.
2. Compare call and put changes.
3. Explain why a dated cash dividend requires a different model treatment.

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.