Plain English
Epsilon is the option’s dividend-yield exposure.
Black–Scholes & Greeks · BSM-12 · Core
Epsilon
What does Epsilon measure, in which units, and what does it not predict?
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.
What you will learn and prerequisites
- Explain Epsilon in plain language.
- Reproduce the unit conversion.
- Compare a local estimate with model revaluation.
Read first: Read the notation
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.
Open the calculation without leaving this lesson
Epsilon across controlled states
Vertical: option-value change per percentage-point yield change
Horizontal: Calendar days remaining
1. Fictional example
Read the graph as a data table
| Calendar days remaining | Fictional example |
|---|---|
| 1 | -0.00138384 |
| 2 | -0.00277922 |
| 3 | -0.00418207 |
| 5 | -0.00700504 |
| 7 | -0.00984669 |
| 10 | -0.0141379 |
| 15 | -0.0213531 |
| 20 | -0.0286345 |
| 30 | -0.0433615 |
| 45 | -0.0657886 |
| 60 | -0.0885496 |
| 90 | -0.134893 |
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
Try the question before revealing its answer. These are reading checks, not trading certification or a site completion record.
What is the displayed Epsilon in this example?
Reveal the worked answer
-0.043361475321 option-value change per percentage-point yield change.
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 Epsilon 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
- Change continuous yield by +1 percentage point with other shocks zero.
- Compare call and put changes.
- Explain why a dated cash dividend requires a different model treatment.
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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