# Veta

**BSM-20 · Black–Scholes & Greeks · Advanced**

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

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

Veta asks how volatility sensitivity changes through time. It changes Vega, not price directly. Adding it to P/L as though it were Theta answers the wrong question.

This lesson displays daily change in Vega already quoted per volatility point.

## Read the derivative as a sentence

Two conversions are needed: divide by 365 for a calendar day and by 100 because the affected Vega is point-scaled. A platform displaying daily change in raw Vega would show a different number.

The sign is state-dependent. Near-money Vega commonly declines toward expiry, but that observation is not a universal statement about every strike or book.

## Use it within its scope

A long-volatility position can own less Vega later without any new trades. Reassess exposures before an event rather than assuming the original hedge and premium sensitivity persist. Keep the valuation date explicit.

## 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 Veta is -68.892452960960. The displayed convention gives -0.001887464465 Vega-per-point change per calendar day. Isolate one calendar day passing while holding every other input fixed.

Starting Vega per volatility point is 0.113997675882. The local estimate after the shock is 0.112110211417; a full-model recomputation gives 0.112093899522. Approximation minus recomputation is +0.000016311895. 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 Veta

```text
Veta_raw=−∂Vega_raw/∂T=Vega_raw[q+d1 A−1/(2T)]; displayed=raw/(100×365)
```

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=-68.892452960960; display multiplier=2.7397260274e-05.

3. Displayed=-0.001887464465.

4. Local affected measure=0.112110211417.

5. Recomputed affected measure=0.112093899522.

**Result:** -0.001887464465 Vega-per-point change per calendar day

## Graph: Veta across controlled states

**Horizontal:** Calendar days remaining  
**Vertical:** Vega-per-point 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-20.js` and displayed as an accessible table in the HTML lesson.

## A correct number can be misread

The stated unit is Vega-per-point 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 Veta in this example?

**Answer:** -0.001887464465 Vega-per-point 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 Veta 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 nothing else changes, how will my volatility sensitivity evolve?

### Relationship

Vega itself has time decay.

### Use

Important when managing volatility exposure through time.

### Sign

Depends on the option state.


## Try it in the working calculator

1. Record Vega/vol point at 30 days and then 29 days.
2. Compare the change with Veta/vol point/day.
3. Check both the /100 volatility scaling and /365 day scaling.

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.