# Delta

**BSM-07 · Black–Scholes & Greeks · Core**

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

**Prerequisites:** BSM-03

Delta is the option's local directional slope. A value near 0.53 says a sufficiently small one-dollar spot rise initially adds about 0.53 per option unit when the other inputs stay fixed. Initially is the important word.

Delta is not tomorrow's forecast and not automatically a real-world probability of success.

## Read the derivative as a sentence

For a standard 100-unit contract, per-unit Delta 0.53 represents approximately 53 share-equivalents. Signed quantity scales that exposure. In the dividend-adjusted model, matching call and put Deltas differ by e^(−qT), not always exactly one.

A neutral hedge cancels the current first-order spot exposure, not Gamma, volatility, time, basis or execution risk.

## Use it within its scope

Use Delta for small shocks and first hedge estimates. Larger moves change Delta through Gamma. If IV moves alongside spot, the observed option change includes more than Delta; disagreement with a simple estimate need not indicate a faulty Greek.

## 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 Delta is 0.527564616403. The displayed convention gives 0.527564616403 option-value change per $1 spot change. Isolate spot +$0.25 while holding every other input fixed.

Starting option value is 2.977574155222. The local estimate after the shock is 3.109465309323; a full-model recomputation gives 3.111195993690. Approximation minus recomputation is -0.001730684367. 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 Delta

```text
Delta=∂V/∂S; call=e^(−qT)N(d1); put=e^(−qT)[N(d1)−1]
```

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=0.527564616403; display multiplier=1.

3. Displayed=0.527564616403.

4. Local affected measure=3.109465309323.

5. Recomputed affected measure=3.111195993690.

**Result:** 0.527564616403 option-value change per $1 spot change

## Graph: Delta across controlled states

**Horizontal:** Spot (USD)  
**Vertical:** option-value change per $1 spot 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-07.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 $1 spot 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 Delta in this example?

**Answer:** 0.527564616403 option-value change per $1 spot 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 Delta 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

Delta is the option’s local directional exposure.

### How to read 0.60

A $1 rise in spot initially adds about $0.60 to the option.

### Typical sign

Calls: positive. Puts: negative.

### Common mistake

Assuming Delta stays fixed after a meaningful spot move.


## Try it in the working calculator

1. Load Lesson example and note Delta.
2. Compare call and put Delta without changing any other input.
3. Run spot +0.25 alone; explain the small difference from full repricing.

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.