Plain English
Delta is the option’s local directional exposure.
Black–Scholes & Greeks · BSM-07 · Core
Delta
What does Delta measure, in which units, and what does it not predict?
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.
What you will learn and prerequisites
- Explain Delta in plain language.
- Reproduce the unit conversion.
- Compare a local estimate with model revaluation.
Read first: Read the notation
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.
Open the calculation without leaving this lesson
Delta across controlled states
Vertical: option-value change per $1 spot change
Horizontal: Spot (USD)
1. Fictional example
Read the graph as a data table
| Spot (USD) | Fictional example |
|---|---|
| 80 | 0.00116971 |
| 81 | 0.00205195 |
| 82 | 0.00347865 |
| 83 | 0.00570678 |
| 84 | 0.00907127 |
| 85 | 0.013989 |
| 86 | 0.0209542 |
| 87 | 0.0305238 |
| 88 | 0.0432903 |
| 89 | 0.0598434 |
| 90 | 0.0807229 |
| 91 | 0.106366 |
| 92 | 0.137057 |
| 93 | 0.17288 |
| 94 | 0.213694 |
| 95 | 0.259114 |
| 96 | 0.308523 |
| 97 | 0.361097 |
| 98 | 0.415854 |
| 99 | 0.471715 |
| 100 | 0.527565 |
| 101 | 0.582324 |
| 102 | 0.635006 |
| 103 | 0.684766 |
| 104 | 0.730933 |
| 105 | 0.77303 |
| 106 | 0.810776 |
| 107 | 0.84407 |
| 108 | 0.872975 |
| 109 | 0.897685 |
| 110 | 0.918493 |
| 111 | 0.935763 |
| 112 | 0.949894 |
| 113 | 0.961298 |
| 114 | 0.970379 |
| 115 | 0.977517 |
| 116 | 0.983056 |
| 117 | 0.987302 |
| 118 | 0.990519 |
| 119 | 0.992926 |
| 120 | 0.994708 |
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
Try the question before revealing its answer. These are reading checks, not trading certification or a site completion record.
What is the displayed Delta in this example?
Reveal the worked answer
0.527564616403 option-value change per $1 spot 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 Delta 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
- Load Lesson example and note Delta.
- Compare call and put Delta without changing any other input.
- Run spot +0.25 alone; explain the small difference from full repricing.
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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