Plain English
Theta is the local effect of the clock.
Black–Scholes & Greeks · BSM-09 · Core
Theta
What does Theta measure, in which units, and what does it not predict?
How to read −0.08/day
One day passing initially removes about $0.08 of value.
Typical sign
Long vanilla options are often negative Theta.
Common mistake
Assuming time decay is linear from now to expiry.
What you will learn and prerequisites
- Explain Theta in plain language.
- Reproduce the unit conversion.
- Compare a local estimate with model revaluation.
Read first: Read the notation
Theta measures the local effect of the clock with other inputs fixed. Remaining maturity T decreases as calendar time passes, so the two derivatives have opposite signs. This course divides the annual derivative by 365 for a calendar-day display.
A daily Theta is not a guaranteed overnight loss: spot and IV may also move.
Read the derivative as a sentence
Put Theta is −S e^(−qT)phi(d1)sigma/(2sqrt(T))−qS e^(−qT)N(−d1)+rK e^(−rT)N(−d2). Long options often have negative Theta, but not universally; some deep-in-the-money European put states can be positive.
Platforms can use different day counts or finite revaluation conventions. Compare definitions before comparing values.
Use it within its scope
Use the same elapsed time in approximation and repricing. A request for thirty days on a seven-day option must not charge thirty days of Theta while the price is silently floored at expiry. Reject or explicitly limit the shared applied horizon. The daily derivative changes as the clock runs.
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 Theta is -18.800737423395. The displayed convention gives -0.051508869653 option-value change per calendar day. Isolate one calendar day passing while holding every other input fixed.
Starting option value is 2.977574155222. The local estimate after the shock is 2.926065285569; a full-model recomputation gives 2.925659116658. Approximation minus recomputation is +0.000406168911. 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
Theta across controlled states
Vertical: option-value change per calendar day
Horizontal: Calendar days remaining
1. Fictional example
Read the graph as a data table
| Calendar days remaining | Fictional example |
|---|---|
| 1 | -0.265085 |
| 2 | -0.188616 |
| 3 | -0.154732 |
| 5 | -0.120741 |
| 7 | -0.102646 |
| 10 | -0.0865074 |
| 15 | -0.0713268 |
| 20 | -0.0622686 |
| 30 | -0.0515089 |
| 45 | -0.042702 |
| 60 | -0.0374355 |
| 90 | -0.0311599 |
A correct number can be misread
The stated unit is option-value 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
Try the question before revealing its answer. These are reading checks, not trading certification or a site completion record.
What is the displayed Theta in this example?
Reveal the worked answer
-0.051508869653 option-value change per calendar day.
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 Theta 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
- Keep all scenario changes zero except one day passing.
- Compare Theta/day with full repricing.
- Try seven days remaining, then exactly seven days passing; distinguish the boundary result from a small local estimate.
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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