How Black Delta Inputs Encode Option Expiry
Summary
The question asks why a QuantLib Black delta calculator appears to convert between delta and strike without using time to expiry. The accepted explanation is that expiry is already reflected in the calculator’s inputs: discount factors for the relevant rates and standard deviation, which incorporates volatility over the option’s life. The displayed formulas for the normal variates therefore do not need a separate expiry argument.
This clarifies that the apparent omission is an input-design issue rather than a general expiry-independent relationship between delta and strike. The note applies to Black pricing inputs, including FX options, but it does not spell out the calculator’s conventions or give a worked numerical example. Users must ensure that the supplied discounts and standard deviation correspond to the option’s expiry and pricing setup.
Key ideas
- Expiry enters through the discount factors and the volatility standard deviation supplied to the calculator.
- The calculator’s formulas need not include a separate time-to-expiry variable when those inputs already embed it.
- Delta-to-strike conversion is therefore not generally independent of expiry.
- The explanation addresses Black model inputs but provides no worked example or discussion of conventions.
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Full text
# Black model: Delta - strike relationship regardless of expiry?
# Black model: Delta - strike relationship regardless of expiry?
While wandering through some QuantLib experimental classes for FX trading, I've found this Black Delta Calculator.
By reading its `.cpp`, it seems that no use of options time to expiry is made at all.
Usual Black, Scholes & Merton $d_{1}$ argument of cumulative Normal distribution has time to expiry as argument: on the contrary, Black Delta Calculator makes a weird use of $d_{1}$ and $d_{2}$, expressing them like at line 222 or 250:
```
d1_ = std::log(forward_/strike)/stdDev_ + 0.5*stdDev_; // .cpp line # 222
d2_ = std::log(forward_/strike)/stdDev_ - 0.5*stdDev_; // .cpp line # 250
```
This is quite different than what usual Itö correction produces over classic GBM process (squared variance and of course time adjustment for annual basis).
By reading that code it seems that going from Delta to strike and back can be made regardless of expiry date, while common sense says that there are a lot of options whose Delta can match any strike if you can search all over implied volatility surface without an expiry boundary.
Questions
- It must be I am missing some important relations which allow for such a simplification: could you show me which one?
- Is this possible relation viable just for FX options or can it be extended to any use of Black model (e.g. interest rates)?
## Answer by Luigi Ballabio (score 6, accepted)
https://quant.stackexchange.com/a/17172
The time to expiry is required, but it's included in the inputs: the two discounts $e^{-rT}$ and $e^{-qT}$ and the standard deviation $\sigma\sqrt{T}$. You might argue it could be documented more clearly, and I might agree with you.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.