Converting FX Option Delta Quotes into Strike Prices
Summary
The document explains why converting an FX option delta into a strike requires the market’s quoting conventions and the correct definition of at-the-money. In the example, the pair uses unadjusted deltas, and the at-the-money strike is delta-neutral: the call and put have equal absolute deltas. This yields a strike above the forward, calculated from the forward, at-the-money volatility, and time to expiry. Using the stated dates to measure maturity in years produces a strike close to the quoted Bloomberg value.
The initial calculation misses because it treats the displayed volatility as if the strike were determined by a generic delta equation, without first accounting for the FX at-the-money convention. The answer also cautions that conventions, including premium adjustment, vary by currency pair. The example’s formula therefore should not be applied universally; traders need to establish the relevant pair’s delta and at-the-money conventions before solving for strike.
Key ideas
- FX option delta conventions differ across currency pairs and may include premium adjustment.
- For the stated unadjusted-delta convention, at-the-money means equal absolute call and put deltas.
- The delta-neutral at-the-money strike is the forward multiplied by an adjustment based on volatility and time to expiry.
- Maturity should be calculated in years using the dates relevant to the option quote.
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Full text
# Strike / delta relationship for FX options
# Strike / delta relationship for FX options
I am trying to find out how to go from delta to strike. If we look at the Bloomberg I am looking at 1M ATM volatility. I have included the Bloomberg data as a picture where we have following information: $f=0.9475$, $r=0.00274-0.02924$, $\sigma =13.32/100$, $T=1/12$, $t=0$.
The strike for $delta=0.4988$ appears at the picture as well, and I try to recreate it. I use definition for the Black Sholes delta and I have this problem now:
$d(k)=\frac{1}{\sigma \sqrt{T-0}}*\text{Log}\left[\frac{f}{k}\right]*\left(r+\frac{\sigma ^2}{2}\right)*(T-t)$, $r_f=0.00274$
$ SOLVE[e^{r_f*(T-t)}*N\left(d_1(k)\right)=0.4988]$ , $ k = 0.950417$
According to bloomberg this correct answer is $k=0.9483$. What went wrong? Personally I believe it's dates and time parameters that's not correct. According to Bjork: Arbitrage Theory in Continuous Time the time parameters need to measured in years. And in general. Is my approach correct?
I have found out about this method by looking at topics that are discussed here: Calculate strike from Black Scholes delta
Black model: Delta - strike relationship regardless of expiry?
## Answer by Gordon (score 7, accepted)
https://quant.stackexchange.com/a/34356
In FX world, the ATM strike is the delta-neutral strike, that is, the absolute delta values of a call and the corresponding put are the same. Moreover, the delta can be premium adjusted or not depending on the particular currency pair. See the linked paper as mentioned by @AntoineConze.
For AUD/USD, the delta is not premium adjusted, and then the delta-neutral ATM strike is determined by the equation \begin{align*} \Phi(d_1) = \Phi(-d_1), \end{align*} that is, \begin{align*} K = Fe^{\frac{1}{2}\sigma_{ATM}^2 T}, \end{align*} where $F$ is the forward, $\sigma_{ATM}$ is the ATM volatility, and $T$ is the maturity. Based on the information you provided, \begin{align*} T&=\frac{\mbox{10-Jul-13} - \mbox{7-Jun-13}}{365} =0.090411,\\ K &= 0.9475\times e^{0.5 \times 0.1332^2 \times 0.090411} = 0.94826. \end{align*} See also Page 51 of the book Foreign Exchange Option Pricing by Iain J. Clark.
## Answer by Antoine Conze (score 2)
https://quant.stackexchange.com/a/34350
There are specific quotation conventions for specifying ATM and deltas for FX options quotes (unadjusted deltas, premium adjusted deltas, etc.) and converting deltas to strikes. These conventions vary across currency pairs.
See this paper https://ideas.repec.org/p/zbw/cpqfwp/20.html for details.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.