# Strike / delta relationship for FX options

I am tryinto find out how to go from delta to strike. If wee 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 $delta=0.488$ 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

• I am not certain that your approach is correct (but I think it is) but I am certain that the time to maturity is not totally correct. Bloomberg calendar takes holidays etc in account so it's not as simple as you did it with: 1/12. One of your links is my own post, and the time is discussed at comments. Have a look – Sanjay May 22 '17 at 19:47
• @Sanjay - BBG only takes holiday calendars into account for determining the expiration date for a given tenor using the foreign ccy calendar, domestic calendar, and the US calendar (Where T+2 in the US has a slightly different rule from T+2 in other calendars! Oh joy!). Given the actual expiration date though, the T is determined by the number of calendar days to expiration / 365 in BBG. In your own system where you have holiday calendars, I would do everything in business days/252 or 260 and convert vols appropriately so that theta and vega will make more sense - especially after weekends. – FinanceGuyThatCantCode May 23 '17 at 15:16

## 2 Answers

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.

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.

• I hate how this is the case. – will May 23 '17 at 22:51