# 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

• 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

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.