compute time from FX forward, how use DEPO rates?

Question

assume I have following delta-term vol data from broker:

Spot 3.4550
                 O/N      1WK      2WK      3WK      1M       6WK      2M
Volatility       7.544    7.7      7.731    7.911    8.025    8.18     8.4
Forward Points   0.0004   0.0021   0.0045   0.0063   0.0079   0.0106   0.0164
EUR Depo Rate    0.405    1.205    1.145    1.128    1.1      1.11     1.13
PLN Depo Rate    4.216    5.028    4.586    4.187    3.558    3.58     3.626
Butterfly        0.157    0.19     0.229    0.268    0.34     0.368    0.44
RiskReversal     0.35     0.45     0.567    0.683    0.9      0.983    1.2

now I want to retrieve strikes from delta-term vol surface. there are few convensions of quoting delta and ATM condition. apart from this, lets assume that my appropriate formula for ATM strike is

$K=fe^{-\frac12\sigma^2\tau}$

because I need to specify time $\tau$ to put into my other calculations and I don't want to count days and wonder what basis etc, I would like to retrieve first $\tau$ from relationship above. it is then given by:

$\tau=-ln({\frac{K}f})\frac2{\sigma^2}$

but still I need rates to compute other strikes. So what should they be:

we know that

$f=S+Forward Points$

and for instance

$f_{ON}=3.4550+ 0.0004=3.4554$

$f_{1M}=3.4550+ 0.0079=3.4629$

and ofcourse

$f=Se^{(r_d-r_f)\tau}$

now the question is how to use DEPO rates in $f(S,r_d,r_f,\tau)$ formula.

so what $r_d$, $r_f$ should be put into $f=Se^{(r_d-r_f)\tau}$ for 1M based on table presented? is this ready tu put into it so for $f_{1M}$ I have $r_d\tau=0.03558\tau$ or do I have to annualize it first or get equal rate in continuous compounding? please be exact and write down exact formula for say $1M$ $r_D$

I have tried just this rates ($r_d\tau=0.03558\tau$) and also rates continuously compounded ($r_{d,cont}=ln(1+r_d)$) but the results for strikes for delta different from ATM are still not exact nevertheless if I use spot, forward, spot p.a. or forward p.a. convention. I use QuantLib.

below are my results as ticks difference between quoted strike and retrieved from delta-term. I am not satisfied with such accuracy, the reason for this discrepancy is not correct use of rates and time convensions.

here is result for EUR/USD, slightly better but still small error

Answer 1

Very old but seems to miss an answer.

If $$K=fe^{-\frac12\sigma^2\tau}$$ is your appropriate formula, you use premium adjusted delta, and get the ATM delta neutral strike. Makes sense for EURPLN which should be Delta premium included by convention. In this case, you have a bigger issue than forward prices. There is no closed form solution and you need a root solver. Yes, there is the closed form formula you gave, but that is NOT from delta; and you wrote you want to retrieve strikes from delta.

FX volatility smile construction from Uwe Wystup and Dimitri Reiswich explains that.

Generally, I would recommend using quoted forwards. Usually the option pricing model needs to be internally consistent (such that it does not matter whether you price with Garman Kohlhagen or Black) and therefore it is common to imply the least liquid interest rate (not the fx forward).

Your main problem will be that one usually lacks consistent data. Do you get spot, ATM DNS, RR and BF as well as depo rates used for all tenors from the same broker? Unlikely. In any case, the daycount and conventions of the rates used must be provided by the source. Ideally, you would also need to adjust for the cross currency basis. Aferwards its easy relatively easy to convert to the continuous analog needed for this exercise.

Side remark, are your BF quoted as smile BF, or quoted as market BF (market strangle, also called broker fly). Also, EURUSD is as much as I know Delta Premium excluded in VOL quotes (and the paper above also seems to confirm that, especially around the time the question was asked). Hence, your above formula will not work and you should use: $$K=fe^{\frac12\sigma^2\tau}$$