I think your answer is actually misleading. Using standard Black Scholes (Garman Kohlhagen) will not give you the values for either premium or delta, using the inputs and trade parameters you provided.
Since FX is all OTC,yzou can never be sure about the details unless you ask your broker. Generally though, in order to maintain liquidity there are a number of standardized conventions. Unfortunately, these conventions vary between currency pairs. Usually, vanilla EURUSD options are not premium adjusted. If you have access to Bloomberg, you can check on OVDV - 92)Settings->Conventions. Wystup and Reiswich, 2009 also have an overview of the most commonly used conventions.
See the following screenshot from Bloomberg's OVML, where the 3rd screenshot has 25D as an input and solves for strike (the exact decimals are shown in white - when you hover over the value in the OVML screen).
The values are quick to replicate:
- For CCY1CCY 2, you have Notional in CCY1 (20MM) and Premium in CCY2 (USD)
- You have a deferred (forward) premium, therefore I use two dates (see here for an explanation)
- all other inputs are given
- The model is just standard Garman Kohlhagen
- all inputs are provided in the question
In Julia, this looks as follows:
#load packages
using Distributions, Dates, DataFrames, PrettyTables
#define helper functions
ppf(x) = quantile(Normal(0.0, 1.0),x)
N(x) = cdf(Normal(0,1),x)
#define GK
function GK(F,K, days_to_expiry, days_to_delivery ,ccy1, ccy2,σ)
d1 = ( log(F/K) + 0.5σ^2days_to_expiry/365 ) / (σsqrt(days_to_expiry/365))
d2 = d1 - σsqrt(days_to_expiry/365)
c = exp(-ccy2days_to_delivery/365)(FN(d1) - KN(d2))
δ_spot = exp(-ccy1days_to_expiry/365) N(d1)
δ_fwd = N(d1)
return c, δ_fwd, δ_spot
end
The inputs are all given, but the days to expiry and delivery are computed. I allow for hours to expiry but that is irrelevant here (the computed price is below the quoted, and delta would increase with increasing time).
# inputs
s = 1.0615
pts = 60.1
fwd_scale = 10^4
f = s + pts / fwd_scale
println("Forward = $f")
k = 1.101
σ = 0.089
ccy1 = 0.0255008 #0.0255 # EUR
ccy2 = 0.0478 # USD
price_dt = Date(2023,3,16)
premium_dt = Date(2023,6,20)
expiry_dt = Date(2023,6,16)
delivery_dt = Date(2023,6,20)
hours = 0 #0.7115 allows to get more accurate pricing but more hours to expiry would be needed (increases delta)
days_to_expiry = (expiry_dt - price_dt).value + hours/24
days_to_delivery = (delivery_dt - premium_dt).value + hours/24
r1_cont = log(1+ccy1days_to_expiry/360)/(days_to_expiry/365)
r2_cont = log(1+ccy2days_to_expiry/360)/(days_to_expiry/365)
I am omitting PrettyTables formatting. Essentially, I compute strike for 25D according to Wystup and Reiswich, 2009 (omitting the call/put flag φ because we only care about calls here):
$$ K = fe^{-N^{-1}(e^{rf\tau} * \delta_{s})*\sigma* \sqrt{t} + \frac{1}{2}*\sigma^{2}*\tau }$$
or for forward delta:
$$ K = fe^{-N^{-1}(\delta_{f})*\sigma* \sqrt{t} + \frac{1}{2}*\sigma^{2}*\tau }$$
δ = 0.25
# compute strike from delta
k_25D = f*exp((1/2)*σ^2*days_to_expiry/365 - ppf(δ*exp(r1_cont *days_to_expiry/365))*σ*sqrt(days_to_expiry/365))
# get option value for computed strike and quoted strike
opt = [GK(f, strike, days_to_expiry, days_to_delivery, r1_cont, r2_cont, σ) for strike in (k, k_25D)]
# get spot premium
premium_dt_spot = Date(2023,3,20)
days_to_delivery_spot = (delivery_dt - premium_dt_spot).value + hours/24
opt2 = [GK(f, strike, days_to_expiry, days_to_delivery_spot, r1_cont, r2_cont, σ) for strike in (k, k_25D)];
Notional = 20_000_000
df = DataFrame("Strike" => [k, k_25D],
"Fwd Premium USD" => [opt[1][1]*Notional, opt[2][1]*Notional ],
"Spot Premium USD" => [opt2[1][1]*Notional ,opt2[2][1]*Notional ],
"Fwd Delta" => [opt[1][2]*100 , opt[2][2]*100 ], "Spot Delta" => [opt[1][3]*100, opt[2][3]*100] )
The output matches Bloomberg exactly:
Using the computed spot delta of 25.0124 returns the strike (1.101) of your broker:
DataFrame("Delta" => [δ, opt[1][3]], "Strike Solved" => [k_25D, f*exp((1/2)*σ^2*days_to_expiry/365 - ppf(opt[1][3]*exp(r1_cont *days_to_expiry/365))*σ*sqrt(days_to_expiry/365))])
Personally, I suspect that there are two things happening here:
- Using Sticky Delta is quite common in FX
- You just paid a slight "mark-up" to the fair price, given the inputs (for instance, rounded to nearest 100)
What is Sticky Delta?
- Sticky Strike is really just Black Scholes Delta computed with Finite Difference.
- Sticky Delta refers to adjusting IV when you bump up and down (because IV is stuck to delta /moneyness).
Therefore, there are 3 possible outcomes relative to Black Scholes Delta:
- a downward sloping IV -> Sticky Delta will be lower
- a flat IV -> Sticky Delta will be identical
- an upward sloping IV -> Sticky Delta will be higher
Now, Bloomberg conveniently displays Sticky Delta as well. As you can see, sticky delta does in fact show 25D.
P.S.
If it were premium included, you would not be able to use a closed form solution and would need to solve for strike numerically - for example, using Brent's root-finding method as suggested by Wystup and Reiswich.
