The approximation (15) begins with the quantity:
$$ \underbrace{c(\sigma^+_\Delta) - p(\sigma_\Delta^-)}_{\text{mkt risk reversal premium}} - \underbrace{ \left(c(\sigma_0) + p(\sigma_0)\right )}_{\text{baseline risk reversal premium atm vol}} $$
The baseline risk reveral premium is required because an X-delta risk reversal does not have zero premium even for a flat curve, and the reason for that is that ATM-delta strike and the ATM-forward strike are not the same. That difference grows with vol and with time to expiry.
I will put some numbers on this for demonstration. First I will use your EUR, USD interest rates to construct interest rate curves and add your FX forward price to configure everything in a consistent framework:
# PYTHON: rateslib >= 1.3.0
from rateslib import *
eur = Curve({dt(2024, 6, 20): 1.0, dt(2024, 9, 28): 1.0}, calendar="tgt")
usd = Curve({dt(2024, 6, 20): 1.0, dt(2024, 9, 28): 1.0}, calendar="nyc")
eurusd = Curve({dt(2024, 6, 20): 1.0, dt(2024, 9, 28): 1.0})
fxr = FXRates({"eurusd": 1.0727}, settlement=dt(2024, 6, 24))
fxf = FXForwards(
fx_rates=fxr,
fx_curves={"eureur": eur, "eurusd": eurusd, "usdusd": usd}
)
pre_solver = Solver(
curves=[eur, usd, eurusd],
instruments=[
IRS(dt(2024, 6, 24), "3m", spec="eur_irs", curves=eur),
IRS(dt(2024, 6, 24), "3m", spec="usd_irs", curves=usd),
FXExchange(pair="eurusd", settlement=dt(2024, 9, 24), curves=[None, eurusd, None, usd])
],
s=[3.77, 5.51, 1.0775], # <- YOUR RATES
fx=fxf,
)
SUCCESS: `func_tol` reached after 8 iters, `f_val`: 3.92e-13, `time`: 0.0102s
Now we can get some option values. These are the FX points premium values (in USD) for each option in the 25D Risk Reversal. Notice that they do not net to zero, even though the RR price in vol points is zero (since both are priced with 6.6% vol):
fx_option_args = dict(
pair="eurusd",
expiry=dt(2024, 9, 20),
curves=[None, eurusd, None, usd],
delivery_lag=2,
payment_lag=dt(2024, 6, 20),
calendar="tgt|nyc",
)
FXCall(strike="25d", **fx_option_args).rate(vol=6.6, fx=fxf)
# <Dual: 52.273662, .., ..>
FXPut(strike="-25d", **fx_option_args).rate(vol=6.6, fx=fxf)
# <Dual: 54.042099, ..,..>
Next the assumption is to approximate with first order the value of the calls and puts relative to their equivalent prices.
$$ c(\sigma_\Delta^+) \approx c(\sigma_0) + c_\sigma(\sigma_0) (\sigma_\Delta^+ - \sigma_0) $$
$$ p(\sigma_\Delta^-) \approx p(\sigma_0) + p_\sigma(\sigma_0) (\sigma_\Delta^- - \sigma_0) $$
Plugging this in gets you the second line of (15). Since it is first order its accuracy deteriates as the market vol price diverge further from the ATM vol.
Is this a reasonable approximation in your case? Lets explore the Put which has the greatest discrepancy. The -25D Put with vol at 7.621% has the quantities:
FXPut(strike="-25d", **fx_option_args).rate(vol=7.621, fx=fxf)
# <Dual: 62.565252, .., ..>
FXPut(strike="-25d", **fx_option_args).analytic_greeks(vol=7.621, fx=fxf)
# strike: 1.0511201224444822
# vega_usd: <Dual: 1702.692382, .., ..>
The same option as priced with 6.6% vol gives the values:
FXPut(strike=1.0511201224444822, **fx_option_args).rate(vol=6.60, fx=fxf)
# <Dual: 45.732276, .., ..>
FXPut(strike=1.0511201224444822, **fx_option_args).analytic_greeks(vol=6.60, fx=fxf)
# vega_usd: <Dual: 1587.730467, .., ..>
# vomma_usd: <Dual: 134.5585, .., ..>
So now, to first order:
$$
45.732276 + 15.877304 * 1.021 = 61.943
$$
Whilst to second order, including vomma:
$$
45.732276 + 15.877304 * 1.021 + 1.345585 * 1.021^2 / 2 = 62.644
$$
So even this reasonably small deviation produces enough error to negate some processes. I suppose it depends what you are doing.
The additional assumption that is used in (15) to derive the final lines is that $c_\sigma(\sigma_0)$ and $p_\sigma(\sigma_0)$ are the same. It is not clear to me the author's intention with notion but, to me, to be consistent with the linear approximation in the first step, $c(.)$ and $p(.)$ must have strikes that are determined with the correct market volatilities. When the strikes are determined in this manner the vega quantities are symmetric, i.e. $c_\sigma(\sigma^+_\Delta) = p_\sigma(\sigma_\Delta^-)$ but it is not true that $c_\sigma(\sigma_0) = p_\sigma(\sigma_0)$
FXCall(strike=1.1003294194069992, **fx_option_args).analytic_greeks(vol=6.6, fx=fxf)
# vega_usd: <Dual: 1759.13, .., ..>
FXPut(strike=1.0511201224444822, **fx_option_args).analytic_greeks(vol=6.60, fx=fxf)
# vega_usd: <Dual: 1587.73, .., ..>
So I too struggle with the validity of the claim.
