FX Option strikes from ATM, RR, BF quotes

Question

I am trying to replicate the results in Consistent Pricing of FX Options, A. Castagna and F. Mercurio. However, when I calculate the strike prices for 25-delta put and call and ATM I cannot get the same result as in the article.

The parameters given in the article (p.5):

• T = 94/365
• S = 1.205
• s(ATM) = 0.0905
• s(RR) = -0.0050
• s(BF) = 0.0013

These result in s(25dPut) = 0.0943 and s(25dCall) = 0.0893 (equations 4 and 5 on pages 2 and 3).

• K(25dPut) = 1.1733
• K(25dCall) = 1.2487

The values I get (equations 6 and 7 on p. 3) are:

• K(25dPut) = 1.16688287...
• K(25dCall) = 1.2421907...

Here is my Python code:

S       = 1.205
tau     = 94.0 / 365.0
iv_v    = 0.0905
rr_v    = -0.005
bf_v    = 0.0013
for_df  = 0.9902752
dom_df  = 0.9945049

vol_call = iv_v + bf_v + 0.5 * rr_v
vol_put = iv_v + bf_v - 0.5 * rr_v

alpha = - scipy.stats.norm.ppf( 0.25 * np.exp( (for_df**(-1) - 1) * tau) )
k1 = S * np.exp( - alpha * vol_put * np.sqrt(tau) + ((dom_df**(-1) - 1) - (for_df**(-1) - 1) + 0.5 * vol_put**(2) ) * tau )
k2 = S * np.exp( alpha * vol_call * np.sqrt(tau) + ((dom_df**(-1) - 1) - (for_df**(-1) - 1) + 0.5 * vol_call**(2) ) * tau )

This code gives wrong results, but I cannot figure out where the error is.

Answer 1

The interest rates (calculated from discount factors) you are using are wrong. Correct formulas: $r_{dom}$ = $-\frac{log(dom_{df})}{T}$

$r_{for}$ = $-\frac{log(for_{df})}{T}$

alpha = - scipy.stats.norm.ppf( 0.25 * np.exp( for_df**(-1) )

k1 = S * np.exp( - alpha * vol_put * np.sqrt(tau) + ($r_{dom}$ - $r_{for}$ + 0.5 * vol_put**(2) ) * tau )

k2 = S * np.exp( alpha * vol_call * np.sqrt(tau) + ($r_{dom}$ - $r_{for}$ + 0.5 * vol_call**(2) ) * tau )