FX Option Pricing Under Basis Adjustment

Question

Given money market rates such as USD LIBOR and EURIBOR and in the context of FX options valuation, I have been reading about the importance to include a so called basis adjustment to one of the respective mm rates.

1. Since interest rate parity seems to break down, what is the economical foundation of such an adjustment?

2. With reference to my previous question, if I use the Black model (in a GBM world) to value, say a one month call option on the EURUSD spot (priced on the forward), is it true that the basis adjustment is already included in the one month forward? So I can simply use the (unadjusted) one month EURIBOR for the domestic interest rate r (the discount factor)?

Answer 1

If the forward fx price is available, it already includes any basis adjustment and you may use the Black model (in a GBM world) to value the option. In that case the only usage of the interest rate input is to discount the payoff.

The reason that the fx forward may not be consistent with the two risk-free interest rates (i.e. appearing to violate interest rate parity) is that in practice, you cannot actually borrow and lend at those two interest rates. That's certainly true for individuals, but it's also true for banks. A bank can't easily borrow dollars at Fed Funds (unsecured) and invest Euros at Eonia separately. What it can do is simultaneously borrow dollars and invest in Euros, in the so called currency basis swap market.

Answer 2

Old question but generally, for indirect quotes like EURUSD, the foreign currency is the base currency (EUR), and the domestic currency is the quote currency (USD).

Standard GK as shown here can simply be written in terms of the forward (applying covered interest rate parity) which yields:

$$exp^{-r_dt}[FN(d_1) - KN(d_2)]$$

where $r_d$ is domestic interest rate (in case of EURUSD, the USD rate). I prefer to use ccy1 and ccy2 (CCY1CCY2) to avoid any potential confusion.

Hence, you would discount with USD not EUR. Generally, BS for FX as defined in GK (see wiki link above) is in terms of notional in CCY1 (EUR) and payoff in CCY2 (USD). All else must be adjusted accordingly.

Many pricers actually imply one rate to make the model internally consistent (no arb is default assumption). So you use covered interest rate parity, and use for example spot, fwd and domestic rate to imply the foreign rate (EUR here). Alternatively, you can also imply forward and so forth. Most tools allow for flexibility (what interest rate curves, what will be implied and so forth).