# How is FX cross rates options are priced?

Say I have market for EUR/USD and also USD/CAD, how would EUR/CAD would be priced and hedged in practice? What are good papers/book chapters to read on that? (Assuming basic knowledge already on option pricing/hedging)

## 1 Answer

FX spot (and by extension FX forward rates) rates can be calculated as ratios of each other.

EUR/CAD (setting notion of number of EUR per CAD) = (EUR/USD) * (USD/CAD) = (EUR * USD) / (USD * CAD). This is the direct calculation of the EUR/CAD FX spot rate. EUR/CAD would then be the underlying variable for the option pricing (usually Black-scholes).

Specifically, let us set FX(t=0) as the FX rate EUR/CAD as seen at time now. The FX forward as projected at time-to-expiry of the option = FX(t=0) * [(1+rf)/(1+rd)]^T of the approriate discount-rates and T = time-to-expiry.

Use FX forward in the Black-Scholes equation : Call = FX(T) * N(d1) - K*e^rt * N(d2) of the usual black-scholes formula.

• but as I understand this has nothing to do with how implied vol of EUR/CAD is determined, and how hedge with EUR/USD and USD/CAD options could be performed – Gleb Yarnykh Sep 30 '18 at 11:07
• you might be talking about the tri- angulation of the ATM vols from cross ccy pairs. The ITM/OTM skew can then be calculated via (usually) bivariate copulas. But these are essentially the modelling of the implied vol surface which I thought was a little bit too detailed. – Kiann Sep 30 '18 at 11:18
• can you give book/paper reference on bivariate copulas with regard to this task? I'm capable of reading those. Thanks!! – Gleb Yarnykh Sep 30 '18 at 11:22