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Pricing and Hedging FX Cross-Currency Options

Article Quant Q&A · Author: tradinggy
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Summary

The document considers how to price and hedge an option on EUR/CAD when EUR/USD and USD/CAD markets are available. It constructs the cross rate by multiplying the two quoted rates, with the quote convention needing careful attention. It then treats that cross rate as the option’s underlying variable.

For pricing, the answer projects the spot cross rate to expiry using the relevant currencies’ interest rates, then applies a Black–Scholes-style call formula using the forward rate. The response provides no worked numerical example and does not explain how to hedge the cross option in practice, despite that being part of the question. It also omits key implementation details such as quote conventions, volatility inputs and their correlation, settlement conventions, and the consistency of discounting and carry assumptions; these would need to be specified for a complete valuation.

Key ideas

  • A cross-currency spot rate can be derived from two FX rates through their shared currency.
  • The quote convention determines whether the component rates should be multiplied or divided.
  • The answer proposes using the cross rate as the underlying and a forward rate for option pricing.
  • A practical hedge and the volatility and correlation inputs needed for valuation are not developed.

Tags

ForexOptionsDerivatives pricing
Full text
# How is FX cross rates options are priced?


# 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)

## Answer by Kiann (score 1)

https://quant.stackexchange.com/a/41982

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.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.