Valuation models

Quanto and compo options

A quanto option pays a foreign asset's payoff converted at a fixed exchange rate, valued by Black-Scholes with the quanto drift adjustment (minus correlation times asset vol times FX vol); a compo option is struck in the payment currency, valued on the asset in that currency with the combined volatility. Value, forward, correlation sensitivity, asset and FX vega.

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Methodology

Conventions

  • Quanto: pays the foreign asset's payoff converted at a fixed rate; the drift falls by ρ σ_S σ_X.
  • Compo: the strike is in the payment currency, so the option is on S X with the combined vol.
  • ρ is between the asset and the FX rate quoted as payment currency per unit of the asset's currency.

References

  • Reiner, Quanto Mechanics (1992)
  • Hull, Options, Futures and Other Derivatives, ch. 30

Formulas

Quanto forward
F = S e^{(r_f − q − ρ σ_S σ_X) T}
Quanto price
V = X̄ D_d(T) [F Φ(d₁) − K Φ(d₂)]
Compo vol
σ = √(σ_S² + σ_X² + 2ρ σ_S σ_X)