Valuation models

KIKO, window barrier and digital FX options

Knock-in knock-out (KIKO) options, window barriers and cash or asset digitals, by Garman-Kohlhagen, Vanna-Volga from ATM, risk reversal and butterfly quotes, Dupire local vol or Heston stochastic local vol Monte Carlo with a Brownian bridge.

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Methodology

Conventions

  • The smile comes from ATM, 25 delta risk reversal and butterfly quotes, by Vanna-Volga.
  • Vanna-Volga adds the cost of hedging vega, vanna and volga with three pillar options, weighted by survival.
  • Window barriers and smile Monte Carlo use daily steps with a Brownian bridge for continuous monitoring.

References

  • Wystup, FX Options and Structured Products
  • Castagna and Mercurio (2007), The vanna-volga method
  • Dupire (1994), Pricing with a smile

Formulas

KIKO
KIKO = KO(B_out) − DKO(B_in, B_out)
Vanna-Volga
X = X_GK + p Σ xᵢ [C(Kᵢ, σᵢ) − C(Kᵢ, σ_ATM)]
Digital
D = −∂C(K, σ(K)) / ∂K