Valuation models

Touches, dual currency deposits, range accruals and FX Asians

One-touch, no-touch and double no-touch options in closed form with a survival-weighted Vanna-Volga smile adjustment and the touch probability; dual currency deposits as a deposit less the sold option, with the fair enhanced rate; range accruals as a strip of daily digitals priced on the smile; and average rate (Asian) options by Levy moment matching, with delta, gamma, vega and PV01.

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Methodology

Conventions

  • One-touch pays at expiry if the spot touches the barrier; one-touch plus no-touch is the discount factor.
  • Vanna-Volga: the smile cost of the vega, vanna and volga hedges, weighted by the survival probability.
  • Dual currency deposit: a deposit plus a sold option; repaid in the other currency at the strike.
  • Range accrual: one digital pair per daily fixing. Asian: Levy moment matching.

References

  • Wystup, FX Options and Structured Products
  • Castagna and Mercurio, The Vanna-Volga Method (2007)
  • Levy, Pricing European Average Rate Currency Options (1992)

Formulas

No-touch
NT = D_q(T) [Φ(·) − (H/S)^{2μ/σ²} Φ(·)]
Vanna-Volga
V = V_{BS} + p_{surv} Σ wᵢ (Cᵢ^{mkt} − Cᵢ^{BS})
Dual currency deposit
V = A D(T)(1 + r τ) − A (1 + r τ) C(K) / K
Range accrual
V = N c D(T) (1/n) Σ P(L ≤ S_{tᵢ} ≤ U)