Vanna-Volga Pricing and Hedging of FX Barrier Options

Scenario

In foreign exchange (FX) options markets, exotic barrier contracts trade with substantial smile risk. The Vanna-Volga pricing methodology hedges an exotic option's Vega, Vanna, and Volga using three liquid vanilla instruments: the ATM Straddle, the 25-Delta Risk Reversal (RR), and the 25-Delta Vega-Neutral Butterfly (BF).

Your Task

1. Formulate the replicating portfolio $\Pi = V_{BS} + w_1 C_{ATM} + w_2 C_{RR} + w_3 C_{BF}$ matching Vega ($\frac{\partial V}{\partial \sigma}$), Vanna ($\frac{\partial^2 V}{\partial S \partial \sigma}$), and Volga ($\frac{\partial^2 V}{\partial \sigma^2}$).
2. Derive the Vanna-Volga pricing adjustment formula: $V_{\text{VV}} = V_{\text{BS}} + \frac{\text{Vanna}(V)}{\text{Vanna}(RR)} (C_{RR}^{\text{mkt}} - C_{RR}^{\text{BS}}) + \frac{\text{Volga}(V)}{\text{Volga}(BF)} (C_{BF}^{\text{mkt}} - C_{BF}^{\text{BS}})$
3. Compute the Vanna-Volga price of an FX touch option when $V_{\text{BS}} = 4.20$, target option $\text{Vanna} = 12.0, \text{Volga} = 35.0$, with benchmark instruments: $\text{Vanna}(RR) = 24.0$, $\text{Volga}(BF) = 70.0$, and market smile premiums $(C_{RR}^{\text{mkt}} - C_{RR}^{\text{BS}}) = 0.30$, $(C_{BF}^{\text{mkt}} - C_{BF}^{\text{BS}}) = 0.40$.

Constraints

• Linear sensitivity matching across Volga and Vanna.

Examples If the target option has zero Vanna and Volga, the smile adjustment is zero and the price equals Black-Scholes.