Vanna-Volga approximation is the quick-n-dirty way traders use to tweak B-S price so that smile/skew effects are taken into account. It is used to price exotic products. It consists of 3 components: X_BS, RR (risk reversal) correction and BF (butterfly) correction. Incidentally, unlike in equity market where the smile is simply quoted from the market, the FX market convention is much more convoluted, with only ATM, RR and strangle volatilities directly quotable (note also the RR and strangle and BF volatilites are not really volatility in the strict sense; they are merely traded entities).
- X_BS is the B-S exotic price
- RR stands for risk-reversal. An RR strategy consists of long OTM call and short OTM put.
- BF stands for butterfly. A BF strategy consists of long strangle and short straddle.
Intuitively, RR and BF have opposite parity about the strike (RR increases monotonically, BF increases either way). Hence they together can produce corrections that fit market prices. In fact, RR strategy has large (small) vanna (volga) exposure, while BF strategy has large (small) volga (vanna) exposure (this can be seen by considering the B-S Vega expression, Vega ~ S phi(d_1)). The Vanna-Volga method ignores the cost of hedging Vega.
*Volga = (d/d sigma) Vega, Vanna = (d/dS) Vega
Reference:
Bossens 2010 - Vanna-Volga method
Reiswich 2010 - Constructing a (quadratic) smile using the market quoted volatilities
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