The Vanna–Volga method is a mathematical tool used in finance. It is a technique for pricing first-generation exotic options in foreign exchange market (FX) derivatives.

Description

It consists of adjusting the Black–Scholes theoretical value (BSTV) by the cost of a portfolio which hedges three main risks associated to the volatility of the option: the Vega ${\displaystyle {\mathcal {V}}}$, the Vanna and the Volga. The Vanna is the sensitivity of the Vega with respect to a change in the spot FX rate:

${\displaystyle {\textrm {Vanna}}={\frac {\partial {\mathcal {V}}}{\partial S}}}$.

Similarly, the Volga is the sensitivity of the Vega with respect to a change of the implied volatility ${\displaystyle \sigma }$:

${\displaystyle {\textrm {Volga}}={\frac {\partial {\mathcal {V}}}{\partial \sigma }}}$.

If we consider a smile volatility term structure ${\displaystyle \sigma (K)}$ with ATM strike ${\displaystyle K_{0}}$, ATM volatility ${\displaystyle \sigma _{0}}$, 25-Delta call/put volatilities ${\displaystyle \sigma (K_{c/p})}$, and where ${\displaystyle K_{c/p}}$ are the 25-Delta call/put strikes (obtained by solving the equations ${\displaystyle \Delta _{call}(K_{c},\sigma (K_{c}))=1/4}$ and ${\displaystyle \Delta _{put}(K_{p},\sigma (K_{p}))=-1/4}$ where ${\displaystyle \Delta _{call/put}(K,\sigma )}$ denotes the Black–Scholes Delta sensitivity) then the hedging portfolio will be composed of the at-the-money (ATM), risk-reversal (RR) and butterfly (BF) strategies:

${\displaystyle {\begin{aligned}{\textrm {ATM}}(K_{0})&={\frac {1}{2}}\left({\textrm {Call}}(K_{0},\sigma _{0})+{\textrm {Put}}(K_{0},\sigma _{0})\right)\\{\textrm {RR}}(K_{c},K_{p})&={\textrm {Call}}(K_{c},\sigma (K_{c}))-{\textrm {Put}}(K_{p},\sigma (K_{p}))\\{\textrm {BF}}(K_{c},K_{p})&={\frac {1}{2}}\left({\textrm {Call}}(K_{c},\sigma (K_{c}))+{\textrm {Put}}(K_{p},\sigma (K_{p}))\right)-{\textrm {ATM}}(K_{0})\end{aligned}}}$

with ${\displaystyle {\textrm {Call}}(K,\sigma )}$ the Black–Scholes price of a call option (similarly for the put).

The simplest formulation of the Vanna–Volga method suggests that the Vanna–Volga price ${\displaystyle X^{VV}}$ of an exotic instrument ${\displaystyle X}$ is given by

${\displaystyle X^{\rm {VV}}=X^{BS}+\underbrace {\frac {{\textrm {X}}_{vanna}}{{\textrm {RR}}_{vanna}}} _{w_{RR}}{RR}_{cost}+\underbrace {\frac {{\textrm {X}}_{volga}}{{\textrm {BF}}_{volga}}} _{w_{BF}}{BF}_{cost}}$

where by ${\displaystyle X^{BS}}$ denotes the Black–Scholes price of the exotic and the Greeks are calculated with ATM volatility and