Vanna–Volga pricing

Overview

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

V

{\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:

Vanna

=

∂

V

∂ S

{\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

σ

{\displaystyle \sigma }

:

Volga

=

∂

V

∂ σ

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

Survival probability

The survival probability

p

s u r v

∈ [ 0 , 1 ]

{\displaystyle p_{surv}\in [0,1]}

refers to the probability that the spot does not touch one or more barrier levels

{

B

i

}

{\displaystyle \{B_{i}\}}

. For example, for a single barrier option we have

p

s u r v

=

E

[

1

S

t

First-exit time

The first exit time (FET) is the minimum between: (i) the time in the future when the spot is expected to exit a barrier zone before maturity, and (ii) maturity, if the spot has not hit any of the barrier levels up to maturity. That is, if we denote the FET by

u (

S

t

, t )

{\displaystyle u(S_{t},t)}

then

u (

S

t

, t ) =

{\displaystyle u(S_{t},t)=}

min

{ ϕ , T }

{\displaystyle \{\phi ,T\}}

where

ϕ =

inf

{ ℓ ∈ [ 0 , T ) }

{\displaystyle \phi ={\textrm {inf}}\{\ell \in [0,T)\}}

References

Frédéric Bossens; Grégory Rayée; Nikos S. Skantzos; Griselda Deelstra (2009). "Vanna-Volga methods applied to FX derivatives : from theory to market practice". arXiv:0904.1074 [q-fin.PR]. Castagna, Antonio; Mercurio, Fabio (1 March 2007). "Vanna-Volga methods applied to FX derivatives : from theory to market practice". Risk Magazine. PDF. Shkolnikov, Yuriy (2009). "Generalized Vanna-Volga Method and its Applications". SSRN Electronic Journal. doi:10.2139/ssrn.1186383. SSRN 1186383. Wystup, Uwe (2006), FX Options and Structured Products, Wiley