Vanna–Volga pricing explained

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

l{V} , the Vannaand the Volga. The Vanna is the sensitivity of the Vega with respect to a change in the spot FX rate:rm{Vanna} = (\partial l{V)/() }.

Similarly, the Volga is the sensitivityof the Vega with respect to a change of the implied volatility

\sigma rm{Volga}= (\partial l{V)/() }.\sigma(K) with ATM strike K _{0} , ATM volatility \sigma _{0} , 25-Delta call/put volatilities \sigma(K _{c/p} ) , and where K _{c/p} are the 25-Deltacall/put strikes (obtained by solving the equations \Delta _{call} (K _{c,\sigma(K} _{c))=1/4} and \Delta _{put} (K _{p,\sigma(K} _{p))=-1/4} where \Delta _{call/put} (K,\sigma) denotes theBlack–Scholes Delta sensitivity) then the hedging portfoliowill be composed of the at-the-money (ATM), risk-reversal (RR) and butterfly (BF)strategies:\begin{align} rm{ATM}(K _{0)} & = (12)/(\left( rm{Call}(K) _{0,\sigma} _{0)} + rm{Put}(K _{0,\sigma} _{0)\right)} \\ rm{RR}(K _{c,K} _{p)} & = rm{Call}(K _{c,\sigma(K} _{c))-rm{Put}(K} _{p,\sigma(K} _{p))} \\ rm{BF}(K _{c,K} _{p)} & = (12)/(\left( rm{Call}(K) _{c,\sigma(K} _{c))} + rm{Put}(K _{p,\sigma(K} _{p))\right)-} rm{ATM}(K _{0) \end{align}}

with

rm{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 theVanna–Volga price

X ^{VV} of an exotic instrument X isgiven byX ^{\rm} = X ^{BS} + \underbrace{ (rm{X)/(_{vanna}) }}_ _ +\underbrace_ _

where by

X denotes the Black–Scholes price of theexotic and the Greeks are calculated with ATM volatility and\begin{align} RR _{cost} & = \left[ rm{Call}(K _{c,\sigma(K} _{c))-rm{Put}(K} _{p,\sigma(K} _{p))} \right] - \left[ rm{Call}(K _{c,\sigma} _{0)-rm{Put}(K} _{p,\sigma} _{0)} \right] \\ BF _{cost} & = (12)/(\left[) rm{Call}(K _{c,\sigma(K} _{c))+rm{Put}(K} _{p,\sigma(K} _{p))} \right] - (12)/(\left[) rm{Call}(K _{c,\sigma} _{0)+rm{Put}(K} _{p,\sigma} _{0)} \right] \end{align}

These quantities represent a smile cost, namely thedifference between the price computed with/without including thesmile effect.

The rationale behind the above formulation of the Vanna-Volga price is that one can extractthe smile cost of an exotic option by measuring thesmile cost of a portfolio designed to hedge its Vanna andVolga risks. The reason why one chooses the strategies BF and RRto do this is because they are liquid FX instruments and theycarry mainly Volga, and respectively Vanna risks. The weightingfactors

w _{RR} and w _{BF} representrespectively the amount of RR needed to replicate the option'sVanna, and the amount of BF needed to replicate the option'sVolga. The above approach ignores the small (but non-zero)fraction of Volga carried by the RR and the small fraction ofVanna carried by the BF. It further neglects the cost of hedgingthe Vega risk. This has led to a more general formulation of theVanna-Volga method in which one considers that within the Black–Scholesassumptions the exotic option's Vega, Vanna and Volga can bereplicated by the weighted sum of three instruments:X _{i} = w { ATM _{i}} + w {RR _{i}} + w _{BF} {BF _{i}} i =vega,vanna,volga

where the weightings are obtained by solving the system:

\vec{x} = A \vec{w}

with

A = \begin{pmatrix} ATM _{vega} & RR _{vega} & BF _{vega} \\ ATM _{vanna} & RR _{vanna} & BF _{vanna} \\ ATM _{volga} & RR _{volga} & BF _{volga} \end{pmatrix} ,\vec{w} = \begin{pmatrix} w _{ATM} \\ w _{RR} \ w _{BF} \end{pmatrix} ,\vec{x} = \begin{pmatrix} X _{vega} \\ X _{vanna} \ X _{volga} \end{pmatrix}

Given this replication, the Vanna–Volga method adjusts the BSprice of an exotic option by the smile cost of the aboveweighted sum (note that the ATM smile cost is zero byconstruction):

\begin{align} X ^{\rm} & = X ^{BS} + w _{RR} ({RR} ^{mkt} - {RR} ^{BS} ) + w _{BF} ({BF} ^{mkt} - {BF} ^{BS} ) \\ & = X ^{BS} + \vec{x} ^{T(A} ^{T)} ^{-1} \vec{I} \\ & = X ^{BS} + X _{vega} \Omega _{vega} + X _{vanna} \Omega _{vanna} + X _{volga} \Omega _{volga} \\ \end{align}

where

\vec{I} = \begin{pmatrix} 0 \\ {RR} ^{mkt} - {RR} ^{BS} \\ {BF} ^{mkt} - {BF} ^{BS} \end{pmatrix}

and

\begin{pmatrix} \Omega _{vega} \\ \Omega _{vanna} \\ \Omega _{volga} \end{pmatrix} = ( A ^{T)} ^{-1} \vec{I}

The quantities

\Omega _{i} can be interpreted as themarket prices attached to a unit amount of Vega, Vanna and Volga,respectively. The resulting correction, however, typically turnsout to be too large. Market practitioners thus modifyX ^{VV} to\begin{align} X ^{\rm} & = X ^{BS} + p _{vanna} X _{vanna} \Omega _{vanna} + p _{volga} X _{volga} \Omega _{volga} \end{align}

The Vega contribution turns out to beseveral orders of magnitude smaller than the Vanna and Volga termsin all practical situations, hence one neglects it.

The terms

p _{vanna} and p _{volga} are put in by-hand and represent factors that ensure the correct behaviour of the price of an exotic option near a barrier:as the knock-out barrier level B of an optionis gradually moved toward the spot level S _{0} , the BSTV price of aknock-out option must be a monotonically decreasing function, convergingto zero exactly at B = S _{0} . Since the Vanna-Volga method is asimple rule-of-thumb and not a rigorous model, there is noguarantee that this will be a priori the case. The attenuation factors are of a different from for the Vanna or the Volgaof an instrument. This is because for barrier values close to the spot they behave differently: the Vanna becomes large while,on the contrary, the Volga becomes small. Hence theattenuation factors take the form:\begin{align} p _{\rm} & = a \gamma \ p _{\rm} & = b + c \gamma \end{align}

where

\gamma\in[ 0 , 1 ] represents some measure of the barrier(s)vicinity to the spot with the features\begin{align} \gamma =0 &{for} S _{0\to} B \\ \gamma =1 &{for} |S _{0-B|\gg} 0 \end{align}

The coefficients

a,b,c are found through calibration of the model to ensure that it reproduces the vanilla smile. Good candidates for \gamma that ensure the appropriate behaviour close to the barriers are the survival probability and the expected first exit time. Both of these quantities offer the desirable property that they vanish close to a barrier.

Survival probability

The survival probability

p _{surv} \in[ 0 , 1 ] refers to theprobability that the spot does not touch one or more barrierlevels \{B _{i\}} . For example, for a single barrier option we havep _{surv} = E [ 1 _{S _{t<B,} t _{rm{tod}} <t<t _{rm{mat} }}] = \mathrm(B) / \mathrm(t_,t_)

where

NT (B) is the value of a no-touch option and DF (t _{rm{tod} },t_) the discount factor between today and maturity. Similarly, for options with two barriersthe survival probability is given through the undiscounted valueof a double-no-touch option.

First-exit time

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

u(S _{t,t)} then u(S _{t,t)=} min\{\phi,T\} where\phi = rm{inf}\{\ell\in [ 0 ,T)\} such that S _{t+\ell} > H orS _{t+\ell} <L where L,H are the 'low' vs 'high' barrier levels andS _{t} the spot of today.

The first-exit time is the solution of the following PDE

(\partial u(S,t))/(\partial t) + (12 \sigma)/(^{2}) S ^{2} (\partial ^{2} u(S,t))/(\partial S ^{2}) + \mu S (\partial u(S,t))/(\partial S) =0

This equation is solved backwardsin time starting from the terminal condition

u(S,T) = T where T is the time to maturity andboundary conditions u(L,t') = u(H,t') = t' . In case of a singlebarrier option we use the same PDE with either H\gg S _{0} or L\ll S _{0} . The parameter \mu represents the risk-neutral drift of the underlying stochastic process.

References

• 0904.1074 . Frédéric Bossens . Grégory Rayée . Nikos S. Skantzos . Griselda Deelstra . Vanna-Volga methods applied to FX derivatives : from theory to market practice . q-fin.PR . 2009 .
• Castagna . Antonio . Mercurio. Fabio . Vanna-Volga methods applied to FX derivatives : from theory to market practice . Risk Magazine . 1 March 2007 . PDF.
• Shkolnikov . Yuriy . Generalized Vanna-Volga Method and its Applications . SSRN Electronic Journal . 2009 . 10.2139/ssrn.1186383 . 1186383.