Vanna–Volga pricing

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 $\backslash mathcal\{V\}$, the Vanna

and the Volga.

The Vanna is the sensitivity of the Vega with respect to a change in the spot FX rate:

$\backslash textrm\{Vanna\}\; =\; \backslash frac\{\backslash partial$

\mathcal{V}}{\partial S}.

Similarly, the Volga is the sensitivity

of the Vega with respect to a change of the implied volatility

$\backslash sigma$:

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

If we consider a smile volatility term structure $\backslash sigma(K)$ with ATM strike $K\_0$, ATM volatility $\backslash sigma\_0$, 25-Delta call/put volatilities $\backslash sigma(K\_\{c/p\})$, and where $K\_\{c/p\}$ are the 25-Delta

call/put strikes (obtained by solving the equations $\backslash Delta\_\{$

call}(K_c,\sigma(K_c))=1/4 and $\backslash Delta\_\{$

put}(K_p,\sigma(K_p))=-1/4 where $\backslash Delta\_\{call/put\}(K,\backslash sigma)$ 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:

$$

\begin{align}

\textrm{ATM}(K_0) &= \frac12 \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) &= \frac12 \left(\textrm{Call}(K_c,\sigma(K_c)) + \textrm{Put}(K_p,\sigma(K_p))\right)- \textrm{ATM}(K_0)

\end{align}

with $\backslash textrm\{Call\}(K,\backslash sigma)$ 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 $X^\{VV\}$ of an exotic instrument $X$ is

given by

$$

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 $X^\{\; BS\}$ denotes the Black–Scholes price of the

exotic and the Greeks are calculated with ATM volatility and

$$

\begin{align}

RR_{cost} &=\left[ \textrm{Call}(K_c,\sigma(K_c))-\textrm{Put}(K_p,\sigma(K_p)) \right] - \left[ \textrm{Call}(K_c,\sigma_0)-\textrm{Put}(K_p,\sigma_0) \right]

\\

BF_{cost} &= \frac12 \left[

\textrm{Call}(K_c,\sigma(K_c))+\textrm{Put}(K_p,\sigma(K_p)) \right] - \frac12 \left[ \textrm{Call}(K_c,\sigma_0)+\textrm{Put}(K_p,\sigma_0) \right]

\end{align}

These quantities represent a smile cost, namely the

difference between the price computed with/without including the

smile effect.

The rationale behind the above formulation of the Vanna-Volga price is that one can extract

the smile cost of an exotic option by measuring the

smile cost of a portfolio designed to hedge its Vanna and

Volga risks. The reason why one chooses the strategies BF and RR

to do this is because they are liquid FX instruments and they

carry mainly Volga, and respectively Vanna risks. The weighting

factors $w\_\{RR\}$ and $w\_\{BF\}$ represent

respectively the amount of RR needed to replicate the option's

Vanna, and the amount of BF needed to replicate the option's

Volga. The above approach ignores the small (but non-zero)

fraction of Volga carried by the RR and the small fraction of

Vanna carried by the BF. It further neglects the cost of hedging

the Vega risk. This has led to a more general formulation of the

Vanna-Volga method in which one considers that within the Black–Scholes

assumptions the exotic option's Vega, Vanna and Volga can be

replicated by the weighted sum of three instruments:

$$

X_i = w_{ ATM}\, { ATM_i} + w_{ RR}\, {RR_i} +

w_{BF}\, {BF_i} \, \, \, \, \, i\text{=vega, vanna, volga}

where the weightings are obtained by solving the system:

$$

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

with

$$

\mathbb{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}

,

$\backslash vec\{x\}=\; \backslash begin\{pmatrix\}\; X\_\{vega\}\; \backslash \backslash$

X_{vanna} \\ X_{volga}

\end{pmatrix}

Given this replication, the Vanna–Volga method adjusts the BS

price of an exotic option by the smile cost of the above

weighted sum (note that the ATM smile cost is zero by

construction):

$$

\begin{align} X^{\rm VV} &= X^{BS} + w_{RR} ({RR}^{mkt}-{RR}^{BS}) +

w_{BF} ({BF}^{mkt}-{BF}^{BS}) \\

&= X ^{BS} + \vec{x}^T(\mathbb{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} = (\mathbb{A}^T)^{-1}\vec{I}

The quantities $\backslash Omega\_i$ can be interpreted as the

market prices attached to a unit amount of Vega, Vanna and Volga,

respectively. The resulting correction, however, typically turns

out to be too large. Market practitioners thus modify

$X^\{VV\}$ to

$$

\begin{align}

X^{\rm VV} &= X ^{BS} + p_{vanna} X_{vanna}

\Omega_{vanna} + p_{volga} X_{volga} \Omega_{volga}

\end{align}

The Vega contribution turns out to be

several orders of magnitude smaller than the Vanna and Volga terms

in 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 option

is gradually moved toward the spot level $S\_0$, the BSTV price of a

knock-out option must be a monotonically decreasing function, converging

to zero exactly at $B=S\_0$. Since the Vanna-Volga method is a

simple rule-of-thumb and not a rigorous model, there is no

guarantee that this will be a priori the case. The attenuation factors are of a different from for the Vanna or the Volga

of 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 the

attenuation factors take the form:

$$

\begin{align}

p_{\rm vanna} &= a \, \gamma \\ p_{\rm volga} &= b + c

\gamma

\end{align}

where $\backslash gamma\backslash in[0,1]$ 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 $\backslash gamma$ 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.

The survival probability $p\_\{surv\}\backslash in[0,1]$ refers to the

probability that the spot does not touch one or more barrier

levels $\backslash \{B\_i\backslash \}$. For example, for a single barrier option we have

$$

p_{surv} = \mathbb{E}[ 1_{S_t

where $\backslash mathrm\{NT\}(B)$ is the value of a no-touch option and $\backslash mathrm\{DF\}(t\_\{\backslash textrm\{tod\}\},t\_\{\backslash textrm\{mat\}\})$ the discount factor between today and maturity. Similarly, for options with two barriers

the survival probability is given through the undiscounted value

of a double-no-touch option.

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

$u(S\_t,t)$ then $u(S\_t,t)=$min$\backslash \{\backslash phi,T\backslash \}$ where

$\backslash phi=\backslash textrm\{inf\}\backslash \{\backslash ell\backslash in\; [0,T)\backslash \}$ such that $S\_\{t+\backslash ell\}\; >\; H$ or

$S\_t$ the spot of today.

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

$$

\frac{\partial

u(S,t) }{\partial t} + \frac12\sigma^2 S^2 \frac{\partial^2 u(S,t) }{\partial

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

This equation is solved backwards

in time starting from the terminal condition $u(S,T)\; =\; T$ where $T$ is the time to maturity and

boundary conditions $u(L,t\text{'})=u(H,t\text{'})=t\text{'}$. In case of a single

barrier option we use the same PDE with either $H\backslash gg\; S\_0$ or $L\backslash ll$

S_0. The parameter $\backslash mu$ represents the risk-neutral drift of the underlying stochastic process.

• {{cite arXiv |eprint=0904.1074 |author1=Frédéric Bossens |author2= Grégory Rayée |author3= Nikos S. Skantzos |author4=Griselda Deelstra |title=Vanna-Volga methods applied to FX derivatives : from theory to market practice |class=q-fin.PR |year=2009 }}
• {{Cite magazine

| last1 = Castagna | first1 = Antonio

| last2 = Mercurio| first2 = Fabio

| title = Vanna-Volga methods applied to FX derivatives : from theory to market practice

| journal = Risk Magazine

| date = 1 March 2007

| url = http://www.risk.net/risk-magazine/technical-paper/1506580/the-vanna-volga-method-implied-volatilities

}} [http://www.javaquant.net/papers/castagna_2007_vanna-volga.pdf PDF].

• {{Cite journal |last1=Shkolnikov |first1=Yuriy |title=Generalized Vanna-Volga Method and its Applications |journal=SSRN Electronic Journal |year=2009 |doi=10.2139/ssrn.1186383 |ssrn=1186383}}

• {{Citation

| last = Wystup

| first = Uwe

| year = 2006

| title = FX Options and Structured Products

| publisher = Wiley

}}

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