The vanna-volga method for implied volatilities

σ is the constant BS implied volatility. 4 In real financial markets,
however, volatility is stochastic and traders hedge the associated
risk by constructing portfolios that are vega-neutral in a BS (flatsmile)
world.
Maintaining the assumption of flat but stochastic implied volatilities,
the presence of three basic options in the market makes it
possible to build a portfolio that zeros out partial derivatives up
to the second order. In fact, denoting respectively by Δ t
and x i
the
units of the underlying asset and options with strikes K i
held at
time t and setting C i
BS
(t) = C BS (t; K i
), under diffusion dynamics
for both S t
and σ = σ t
, we have by Itô’s lemma:
dC BS
( t; K ) − ∆ t dS t − x i dC BS i t
( )
⎡
= ∂C BS 3
t; K ∂C BS
− x i t ⎤
⎢ ∑ i ⎥ dt
⎣ ∂t i=1 ∂t ⎦
⎡
+ ∂C BS 3
( t; K )
∂C BS
− ∆ t − x i ( t ) ⎤
⎢
∑ i ⎥ dS t
⎣ ∂S
i=1 ∂S ⎦
⎡
+ ∂C BS 3
( t; K ) ∂C BS
− x i ( t ) ⎤
⎢ ∑ i ⎥ dσ t
(2)
⎣ ∂σ i=1 ∂σ ⎦
+ 1 ⎡ ∂ 2 C BS
3
( t; K ) ∂ 2 C BS
− x i ( t ) ⎤
⎢ ∑
2 ∂S 2 i
⎣
∂S 2 ⎥( dS t ) 2
i=1
⎦
+ 1 ⎡ ∂ 2 C BS
3
( t; K ) ∂ 2 C BS
− x i ( t ) ⎤
⎢ ∑
2 ∂σ 2 i
⎣
∂σ 2 ⎥( dσ t ) 2
i=1
⎦
⎡
+ ∂2 C BS 3
( t; K ) ∂ 2 C BS
− x i ( t ) ⎤
⎢ ∑ i ⎥ dS t dσ t
⎣ ∂S∂σ i=1 ∂S∂σ ⎦
Choosing Δ t
and x i
so as to zero out the coefficients of dS t
, dσ t
,
(dσ t
) 2 and dS t
dσ t
, 5 the portfolio comprises a long position in
the call with strike K, and short positions in x i
calls with strike
K i
and short the amount Δ t
of the underlying, and is locally
risk-free at time t, in that no stochastic terms are involved in
its differential 6 :
dC BS
= r d ⎡
⎢C BS t; K
⎣
3
∑
i=1
( )
( )
( t; K ) − ∆ t dS t − x i dC BS i t
3
∑
i=1
3
( )
( ) − ∆ t S t − ∑ x i C BS i ( t )
i=1
(3)
⎤
⎥ dt ⎦
Therefore, when volatility is stochastic and options are valued
with the BS formula, we can still have a (locally) perfect hedge,
provided that we hold suitable amounts of three more options to
rule out the model risk. (The hedging strategy is irrespective of
the true asset and volatility dynamics, under the assumption of
no jumps.)
■ Remark 1. The validity of the previous replication argument
may be questioned because no stochastic-volatility model can
produce implied volatilities that are flat and stochastic at the same
time. The simultaneous presence of these features, though inconsistent
from a theoretical point of view, can however be justified
on empirical grounds. In fact, the practical advantages of the BS
paradigm are so clear that many forex option traders run their
books by revaluing and hedging according to a BS flat-smile
model, with the ATM volatility being continuously updated to
the actual market level. 7
The first step in the VV procedure is the construction of the
above hedging portfolio, whose weights x i
are explicitly calculated
in the following section.
Calculating the VV weights
We assume hereafter that the constant BS volatility is the ATM
one, thus setting σ = σ 2
(= σ ATM
). We also assume that t = 0, dropping
accordingly the argument t in the call prices. From equation
(2), we see that the weights x 1
= x 1
(K), x 2
= x 2
(K) and x 3
= x 3
(K),
for which the resulting portfolio of European-style calls with
maturity T and strikes K 1
, K 2
and K 3
has the same vega, ∂Vega/
∂Vol and ∂Vega/∂Spot as the call with strike K, 8 can be found by
solving the following system:
∂C BS
∂σ
3
( K ) = x i K
∑
i=1
3
i=1
3
( )
∂C BS
∂σ
∂ 2 C BS
∂
( K ) = ∑ x
∂σ 2
i ( K )
2 C BS
∂σ 2
∂ 2 C BS
∂
( K ) = ∑ x i ( K )
2 C BS
∂σ∂S 0 i=1 ∂σ∂S 0
( K i )
( K i )
( K i )
Denoting by V(K) the vega of a European-style option with
maturity T and strike K:
BS
∂C
V ( K ) = ( ∂σ
K ) = S 0 e− r f T
d 2
( )
T ϕ d 1 ( K )
(5)
d 1 ( K ) = ln S 0
+ r d − r f + 1 σ 2
K ( 2 )T
σ T
where ϕ(x) = Φʹ(x) is the normal density function, and calculating
the second-order derivatives:
∂ 2 C BS
( K ) = V ( K )
∂σ 2 σ d ( 1 K )d ( 2 K )
∂ 2 C BS
( K ) = − V ( K )
∂σ∂S 0 S 0 σ T d ( 2 K )
( K ) = d 1 ( K ) − σ T
we can prove the following.
■ Proposition 1. The system (4) admits a unique solution, which
is given by:
x 1
x 2
x 3
( )
( )
( K ) = V K
V K 1
( )
( )
( K ) = V K
V K 2
( )
( )
( K ) = V K
V K 3
ln K 2
ln K 3
K K
ln K 2
K 1
ln K 3
K 1
ln K K 1
ln K 3
K
ln K 2
ln K 3
K 1 K 2
ln K ln K K 1 K 2
ln K 3
K 1
ln K 3
K 2
In particular, if K = K j
then x i
(K) = 1 for i = j and zero otherwise.
The VV option price
We can now proceed to the definition of an option price that is
consistent with the market prices of the basic options.
5
The coefficient of (dS t
) 2 will be zeroed accordingly, due to the relation linking an option’s gamma and
vega in the BS world
6
We also use the BS partial differential equation
7
‘Continuously’ typically means a daily or slightly more frequent update
8
This explains the name assigned to the smile-construction procedure, given the meaning of the terms
vanna and volga
(4)
(6)
risk.net 107