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Consistent modeling and e�cient pricing of
VIX options
A. Barletta
Department of Economics and Business Economics
Aarhus University
London, 14 March 2016
The research leading to these results has received funding from the European Union SeventhFramework Programme (FP7/2007-2013) under grant agreement n◦289032.
.
2 of 16 • Introduction 〉
Introduction
The Volatility Index (VIX) is a measure of market
volatility solely based on OTM option prices on the S&P
500.
It is determined as the square root of the value of a
portfolio that replicates the "fair" price of a 30-days (=τ)
variance swap.
The continuous time version of the VIX is the square root
of the log-price conditional annualized quadratic variation
VIX2t =
1
τEt
[[X ]t+τt
]:=
1
τlim
N→+∞Et
[N∑n=0
(log(St+n τ
N/St+(n−1) τ
N))2]
.
3 of 16 • Introduction 〉 Modeling VIX under SV
Modeling VIX under stochastic volatility
In our setting today's price of a VIX call option with strike K
and maturity T is always
C (K ,T ) = e−rTE[(VIXT −K )+
],
therefore we model the risk-neutral dynamics of the VIX.
The VIX has no explicit dependence on the stock price,
but they can be linked through the instantaneous variance
v .
A starting point is the Heston model
d log(St ) =(r −
vt
2
)dt +
√vtdZt
dvt = κ(θ− vt )dt + ε√vtdWt ,
d 〈Z ,W 〉t = ρdt .
4 of 16 • Introduction 〉 Modeling VIX under SV
Advantages of the Heston model
We can express VIX2t as an a�ne function of vt
VIXt = 100 ·√avt + b.
The density of vt is available in closed-form.
A pitfall of the Heston model
It is known that the Heston model fails in matching the upward
sloping skew of VIX implied volatilities.
5 of 16 • Introduction 〉 Modeling VIX under SV
Possible generalizations
Adding jumps in stock and volatility (AJD framework).
> The a�ne relation between VIX2 and v is preserved.
> The Laplace transform of v is known in semi-closed form.
> The moments of v can be determined very e�ciently by
the matrix-based technique of Cuchiero et al. (2010).
Introducing additional volatility factors (double Heston,double mean reverting, etc.).
> The a�ne relation between VIX2 and v is preserved as
long as all the volatility factors have linear drift.
> Remaining in pure-di�usion allows for passing through
PDEs.
6 of 16 • SP1 - Affine jump-diffusion 〉
Subproject 1 - Pricing VIX under AJD
The AJD framework
Under the a�ne jump-di�usion (AJD) framework we have
d log(St ) =
(r − λµ−
1
2vt
)dt +
√vtdZt + dJXt ,
dvt = κ(θ− vt )dt + ε√vtdWt + dJ vt ,
d 〈Z ,W 〉t = ρdt ,
where J = (JX ,J v ) is a pure jump process taking values in
R× R+ and µ is the compensator of JX .
7 of 16 • SP1 - Affine jump-diffusion 〉 Orthogonal polynomial expansions
Orthogonal polynomial expansions
Denoting by f the risk-neutral density related to v we approxi-
mate f as follows
f (x ) ≈ φ(x )
(1+
n∑k=0
ckhk (x )
)
where φ (kernel) represents a "tractable initial guess" for f and
hk are polynomials that only depend on φ,
ck are corrective factors embedding all the information on
f .
8 of 16 • SP1 - Affine jump-diffusion 〉 Orthogonal polynomial expansions
The polynomials (hk )n∈N are chosen to be orthogonal with respect to
φ, i.e.
(i) µk :=∫+∞−∞ x kφ(x )dx < +∞, ∀k ∈ N
(ii)∫+∞−∞ hk (x )h`(x )φ(x )dx = 0, ∀k 6= `
Why using orthogonal polynomials?
If (hk )k∈N are orthogonal to φ then ck are linear combinations of mo-
ments of f
ck =1
Ck
k∑i=0
wki
∫+∞−∞ x i f (x )dx .
with hk (x ) =∑
k
i=0wkix
i and Ck =√∫+∞
−∞ h2
k(x )φ(x )dx
9 of 16 • SP1 - Affine jump-diffusion 〉 Orthogonal polynomial expansions
The eGIG kernel
We consider an enriched generalized inverse Gaussian (eGIG) kernel
φ(x ) = xα−1e−(βx p+γx−1)1[0,+∞)(x ), α, β, γ > 0, p ∈ (1
2, 1].
We focus on three subcases:
the gamma (p = 1, γ = 0) kernel
the GIG (p = 1) kernel
the gW (γ = 0) kernel
0.87 1 1.31 1.53 1.75 1.97 2.190.3
0.4
0.5
0.6
0.7
0.8
0.9T = 2 months
0.87 1 1.31 1.53 1.75 1.96 2.18
0.3
0.4
0.5
0.6
T = 4 months
0.87 1 1.31 1.53 1.74 1.96 2.18
0.25
0.3
0.35
0.4
0.45
0.5
T = 6 months
Speci�cation 1: Normal Poisson jumps in log(S) and exponential Poisson jumps in v .
0.87 1 1.31 1.53 1.75 1.96 2.180.3
0.4
0.5
0.6
0.7
T = 2 months
0.87 1 1.31 1.52 1.74 1.96 2.180.25
0.3
0.35
0.4
0.45
0.5
0.55
T = 4 months
0.87 1 1.3 1.52 1.74 1.95 2.17
0.25
0.3
0.35
0.4
0.45
T = 6 months
Speci�cation 2: Normal Poisson jumps in log(S) and IG Poisson jumps in v .
Figure: VIX implied volatilities/moneyness under AJD with Poisson jumps.
Efficiency
Under Speci�cation 2 we compare
Our technique (orders ranging from 10 to 20).
Laplace integration optimized formula of Lian and Zhu (2013).
Monte Carlo with Euler scheme on a time grid of 103 points/month.
10 11 12 13 14 15 16 17 18 19 20
Expansion order
10-1
100
101
102
103
Com
puta
tion
time
(sec
onds
)
Gamma kernel GW kernel GIG kernel Lian-Zhu MC (105 paths) MC (106 paths)
12 of 16 • SP2 - Multi-factor models 〉
Subproject 2 - Pricing VIX under
multi-factor pure diffusion
We consider the following multi-factor SV model
d log(St ) =
(r −
1
2
q∑i=1
v it
)dt +
q∑i=1
√v it dZ
it ,
dvt = (αvt + β) dt + η (t , vt ) dWt , vt ∈ Rm , m ≥ q ,
where α ∈ Rq×q , β ∈ Rq+ , and η = diag(η1, . . . , ηq) ∈ Rq×q is a
diagonal matrix of functions.
If Z have no correlation we have
VIXt = 100 ·√〈a , vt 〉+ b
for some a ∈ Rm , b ∈ R.
13 of 16 • SP2 - Multi-factor models 〉
Some history:
Hagan and Woodward (1999) found asymptotics for the
implied volatilities under a CEV model based on a
PDE-technique.
Lorig et al. (2015) have recently extended this technique
to a broader pure-di�usion setting.
Based on this, Pagliarani and Pascucci (2016) have
extended these results to include the implied volatility
sensitivities.
We extend these results to a context where the underlying
(the VIX futures) dynamics are not explicit.
All these results apply to short-time and near ATM
options.
14 of 16 • Examples 〉
Examples
Introducing an elasticity coefficient in the
vol-of-vol
d log(St ) = −1
2vtdt +
√vtdW
Xt
dvt = k(vt − θ
h)dt + εhv δt dW
Yt
We have
limK→VIX0T→0
∂
∂Kσimp(T ,K ) =
εvδ− 1
20
(2δVIX2
0 − 2av0)
4VIX20
√v0
.
The slope is controlled by
2δVIX20 − 2av0 = 2(δ− 1)av0 + 2δb
15 of 16 • Examples 〉
Randomizing the vol-of-vol size (SVV model)
d log(St ) = −1
2v1t dt +
√v1t dZt
dv1t = κ1(v1t − θ1
)dt + v2t
√v1t dW
1t
dv2t = κ2(v2t − θ2
)dt + η
(v2t)dW 2
t
d⟨W 1,W 2
⟩t= ρdt
We have
limK→VIX0T→0
∂
∂Kσimp(T ,K ) =
v20(VIX2
0 − 2a1v10
)4√v10VIX
20
+ ρη(v20 )
2v20.
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