Robust Calibration For SVI Model Arbitrage Free

Overview
Objectives
Stochastic Volatility Inspired SVI
Characterisation of Static Arbitrage
SVI Calibration
Numerical Applications
Non-parametric Interpolation
Implied Volatility Models
Tahar FERHATI
MSc Probability and Finance
Sorbonne Université, École polytechnique X
tahar.ferhati@gmail.com
September 27, 2019
Tahar FERHATI MSc Probability and Finance Sorbonne Université, École polytechnique X tahar.ferhati@gmail.comImplied Volatility Models

Overview
Objectives
SVI Calibration
Outline
Overview
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SVI Calibration

Overview
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SVI Calibration
Overview
Subject: study the interpolation and extrapolation methods for the
implied volatility slice and multi-slices under arbitrage free
conditions.
Many authors in the past such as Shimko (1993), Kahalé
(2004), Fengler (2009) Jäckel (2014) tried to model implied
volatility.
Two interpolation techniques: parametric interpolation models
such as: SABR, Heston, and non-parametric interpolation like
cubic spline, shape-preserving, natural smoothing splines...etc.

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SVI Calibration
Figure: Implied volatility models overview

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SVI Calibration
Overview
Kahalé (2004) interpolates the call price using piece-wise
convex polynomials, then he calculates the implied volatility
and he interpolates linearly the total implied variance.
Jim Gatheral (2004) presented for the first time the
Stochastic Volatility Inspired model (SVI) in Madrid.
Benko et al.(2007)applied non-parametric smoothing
methods to estimate the impliedvolatility (IV).
Fengler (2009) uses the natural smoothing splines under
suitable shape constraints.
Andreasen-Huge (2010) presented a method based on one
step implicit finite difference Euler scheme to local volatility.
Fingler-Hin (2013) uses semi-nonparametric estimator for
the entire call price surface basedon a tensor-product B-spline.

Overview
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SVI Calibration
Select Model
We choose Stochastic Volatility Inspired model (SVI) model
presented by Gatheral & Jacquier (2014), Why ?
It’s a parametric model that ﬁts very well the input data (total
implied variance) in the equity market.
Model with closed formula.
SVI is a good ﬁt for the slice.
Explicit formula to pass from implied volatility (slice) to
multi-slices (surface).

Overview
Objectives
SVI Calibration
Stochastic Volatility Inspired Model (SVI)
A parametric model with parameters set χR = {a, b, ρ, m, σ}, the
raw SVI parameterization of total implied variance is
w (k; χR) = σ2
imp (k; χR) T = a + b{ρ(k − m) + (k − m)2 + σ2}
Where a ∈ R, b ≥ 0, |ρ| < 1, m ∈ R, σ > 0 and the positivity
condition a + bσ 1 − ρ2 ≥ 0 that ensure w (k, χR) ≥ 0
With k is the log forward moneyness k := log K
FT

Overview
Objectives
SVI Calibration
Stochastic Volatility Inspired (SVI)
Figure: Stochastic Volatility Inspired Model (SVI) with
{a, b, ρ, m, σ} = (1, 0.3, −0.2, 0.001, 0.5)

Overview
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SVI Calibration
SVI Parameters Interpretation
a: determines the overall level of variance: an increasing a
increases the general level of variance, a vertical translation of
the smile.

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SVI Calibration
b: controls the angle between the left and right asymptotes:
Increasing b increases the slopes of both the put and call
wings, tightening the smile.

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SVI Calibration
ρ: determines the orientation of the smile: increasing ρ
decreases the slope of the left wing, and increases the slope of
the right wing a counter-clockwise rotation of the smile.

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SVI Calibration
m: translates the graph: increasing m translates the smile to
the right.

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SVI Calibration
σ: determines how smooth the vertex is: increasing σ reduces
the at-the-money (ATM) curvature of the smile.

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SVI Calibration
w (k; χN) for Large Strikes
The total implied variance w (k; χN) has the left and right
asymptotes that respect the assumption of linear wings:
w (k; χR) = a − b(1 − ρ)(k − m) k → −∞ (1)
w (k; χR) = a + b(1 + ρ)(k − m) k → ∞ (2)
SVI is consistent with the Roger Lee’s moment formula.

Overview
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SVI Calibration
Convergence From Heston Model to SVI
Gatheral and Jacquier (2010) show the large-time asymptotic
convergence of the Heston implied volatility to SVI.
We consider the Heston model where (St)t≥0 follow the process
dSt =
√
vtStdWt, S0 ∈ R∗
+
dvt = κ (θ − vt) dt + η
√
vtdZt, v0 ∈ R∗
+
d W , Z t = ˜ρdt
(3)
With ˜ρ ∈ [−1, 1], κ, θ, η and v0 are strictly positive and 2κθ ≥ η2
(the Feller condition).

Overview
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SVI Calibration
Convergence From Heston Model To SVI
Proposition: If κ − ˜ρη > 0 then σ2
SVI (x) = σ2
∞(x) for all x ∈ R
and (T −→ ∞)
The SVI parameters in function of the Heston parameters model
are:
ω1 :=
4κθ
η2 (1 − ˜ρ2)
(2κ − ˜ρη)2 + η2 (1 − ˜ρ2) − (2κ − ˜ρη) , and
ω2 := η
κθ
and we ﬁnd the equivalent SVI parameters (a, b, ρ, m, σ);
a =
ω1
2
1 − ρ2
, b =
ω1ω2
2T
, ρ = ˜ρ
m = −
ρT
ω2
, σ =
1 − ρ2T
ω2
(4)

Overview
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SVI Calibration
The SVI Jump-Wings (SVI-JW)
In order to be more intuitive for traders, SVI-JW with parameters
set χJ = {vt, ψt, pt, ct, vt} is deﬁned from the raw SVI parameters
by
vt =
a + b{−ρm +
√
m2 + σ2}
t
ψt =
1
√
wt
b
2
−
m
√
m2 + σ2
+ ρ
pt =
1
√
wt
b(1 − ρ)
ct =
1
√
wt
b(1 + ρ)
˜vt = a + bσ 1 − ρ2 /t

Overview
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SVI Calibration
Interpretation of SVI-JW parameters
The SVI-JW parameters have the following interpretations:
Vt gives the ATM variance;
ψt gives the ATM skew;
Pt gives the slope of the left (put) wing;
Ct gives the slope of the right (call) wing;
vt is the minimum implied variance.

Overview
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SVI Calibration
Definition:
A volatility surface is free of static arbitrage if and only if the
following conditions are satisfied:
1. It is free of calendar spread arbitrage;
2. Each time slice is free of butterfly arbitrage.

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SVI Calibration
Calendar Spread Arbitrage (Expiry T axis)
Deﬁnition: the volatility surface w is free of calendar spread
arbitrage if and only if
∂tw(k, t) ≥ 0, for all k ∈ R and t > 0
There is no calendar spread arbitrage if there are no crossed lines
on a total variance plot (ﬁg. right).

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SVI Calibration
Butterfly Arbitrage (strike k axis)
Butterfly arbitrage is related to the convexity of the (call/put)
price with respect to the strike k.
We can find an equivalent condition to the (call/put)
price convexity in terms of the total implied variance w(k).

Overview
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SVI Calibration
Deﬁnition: A slice is said to be free of butterﬂy arbitrage if the
corresponding density is non-negative.
Breeden and Litzenberger, derive an expression of the
discounted risk neutral density p(k) as function of
the second derivative of the call price C(K) with respect to
the strike K.
p(k) =
∂2C(k)
∂K2
K=Ft ek
=
∂2CBS(k, w(k))
∂K2
K=Ft ek
, k ∈ IR (5)
p(k) =
g(k)
2πw(k)
exp −
d−(k)2
2
(6)

Overview
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SVI Calibration
Butterfly Arbitrage (Strike k axis)
Where the function g : IR → IR
g(k) := 1 −
kw (k)
2w(k)
2
−
w (k)2
4
1
w(k)
+
1
4
+
w (k)
2
(7)
Lemma: A slice is free of butterfly arbitrage if and only if
g(k) ≥ 0 for all k ∈ R.
limK→+∞ CBS(T, k) = 0 ⇐⇒ limk→+∞ d+(k) = −∞
Problem: the nonlinear behavior of g(k) makes it difficult to find
general conditions on the parameters that would eliminate butterfly
arbitrage.

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SVI Calibration
SVI’s Parameters Boundaries
We determine Lower and Upper boundaries of each of the SVI
parameters (a, b, ρ, m, σ).
We have some restrictions on the parameters that follow from the
parameterization of the model such as:
b ≥ 0; | ρ |< 1; σ > 0.
Parameter a and SVI Minimum
wmin(k∗
) = a + b σ 1 − ρ2 > 0 ⇐⇒ (a > 0)
0 < a ≤ max(wmarket
)
Parameter b and left wing: for the right wing the slop is
b(ρ + 1) and it should not exceed to 2 which is consistent with
Roger Lee formula.

Overview
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SVI Calibration
lim
K→+∞
CBS(T, k) = 0 ⇐⇒ lim
k→+∞
d+(k) = −∞
Is satisﬁed for a function w(k) if
lim sup
k→∞
w(k)
2k
< 1.
Or we have,
lim sup
k→∞
w(k)
k
= b(ρ + 1)
Finally, we obtain
b(ρ + 1) < 2, for | ρ |< 1 (8)
The b boundaries are: 0 < b < 1

Overview
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SVI Calibration
Correlation Parameter ρ
−1 < ρ < 1
We can summarize the obvious boundaries for the SVI raw
parameters as following



0 < amin = 10−5
≤ a ≤ max(wmarket
)
0 < bmin = 0.001 < b < 1
−1 < ρ < 1
2 min
i
ki ≤ m ≤ 2 max
i
ki
0 < σmin = 0.01 ≤ σ ≤ σmax = 1
(9)

Overview
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SVI Calibration
Sequential Least-Squares Quadratic Programming (SLSQP)
To calibrate the SVI, we apply a Sequential Quadratic
Programming (SQP) algorithm which is a non-linearly
constrained gradient-based optimization.
SQP algorithm is proposed for the ﬁrst time by Wilson in his
PhD thesis (1963).
Deﬁnition: we consider xk
k∈N0
a sequence of iterates
converging to x∗, the sequence is said to convergence quadratically,
if there exist c > 0 and kmax ≥ 0 such that for all k ≥ kmax
xk+1
− x∗
≤ c xk
− x∗
2

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SVI Calibration
Some Classiﬁcation of QP’s

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SVI Calibration
The Objective Function
The least-Squares objective function to optimize is f (k; χR), where
χR = {a, b, ρ, m, σ} is the set of the parameters model, for an
expiry time ﬁx T.
f (k; χR) =
n
i=1
ωmodel
SVI(i) − ωmarket
Total(i)
2
f (k; χR) =
n
i=1
a + b ρ(k − m) + (k − m)2 + σ2 − ωmarket
Total(i)
2
Where ki := log Ki
FT

Overview
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SVI Calibration
The Non Linear Problem (NLP)
We optimize the objective function subject to constraints then the
problem is reduced to ﬁnd the optimal parameters
χR = (a∗, b∗, ρ∗, m∗, σ∗) s.t:



(NLP) : min
x∈R5
f (k; χR)
ad ≤ a ≤ au
bd ≤ b < bu
ρd < ρ < ρu
md ≤ m ≤ mu
σd ≤ σ ≤ σu
g (k; χR) > = constant
(10)

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SVI Calibration
Lagrangian of the NLP
The problem deﬁne in (10) is a Quadratic Problem with
inequality and bound constraints.
The Lagrange function of the (NLP) SVI optimization problem
is
L (k; χR) = f (k; χR) −
m
j=1
λj [gj (k; χR) − ] (11)
Where λj is the Lagrange multiplier.

Overview
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SVI Calibration
Sequential Quadratic Programming (SQP) in Brief
Given a constrained optimization problem
A rough scheme for the SQP algorithm:
Step 0: start from x0
Step k: xk+1 = xk + αkdk
where αk the step-length and dk is the search-direction.
To ﬁnd the search direction dk, we reformulate the original (NLP)
problem to a Quadratic Programming Sub-problem by
A quadratic approximation of the Lagrange function L (k; χR).
A linear approximation of the constraints function gj (k; χR).

Overview
Objectives
SVI Calibration
Sequential Quadratic Programming (SQP) in Brief
The search-direction dk: is computed using a Quadratic
Programming Sub-problem:
(QP) : min
d∈Rn
1
2
dT
Bk
d + f xk
d
Subject to
gj xk
d + gj xk
≥ , j = 1, . . . , m
Where B := 2
xx L(x, λ) is the Hessian of the Lagrangian.
The step-length αk: is determine by a 1D minimization of a
merit function M xk + αdk a function that guarantees a
suﬃcient decrease in the objective f (x) and satisfaction of
constraints along dk with an appropriate step length αk

Overview
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SVI Calibration
Some Arbitrage Examples
Example 01: Axel Vogt From Wilmott.com
Using Sequential Least-Squares Quadratic Programming
(SLSQP), we are able to tackle this problem and to calibrate
the SVI with arbitrage free.
We consider the following raw SVI parameters:
(a, b, m, ρ, σ) = (−0.0410, 0.1331, 0.3586, 0.3060, 0.4153)
With T = 1
Then the new SVI parameters arbitrage free are;
(a, b, m, ρ, σ) = (10−5
, 0.08414, 0.24957, 0.16962, 0.1321)

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SVI Calibration
Example 01 Axel Vogt
Figure: Plots of the total variance (left) and the function g(k) (right),
with and without arbitrage for g(k) > = 0.05

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SVI Calibration
Arbitrage Examples 02
Let’s consider the following example
(a, b, m, ρ, σ) = (0.001, 0.6, −0.5, 0.07, 0.1)
Some thing interesting in our example is that the SVI
parameters even when they respect the boundaries conditions
(14), we can have an arbitrage.
Then the new SVI parameters with arbitrage free after
calibration are
(a, b, m, ρ, σ) = (10−5
, 0.5691, −0.4345, 0.1318, 0.1428)

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SVI Calibration
Arbitrage Examples 02
Figure: Plots of the total variance (left) and the function g(k) (right),
with and without arbitrage

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SVI Calibration
Equity Indexes Calibration
After testing our algorithm in two arbitrage examples, we test
the performance of our algorithm.
The input data is the options prices (Call and Put) listed on
23 indexes (14 maturities each one) such as: EURO STOXX
50, CAC 40, NIKKEI 225, FTSE Mid 250 Index, SWISS
MARKET IND, Hang Seng, NASDAQ 100, FTSE 100, MSCI
world TR Index, Sao Paulo SE Bovespa Index,...etc

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SVI Calibration
SVI Calibration for CAC40
Figure: SVI ﬁts for the total implied variance CAC40 on April 05, 2019

Overview
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SVI Calibration
Multi-Slices SVI Calibration
We apply the Sequential Least-Squares Quadratic
Programming (SLSQP) method to calibrate the SVI model in
both axes: strikes andmaturities.
This calibration will respect the SVI’s bounds and mostly both
type of arbitrage: butterﬂy ( in the strikes axis) and calendar
spread (in the maturity axis).
The objective function f (k; χR) as previously, where
χR = {a, b, ρ, m, σ} is the set of the model’s parameters, for
an expiry time ﬁx T.
f (k; χR) =
n
i=1
ωmodel
SVI(i) − ωmarket
Total(i)
2
(12)

Overview
Objectives
SVI Calibration



(NLP) : min
x∈R5
f (k; χR)
ad ≤ a ≤ au
bd ≤ b < bu
ρd < ρ < ρu
md ≤ m ≤ mu
σd ≤ σ ≤ σu
g (k; χR) > = constant > 0
∂T w(k, T) ≥ , ∀k ∈ IR, T > 0, > 0
(13)

Overview
Objectives
SVI Calibration
In practice, we start the calibration with SVI slice corresponding to
the lowest maturity and more we move up to the next maturity
more we add the constraint of non-crossing slices as bellow



w(k, T0) > 0, > 0
w(k, T1) > w(k, T0)
...
w(k, Ti ) > w(k, Ti−1) 1 ≤ i ≤ n
(14)

Overview
Objectives
SVI Calibration
The Lagrange function of the (NLP) SVI optimization problem is
L (k; χR) = f (k; χR) −
m
j=1
λj [gj (k; χR) − ]
−
n
i=1
m
j=1
νj [wj (k, Ti ) − wj (k, Ti−1) − ]
Applying the SLSQP algorithm for calibration allows to
eliminate both arbitrages: calendar spread and butterﬂy during
the calibration step. Our SVI’s parameters are arbitrage free.

Overview
Objectives
SVI Calibration
The Lagrange function of the (NLP) SVI optimization problem is
L (k; χR) = f (k; χR) −
m
j=1
λj [gj (k; χR) − ]
−
n
i=1
m
j=1
νj [wj (k, Ti ) − wj (k, Ti−1) − ]
The continues lines represent the ﬁt of the SVI model, and we note
that this lines are separated as following.

Overview
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SVI Calibration
Figure: SVI calibration arbitrage free (butterﬂy & calendar spread) for
(SP ASX 200), dots (input data), continue line (SVI model)

Overview
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SVI Calibration
Figure: SVI calibration for the index (DJ Stoxx 600 Utilities Rt Inde)
(left) and (Swiss Market Ind) (right), with both butterﬂy & calendar
spread arbitrage free. Dots (input data) and continue line is SVI model.

Overview
Objectives
SVI Calibration
SVI Calibration without Weights
The SVI calibration ﬁts all the input data points with the same
weight which is one.
Figure: SVI ﬁts without weights for the TOPIX Stock Index

Overview
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SVI Calibration
SVI Calibration with Weights
In practice, the very important and liquid zone is At The
Money zone (ATM), hence, giving more weights in this zone is
very important comparing to the wings zone.
The most appropriate method is to give weights to the loss
function that gradually decreases from the ATM to the wings.
f (k; χR) =
n
i=1
wi SVImodel
(i) − SVImarket
Total(i)
2
W = (..., wi−3, wi−2, wi−1, wATM, wi , wi+1, wi+2, ...)

Overview
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SVI Calibration
SVI Calibration with Weights
Figure: SVI ﬁts with weights for the TOPIX Stock Index

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SVI Calibration
Arbitrage-Free Smoothing by Fengler
Fengler presented an approach for smoothing implied volatility
interpolation.
Method: natural (cubic) splines to interpolate call price under
suitable shape constraints.
The input data: don’t need to be arbitrage free.
These splines ﬁt the input data backwards in time axis starting
with the largest maturity.

Overview
Objectives
SVI Calibration
Let’s consider yi is the call prices observed in the market with
respect to strikes a = u0, u1, ..., un+1 = b.
g is the natural cubic spline function (call price model)
twice diﬀerentiable C2([a, b]) and deﬁned on [a, b] by
g(u) =
n
i=0
1 {[ui , ui+1)} si (u) (15)
With;
si (u)
def
= di (u − ui )3
+ ci (u − ui )2
+ bi (u − ui ) + ai
For i = 0, . . . , n and given constants parameters ai , bi , ci , di .

Overview
Objectives
SVI Calibration
Problem Optimization
We consider the optimization problem subject to a number of linear
inequality constraints
n
i=1
{yi − g (ui )}2
+ λ
b
a
g (v) dv (16)
The smoothness of g can be determined by varying the
parameter λ > 0.
The optimization problem in (16) can be written as the
solution of the quadratic program.
min
x
− y x +
1
2
x Bx
subject to A x = 0
(17)

Overview
Objectives
SVI Calibration
No Arbitrage Constraints
The convexity constraint of the call price
g (ui ) = γi ≥ 0, γ1 = γn = 0
The monotonicity & positivity
g2 − g1 ≥ −e−rt,τ τ
(u2 − u1)
gn−1 − gn ≥ 0
The no-arbitrage constraints on the call price
e−δt,r τ
St − e−rt,τ τ
u1 ≤ g1 ≤ e−δt,τ τ
St
gn ≥ 0

Overview
Objectives
SVI Calibration
Conclusion
In this thesis we studied the Stochastic Volatility Inspired
model (SVI) as implied volatility model and we gave an
overview for some non-parametric interpolation methods.
We established the characterization of static arbitrage
(calendar spread & butterﬂy).
We provided the SVI’s parameters boundaries and the
initial guess.
The main result in this thesis is: a new robust calibration
method for the SVI model using Sequential Quadratic
Programming (SQP) optimization method that allows
automatic elimination of arbitrage (butterﬂy and calendar
spread) during the calibration.

Overview
Objectives
SVI Calibration
Conclusion
We illustrated the performance of our algorithm in two
numerical examples with arbitrage, one of them is the famous
Axel Vogt example.
We applied the method to calibrate the implied volatility for 23
indexes with 14 maturities each (322 slices).
We presented calibration with weights to give more
importance to the ATM zone rather than the wings.
The prospects results in the future: use the SVI calibration
model in the FX market and also to price diﬀerent interest
rates derivatives such as: swaptions, cap and ﬂoor...

Overview
Objectives
SVI Calibration
Conclusion
We plan to make an asymptotic study of the function
g(k), and to ﬁnd an analytic expression that guarantees the
positivity of this function.
We could also apply the same calibration method (SLSQP) to
the SABR model which will could guarantee calibration
arbitrage free and to ﬁx the arbitrage problem in this model.

Overview
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SVI Calibration

Overview
Objectives
SVI Calibration
ANNEX - Solution of The NLP
There are two classes of algorithms:
The active-set method (ASM)
The interior point method (IPM).
Active Set Method: the solution of the (NLP) problem is
iteratively, we start with initial value of the parameter’s vector x0,
and the (k+1) interation of xk+1 will be obtained from the previous
one of xk.
xk+1
:= xk
+ αk
dk
(18)
Where;
dk is the search direction in the kth step.
αth is the step length.

Overview
Objectives
SVI Calibration
The search direction
To ﬁnd the search direction dk, we reformulate the original (NLP)
problem to a Quadratic Programming Sub-problem by
A quadratic approximation of the Lagrange function L (k; χR).
A linear approximation of the constraints function gj (k; χR).

Overview
Objectives
SVI Calibration
The SQP algorithm replace the objective function by its local
quadratic approximation,
f (x) ≈ f xk
+ f xk
x − xk
+
1
2
x − xk
T
Hf xk
x − xk
and similarly the constraint function will be replaced by linear
approximation,
g(x) ≈ g xk
+ g xk
x − xk
We deﬁne,
d(x) := x − xk
, Bk
:= Hf xk
Where H is the Hessian matrix.

Overview
Objectives
SVI Calibration
The formulation of the subproblem will be
(QP) : min
d∈Rn
1
2
dT
Bk
d + f xk
d
Subject to
gj xk
d + gj xk
≥ , j = 1, . . . , m
Where ; f (x) and gj (x) are the gradients of the functions f and
g respectively and B is the search direction proposed for the ﬁrst
time by Wilson in 1963 and deﬁnes by B := 2
xx L(x, λ)

Overview
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SVI Calibration
Quadratic Programming Sub-problem



(EQP) : min
d∈Rn
1
2
dT
Bk
d + f xk
d
S.t ad ≤ a ≤ au
bd ≤ b < bu
ρd < ρ < ρu
md ≤ m ≤ mu
σd ≤ σ ≤ σu
gj xk
d + gj xk
≥ j = 1, . . . , m
(19)

Overview
Objectives
SVI Calibration
Active Set Method
In order to solve this problem, we consider the solution dk, λk T
and we move in this direction
xk+1
:= xk
+ αk
dk
respecting the restrictions
f xk
+ αk
dk
< f xk
and
α ≤ ˆαk
=



min
cj −aT
j xk
aT
j dk , if aT
j dk < 0 for some j /∈ Ik
a
+∞, if aT
j dk ≥ 0 for all j /∈ Ik
a
(20)

Overview
Objectives
SVI Calibration
Active Set Method
ˆαk is positive because the index j does not belong to the active set
and the condition (20) means:
If aT
j dk ≥ 0 all step along dk will not violate the inactive
constraint j.
If aT
j dk < 0 there exist a step αj in which the activates
constraint j : cj − aT
j xk + αj dk = 0

Overview
Objectives
SVI Calibration
References
[1] Bruno Dupire. Pricing with a smile. 1994.
[2] Emanuel Derman and Iraj Kani. Riding on a smile.Risk, 7, 01
1994.
[3] Nabil Kahale. An arbitrage-free interpolation of volatilities.Risk,
17, 04 2003.
[4] Wolfgang Karl Härdle, M Benko, Matthias Fengler, and Milos
Kopa. On extracting informa-tion implied in options.Computational
Statistics, 22:543–553, 02 2007.
[5] Matthias Fengler. Arbitrage-free smoothing of the implied
volatility surface.QuantitativeFinance, 9(4):417–428, June 2009.
[6] Jesper Andreasen and Brian Huge. Volatility interpolation.Risk
Magazine, 03 2010.

Overview
Objectives
SVI Calibration
References
[7] Judith A. Glaser and P. Heider. Arbitrage-free approximation of
call price surfaces and inputdata risk. 2012.
[8] Matthias Fengler and Lin-Yee Hin. Semi-nonparametric
estimation of the call price surfaceunder strike and time-to-expiry
no-arbitrage constraints. Economics Working Paper Series1136,
University of St. Gallen, School of Economics and Political Science,
September 2011.
[9] Jim Gatheral and Antoine Jacquier. Arbitrage-free SVI volatility
surfaces.arXiv e-prints,page arXiv:1204.0646, Apr 2012.
[10] Jim Gatheral. Stochastic volatility and local volatility.Merrill
Lynch, 2003.

Overview
Objectives
SVI Calibration
References
[11] Douglas T. Breeden and Robert H. Litzenberger. Prices of
state-contingent claims implicit inoption prices.The Journal of
Business, 51(4):621–651, 1978.
[12] Jim Gatheral. The volatility surface; a practitioner’s guide.
2006.
[13] Roger W. Lee. The moment formula for implied volatility at
extreme strikes. volume 14.3,pages 469–480, 2004.
[14] Patrick Hagan, Deep Kumar, Andrew Lesniewski, and Diana
E. Woodward. Managing smilerisk.Wilmott Magazine, 1:84–108, 01
2002.
[15] John C. Cox. The constant elasticity of variance option pricing
model.The Journal of PortfolioManagement, 23(5):15–17, 1996.

Overview
Objectives
SVI Calibration
References
[16] Patrick S. Hagan and Diana E. Woodward. Equivalent black
volatilities.Applied MathematicalFinance, 6(3):147–157, 1999.
[17] Jim Gatheral and Antoine Jacquier. Convergence of Heston to
SVI.arXiv e-prints, pagearXiv:1002.3633, Feb 2010.
[18] Zeliade White and Millard B. Stahle. Quasi-explicit calibration
of gatheral ’ s svi model. 2009.
[19] Gaoyue Guo, Antoine Jacquier, Claude Martini, and Leo
Neufcourt. Generalised arbitrage-freeSVI volatility surfaces.arXiv
e-prints, page arXiv:1210.7111, Oct 2012.
[20] Peter Carr and Dilip B. Madan. A note on suﬃcient
conditions for no arbitrage. 2005.
[21] Fabrice Douglas Rouah. Using the risk neutral density to verify
no arbitrage in impliedvolatility.http://www.frouah.com/

Overview
Objectives
SVI Calibration
References
[22] Abebe Geletu, Quadratic programming problems - a review on
algorithms and applications (Active-set and interior point methods),
Ilmenau University of Technology.
[23] Lykke Rasmussen, Computational Finance - on the search for
performance, PhD Thesis, School of The Faculty of Science,
University of Copenhagen, June 2016.

Robust Calibration For SVI Model Arbitrage Free

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Robust Calibration For SVI Model Arbitrage Free