Volatility Smile - Heston, SABR

Introduction
Heston Model
SABR Model
Conclusio
Volatility Smile
Heston, SABR
Nowak, Sibetz
April 24, 2012
Nowak, Sibetz Volatility Smile

Table of Contents
1 Introduction
Implied Volatility
Introduction
Heston Model
SABR Model
Conclusio
2 Heston Model
Derivation of the Heston Model
Summary for the Heston Model
FX Heston Model
Implied Volatility
Calibration of the FX Heston
Model
3 SABR Model
Definition
Derivation
SABR Implied Volatility
Calibration
4 Conclusio
Nowak, Sibetz Volatility Smile

Introduction
Heston Model
SABR Model
Conclusio
Black Scholes Framework
Implied Volatility
Black Scholes SDE
The stock price follows a geometric Brownian motion with
constant drift and volatility.
dSt = µS dt + σS dWt
Under the risk neutral pricing measure Q we have µ = rf
One can perfectly hedge an option by buying and selling the
underlying asset and the bank account dynamically
The BSM option’s value is a monotonic increasing function of
implied volatility c.p.
⎛
S
σ2
ln( ) + (r + )(T − t)
Ct = StΦ ⎝ K 2
σ √ ⎞
⎠−Ke
T − t
−r(T −t) ⎛
S
σ2
ln( ) + (r − )(T − t)
Φ ⎝ K 2
σ √ ⎞
⎠
T − t
Nowak, Sibetz Volatility Smile

Introduction
Heston Model
SABR Model
Conclusio
Black Scholes Implied Volatility
Implied Volatility
The implied volatility σimp is that the Black Scholes option
model price C BS equals the option’s market price C mkt .
C BS (S, K, σimp, rf , t, T ) = C mkt
Nowak, Sibetz Volatility Smile

Table of Contents
1 Introduction
Implied Volatility
Introduction
Heston Model
SABR Model
Conclusio
2 Heston Model
Derivation of the Heston Model
Summary for the Heston Model
FX Heston Model
Derivation of the Heston Model
Summary for the Heston Model
FX Heston Model
Calibration of the FX Heston Model
Calibration of the FX Heston
Model
3 SABR Model
Definition
Derivation
SABR Implied Volatility
Calibration
4 Conclusio
Nowak, Sibetz Volatility Smile

Definition
Stochastic Volatility Model
Introduction
Heston Model
SABR Model
Conclusio
Derivation of the Heston Model
Summary for the Heston Model
FX Heston Model
Calibration of the FX Heston Model
dSt = µStdt + √ νtStdW S t
dνt = κ(θ − νt)dt + σ √ νtdW ν t
dW S t dW ν t = ρdt
The parameters in this model are:
µ the drift of the underlying process
κ the speed of mean reversion for the variance
θ the long term mean level for the variance
σ the volatility of the variance
ν0 the initial variance at t = 0
ρ the correlation between the two Brownian motions
Nowak, Sibetz Volatility Smile

Sample Paths
Introduction
Heston Model
SABR Model
Conclusio
Derivation of the Heston Model
Summary for the Heston Model
FX Heston Model
Calibration of the FX Heston Model
Path simulation of the Heston model and the geometric Brownian
motion.
FX rate
1.8 2.0 2.2 2.4 2.6 2.8
Heston
GBM
0 200 400 600
Volatility
0.10 0.15 0.20 0.25 0.30 0.35
Nowak, Sibetz Volatility Smile
0 200 400 600

Introduction
Heston Model
SABR Model
Conclusio
Derivation of the Heston Model
Derivation of the Heston Model
Summary for the Heston Model
FX Heston Model
Calibration of the FX Heston Model
As we know the payoff of a European plain vanilla call option to be
CT = (ST − K) +
we can generally write the price of the option to be at any time
point t ∈ [0, T ]:
Ct = e −r(T −t) E (ST − K) + =
Ft e −r(T −t) E =
(ST − K)1
(ST >K) Ft
e −r(T −t) E
ST 1 Ft
(ST >K) − e
=:(∗)
−r(T −t) KE
1 Ft
(ST >K)
=:(∗∗)
Nowak, Sibetz Volatility Smile

Introduction
Heston Model
SABR Model
Conclusio
Derivation of the Heston Model
Summary for the Heston Model
FX Heston Model
Calibration of the FX Heston Model
With constant interest rates the stochastic discount factor using the bank
account Bt then becomes 1/Bt = e − t
0 rsds −rt
= e . We now need to perform
a Radon-Nikodym change of measure.
Zt = dQ St BT
Ft =
dP
Thus the first term (∗) gets
Bt ST
(∗) = e −r(T −t) E P ST 1(S T >K)
Ft
Ft
= Bt
E
BT
P ST 1(ST >K)
= Bt
E
BT
Q
ZtST 1(S
T >K) Ft
= Bt
E
BT
Q
St BT
ST 1(ST >K)
Bt ST
= E Q
St1(S
Ft
T >K)
Ft = StE Q 1(S T >K)
= StQ (ST > K|Ft)
Nowak, Sibetz Volatility Smile
Ft

Introduction
Heston Model
SABR Model
Conclusio
Get the distribution function
How to do ...
Find the characteristic function
Derivation of the Heston Model
Summary for the Heston Model
FX Heston Model
Calibration of the FX Heston Model
Fourier Inversion theorem to get the probability distribution function
We apply the Fourier Inversion Formula on the characteristic function
FX(x) − FX(0) = lim
T →∞
T
1 e
2π −T
iux − 1
−iu ϕX(u)du
and use the solution of Gil-Pelaez to get the nicer real valued solution of
the transformed characteristic function:
P(X > x) = 1 − FX(x) = 1
∞ −iux
1 e
+ ℜ
2 π iu ϕX(u)
du
Nowak, Sibetz Volatility Smile
0

The Heston PDE
Introduction
Heston Model
SABR Model
Conclusio
We apply the Ito-formula to expand dU(S, ν, t):
Derivation of the Heston Model
Summary for the Heston Model
FX Heston Model
Calibration of the FX Heston Model
dU = Utdt + USdS + Uνdν + 1
2 USS(dS) 2 + USν(dSdν) + 1
2 Uνν(dν)2
With the quadratic variation and covariation terms expanded we get
(dS) 2 = d 〈S〉 = νS 2
d W S
= νS 2 dt,
(dSdν) =
d 〈S, ν〉 = νSσd W S , W ν
= νSσρdt, and
(dν) 2 = d 〈ν〉 = σ 2 νd 〈W ν 〉 = σ 2 νdt.
The other terms including d 〈t〉 , d 〈t, W ν 〉 , d t, W S are left out, as the quadratic
variation of a finite variation term is always zero and thus the terms vanish. Thus
dU = Utdt + USdS + Uνdν + 1
2 USSνSdt + USννSσρdt + 1
2 Uννσ2 νdt
= Ut + 1
2 USSνS + USννSσρ + 1
2 Uννσ2
ν dt + USdS + Uνdν
Nowak, Sibetz Volatility Smile

The Heston PDE
Introduction
Heston Model
SABR Model
Conclusio
Derivation of the Heston Model
Summary for the Heston Model
FX Heston Model
Calibration of the FX Heston Model
As in the BSM portfolio replication also in the Heston model you get
your portfolio PDE via dynamic hedging, but we have a portfolio
consisting of:
one option V (S, ν, t)
a portion of the underlying ∆St and
a third derivative to hedge the volatility φU(S, ν, t).
1
2 νUXX + ρσνUXν + 1
2 σ2
νUνν + r − 1
2 ν
+
κ(θ − νt) − λ0νt Uν − rU − Uτ = 0
where λ0νt is the market price of volatility risk.
Nowak, Sibetz Volatility Smile
UX +

Introduction
Heston Model
SABR Model
Conclusio
Characteristic Function PDE
Derivation of the Heston Model
Summary for the Heston Model
FX Heston Model
Calibration of the FX Heston Model
Heston assumed the characteristic function to be of the form
ϕ i xτ (u) = exp (Ci(u, τ) + Di(u, τ)νt + iux)
The pricing PDE is always fulfilled irrespective of the terms in the call contract.
S = 1, K = 0, r = 0 ⇒ Ct = P1
S = 0, K = 1, r = 0 ⇒ Ct = −P2
We have to set up the boundary conditions we know to solve the PDE:
C(T, ν, S) = max(ST − K, 0)
C(t, ∞, S) =
−r(T −t)
Se
∂C
(t, ν, ∞)
∂S
= 1
C(t, ν, 0)
rC(t, 0, S)
=
=
0
rS ∂C
∂C ∂C
+ κθ + (t, 0, S)
∂S ∂ν ∂t
The Feynman-Kac theorem ensures that then also the characteristic function follows
the Heston PDE.
Nowak, Sibetz Volatility Smile

Heston Model Steps
Introduction
Heston Model
SABR Model
Conclusio
Derivation of the Heston Model
Summary for the Heston Model
FX Heston Model
Calibration of the FX Heston Model
Recall that we have a pricing formula of the form
Ct = StP1(St, νt, τ) − e −r(T −t) KP2(St, νt, τ)
where the two probabilities Pj are
Pj = 1 1
+
2 π
∞
0
e−iux ℜ
iu ϕj
X (u)
du
with the characteric function being of the form
ϕj(u) = e Cj(τ,u)+Dj(τ,u)νt+iux .
Nowak, Sibetz Volatility Smile

Introduction
Heston Model
SABR Model
Conclusio
FX Black Scholes Framework
Derivation of the Heston Model
Summary for the Heston Model
FX Heston Model
Calibration of the FX Heston Model
The exchange rate process Qt is the price of units of domestic currency for 1
unit of the foreign currency and is described under the actual probability
measure P by
dQt = µQtdt + σQtdWt
Let us now consider an auxiliary process Q ∗ t := QtB f
t /B d t which then of course
satisfies
Q ∗ t = QtBf
t
Bd t
µ− σ2
t+σWt
2
e (rf −rd)t
µ+rf −rd− σ2
t+σWt
2
= Q0e
= Q0e
Thus we can clearly see that Q ∗ t is a martingale under the original measure P iff
µ = rd − rf .
Nowak, Sibetz Volatility Smile

FX Option Price
Introduction
Heston Model
SABR Model
Conclusio
Derivation of the Heston Model
Summary for the Heston Model
FX Heston Model
Calibration of the FX Heston Model
If we now assume that the underlying process (Qt) is now the
exchange rate we still have the final payoff for a Call option of the
form
F XCT = max(QT − K, 0)
and following the Garman-Kohlhagen model we know that the
price of the FX option gets
F XCt = e −rf (T −t) F X
QtP1 (Qt, νt, τ) − e −rd(T −t) F X
KP2 (Qt, νt, τ)
Nowak, Sibetz Volatility Smile

Introduction
Heston Model
SABR Model
Conclusio
FX Option Volatility Surface
Risk Reversal:
Risk reversal is the difference between the
volatility of the call price and the put
price with the same moneyness levels.
RR25 = σ25C − σ25P
Butterfly:
Butterfly is the difference between the
avarage volatility of the call price and put
price with the same moneyness level and
at the money volatility level.
BF25 = (σ25C + σ25P )/2 − σAT M
Derivation of the Heston Model
Summary for the Heston Model
FX Heston Model
Calibration of the FX Heston Model
FX volatility smile with the 3-point
market quotation
Implied Volatility
Nowak, Sibetz Volatility Smile
●
●
FX Volatility Smile
●
10C 25C ATM 25P 10P
Delta
BF10
ATM
●
●
RR10

Introduction
Heston Model
SABR Model
Conclusio
Bloomberg FX Option Data
Nowak, Sibetz Volatility Smile
Derivation of the Heston Model
Summary for the Heston Model
FX Heston Model
Calibration of the FX Heston Model

Introduction
Heston Model
SABR Model
Conclusio
Bloomberg FX Option Data
Derivation of the Heston Model
Summary for the Heston Model
FX Heston Model
Calibration of the FX Heston Model
USD/JPY and EUR/JPY volatility surface
Implied Volatility
0.09 0.10 0.11 0.12 0.13 0.14 0.15
USDJPY FX Option Volatility Smile
10C 25C ATM 25P 10P
Delta
Implied Volatility
0.12 0.14 0.16 0.18 0.20 0.22
Nowak, Sibetz Volatility Smile
EURJPY FX Option Volatility Smile
10C 25C ATM 25P 10P
Delta

Introduction
Heston Model
SABR Model
Conclusio
Derivation of the Heston Model
Summary for the Heston Model
FX Heston Model
Calibration of the FX Heston Model
Calibration to the Implied Volatility Surface
Implement the Heston Pricing procedure
Characteristic function
Numerical integration algorithm
Heston pricer
BSM implied volatility from Heston prices
Sum of squared errors minimisation algorithm
compare the market implied volatility ˆσ with the volatility returned
by the Heston model σ(κ, θ, σ, ν0, ρ)
⎛
min ⎝
ˆσ − σ(κ, θ, σ, ν0, ρ) ⎞
2⎠
θ,σ,ρ
i,j
Nowak, Sibetz Volatility Smile

Parameter Impacts
Recall
⇒ set √ ν0 = σAT M .
Implied Volatility
0.08 0.10 0.12 0.14 0.16
Introduction
Heston Model
SABR Model
Conclusio
Derivation of the Heston Model
Summary for the Heston Model
FX Heston Model
Calibration of the FX Heston Model
dSt = µStdt + √ νtStdW S t
dνt = κ(θ − νt)dt + σ √ νtdW ν t
dW S t dW ν t = ρdt
Parameter Analysis − theta
theta = 0.03
theta = 0.05
theta = 0.07
10C 25C ATM 25P 10P
Delta
Implied Volatility
Nowak, Sibetz Volatility Smile
0.10 0.12 0.14 0.16 0.18
Parameter Analysis − nu0
nu0 = 0.01
nu0 = 0.02
nu0 = 0.03
10C 25C ATM 25P 10P
Delta

Parameter Impacts 2
Implied Volatility
0.09 0.10 0.11 0.12 0.13 0.14 0.15
Introduction
Heston Model
SABR Model
Conclusio
Derivation of the Heston Model
Summary for the Heston Model
FX Heston Model
Calibration of the FX Heston Model
dSt = µStdt + √ νtStdW S t
dνt = κ(θ − νt)dt + σ √ νtdW ν t
dW S t dW ν t = ρdt
Parameter Analysis − sigma
sigma = 0.20
sigma = 0.30
sigma = 0.40
10C 25C ATM 25P 10P
Delta
Implied Volatility
0.10 0.11 0.12 0.13 0.14 0.15
Parameter Analysis − kappa
kappa = 0.5
kappa = 1.5
kappa = 3.0
10C 25C ATM 25P 10P
⇒ use for κ fixed values depending on curvature. E.g. 0.5, 1.5, or 3.
Nowak, Sibetz Volatility Smile
Delta

Parameter Impacts 3
The skew parameter ρ:
Introduction
Heston Model
SABR Model
Conclusio
Derivation of the Heston Model
Summary for the Heston Model
FX Heston Model
Calibration of the FX Heston Model
dSt = µStdt + √ νtStdW S t
dνt = κ(θ − νt)dt + σ √ νtdW ν t
dW S t dW ν t = ρdt
Implied Volatility
0.08 0.10 0.12 0.14
Parameter Analysis − rho
rho = −0.25
rho = 0.05
rho = 0.30
10C 25C ATM 25P 10P
Delta
Nowak, Sibetz Volatility Smile

Introduction
Heston Model
SABR Model
Conclusio
FX Option Data Calibration
Derivation of the Heston Model
Summary for the Heston Model
FX Heston Model
Calibration of the FX Heston Model
USD/JPY and EUR/JPY volatility surface calibration
optim NM optim BFGS nlmin constr.
theta 0.03423300 0.03423542 0.03423272
vol 0.27744796 0.27746901 0.27745127
rho -0.01206708 -0.01208952 -0.01204884
Implied Volatility
0.09 0.10 0.11 0.12 0.13 0.14 0.15
USDJPY FX Option Volatility Smile
10C 25C ATM 25P 10P
Delta
optim NM optim BFGS nlmin constr.
theta 0.0508903 0.0508923 0.0508911
vol 0.4366006 0.4366059 0.4365979
rho -0.3715149 -0.3715445 -0.3715368
Implied Volatility
0.10 0.12 0.14 0.16 0.18 0.20 0.22
Nowak, Sibetz Volatility Smile
EURJPY FX Option Volatility Smile
10C 25C ATM 25P 10P
Delta

Table of Contents
1 Introduction
Implied Volatility
Introduction
Heston Model
SABR Model
Conclusio
2 Heston Model
Derivation of the Heston Model
Summary for the Heston Model
FX Heston Model
Definition
Derivation
SABR Implied Volatility
Calibration
Calibration of the FX Heston
Model
3 SABR Model
Definition
Derivation
SABR Implied Volatility
Calibration
4 Conclusio
Nowak, Sibetz Volatility Smile

Definition
Stochastic Volatility Model
The parameters are
Introduction
Heston Model
SABR Model
Conclusio
d ˆ F = ˆα ˆ F β dW1,
Definition
Derivation
SABR Implied Volatility
Calibration
ˆ F (0) = f
dˆα = ν ˆαdW2, ˆα(0) = α
dW1dW2 = ρdt
α the initial variance,
ν the volatility of variance,
β the exponent for the forward rate,
ρ the correlation between the two Brownian motions.
Nowak, Sibetz Volatility Smile

Derivation
Introduction
Heston Model
SABR Model
Conclusio
Definition
Derivation
SABR Implied Volatility
Calibration
The derivation is based on small volatility expansions, ˆα and ν, re-written
to ˆα → ɛˆα and ν → ɛν such that
d ˆ F = ɛˆαC( ˆ F )dW1,
dˆα = ɛν ˆαdW2
with dW1dW2 = ρdt in the distinguished limit ɛ ≪ 1 and C( ˆ F )
generalized. The probability density is defined as
p(t, f, α; T, F, A)dF dA = P rob F < ˆ F (T ) < F + dF, A < ˆα(T ) < A + dA
| ˆ
F (t) = f, ˆα(t) = α .
Then the density at maturity T is defined as
with
T
p(t, f, α; T, F, A) = δ(F − f)δ(A − α) +
t
p (t, f, α; T, F, A)dT
T
p =
T 1
2 ɛ2 2 ∂2
A
∂F 2 C2 (F )p + ɛ 2 ρν ∂2
∂F ∂A A2C 2 (F )p + 1
2 ɛ2 2 ∂2
ν
∂A2 A2p. Nowak, Sibetz Volatility Smile

Derivation
Introduction
Heston Model
SABR Model
Conclusio
Definition
Derivation
SABR Implied Volatility
Calibration
Let V (t, f, α) then be the value of an European call option at t at
above defined state of economy:
V (t, f, α) = E [ ˆ F (T ) − K] + | ˆ
F (t) = f, ˆα(t) = α
.
=
∞ ∞
−∞
K
= [f − K] + +
= [f − K] + + ɛ2
2
(F − K)p(t, f, α; T, F, A)dF dA
T ∞ ∞
t −∞ K
T ∞ ∞
= [f − K] + + ɛ2 C 2 (K)
2
= [f − K] + + ɛ2 C 2 (K)
2
t
−∞ K
T ∞
t
τ
t
(F − K)p T (t, f, α; T, F, A)dT
A 2 (F − K) ∂2
∂F 2 C2 (F )p dF dAdT
−∞
A 2 p(t, f, α; T, K, A)dAdT
P (τ, f, α; K)dτ
Nowak, Sibetz Volatility Smile

Derivation
Where
Introduction
Heston Model
SABR Model
Conclusio
P (t, f, α; T, K) =
∞
−∞
and P (τ, f, α; K) is the solution of
Definition
Derivation
SABR Implied Volatility
Calibration
A 2 p(t, f, α; T, K, A)dA
Pτ = 1
2 ɛ2α 2 C 2 (f) ∂2P ∂f 2 + ɛ2ρνα 2 C(f) ∂2P 1
+
∂f∂α 2 ɛ2ν 2 α 2 ∂ 2 P
, for τ > 0,
∂α2 P = α 2 δ(f − K), for τ = 0.
with τ = T − t.
Given these results one could obtain the option formula directly.
However more useful formulas can be derived through
1 Singular perturbation expansion
2 Equivalent normal volatility
3 Equivalent Black volatility
4 Stochastic β model
Nowak, Sibetz Volatility Smile

Introduction
Heston Model
SABR Model
Conclusio
Singular perturbation expansion
Definition
Derivation
SABR Implied Volatility
Calibration
The goal is to use perturbation expansion methods which yield a
Gaussian density of the form
α
P = e
2πɛ2C 2 )K)τ −
(f−K)2
2ɛ2α2 C2 (K)τ {1+··· } .
Consiquently, the singular perturbation expansion yields a
European call option value
V (t, f, α) = [f − K] + +
| f − K |
4 √ π
∞
x 2
2τ −ɛ2 θ
e −q
dq
q3/2 with
x = 1
ɛν log
1 − 2ɛρνz + ɛ2ν 2z2 − ρ + ɛνz
, z =
1 − ρ
1
f
df
ɛα K
′
C(f ′ ) ,
ɛ 2
ɛαz 1/2
xI (ɛνz)
θ = log
B(0)B(ɛαz) + log
+
f − K
z
1
4 ɛ2ρναb1z 2 .
Nowak, Sibetz Volatility Smile

Introduction
Heston Model
SABR Model
Conclusio
Equivalent normal volatility
Definition
Derivation
SABR Implied Volatility
Calibration
Suppose the previous analysis is repeated under the normal model
d ˆ F = σ N dW, ˆ F (0) = f.
with σ N constant, not stochastic. The option value would then be
V (t, f) = [f − K] + +
| f − K |
4 √ π
∞
(f−K) 2
2σ 2 N τ
e −q
dq
q3/2 for C(f) = 1, ɛα = σN and ν = 0. Integration yields then
f − K √ f − K
V (t, f) = (f − K)Φ √ + σN τG √
σN τ
σN τ
with the Gaussian density G
G(q) = 1
√ 2π e −q2 /2 .
Nowak, Sibetz Volatility Smile

Introduction
Heston Model
SABR Model
Conclusio
Equivalent normal volatility
Definition
Derivation
SABR Implied Volatility
Calibration
The option price under the normal model matches the option price
under the SABR model, iff σN is chosen the way that
f − K
σN = 1 + ɛ
x
2
θ
τ + · · ·
x2 through O(ɛ2 ). Simplifying yields the the implied normal volatility
ɛα(f − K)
σN (K) = f df ′
ζ
ˆx(ζ)
with
· 1 +
K C(f ′ )
2γ2 − γ 2 1
24
α 2 C 2 (fav) + 1
4
2 − 3ρ2
ρναγ1C(fav) + ν
24
2
ɛ 2
τ + · · ·
fav = fK, γ1 = C′ (fav)
C(fav) , γ2 = C′′ (fav)
C(fav)
ζ =
ν(f − K)
, ˆx(ζ) = log
αC(fav)
Nowak, Sibetz Volatility Smile
1 − 2ρζ + ζ2 − ρ + ζ
.
1 − ρ

Introduction
Heston Model
SABR Model
Conclusio
Equivalent Black volatility
Definition
Derivation
SABR Implied Volatility
Calibration
To derive the implied volatility consider again Black’s model
d ˆ F = ɛσ B ˆ F dW, ˆ F (0) = f
with ɛσ B for consistency of the analysis. The implied normal
volatility for Black’s model for SABR can be obtained by setting
C(f) = f and ν = 0 in previous results such that
σ (K) =
N ɛσ (f − K)
B
log f
{1 −
K
1
24 ɛ2σ 2
τ + · · · }.
B
through O(ɛ 2 ). Solving the equation for σ B yields
f
α log
K
σ (K) =
B f df
K
′
C(f ′
ζ
ˆx(ζ)
)
⎧ ⎡
⎪⎨ 2γ2 − γ
⎢
· 1 + ⎣
⎪⎩ 2 1 + 1
f2 av
α
24
2 C 2 (fav) + 1
4 ρναγ1C(fav)
2 − 3ρ2
+ ν
24
2
⎤
⎥
⎦ ɛ 2 ⎫
⎪⎬
τ + · · ·
⎪⎭ .
Nowak, Sibetz Volatility Smile

Stochastic β model
Introduction
Heston Model
SABR Model
Conclusio
Definition
Derivation
SABR Implied Volatility
Calibration
Finally, let’s look at the original state with C(f) = f β . Making the
substitutions as previously and following approximations
f − K = fK log f/K{1 + 1
24 log2 f/K + 1
1920 log4 f/K + · · · },
f 1−β − K 1−β = (1 − β)(fK) (1−β)/2 (1 − β)2
log f/K{1 + log
24
2 (1 − β)4
f/K + log
1920
4 f/K + · · · },
the implied normal volatility reduces to
σ (K) = ɛα(fK)
N β/2 1 + 1 24 log2 f/K + 1
1920 log4 f/K + · · ·
1 + (1−β)2
log
24
2 f/K + (1−β)4
log
1920
4
ζ
f/K + · · · ˆx(ζ)
2
−β(2 − β)α ρανβ 2 − 3ρ2
· 1 +
+
+ ν
24(fK) 1−β 4(fK) (1−β)/2 24
2
ɛ 2
τ + · · ·
with ζ = ν
α (fK)(1−β)/2 log f/K. Setting ɛ = 1 one gets ...
Nowak, Sibetz Volatility Smile

Introduction
Heston Model
SABR Model
Conclusio
SABR Implied Volatility - General
Definition
Derivation
SABR Implied Volatility
Calibration
The implied volatility σ B (f, K) is given by
σ B (K, f) =
1 +
(fK) (1−β)/2
(1 − β) 2
24
where z is defined by
1 + (1−β)2
24
α
α 2
1
+
(fK) 1−β 4
2 f (1−β)4 f
log + log4 K 1920 K
z = ν
α (fK)(1−β)/2 log f
K
z
· ·
x(z)
ρβνα 2 − 3ρ2
+ ν
(fK) (1−β)/2 24
2
T
and x(z) is given by
1 − 2ρz + z2 + z − ρ
x(z) = log
.
1 − ρ
Nowak, Sibetz Volatility Smile

Introduction
Heston Model
SABR Model
Conclusio
SABR Implied Volatility - ATM
Definition
Derivation
SABR Implied Volatility
Calibration
For at-the-money options (K = f) the formula reduces to
σ B (f, f) = σ AT M such that
σ AT M =
(1−β) 2
α 1 + 24
α2 f 2−2β + 1 ρβνα
4 f (1−β) + 2−3ρ2
24 ν2
f (1−β)
Nowak, Sibetz Volatility Smile
T
.

Model Dynamics
Introduction
Heston Model
SABR Model
Conclusio
Approximate the model with λ = ν
σB (K, f) = α
f 1−β
+ 1
12
The SABR model is then described with
Backbone: α
f 1−β
Skew: − 1
2
Smile: 1
12 (2 − 3ρ2 ) log
Definition
Derivation
SABR Implied Volatility
Calibration
α f 1−β such that
1 − 1
K
(1 − β − ρλ) log
2 f
2 2 2
(1 − β) + (2 − 3ρ )λ log 2
K
,
f
(1 − β − ρλ) log K
2 K
f
f
1 , 12 (1 − β)2 2 K log f
Nowak, Sibetz Volatility Smile

Backbone
Introduction
Heston Model
SABR Model
Conclusio
Definition
Derivation
SABR Implied Volatility
Calibration
Nowak, Sibetz Volatility Smile

Parameter Estimation
Introduction
Heston Model
SABR Model
Conclusio
Definition
Derivation
SABR Implied Volatility
Calibration
For estimation of the SABR model the estimation of β is used as a
starting point.
With β estimated, there are two possible choices to continue
calibration:
1 Estimate α, ρ and ν directly, or
2 Estimate ρ and ν directly, and infer α from ρ, ν and the
at-the-money.
In general, it is more convenient to use the ATM volatility
σ AT M , β, ρ and ν as the SABR parameters instead of the original
parameters α, β, ρ and ν.
Nowak, Sibetz Volatility Smile

Estimation of β
Introduction
Heston Model
SABR Model
Conclusio
Definition
Derivation
SABR Implied Volatility
Calibration
For estimation of β the at-the money volatility σ AT M from
equation is used
log σAT M = log α − (1 − β) log f +
(1 − β) 2 α
log 1 +
24
2 1 ρβνα 2 − 3ρ2
+ + ν
f 2−2β 4 f (1−β) 24
2
T
≈ log α − (1 − β) log f
Alternatively, β can be chosen from prior beliefs of the appropriate
model:
β = 1: stochastic log-normal, for FX option markets
β = 0: stochstic normal, for markets with zero or negative f
β = 1
2
: CIR model, for interest rate markets
Nowak, Sibetz Volatility Smile

Introduction
Heston Model
SABR Model
Conclusio
Estimation of α, ρ and ν
Definition
Derivation
SABR Implied Volatility
Calibration
Estimation of all three parameters by minimization of the errors
between the model and the market volatilities σmkt i at identical
maturity T .
Using the sum of squared errors (SSE)
is produced.
(ˆα, ˆρ, ˆν) = arg min
α,ρ,ν
i
σ mkt
i
Nowak, Sibetz Volatility Smile
− σ B (fi, Ki; α, ρ, ν) 2 .

Estimation of ρ and ν
Introduction
Heston Model
SABR Model
Conclusio
Definition
Derivation
SABR Implied Volatility
Calibration
The number of parameters can be reduced by extracting α directly
from σ AT M . Thus, by inverting the equation the cubic equation is
received
(1 − β) 2 T
24f 2−2β
α 3
1
+
ρβνT
4
f (1−β)
α 2
+ 1 +
2 − 3ρ2
ν
24
2
T α − σAT M f (1−β) = 0.
As it it possible to receive more than one single real root, it is
suggested to select the smallest positive real root.
Given α the SSE
(ˆα, ˆρ, ˆν) = arg min
α,ρ,ν
i
σ mkt
i
has to be minimized for the ρ and ν.
Nowak, Sibetz Volatility Smile
− σ B (fi, Ki; α(ρ, ν), ρ, ν) 2

Calibration
Implied Volatility
0.14 0.16 0.18 0.20 0.22
Introduction
Heston Model
SABR Model
Conclusio
Calibration for a fictional
data set, with 15 implied
market volatilities at
maturity T = 1.
Market Implied Volatilities
0.05 0.06 0.07 0.08 0.09 0.10 0.11
Strike
Definition
Derivation
SABR Implied Volatility
Calibration
1.Param. 2.Param.
α 0.139 0.136
ρ -0.069 -0.064
ν 0.578 0.604
SSE 2.456 · 10 −4
Nowak, Sibetz Volatility Smile
Implied Volatility
0.14 0.16 0.18 0.20 0.22
●
●
●
Difference by Parametrization
●
●
●
●
●
0.05 0.06 0.07 0.08 0.09 0.10 0.11
Strike
●
●
●
●
2.860 · 10 −4
●
●
●
0.0005 0.0010 0.0015 0.0020 0.0025 0.0030
Error

Introduction
Heston Model
SABR Model
Conclusio
Parameter dynamics - β, α
Implied Volatility
0.14 0.16 0.18 0.20 0.22
Dynamics of β
0.05 0.06 0.07 0.08 0.09 0.10 0.11
Strike
β=1
β= 1
2
β=0
Implied Volatility
Definition
Derivation
SABR Implied Volatility
Calibration
0.14 0.16 0.18 0.20 0.22
Nowak, Sibetz Volatility Smile
Dynamics of α
0.05 0.06 0.07 0.08 0.09 0.10 0.11
Strike
α
α+5%
α−5%

Introduction
Heston Model
SABR Model
Conclusio
Parameter dynamics - ρ, ν
Implied Volatility
0.14 0.16 0.18 0.20 0.22
Dynamics of ν
0.05 0.06 0.07 0.08 0.09 0.10 0.11
Strike
ν
ν+15%
ν−15%
Implied Volatility
Definition
Derivation
SABR Implied Volatility
Calibration
0.14 0.16 0.18 0.20 0.22
Nowak, Sibetz Volatility Smile
Dynamics of ρ
0.05 0.06 0.07 0.08 0.09 0.10 0.11
Strike
ρ=− 0.06
ρ=0.25
ρ=− 0.25

Introduction
Heston Model
SABR Model
Conclusio
SABR and FX Options- EUR/JPY
Implied Volatility
0.12 0.14 0.16 0.18 0.20
EURJPY FX Option Volatility Smile
10C 25C ATM 25P 10P
Delta
Implied Volatility
Definition
Derivation
SABR Implied Volatility
Calibration
0.12 0.14 0.16 0.18 0.20
Nowak, Sibetz Volatility Smile
EURJPY FX Option Volatility Smile
10C 25C ATM 25P 10P
Delta

Introduction
Heston Model
SABR Model
Conclusio
SABR and FX Options - USD/JPY
Implied Volatility
0.10 0.11 0.12 0.13
USDJPY FX Option Volatility Smile
10C 25C ATM 25P 10P
Delta
Implied Volatility
Definition
Derivation
SABR Implied Volatility
Calibration
0.10 0.11 0.12 0.13
Nowak, Sibetz Volatility Smile
EURJPY FX Option Volatility Smile
10C 25C ATM 25P 10P
Delta

Table of Contents
1 Introduction
Implied Volatility
Introduction
Heston Model
SABR Model
Conclusio
2 Heston Model
Derivation of the Heston Model
Summary for the Heston Model
FX Heston Model
Calibration of the FX Heston
Model
3 SABR Model
Definition
Derivation
SABR Implied Volatility
Calibration
4 Conclusio
Nowak, Sibetz Volatility Smile

Introduction
Heston Model
SABR Model
Conclusio
Observations and Facts
Heston
its volatility structure permits
analytical solutions to be
generated for European options
this model describes important
mean-reverting property of
volatility
allows price dynamics to be of
non-lognormal probability
distributions
the model does not perform well
for short maturities
parameters after calibration to
market data turn out to be
non-constant
SABR
Nowak, Sibetz Volatility Smile
simple stochastic volatility
model; as only one formula
no derivation of prices,
comparision directly via implied
volatility
no time dependency
implemented
interpolation errenous and
inaccurate (e.g. shifts)