Volatility Smile Heston, SABR

Introduction Heston Model SABR Model Conclusio

Nowak, Sibetz

April 24, 2012

Nowak, Sibetz Volatility Smile Introduction Heston Model Implied Volatility SABR Model Conclusio Table of Contents

Calibration of the FX Heston 1 Introduction Model Implied Volatility 3 SABR Model Deﬁnition 2 Heston Model Derivation Derivation of the Heston Model SABR Implied Volatility Summary for the Heston Model Calibration FX Heston Model 4 Conclusio

Nowak, Sibetz Volatility Smile Introduction Heston Model Implied Volatility SABR Model Conclusio Black Scholes Framework

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   S σ2  ln( K ) + (r + 2 )(T − t) −r(T −t) ln( K ) + (r − 2 )(T − t) Ct = StΦ  √ −Ke Φ  √  σ T − t σ T − t

Nowak, Sibetz Volatility Smile Introduction Heston Model Implied Volatility SABR Model Conclusio Black Scholes Implied Volatility

The implied volatility σimp is that the Black Scholes option model price CBS equals the option’s market price Cmkt. BS mkt C (S, K, σimp, rf , t, T ) = C

Stochastic Volatility Model

√ S dSt = µStdt + νtStdWt √ ν dνt = κ(θ − νt)dt + σ νtdWt S ν dWt dWt = 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

Path simulation of the Heston model and the geometric Brownian motion.

Heston

GBM 0.35 2.8 0.30 2.6 0.25 2.4 FX rate Volatility 0.20 2.2 0.15 2.0 0.10 1.8

0 200 400 600 0 200 400 600

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 ]:

−r(T −t) + Ct = e E (ST − K) Ft −r(T −t) = e E (ST − K)1(ST >K) Ft −r(T −t) −r(T −t) = e E ST 1(ST >K) Ft − e KE 1(ST >K) Ft | {z } | {z } =:(∗) =:(∗∗)

Nowak, Sibetz Volatility Smile Introduction Derivation of the Heston Model Heston Model Summary for the Heston Model SABR Model FX Heston Model Conclusio Calibration of the FX Heston Model With constant interest rates the stochastic discount factor using the bank R t − rsds −rt account Bt then becomes 1/Bt = e 0 = e . We now need to perform a Radon-Nikodym change of measure.

dQ St BT Zt = Ft = dP Bt ST Thus the ﬁrst term (∗) gets

−r(T −t) P (∗) = e E ST 1(ST >K) Ft

Bt P = E ST 1(ST >K) Ft BT

Bt Q = E ZtST 1(ST >K) Ft BT Bt Q St BT = E ST 1(ST >K) Ft BT Bt ST Q = E St1(ST >K) Ft Q = StE 1(ST >K) Ft = StQ (ST > K|Ft)

How to do ... Find the characteristic function Fourier Inversion theorem to get the probability distribution function We apply the Fourier Inversion Formula on the characteristic function

1 Z T eiux − 1 FX (x) − FX (0) = lim ϕX (u)du T →∞ 2π −T −iu and use the solution of Gil-Pelaez to get the nicer real valued solution of the transformed characteristic function: 1 1 Z ∞ e−iux P(X > x) = 1 − FX (x) = + < ϕX (u) du 2 π 0 iu

We apply the Ito-formula to expand dU(S, ν, t):

1 2 1 2 dU = Utdt + US dS + Uν dν + USS (dS) + USν (dSdν) + Uνν (dν) 2 2 With the quadratic variation and covariation terms expanded we get D E (dS)2 = d hSi = νS2d W S = νS2dt, D E (dSdν) = d hS, νi = νSσd W S ,W ν = νSσρdt, and

(dν)2 = d hνi = σ2νd hW ν i = σ2νdt.

The other terms including d hti , d ht, W ν i , 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

1 1 2 dU = Utdt + US dS + Uν dν + USS νSdt + USν νSσρdt + Uνν σ νdt 2 2 1 1 2 = Ut + USS νS + USν νSσρ + Uνν σ ν dt + US dS + Uν dν 2 2

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 1 1 νU + ρσνU + σ2νU + r − ν U + 2 XX Xν 2 νν 2 X + κ(θ − νt) − λ0νt Uν − rU − Uτ = 0

where λ0νt is the market price of volatility risk.

Heston assumed the characteristic function to be of the form

i ϕxτ (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) = Se−r(T −t) ∂C (t, ν, ∞) = 1 ∂S C(t, ν, 0) = 0 ∂C ∂C ∂C rC(t, 0,S) = rS + 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 Introduction Derivation of the Heston Model Heston Model Summary for the Heston Model SABR Model FX Heston Model Conclusio Calibration of the FX Heston Model Heston Model Steps

Recall that we have a pricing formula of the form

−r(T −t) Ct = StP1(St, νt, τ) − e KP2(St, νt, τ)

where the two probabilities Pj are

Z ∞ −iux 1 1 e j Pj = + < ϕX (u) du 2 π 0 iu with the characteric function being of the form

Cj (τ,u)+Dj (τ,u)νt+iux ϕj(u) = e .

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 ∗ f d Let us now consider an auxiliary process Qt := QtBt /Bt which then of course satisﬁes

f ∗ QtBt Qt = d Bt σ2 µ− t+σWt 2 (rf −rd)t = Q0e e σ2 µ+rf −rd− 2 t+σWt = Q0e

∗ Thus we can clearly see that Qt is a martingale under the original measure P iﬀ µ = rd − rf .

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 FXCT = max(QT − K, 0) and following the Garman-Kohlhagen model we know that the price of the FX option gets

−rf (T −t) FX −rd(T −t) FX FXCt = e QtP1 (Qt, νt, τ) − e KP2 (Qt, νt, τ)

Risk Reversal: FX volatility smile with the 3-point Risk reversal is the diﬀerence between the market quotation volatility of the call price and the put price with the same moneyness levels. FX Volatility Smile

RR25 = σ25C − σ25P

●

Butterﬂy: RR10

Butterﬂy is the diﬀerence between the ● avarage volatility of the call price and put ● BF10 price with the same moneyness level and ● Implied Volatility at the money volatility level. ●

ATM

BF25 = (σ25C + σ25P )/2 − σAT M 10C 25C ATM 25P 10P

Delta

USD/JPY and EUR/JPY volatility surface

USDJPY FX Option Volatility Smile EURJPY FX Option Volatility Smile 0.15 0.22 0.14 0.20 0.13 0.18 0.12 0.16 Implied Volatility Implied Volatility 0.11 0.14 0.10 0.12 0.09

10C 25C ATM 25P 10P 10C 25C ATM 25P 10P

Delta Delta

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 σˆ with the volatility returned by the Heston model σ(κ, θ, σ, ν0, ρ)   X 2 min  σˆ − σ(κ, θ, σ, ν0, ρ)  θ,σ,ρ i,j

Recall

√ S dSt = µStdt + νtStdWt √ ν dνt = κ(θ − νt)dt + σ νtdWt S ν dWt dWt = ρdt

Parameter Analysis − theta Parameter Analysis − nu0

theta = 0.03 nu0 = 0.01 0.18 theta = 0.05 nu0 = 0.02 theta = 0.07 nu0 = 0.03 0.16 0.16 0.14 0.14 0.12 Implied Volatility Implied Volatility 0.12 0.10 0.08 0.10

10C 25C ATM 25P 10P 10C 25C ATM 25P 10P

Delta Delta √ ⇒ set ν0 = σAT M .

√ S dSt = µStdt + νtStdWt √ ν dνt = κ(θ − νt)dt + σ νtdWt S ν dWt dWt = ρdt

Parameter Analysis − sigma Parameter Analysis − kappa

sigma = 0.20 kappa = 0.5 0.15 0.15 sigma = 0.30 kappa = 1.5 sigma = 0.40 kappa = 3.0 0.14 0.14 0.13 0.13 0.12 0.12 Implied Volatility Implied Volatility 0.11 0.11 0.10 0.10 0.09

10C 25C ATM 25P 10P 10C 25C ATM 25P 10P

Delta Delta ⇒ use for κ ﬁxed values depending on curvature. E.g. 0.5, 1.5, or 3.

The skew parameter ρ: √ S dSt = µStdt + νtStdWt √ ν dνt = κ(θ − νt)dt + σ νtdWt S ν dWt dWt = ρdt

Parameter Analysis − rho

rho = −0.25 rho = 0.05 rho = 0.30 0.14 0.12 Implied Volatility 0.10 0.08

10C 25C ATM 25P 10P

Delta

USD/JPY and EUR/JPY volatility surface calibration

optim NM optim BFGS nlmin constr. optim NM optim BFGS nlmin constr. theta 0.03423300 0.03423542 0.03423272 theta 0.0508903 0.0508923 0.0508911 vol 0.27744796 0.27746901 0.27745127 vol 0.4366006 0.4366059 0.4365979 rho -0.01206708 -0.01208952 -0.01204884 rho -0.3715149 -0.3715445 -0.3715368

USDJPY FX Option Volatility Smile EURJPY FX Option Volatility Smile 0.15 0.22 0.14 0.20 0.13 0.18 0.16 0.12 Implied Volatility Implied Volatility 0.14 0.11 0.12 0.10 0.10 0.09 10C 25C ATM 25P 10P 10C 25C ATM 25P 10P

Delta Delta

Nowak, Sibetz Volatility Smile Introduction Deﬁnition Heston Model Derivation SABR Model SABR Implied Volatility Conclusio Calibration Table of Contents

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

Stochastic Volatility Model

β dFˆ =α ˆFˆ dW1, Fˆ(0) = f

dαˆ = ναdWˆ 2, αˆ(0) = α

dW1dW2 = ρdt

The parameters are 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 Introduction Deﬁnition Heston Model Derivation SABR Model SABR Implied Volatility Conclusio Calibration Derivation

The derivation is based on small volatility expansions, αˆ and ν, re-written to αˆ → αˆ and ν → ν such that

dFˆ = αCˆ (Fˆ)dW1,

dαˆ = ναdWˆ 2

with dW1dW2 = ρdt in the distinguished limit 1 and C(Fˆ) generalized. The probability density is deﬁned as n p(t, f, α; T,F,A)dF dA = P rob F < Fˆ(T ) < F + dF, A < αˆ(T ) < A + dA o | Fˆ(t) = f, αˆ(t) = α .

Then the density at maturity T is deﬁned as

Z T p(t, f, α; T,F,A) = δ(F − f)δ(A − α) + p (t, f, α; T,F,A)dT T t with 1 ∂2 ∂2 1 ∂2 p = 2A2 C2(F )p + 2ρν A2C2(F )p + 2ν2 A2p. T 2 ∂F 2 ∂F ∂A 2 ∂A2

Nowak, Sibetz Volatility Smile Introduction Deﬁnition Heston Model Derivation SABR Model SABR Implied Volatility Conclusio Calibration Derivation 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 [Fˆ(T ) − K] | Fˆ(t) = f, αˆ(t) = α Z ∞ Z ∞ = (F − K)p(t, f, α; T,F,A)dF dA −∞ K Z T Z ∞ Z ∞ + = [f − K] + (F − K)pT (t, f, α; T,F,A)dT t −∞ K 2 Z T Z ∞ Z ∞ 2 + 2 ∂ 2 = [f − K] + A (F − K) 2 C (F )p dF dAdT 2 t −∞ K ∂F 2C2(K) Z T Z ∞ = [f − K]+ + A2p(t, f, α; T,K,A)dAdT 2 t −∞ . . 2C2(K) Z τ = [f − K]+ + P (τ, f, α; K)dτ 2 t

Nowak, Sibetz Volatility Smile Introduction Deﬁnition Heston Model Derivation SABR Model SABR Implied Volatility Conclusio Calibration Derivation Where Z ∞ P (t, f, α; T,K) = A2p(t, f, α; T,K,A)dA −∞ and P (τ, f, α; K) is the solution of 1 ∂2P ∂2P 1 ∂2P P = 2α2C2(f) + 2ρνα2C(f) + 2ν2α2 , for τ > 0, τ 2 ∂f 2 ∂f∂α 2 ∂α2 P = α2δ(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 Deﬁnition Heston Model Derivation SABR Model SABR Implied Volatility Conclusio Calibration Singular perturbation expansion The goal is to use perturbation expansion methods which yield a Gaussian density of the form

2 α − (f−K) {1+··· } P = e 22α2C2(K)τ . p2π2C2)K)τ Consiquently, the singular perturbation expansion yields a European call option value | f − K | Z ∞ e−q + √ V (t, f, α) = [f − K] + 3/2 dq 4 π x2 2 q 2τ − θ with

p ! f 1 1 − 2ρνz + 2ν2z2 − ρ + νz 1 Z df 0 x = log , z = 0 , ν 1 − ρ α K C(f ) αz xI1/2(νz) 1 2θ = log pB(0)B(αz) + log + 2ρναb z2. f − K z 4 1

Nowak, Sibetz Volatility Smile Introduction Deﬁnition Heston Model Derivation SABR Model SABR Implied Volatility Conclusio Calibration Equivalent normal volatility

Suppose the previous analysis is repeated under the normal model ˆ ˆ dF = σN dW, F (0) = f.

with σN constant, not stochastic. The option value would then be | f − K | Z ∞ e−q V (t, f) = [f − K]+ + √ dq 4 π (f−K)2 q3/2 2σ2 τ N

for C(f) = 1, α = 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

1 2 G(q) = √ e−q /2. 2π

Nowak, Sibetz Volatility Smile Introduction Deﬁnition Heston Model Derivation SABR Model SABR Implied Volatility Conclusio Calibration Equivalent normal volatility The option price under the normal model matches the option price

under the SABR model, iﬀ σN is chosen the way that f − K θ σ = 1 + 2 τ + ··· N x x2 through O(2). Simplifying yields the the implied normal volatility α(f − K) ζ σ (K) = N R f df 0 xˆ(ζ) K C(f 0) 2γ − γ2 1 2 − 3ρ2 · 1 + 2 1 α2C2(f ) + ρναγ C(f ) + ν2 2τ + ··· 24 av 4 1 av 24 with 0 00 p C (fav) C (fav) fav = fK, γ1 = , γ2 = C(fav) C(fav) p ! ν(f − K) 1 − 2ρζ + ζ2 − ρ + ζ ζ = , xˆ(ζ) = log . αC(fav) 1 − ρ

Nowak, Sibetz Volatility Smile Introduction Deﬁnition Heston Model Derivation SABR Model SABR Implied Volatility Conclusio Calibration Equivalent Black volatility To derive the implied volatility consider again Black’s model

dFˆ = σ Fˆ dW, Fˆ(0) = f B

with σB 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 (f − K) 1 σ (K) = B {1 − 2σ2 τ + ···}. N f 24 B log K

2 through O( ). Solving the equation for σB yields

α log f ζ ! σ (K) = K B R f df0 xˆ(ζ) K C(f0)   2 1   2γ − γ + 2  2 1 f2 1 2 − 3ρ  · 1 +  av α2C2(f ) + ρναγ C(f ) + ν2 2τ + ··· .  av 1 av   24 4 24 

Nowak, Sibetz Volatility Smile Introduction Deﬁnition Heston Model Derivation SABR Model SABR Implied Volatility Conclusio Calibration Stochastic β model

Finally, let’s look at the original state with C(f) = f β. Making the substitutions as previously and following approximations

1 1 f − K = pfK log f/K{1 + log2 f/K + log4 f/K + ···}, 24 1920 (1 − β)2 (1 − β)4 f 1−β − K1−β = (1 − β)(fK)(1−β)/2 log f/K{1 + log2 f/K + log4 f/K + ···}, 24 1920 the implied normal volatility reduces to

1 + 1 log2 f/K + 1 log4 f/K + ··· ζ ! σ (K) = α(fK)β/2 24 1920 N 2 4 (1−β) 2 (1−β) 4 xˆ(ζ) 1 + 24 log f/K + 1920 log f/K + ··· ( " −β(2 − β)α2 ρανβ 2 − 3ρ2 # ) · 1 + + + ν2 2τ + ··· 24(fK)1−β 4(fK)(1−β)/2 24

ν (1−β)/2 with ζ = α (fK) log f/K. Setting = 1 one gets ...

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

The implied volatility σB (f, K) is given by α z σ (K, f) = · · B n 2 4 o (1−β)/2 (1−β) 2 f (1−β) 4 f x(z) (fK) 1 + 24 log K + 1920 log K (1 − β)2 α2 1 ρβνα 2 − 3ρ2 1 + + + ν2 T 24 (fK)1−β 4 (fK)(1−β)/2 24 where z is deﬁned by ν f z = (fK)(1−β)/2 log α K and x(z) is given by ( ) p1 − 2ρz + z2 + z − ρ x(z) = log . 1 − ρ

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

For at-the-money options (K = f) the formula reduces to

σB (f, f) = σAT M such that

n h 2 2 2 i o α 1 + (1−β) α + 1 ρβνα + 2−3ρ ν2 T 24 f 2−2β 4 f (1−β) 24 σ = . AT M f (1−β)

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

ν 1−β Approximate the model with λ = α f such that α 1 K σ (K, f) = 1 − (1 − β − ρλ) log B f 1−β 2 f 1 K + (1 − β)2 + (2 − 3ρ2)λ2 log2 , 12 f

The SABR model is then described with α Backbone: f 1−β 1 K 1 2 2 K Skew: − 2 (1 − β − ρλ) log f , 12 (1 − β) log f 1 2 2 K Smile: 12 (2 − 3ρ ) log f

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

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

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

σAT M , β, ρ and ν as the SABR parameters instead of the original parameters α, β, ρ and ν.

Nowak, Sibetz Volatility Smile Introduction Deﬁnition Heston Model Derivation SABR Model SABR Implied Volatility Conclusio Calibration Estimation of β

For estimation of β the at-the money volatility σAT M from equation is used

log σAT M = log α − (1 − β) log f + (1 − β)2 α2 1 ρβνα 2 − 3ρ2 log 1 + + + ν2 T 24 f 2−2β 4 f (1−β) 24 ≈ 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 1 β = 2 : CIR model, for interest rate markets Nowak, Sibetz Volatility Smile Introduction Deﬁnition Heston Model Derivation SABR Model SABR Implied Volatility Conclusio Calibration Estimation of α, ρ and ν

Estimation of all three parameters by minimization of the errors mkt between the model and the market volatilities σi at identical maturity T .

Using the sum of squared errors (SSE)

X mkt 2 (ˆα, ρ,ˆ νˆ) = arg min σi − σB (fi,Ki; α, ρ, ν) . α,ρ,ν i is produced.

Nowak, Sibetz Volatility Smile Introduction Deﬁnition Heston Model Derivation SABR Model SABR Implied Volatility Conclusio Calibration Estimation of ρ and ν

The number of parameters can be reduced by extracting α directly

from σAT M . Thus, by inverting the equation the cubic equation is received (1 − β)2T 1 ρβνT 2 − 3ρ2 α3 + α2 + 1 + ν2T α − σ f (1−β) = 0. 24f 2−2β 4 f (1−β) 24 AT M

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

X mkt 2 (ˆα, ρ,ˆ νˆ) = arg min σi − σB (fi,Ki; α(ρ, ν), ρ, ν) α,ρ,ν i has to be minimized for the ρ and ν.

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

Calibration for a ﬁctional 1.Param. 2.Param. data set, with 15 implied α 0.139 0.136 ρ -0.069 -0.064 market volatilities at ν 0.578 0.604 maturity T = 1. SSE 2.456 · 10−4 2.860 · 10−4

Market Implied Volatilities Difference by Parametrization 0.0030 0.22 0.22

● 0.0025 ● 0.20 0.20

● 0.0020

● 0.18 0.18

● Error 0.0015 ●

Implied Volatility Implied Volatility ● ● ● ● 0.16 0.16

● 0.0010 ●

● ● ● 0.14 0.14 0.0005

0.05 0.06 0.07 0.08 0.09 0.10 0.11 0.05 0.06 0.07 0.08 0.09 0.10 0.11

Strike Strike Nowak, Sibetz Volatility Smile Introduction Deﬁnition Heston Model Derivation SABR Model SABR Implied Volatility Conclusio Calibration Parameter dynamics - β, α

Dynamics of β Dynamics of α

0.22 0.22 α β=1 α+5% α−5% 1 β= 2

0.20 β=0 0.20 0.18 0.18 Implied Volatility Implied Volatility 0.16 0.16 0.14 0.14

0.05 0.06 0.07 0.08 0.09 0.10 0.11 0.05 0.06 0.07 0.08 0.09 0.10 0.11

Strike Strike

Nowak, Sibetz Volatility Smile Introduction Deﬁnition Heston Model Derivation SABR Model SABR Implied Volatility Conclusio Calibration Parameter dynamics - ρ, ν

Dynamics of ν Dynamics of ρ

ν ρ −

0.22 0.22 = 0.06 ν+15% ρ=0.25 ν−15% ρ=− 0.25 0.20 0.20 0.18 0.18 Implied Volatility Implied Volatility 0.16 0.16 0.14 0.14

0.05 0.06 0.07 0.08 0.09 0.10 0.11 0.05 0.06 0.07 0.08 0.09 0.10 0.11

Strike Strike

Nowak, Sibetz Volatility Smile Introduction Deﬁnition Heston Model Derivation SABR Model SABR Implied Volatility Conclusio Calibration SABR and FX Options- EUR/JPY

EURJPY FX Option Volatility Smile EURJPY FX Option Volatility Smile 0.20 0.20 0.18 0.18 0.16 0.16 Implied Volatility Implied Volatility 0.14 0.14 0.12 0.12

10C 25C ATM 25P 10P 10C 25C ATM 25P 10P

Delta Delta

Nowak, Sibetz Volatility Smile Introduction Deﬁnition Heston Model Derivation SABR Model SABR Implied Volatility Conclusio Calibration SABR and FX Options - USD/JPY

USDJPY FX Option Volatility Smile EURJPY FX Option Volatility Smile 0.13 0.13 0.12 0.12 0.11 0.11 Implied Volatility Implied Volatility 0.10 0.10

10C 25C ATM 25P 10P 10C 25C ATM 25P 10P

Delta Delta

Nowak, Sibetz Volatility Smile Introduction Heston Model SABR Model Conclusio Table of Contents

Nowak, Sibetz Volatility Smile Introduction Heston Model SABR Model Conclusio Observations and Facts

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

Nowak, Sibetz Volatility Smile