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Andrew Matytsin New York, 29 January 2000
(212) 648 0820
Perturbative Analysis of Volatility Smiles
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This report represents only the personal opinions of the authorand not those of J.P. Morgan, its subsidiaries or affiliates.
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700
800
900
1000
1100
1200
1300
05-Jan-98 15-Apr-98 24-Jul-98 01-Nov-98 09-Feb-99
Main Risks in Options Markets
Volatility changes in time
Most models ignore at least one of these risks
Markets jump
l Index volatility is mean-reverting
l It is negatively correlated with the price
l A jump in price often entails a volatility jump
10%
20%
30%
40%
50%
60%
3M ATM volatility
S&P500
900
1000
1100
1200
1300
05-Jan-98 15-Apr-98 24-Jul-98 01-Nov-98
0%
10%
20%
30%
40%
50%
S&P500
3M ATM volatility
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What Is the Shape of Smile?
What underlying processes produce such skews?
Implied volatility vs. standard deviation
l Implied volatility decreases with strike price
l The skew slope is the greatest for short maturities
85 95
105
115
0.2
1.0
3.0
5.0
0.05
0.1
0.15
0.2
S&P500 volatility surfaceon January 11, 1996
maturity
implied volatility
strike0%
10%
20%
30%
40%
50%
-10 -5 0 5 10
5
Is the Skew Due to Jumps?
The jump diffusion model works well for short maturities
l Jump Diffusion model
– between jumps
– in a jump
– jumps arrive with rate
l For S&P 500
÷∝−∝
∝
15.005.02.0
5.0
δγλ
)(/ tdzdtSdS tt σµ += ∗
)1,0(NeSS tt ∝→ + εδεγ
λT
n
n enT
p λλ −=!)(
(creates skew)
(creates curvature)
implied volatility
strikematurity
Volatility surfacefrom jump diffusion
70
85
100
115 30-Dec-99
30-Nov-00
30-Nov-060%
10%
20%
30%
40%
50%
60%
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What Happens at Longer Maturities?
Stochastic volatility models work well for long maturities
l Stochastic volatility
l For S&P 500
(creates skew)
(creates curvature)
+−=+=
)()(
)(/
tdzvvdv
tdzvdtSdS
vttt
sttt
σθκµ
0),( <= ρvs dzdzCorr
÷∝−∝
∝
317.0
8.0
κρσ
Volatility surface froma stochastic volatility model
implied volatility
strikematurity
70
85
100
115 30-Dec-99
30-Nov-00
30-Nov-060%5%
10%15%20%
25%
30%
35%
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How to Combine Stochastic Volatilityand Jump Diffusion ?
European option prices can be computed analytically
l between jumps ρ
σθκµ
=
+−=+=
),()(
/21
2
1 dzdzCorrdzvdtvdv
dzvdtSdS
l market crashes form a Poisson process with rate λ
l the option price obeys the equation
+→++→
v
ss
vvSS
γεδγloglog )1,0(N∝ε
[ ] rfvSfvSefE
vSfS
vf
SfSv
vfv
SfS
tf
vss =−++
∂∂∂+
∂∂+
∂∂+
∂∂−+
∂∂+
∂∂
+ ),(),(
221)(
*
2
2
22
2
22*
γλ
ρσσθκµ
εδγ
8
What Is the Distribution of Stock Prices?
This model accounts for the main risks of options markets
C S P K e PrT= − −1 0l Call prices equal
vtTDtTCn eP ),(),(ˆ ϕϕ −+−=l Use the affine ansatz to derive
l Find the characteristic functional
∫∫−
−+
−
−+ ++−==
),(
00
),(
)/1)(/1(21
)(ϕτγτ
ϕγ γτ
τD z
vtDv
v
pzpzdze
ppdteI
)(2 dibp v ±−=± ρσϕσγ[ ]
=−+= −
),(),(1)(),(),( 2/22
ϕτϕττλτϕτϕτ δϕϕγ
H
iH
DDIeCC ss
[ ] 0)/ln(* of TransformFourier ),( PeEtf FSi ′== φφ
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Calibration errors
Does the Model Fit the Smile?
S&P500 volatility surfaceon June 11, 1997
The whole volatility surface is described byone set of constant parameters
85 95
105
115
0.08
0.25
1.003.00
5.00
-60-40-20
020406080
85 90 95
100
105
110
0.17
0.51
2.01
4.01
0.15
0.17
0.19
0.21
0.23
0.25
0.27
maturity
strike
error, bps
maturity
strike
implied volatility
10
-0.2
-0.15
-0.1
-0.05
0
0
0.5
1
1.5
2
28-Oct-95 15-May-96 01-Dec-96 19-Jun-97 05-Jan-98 24-Jul-98 09-Feb-99
0
0.5
1
1.5
2
2.5
28-Oct-95 15-May-96 01-Dec-96 19-Jun-97 05-Jan-98 24-Jul-98 09-Feb-99
Are Smile Parameters Stable Over Time?
Mean reversion, correlation and crash size are constant
l Volatility parameters:
– current volatility– correlation– vol of vol– long run volatility– mean reversion rate
l Market crash parameters:
– crash rate– crash magnitude– vol jump magnitude
ρv
σθκ
λSγ
vγ
ρ
κ-1.2
-1
-0.8
-0.6
-0.4
-0.2
0
λ
Sγ
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Patterns in Stochastic Volatility Parameters
00.050.1
0.150.2
0.250.3
0.35
28-Oct-95 15-May-96 01-Dec-96 19-Jun-97 05-Jan-98 24-Jul-98 09-Feb-990
0.02
0.04
0.06
0.08
0.1
28-Oct-95 15-May-96 01-Dec-96 19-Jun-97 05-Jan-98 24-Jul-98 09-Feb-99
00.20.40.60.8
11.21.4
28-Oct-95 15-May-96 01-Dec-96 19-Jun-97 05-Jan-98 24-Jul-98 09-Feb-990
0.05
0.1
0.15
0.2
28-Oct-95 15-May-96 01-Dec-96 19-Jun-97 05-Jan-98 24-Jul-98 09-Feb-99
vol of vol
current volatility variance jump
long run vol
Long run diffusion volatility is relatively stable
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What Is the Intuition?
How to construct the weak smile expansion?
l How does each source of risk affect the smile slope– at long maturities– at short maturities
l What is its effect on– ATM volatility– smile curvature
l For many models, the “weak smile expansion” is a goodguide.
l However, the natural expansion is for the characteristicfunctional, not the implied volatilities.
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Linking Characteristic Functionals toImplied Volatilities
l The characteristic functional
l Introduce implied standard deviation
l Then
The probability distribution
Parametrize where
∫∞
=0
)/ln()()( dKeKpF FKit
ηη
2
2
)(KCeKp rT
∂∂=
TTK ),(σϕ =)( zϕϕ =
2)/ln(
2ϕ
ϕ−=≡ KF
dz 2/2
),/ln( ϕ−== FeMKM
z∂∂≡ ϕϕ&
++∂∂−
+++−′=
ϕϕϕϕϕϕ zzzz
dzzNdKKp&& /
1/
1)()(
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Linking Characteristic Functionals toImplied Volatilities
l Changing the integration variable toand integrating by parts
l In terms of
is related to the analytic continuation of
( )ϕηηϕηϕ
&iezNdzFzi
+′=
+−+ ∞
∞−∫ 1)()( 2
1
z
)( zizw ηϕ+≡
)()(21 2
)()(wi
ewNdwFϕηη
η+−+ ∞
∞−∫ ′=
)(21
)()/ln(
wiw
KFw ϕη
ϕ
−+=
)(ηF ϕ
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What Is the First Order Perturbation?
l Assume
l Then
l As a result
The smile slope is a simple integral of
with independent of .
),()( 120
2 ww ηψϕϕ +≅
{ })(1)( 1
)(21 2
0 ηηηηϕ
FeFi
+=+−
)()(
2),()( 11 ηηη
ηψ Fi
wwNdw+
−=′∫+ ∞
∞−
)(21
100
ψηϕϕ
Oix
w +
−+−=
∫+ ∞
∞−
−−
+−=
*2*202
0
2
21
2*
**120
1
41
)(2
2)(ηηϕϕ
η
ηηπ
ϕψix
x
edFex 2/* i+=ηη
)( *1 ηF
0ϕ w
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What Is the Effect of Price Jumps?l In the Merton model
– stable calibration of expected loss– more noise in and
l The smile curvature
– at small , very straight skews– strong dependence on when is large
The ATM smile slope
)1(20
1 −=∂
∂=
γλψeT
x x 00
)1(σ
λσ γ−=⇒=
edxd
x
20
4
02
12
4ϕδλψ T
xx
=∂∂
=Tx
30
24
0 16 σδλσσ +≈
)1( −γλ eλ
δ
)0( =γ
δ δ
γ
17
What Is the Effect of Stochastic Volatility?l Black-Scholes variance
l As ,
l As ,
– calibration of more stable– long run skew often too flat
Hence the smile slope (in stdev space)
The long run Heston smile is often too flat
)1(20
Tev
T κ
κθθϕ −−−+≡
0→T Tx ρσψ21)(1 =′
16.04
1
00
∝== Tvdx
dTdzd ρσσσ
σ
∞→T κρσψ =′ )(1 x
06.02
1
0
∝=Tdz
dθκ
ρσσσ
ρσ
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How Jumps in Volatility Change the Picture?l If , only the change in volatility level
l Interaction of vol jumps with price jumps
Volatility jumps significantly affect the skew
for the jump from 15% to 35%
0== δγ
21 )(1
))(()(T
eTTTx
T
v κκγλψ
κ−−−−=∞→→ TTv as
κλγ
0 as ))(( 21 →→ TTT vγλ
220
1 )(1
2))(()(
TeTTTx
Tv
κκ
ϕγγλψ
κ−−−=
00
0
)1( ,0σλγα
σλγσ +→=′→ xT 2
00 4σ
γα v=
0.110.0 0 ∝⇒∝ αγv
κρσ
κθλγγψ vs.)( , 1
vxT =′∞→ 9.0∝=⇒ ∞ ρσθλγγα v
–as
–as
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What Is the Delta?l When the spot moves, the smile can move too
l Three regimes
Stochastic volatility and jump diffusion yield relative smiles
10%
15%
20%
25%
30%
70 90 110 13010%
15%
20%
25%
30%
70 90 110 13010%
15%
20%
25%
30%
70 90 110 130
SC
SC
SC
δσδ
σδδ
∂∂+
∂∂==∆Thus
Black-Scholessmile stays fixed
Fixed Implied Tree unchanged
Relative Smilesmile floats with spot ),( tSσ
vol
spot
BSRS ∆>∆ BSIT ∆<∆BS∆=∆
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How to Minimize the P(L) Variance?
l Given and ,
l In a stochastic volatility model
l Since
Hedge with shares:
Minimize with respect to :
Optimal “risk management” delta <
Sδ vδ vSC δδδ Λ+∆=
SyCLP δδ −=/
)(Var),(Cov)(2)(Var)()/(Var 22 vvSySyLP δδδδ Λ+Λ−∆+−∆=
∆<Λ+∆= Sy /ρσ
SxBS /)(2 ′Λ−∆=∆ σ
Sy BS 2/Λ+∆= ρσ
y
BS∆
y
21
What Is the Meaning of the Implied Tree?
l Imagine the world is described by a stochastic volatility model,but we hedge with the implied tree model
l Then the smile slope
l When we move the spot, keeping the implied tree fixed,
Implied tree delta mimicks the risk management delta
σρσσ4
=dxd
ySBSIT =Λ+∆=∆ 2/ρσThus
∫
∂
∂==
T
tBB
BB
tEdt
TxTx
0 )(
22 ),(1),(
ξξξξσ
δσδ
22
Summary and Overview
l Stochastic volatility and market jumps produce a skewedsurface of implied volatilities
l The effect of volatility jumps on the skew is highly significant
l Perturbative expansions are a useful tool for understandingthe smile
l The optimal delta depends on the dynamics of volatility
