Perturbative Analysis of Volatility Smiles
Andrew Matytsin
New York, 29 January 2000
(212) 648 0820
1
This report represents only the personal opinions of the author
and not those of J.P. Morgan, its subsidiaries or affiliates.
2
Main Risks in Options Markets
Volatility changes in time
60%
S&P500
50%
1300
50%
1200
40%
1100
40%
Markets jump
1300
3M ATM volatility
1200
30%
1000
30%
900
20%
800
3M ATM volatility
700
05-Jan-98 15-Apr-98 24-Jul-98 01-Nov-98 09-Feb-99
10%
1100
20%
1000
10%
S&P500
0%
05-Jan-98
15-Apr-98
l
Index volatility is mean-reverting
l
It is negatively correlated with the price
l
A jump in price often entails a volatility jump
24-Jul-98
900
01-Nov-98
Most models ignore at least one of these risks
3
What Is the Shape of Smile?
S&P500 volatility surface
on January 11, 1996
Implied volatility vs.
standard deviation
50%
implied 0.2
volatility
40%
0.15
30%
5.0
0.1
20%
3.0
1.0
maturity
0%
0.2
115
strike
105
95
85
0.05
10%
-10
-5
0
5
l
Implied volatility decreases with strike price
l
The skew slope is the greatest for short maturities
10
What underlying processes produce such skews?
4
Is the Skew Due to Jumps?
Jump Diffusion model
St → St eγ+ δε
ε ∝ N ( 0 ,1)
– jumps arrive with rate λ
l
For S&P 500
λ ∝ 0 . 5

γ∝ − 0 .2
δ∝ 0 . 05 ÷ 0 . 15

60%
50%
40%
implied
volatility
30%
20%
10%
30-Nov-06
0%
70
( λT ) n − λT
pn =
e
n!
Volatility surface
from jump diffusion
(creates skew)
(creates curvature)
30-Nov-00
strike
115
– in a jump
dS t / S t = µ ∗dt + σ dz ( t )
100
– between jumps
85
l
30-Dec-99
maturity
The jump diffusion model works well for short maturities
5
What Happens at Longer Maturities?
Stochastic volatility

dSt / St = µ dt + vt dz s ( t )


dvt = κ(θ − vt ) + σ vt dz v ( t )
σ ∝ 0 . 8

ρ ∝ − 0 . 7
κ ∝ 1 ÷ 3

25%
implied
volatility
(creates curvature)
20%
15%
10%
5%
0%
30-Nov-06
(creates skew)
30-Nov-00
strike
115
For S&P 500
30%
70
l
35%
100
Corr ( dz s , dzv ) = ρ < 0
Volatility surface from
a stochastic volatility model
85
l
30-Dec-99
maturity
Stochastic volatility models work well for long maturities
6
How to Combine Stochastic Volatility
and Jump Diffusion ?
l
between jumps 
dS / S
l
market crashes form a Poisson process with rate
= µdt +
vdz1
Corr( dz1 , dz2 ) = ρ


dv = κ(θ − v) dt + σ vdz2
log S → log S + γs + δs ε

v → v + γv
l
λ
ε ∝ N ( 0 ,1)
the option price obeys the equation
2
2
2

∂f
∂
f
∂
f
1
∂
f
∂
f
∂
f 
*
2
2
+ µ S
+ κ(θ − v) + v S
+σ
+ 2 ρσS

2
2
∂t
∂S
∂v 2  ∂S
∂v
∂S∂v 
+ λE * f ( Seγs + δsε , v + γv ) − f ( S , v) = rf
[
]
European option prices can be computed analytically
7
What Is the Distribution of Stock Prices?
l Call
prices equal
l Find
C = S P1 − K e − rT P0
the characteristic functional
[
]
f ( t , φ) = E * eiφln( S / F ) = Fourier Transform of P0′
l Use
the affine ansatz
Pˆn = e C ( T − t ,ϕ ) + D ( T − t ,ϕ ) v
[
]
to derive
iϕγ− ϕ δ / 2

γv
C (τ , ϕ ) = C H (τ , ϕ ) + λτ e s s I (τ ) − 1
p
=
(b −
±

σ2

D(τ , ϕ ) = DH (τ , ϕ )
− γD (τ ,ϕ )
τ
1 γD ( t ,ϕ )
2γv
e − z dz
I (τ ) = ∫e
dt = −
∫
τ0
p+ p−
(1 + z / p+ )(1 + z / p− )
0
2
2
ρσϕ i ± d )
v
v
This model accounts for the main risks of options markets
8
Does the Model Fit the Smile?
S&P500 volatility surface
on June 11, 1997
implied volatility
Calibration errors
error, bps
0.27
80
0.25
60
2.01
0.17
0.51
0.17
110
105
strike
100
95
90
85
0.15
5.00
3.00
1.00
0.25
strike
0.08
115
0.19
20
0
-20
-40
-60
105
4.01
95
0.21
maturity
40
85
maturity
0.23
The whole volatility surface is described by
one set of constant parameters
9
Are Smile Parameters Stable Over Time?
l
Volatility parameters:
–
–
–
–
–
l
current volatility v
correlation ρ
vol of vol σ
long run volatility θ
mean reversion rate κ
Market crash parameters:
2.5
2
0
ρ
-0.2
-0.4
1.5
-0.6
1
0.5
κ
-0.8
-1
0
-1.2
28-Oct-95 15-May-96 01-Dec-96 19-Jun-97 05-Jan-98 24-Jul-98 09-Feb-99
0
2
-0.05
– crash rate λ
-0.15
γ
– crash magnitude S
γS
-0.2
– vol jump magnitude γv 28-Oct-95 15-May-96 01-Dec-96 19-Jun-97 05-Jan-98
-0.1
λ
1.5
1
0.5
0
24-Jul-98 09-Feb-99
Mean reversion, correlation and crash size are constant
10
Patterns in Stochastic Volatility Parameters
0.35
current volatility
0.3
0.1
variance jump
0.08
0.25
0.2
0.06
0.15
0.04
0.1
0.02
0.05
0
0
28-Oct-95 15-May-96 01-Dec-96 19-Jun-97 05-Jan-98 24-Jul-98 09-Feb-99 28-Oct-95 15-May-96 01-Dec-96 19-Jun-97 05-Jan-98 24-Jul-98 09-Feb-99
1.4
1.2
vol of vol
1
0.8
0.6
0.2
0.15
0.1
0.4
0.05
0.2
0
0
28-Oct-95 15-May-96 01-Dec-96 19-Jun-97 05-Jan-98 24-Jul-98 09-Feb-99
long run vol
28-Oct-95 15-May-96 01-Dec-96 19-Jun-97 05-Jan-98 24-Jul-98 09-Feb-99
Long run diffusion volatility is relatively stable
11
What Is the Intuition?
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 good
guide.
l
However, the natural expansion is for the characteristic
functional, not the implied volatilities.
How to construct the weak smile expansion?
12
Linking Characteristic Functionals to
Implied Volatilities
l
The characteristic functional
∞
Ft (η ) = ∫p ( K ) eiη ln( K / F ) dK
0
The probability distribution
∂2 C
p( K ) = e
∂K 2
rT
l
Introduce implied standard deviation ϕ = σ ( K , T ) T
l
Parametrize ϕ = ϕ ( z )
where
ln( F / K ) ϕ
z ≡ d2 =
−
= ln( M / K ),
ϕ
2
Then
M = Fe
− ϕ2 /2


z
∂
1


p ( K ) dK = N ′
( z ) dz − 1 +
−


&
&
ϕ
/
ϕ
+
z
+
ϕ
∂
z
ϕ
/
ϕ
+
z
+
ϕ



ϕ& ≡
∂ϕ
∂z
13
Linking Characteristic Functionals to
Implied Volatilities
l
Changing the integration variable to
and integrating by parts
z
+∞
F (η ) = ∫dz N ′
( z) e
l
In terms of
w ≡ z + iηϕ ( z )
+∞
F (η ) = ∫dw N ′
( w) e
1

− iηϕ  ϕ + z 
2

(1 +
iηϕ&)
−∞
1
− η (η + i ) ϕ 2 ( w )
2
−∞
w=
ln( F / K ) 
1
+ iη − ϕ ( w)
ϕ ( w)
2

F (η ) is related to the analytic continuation of ϕ
14
What Is the First Order Perturbation?
l
Assume ϕ 2 ( w) ≅ϕ 02 + ψ 1 (η , w)
with
l
ϕ 0 independent of w .
Then F (η ) = e
1
− ϕ 02 η (η + i )
2
+∞
{1 +
F1 (η )}
( w) ψ 1 (η , w) = −
∫dw N ′
−∞
w=−
l
2
F1 (η )
η (η + i )
x
1

+ ϕ 0 iη − + O (ψ 1 )
ϕ0
2

As a result
2ϕ 0
ψ 1 ( x) = −
e
2π
x2 + ∞
2 ϕ 02
F1 (η * ) dη * − 2 ϕ 02η *2 − ixη *
e
∫
1
− ∞ η *2 +
4
1
η* = η + i / 2
*
F
(
η
The smile slope is a simple integral of 1 )
15
What Is the Effect of Price Jumps?
l
In the Merton model
The ATM smile slope
∂ψ 1
∂x
γ
= 2 λT ( e − 1)
⇒
x =0
dσ
dx
x =0
λ( eγ− 1)
=
σ0
– stable calibration of expected loss λ( eγ− 1)
– more noise in λ and γ
l
The smile curvature (γ= 0 )
∂2ψ 1
∂x 2
x =0
λTδ4
=
4ϕ 02
λ δ4 x 2
σ ≈σ0 +
16 σ03T
– at small δ, very straight skews
– strong dependence on δwhen δis large
16
What Is the Effect of Stochastic Volatility?
l
Black-Scholes variance ϕ 02 ≡ θT +
1
l
v− θ
(1 − e − κT )
κ
( x) = ρσT
ψ 1′
As T → 0 ,
2
Hence the smile slope (in stdev space)
1 dσ
dσ
ρσ
= T
=
σ0 dz
dx 4 v0
l
As T → ∞ ,
T ∝ 0 . 16
ρσ
κ
ρσ
1 dσ
=
∝ 0 .06
σ0 dz 2κ θT
ψ 1′
( x) =
– calibration of ρσ more stable
– long run skew often too flat
The long run Heston smile is often too flat
17
How Jumps in Volatility Change the Picture?
l
l
If γ= δ= 0 , only the change in volatility level
1
− κT
→
( λT )(γvT )
1 − κT − e
2
ψ 1 ( x) = − ( λT )(γvT )
2
(κT )
λγv
→
T
κ
Interaction of vol jumps with price jumps
( λT )(γvT ) γ 1 − κT − e − κT
ψ 1 ( x) =
ϕ 02
2
(κT ) 2
λγ
λγ
→ (1 + α 0 )
–as T → 0 , σ x′=
σ0
σ0
for the jump from 15% to 35%
–as T → ∞ , ψ 1′
( x) =
λγ
γv
ρσ
vs.
κθ
κ
α0 =
as T → 0
as T → ∞
γv
4σ02
γv ∝ 0 . 10 ⇒ α 0 ∝ 1 . 0
⇒ α∞ =
λγ
γv
∝ 0 .9
ρσθ
Volatility jumps significantly affect the skew
18
What Is the Delta?
l
When the spot moves, the smile can move too
δC ∂C ∂C δσ
∆=
=
+
δS ∂S ∂σ δS
Thus
l
vol
Three regimes
Black-Scholes
smile stays fixed
30%
∆ = ∆ BS
25%
20%
15%
spot
10%
Relative Smile
smile floats with spot
30%
∆ RS > ∆ BS
25%
90
110
130
30%
∆ IT < ∆ BS
25%
20%
20%
15%
15%
10%
70
Fixed Implied Tree
σ( S, t ) unchanged
10%
70
90
110
130
70
90
110
130
Stochastic volatility and jump diffusion yield relative smiles
19
How to Minimize the P(L) Variance?
l
Hedge with
l
δC = ∆ δS + Λδv
Given δ
S and δ
v,
P / L = δC − y δS
y shares:
In a stochastic volatility model
Var ( P / L) = ( ∆ − y ) 2 Var (δS ) + 2 ( ∆ − y ) ΛCov (δS , δv ) + Λ2 Var (δv)
Minimize with respect to
l
Since
y:
y = ∆ + ρσΛ/ S < ∆
∆ = ∆ BS − Λσ 2 ( x) ′
/S
y = ∆ BS + ρσΛ/ 2 S
Optimal “risk management” delta < ∆ BS
20
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,
T
∂σ 2 (ξ , t )

δσ 2 ( x, T ) 1
= ∫dt E BB 

δx
T 0
∂ξ

ξ =ξBB ( t ) 


Thus
dσ ρσ
=
dx 4σ
∆ IT = ∆ BS + ρσΛ/ 2 S = y
Implied tree delta mimicks the risk management delta
21
Summary and Overview
l
Stochastic volatility and market jumps produce a skewed
surface of implied volatilities
l
The effect of volatility jumps on the skew is highly significant
l
Perturbative expansions are a useful tool for understanding
the smile
l
The optimal delta depends on the dynamics of volatility
22