Leverage Effect, Volatility Feedback, andSelf-Exciting Market Disruptions
Disentangling the Multi-dimensional Variations in S&P 500 Index Options
Peter Carr and Liuren Wu
Bloomberg LP and Baruch College
January 23, 2009
Peter Carr & Liuren Wu Leverage Effect, Volatility Feedback, & Self-Exciting Disruptions
Co-movements between stock index and index volatilities
Equity index and volatilities show negative co-movements.
Mechanisms that can generate such co-movements:
Add negative instantaneous correlation between innovations inindex return and return variance, e.g., Heston (1993).
Scale-free dynamics: Changing the units/scale of the index doesnot change the dynamics.
Model the volatility as a function of the index level.
The local volatility model of Dupire (1994):dSt = (r − q)Stdt + σ(St , t)dW .
The constant elasticity of variance: σ(St , t) = δS1−pt , p > 0.
Scaling the price level alters the dynamics.
Evidence:
Derman (1999): Data show different regimes, under which theimplied volatility and the equity index show different dependencestructures.
Peter Carr & Liuren Wu Leverage Effect, Volatility Feedback, & Self-Exciting Disruptions
Our take: multiple channels of interactions
Index volatility varies through at least three distinct channels:
1 The dependence is on the level of financial leverage:Holding the business risk fixed, an increase in financial leverage levelleads to an increase in equity volatility level.
A financial leverage increase can come from stock price declinewhile the debt level is fixed — Black (76)’s classic leverage story.
It can also come from active leverage management.
2 There is a separate “volatility feedback” effect on asset valuation,regardless of the level of financial leverage:
A positive shock to business risk increases the discounting of futurecash flows, and reduces the asset value.
3 In addition, there are self-exciting market disruptions:
A downside jump in the index leads to an upside spike in thechances of having more of the same.
Peter Carr & Liuren Wu Leverage Effect, Volatility Feedback, & Self-Exciting Disruptions
What we do
We propose a model for the stock index dynamics that captures all threechannels of interactions,
by modeling the dynamics of the asset value and the dynamics ofthe financial leverage separately.
We propose a tractable way of pricing options under the specifieddynamics.
We estimate the model on 12 years of S&P 500 index options.
and compare its performance with that of the state-of-the-artreduced-form benchmark.
We explore the implications of the different interaction channels on thevariation of the implied volatility surface.
Peter Carr & Liuren Wu Leverage Effect, Volatility Feedback, & Self-Exciting Disruptions
The model
Decompose the forward value of the equity index Ft into a product ofthe asset value At and the equity-to-asset ratio Xt ,
Ft = AtXt . ⇐ This is just a tautology,
But it allows us to separately model the dynamics of the Xt and At :
Model Xt as a CEV process: dXt/Xt = δX−pt dWt , p > 0.
Leverage effect: A decline in X reduces equity value, increasesleverage, and raises equity volatility.
Model the asset value At as an exponential martingale,
dAt
At=
√vZt dZt +
∫∞0
(ex − 1)(µ+(dx , dt)− πJ+ (x)dxv J
t dt)
+∫ 0
−∞ (ex − 1)(µ−(dx , dt)− πJ−(x)dxv J
t dt),
dvZt = κZ
(θZ − vZ
t
)dt + σZ
√vtdZ v
t , E [dZ vt dZt ] = ρdt,
dv Jt = κJ
(θJ − v J
t
)dt−σJ
∫ 0
−∞ x(µ−(dx , dt)− πJ−(x)dxv J
t dt).
Volatility feedback — ρ < 0.Self-exciting crashes — Negative jumps in asset return areassociated with positive jumps in the arrival rate of jumps v J
t .
Peter Carr & Liuren Wu Leverage Effect, Volatility Feedback, & Self-Exciting Disruptions
Why separately model asset value and leverage?
Financial leverages may not increase when stock prices are down.
Firms actively manage their leverages (Adrian & Shin (2008)):
Commercial banks try to keep their leverage constant.Investment banks take larger leverages during booming periods andde-lever during recessions.
The traditional leverage story (on the negative relation between stockreturns and volatilities) work better for manufacturing companies withthe level of debt relatively fixed.
The state of the art in option pricing: Directly model the equity indexwith two stochastic volatility factors: Bates (2000), Jones (2003),Huang & Wu (2004), Carr & Wu (2007):
By setting X = 1, and Ft = At , we generate a reduced-formbenchmark that supercedes all the above:
dFt
Ft=
√vZt dZt +
∫∞0
(ex − 1)(µ+(dx , dt)− πJ+ (x)dxv J
t dt)
+∫ 0
−∞ (ex − 1)(µ−(dx , dt)− πJ−(x)dxv J
t dt).
Peter Carr & Liuren Wu Leverage Effect, Volatility Feedback, & Self-Exciting Disruptions
A different notation for the asset value dynamics
We can write the log asset return ln AT/At as a summation of twotime-changed Levy processes,
ln AT/At =
[ZT Z
t,T− 1
2T Z
t,T
]+[JT J
t,T− kJ(1)T J
t,T
],
Start with two types of movements as captured by the Levycomponents Zt(Brownian motion) and Jt (jumps).Apply separate time changes (T Z
t,T , T Jt,T ) to the Levy components.
The time changes are defined via their respective activity rates,
T Zt,T ≡
∫ T
t
vZs ds, T J
t,T ≡∫ T
t
v Js ds,
Brownian innovations in vZt (variance rate) are negatively
correlated with Brownian innovations in return (dZt).Jumps in v J
t (arrival rate) are driven by downside jumps in return.
Jump specification: Variance gamma with Levy densities:πJ+ (x) = e−x/vJ+ x−1, πJ−(x) = e−|x|/vJ− |x |−1.
Peter Carr & Liuren Wu Leverage Effect, Volatility Feedback, & Self-Exciting Disruptions
The equity index dynamics with index level dependence
The equity index dynamics:
dFt/Ft = δ(
Ft
At
)−p
dWt +√
vZt dZt
+∫
R0(ex − 1)
(µ(dx , dt)− πJ(x)dxv J
t dt).
The index return variance depends on the index level.
With p > 0, the return variance increases with declining index level.
Scaling Ft by At (both in dollars) makes the return variance aunitless quantity,
and renders the dynamics scale free and stable in the presence ofsplits or trends.
We cannot expect the long-run index level to be stable, but we canexpect the level of financial leverage to be stationary.
In addition to the level dependence, (At , vZt , v
Jt ) add separate variations
to the index return variance.
Peter Carr & Liuren Wu Leverage Effect, Volatility Feedback, & Self-Exciting Disruptions
The equity index dynamics without index level dependence
An alternative representation:
dFt/Ft = δX−pt dWt +
√vZt dZt
+∫
R0(ex − 1)
(µ(dx , dt)− πJ(x)dxv J
t dt).
Index return variance is driven by three state variables (Xt , vZt , v
Jt ),
with no additional index level dependence.
Let vXt = δ2X−2p
t , we obtain a three-factor stochastic volatility model:
dFt/Ft =√
vXt dWt +
√vZt dZt
+∫
R0(ex − 1)
(µ(dx , dt)− πJ(x)dxv J
t dt).
where
dvXt = κX (vX
t )2dt − σX (vXt )3/2dWt ,⇐ 3/2-process.
with κX = p(2p + 1) and σX = 2p. Henceforth, normalize δ = 1.
The model can be represented either as a local vol model with indexlevel dependence or a pure stochastic volatility model with no index leveldependence — unifying the two strands of literature.
Peter Carr & Liuren Wu Leverage Effect, Volatility Feedback, & Self-Exciting Disruptions
Option pricing
Consider the forward value of a European call option:
c (Ft ,K ,T ) = Et
[(FT − K )+
]= Et
[Et
[(XAT − K )+
∣∣∣ (XT = X )]∣∣∣Xt
]= Et [X · C(At ,K/X ,T )|Xt ]
where C(At ,K ,T ) ≡ E [(AT − K )+] is the forward call value on asset.
Option valuation follows a two-step procedure:
Derive the Fourier transform of the asset return. Apply fast Fourierinversion (FFT) to compute the call value on asset C. — OrderN ln(N) computation.
Integrate the call value XC over the known density of XT = Xconditional on Xt :
f (XT = X ,Xt) = X 2p− 32 X
12
t
p(T−t) exp(− X 2p
t +X 2p
2p2(T−t)
)Iv(
X pt X
p
p2(T−t)
).
— Quadrature method.
Peter Carr & Liuren Wu Leverage Effect, Volatility Feedback, & Self-Exciting Disruptions
Fourier transform on asset return
ln AT/At =
(ZT Z
t,T− 1
2T Z
t,T
)+(
JT Jt,T− T J
t,T kJ(1)).
The Fourier transform is exponential affine in (vZt , v
Jt ):
φ(u) ≡ Et [e iu ln AT/At ] = Et
[e
iu
(ZT Z
t,T− 1
2TZ
t,T
)+iu
(JT J
t,T−kJ (1)T J
t,T
)]= exp
(−aZ (τ)− bZ (τ)vZ
t − aJ(τ)− bJ(τ)v Jt
), τ = T − t,
where the coefficients satisfy the following ODEs:
b′Z (τ) = ψZ (u)− κMZ bZ (τ)− 1
2σ2Z bZ (τ)2, a′Z (τ) = bZ (τ)κZθZ ,
b′J(τ) = ψJ(u)− κJbJ(τ)− kMJ−(bJ(τ)σJ), a′J(τ) = bJ(τ)κJθJ ,
starting at aZ (0) = bZ (0) = aJ(0) = bJ(0) = 0, and
κMZ = κZ − iuρσZ , kM
J−(s) = − ln(1 + svMJ−), vM
J− =vJ−
1 + iuvJ−.
Peter Carr & Liuren Wu Leverage Effect, Volatility Feedback, & Self-Exciting Disruptions
FFT for option pricing
Let c(k) = C(At ,K ,T )/At with k = ln K/At .
Derive the Fourier transform of c(k) in terms of φ(u) (asset return):
χ(u) ≡∫ ∞−∞
e iukc(k)dk =φ (u − i)
(iu) (iu + 1), u = ur − iν, ν > 0.
The inversion:
c(k) =e−νk
π
∫ ∞0
e−iurkχ(ur−iν)dur ≈e−νk
π
N∑m=0
δme−iumkχ(um−iν)∆u.
FFT: dj = 1N
∑N−1m=0 fme−jm 2π
N i . If N is power of 2, N2 → (N/2) log2 N.
Map the inversion to the FFT form by setting η = ∆u and um = ηm,kj = −b + λj with λ = 2π/(ηN) and b = λN/2.
c(kj) ≈1
N
N−1∑m=0
fme jm 2πN i , fm = δm
N
πe−νkj +iumbχ(um − iν)η. (1)
Peter Carr & Liuren Wu Leverage Effect, Volatility Feedback, & Self-Exciting Disruptions
Quadrature integration
Approximate the integration with summation
c(Ft ,K ,T ) =
∫ ∞0
f (X|Xt)XC(At ,K/X ,T )dX ≈M∑
j=1
WjXjC(At ,K/Xj ,T ),
Choose the points Xj and weights Wj based on Gauss-Hermitequadrature rule.
Given the quadrature nodes and weights, {xi ,wj}Mj=1, we set
Xj = Xte√
2VX xj− 12 VX , VX = X−2p
t (T − t),
based on a log-normal approximation of the CEV dynamics.
The summation weights are
Wj =f (Xj |Xt)X ′(xj)
e−x2j
wj =f (Xj |Xt)Xj
√2VX
e−x2j
wj .
Peter Carr & Liuren Wu Leverage Effect, Volatility Feedback, & Self-Exciting Disruptions
Market prices of risks and statistical dynamics
The specifications so far are on the risk-neutral dynamics:
The forward index level, forward asset value, and leverage ratio Xt
are all martingales under Q.
Their deviations from the P dynamics reflect the market prices of risks.
We postulate that managerial decisions on financial leverage depend onrisk levels:
dXt = X 1−pt
(aX − κXX Xt − κXZ vZ
t − κXJv Jt
)dt + X 1−p
t dW Pt .
Market price of Wt risk is γXt = aX − κXX Xt − κXZ vZ
t − κXJv Jt .
κXX : Mean reversion, leverage level targeting.κXZ : How leverage responds to diffusion business risk.κXJ : How leverage responds to jump business risk.
Constant market prices on diffusion variance risk (γv ) and jump risk(γJ).
Peter Carr & Liuren Wu Leverage Effect, Volatility Feedback, & Self-Exciting Disruptions
Data analysis
OTC implied volatility quotes on SPX options, January 1996 to March2008, 583 weeks.
40 time series on a grid of
5 relative strikes: 80, 90, 100, 110, 120% of spot.
8 fixed time to maturities: 1m, 3m, 6m, 1y, 2y, 3y, 4y, 5y.
Listed market focuses on short-term options (within 3 years).OTC market is very active on long-dated options.
At one maturity, an implied volatility smile/skew can be generatedby many different mechanisms: jumps, leverage, volatility feedback,self-exciting crashes...
To distinguish the different roles played by the differentmechanisms, we need to look at how these smiles/skews evolveover a wide range of maturities.
Peter Carr & Liuren Wu Leverage Effect, Volatility Feedback, & Self-Exciting Disruptions
The mean implied volatility surface and skew
8090
100110
1201
23
45
15
20
25
30
Maturity, yearsRelative strike, %
Impl
ied
vola
tility
, %
0 1 2 3 4 51
2
3
4
5
6
7
8
9
Maturity, years
Mea
n im
plie
d vo
latil
ity s
kew
, %
Implied volatilities show a negatively sloped skew along strike.
The skew slope becomes flatter as maturity increases due to scaling:80% strike at 5-yr maturity is not nearly as out of money as80% strike at 1-month maturity.
When measured against a standardized moneyness measured = ln(K/100)/(IV
√τ), the skew defined as,
SKt,T =IVt,T (80%)−IVt,T (120%)|dt,T (80%)−dt,T (120%)| , does not flatten as maturity increases.
Peter Carr & Liuren Wu Leverage Effect, Volatility Feedback, & Self-Exciting Disruptions
Standard deviation and autocorrelation
80
90
100
110
120 01
23
45
4
4.5
5
5.5
6
6.5
7
Maturity, yearsRelative strike, %
Sta
ndar
d de
viat
ion,
%
80
90
100
110
120 01
23
45
93
94
95
96
97
98
99
Maturity, yearsRelative strike, %
Wee
kly
auto
corr
elat
ion,
%Downward sloping std term structure: Presence of a highlymean-reverting volatility factor.
Upward sloping auto term structure: Presence of multiple factors withdifferent persistence.
Peter Carr & Liuren Wu Leverage Effect, Volatility Feedback, & Self-Exciting Disruptions
Principal component analysis
1 2 3 4 5 6 7 8 9 100
10
20
30
40
50
60
70
80
Principal component
Nor
mal
ized
eig
enva
lue,
%
5 10 15 20 25 30 35 40
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
0.5
(5 strikes) x 8 maturities
Fac
tor
load
ing
P1P2P3
3 PCs explain 96.6% of variation: 85.1%, 8.2%, 3.3%.
The 1st PC (blue solid line) — the average volatility level variation.The 2nd PC (green dashed) — the variation in the term structure.The 3rd PC (red dash-dotted) — the variation along strike.
The ranking of the 2nd & 3rd PCs can switch for listed options data asthe listed market has more quotes along strikes than maturities.
Peter Carr & Liuren Wu Leverage Effect, Volatility Feedback, & Self-Exciting Disruptions
Principal component loadings on the implied vol surface
80
90
100
110
120 01
23
45
−0.4
−0.2
0
0.2
0.4
0.6
Maturity, yearsRelative strike, %
Fa
cto
r lo
ad
ing
on
P2
80
90
100
110
120 01
23
45
−0.4
−0.2
0
0.2
0.4
0.6
Maturity, yearsRelative strike, %
Fa
cto
r lo
ad
ing
on
P3
Peter Carr & Liuren Wu Leverage Effect, Volatility Feedback, & Self-Exciting Disruptions
Cross-correlogram with the index return
−20 −15 −10 −5 0 5 10 15 20−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
Number of lags in weeks
Sa
mp
le c
ross c
orr
ela
tio
n
[SPX return (Lag), ∆ ATMV]
−20 −15 −10 −5 0 5 10 15 20−0.8
−0.7
−0.6
−0.5
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
Number of lags in weeks
Sa
mp
le c
ross c
orr
ela
tio
n
[SPX return (Lag), ∆ SK]
−20 −15 −10 −5 0 5 10 15 20−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
Number of lags in weeks
Sa
mp
le c
ross c
orr
ela
tio
n
[SPX return (Lag), ∆ TS]
An negative shock to index return is instantaneously associated with
a positive shock to the volatility level (−0.8114)a steepening of the skew (−0.707)a flattening of the term structure [TS=5y-1mATMV] (0.7643)
Over-reaction — Reversion in ATMV, SK, and TS one week later.
Long-run prediction — High vol/skew predicts high return in 2 months.Peter Carr & Liuren Wu Leverage Effect, Volatility Feedback, & Self-Exciting Disruptions
Asymmetric interactions
−10 −5 0 5 10−5
−4
−3
−2
−1
0
1
2
3
4
Weekly index return, %
We
ekl
y im
plie
d v
ola
tility
leve
l ch
an
ge
, %
−10 −5 0 5 10
−1
−0.5
0
0.5
1
Weekly index return, %W
ee
kly
im
plie
d v
ola
tilit
y s
ke
w c
ha
ng
e,
%
Self-exciting behavior: Implied volatility and skew respond more to largedownside index jumps than upside index jumps.
Peter Carr & Liuren Wu Leverage Effect, Volatility Feedback, & Self-Exciting Disruptions
Time series behavior
97 98 99 00 01 02 03 04 05 06 07 08 09700
800
900
1000
1100
1200
1300
1400
1500
1600
The
S&
P 5
00 In
dex
97 98 99 00 01 02 03 04 05 06 07 08 095
10
15
20
25
30
35
40
45
ATM
impl
ied
vola
tility
, %
1 month5 years
97 98 99 00 01 02 03 04 05 06 07 08 09−15
−10
−5
0
5
10
15
Impl
ied
vola
tility
term
stru
ctur
e, %
97 98 99 00 01 02 03 04 05 06 07 08 091
2
3
4
5
6
7
8
9
10
sIm
plie
d vo
latil
ity s
kew
, %
1 month5 years
Peter Carr & Liuren Wu Leverage Effect, Volatility Feedback, & Self-Exciting Disruptions
Model estimation, with dynamic consistency
The model has 10 parameters (p, κZ , θZ , σZ , ρ, κJ , θJ , σJ , vJ+ , vJ−) andthree hidden state variables (Xt , v
Zt , v
Jt ) to price 40 options each date.
The PCA suggests that the majority of the implied volatility surfacevariation can be captured by three properly placed components.
We fix the model parameters over time and use the 3 state variables tocapture the variation of the 5× 8 implied volatility surface.
We cast the model into a state-space form:
Let Vt ≡ [Xt , vZt , v
Jt ] be the state. State propagation equation:
Vt = f (Vt−1; Θ) +√
Qt−1 εt .— 6 additional parameters (a, κXX , κXZ , κXJ , γ
v , γJ) to control thestatistical dynamics.
Let the 40 option series be the observation. Measurementequation: yt = h(Vt ; Θ) +
√Ret , (40× 1)
y : OTM option prices scaled by the BS vega of the option.Assume that the pricing errors on the scaled option series are iid.
Estimation 17 parameters over 23,320 options.
Peter Carr & Liuren Wu Leverage Effect, Volatility Feedback, & Self-Exciting Disruptions
The Classic Kalman filter
Under linear-Gaussian state-space setup,
State : Vt = FVt−1 +√
Qεt ,
Measurement : yt = HVt +√
Ret ,
Kalman filter (KF) generates efficient forecasts and updates.
The ex ante predictions are
V t = F Vt−1, ΣVV ,t = F ΣVV ,t−1F> + Q,y t = HX t , Σyy ,t = HΣVV ,tH> + R.
The ex post filtering updates are (Bayes rule),
Kt = ΣVV ,tH>(Σyy ,t
)−1= ΣVy ,t
(Σyy ,t
)−1,→ Kalman gain
Vt = V t +Kt (yt − y t) ,
ΣVV ,t = ΣVV ,t −KtΣyy ,tK>t .
We can build the log likelihood on the forecasting errors,
lt = − 12 log
∣∣Σyy ,t
∣∣− 12
((yt − y t)>
(Σyy ,t
)−1(yt − y t)
).
Peter Carr & Liuren Wu Leverage Effect, Volatility Feedback, & Self-Exciting Disruptions
Approximating the distribution
Nonlinear measurement : yt = h(Vt) +√
Ret
Extended KF: Linearly approximate the measurement equation:h(Vt) ≈ HtVt .
Particle filter: Draw a large amount of random numbers and propagatethese numbers using Bayes rule.
Unscented filter: Use a few deterministically chosen “sigma” points toapproximate the distribution.
More accurate than EKF, faster than UKF.
Let k be the number of states and η > 0 be a control parameter,we can generate a set of 2k + 1 sigma points χ based on the meanV and covariance ΣVV of the state:
χ0 = V , χi = V±√
(k + η)(ΣVV )j , j = 1, · · · , k; i = 1, · · · , 2k ,
with weights wi given by w0 = η/(k + η), wi = 1/[2(k + η)].
We can regard these sigma points as forming a discrete distributionwith wi as the corresponding probabilities.
Peter Carr & Liuren Wu Leverage Effect, Volatility Feedback, & Self-Exciting Disruptions
The unscented Kalman filter
At each time t, generate a set of 2k + 1 sigma points χt−1 based on
time (t − 1) updated state mean and covariance Vt−1 & ΣVV ,t−1.
Given the sigma points, predict the time-t state mean and covariance:
χt,i = f (χt−1,i ; Θ), V t =∑2k
i=0 wi χt,i ,
ΣVV ,t =∑2k
i=0 wi (χt,i − V t)(χt,i − V t)> + Qt−1.
Re-generate sigma points χt based on the forecasted state mean andcovariance V t & ΣVV ,t .
Compute the forecasted mean and covariances of the measurements,
ξt,i = h(χt,i ; Θ), y t =∑2k
i=0 wiξt,i ,
Σyy ,t =∑2k
i=0 wi
(ξt,i − y t
) (ξt,i − y t
)>+ R,
ΣVy ,t =∑2k
i=0 wi
(χt,i − V t
) (ξt,i − y t
)>.
With the moment conditions, perform the filtering step the same as inKalman filter.
Peter Carr & Liuren Wu Leverage Effect, Volatility Feedback, & Self-Exciting Disruptions
Pricing performance: Root mean squared error
K/τ 1 3 6 12 24 36 48 60
Reduced-form benchmark80 3.904 2.155 1.515 1.301 1.131 0.902 0.908 1.18190 2.252 1.112 0.852 0.810 0.708 0.591 0.711 1.056
100 1.290 0.601 0.641 0.678 0.528 0.425 0.592 0.966110 2.486 0.982 1.067 1.025 0.711 0.488 0.554 0.901120 3.836 1.488 1.299 1.327 1.014 0.663 0.583 0.856
Full model80 2.216 1.103 1.050 0.836 0.607 0.550 0.770 1.06490 1.225 0.727 0.701 0.641 0.445 0.279 0.445 0.758
100 1.111 0.409 0.474 0.555 0.418 0.286 0.377 0.657110 1.499 0.720 0.720 0.717 0.551 0.436 0.465 0.674120 4.014 1.081 1.057 0.984 0.714 0.561 0.572 0.735
Average rmse (in vol points) is 1.187 for benchmark and 0.83 for full model.
Peter Carr & Liuren Wu Leverage Effect, Volatility Feedback, & Self-Exciting Disruptions
Pricing performance: Explained variation
K/τ 1 3 6 12 24 36 48 60
Reduced-form benchmark80 0.672 0.904 0.919 0.932 0.948 0.960 0.954 0.93190 0.869 0.962 0.972 0.975 0.981 0.982 0.973 0.951
100 0.957 0.988 0.983 0.981 0.990 0.991 0.981 0.962110 0.795 0.966 0.949 0.949 0.978 0.988 0.983 0.968120 0.289 0.878 0.907 0.904 0.947 0.976 0.980 0.969
Average 0.717 0.940 0.946 0.948 0.969 0.979 0.974 0.956
Full model80 0.897 0.972 0.961 0.970 0.983 0.987 0.971 0.93690 0.965 0.985 0.982 0.982 0.990 0.996 0.989 0.965
100 0.968 0.994 0.991 0.986 0.991 0.996 0.992 0.977110 0.930 0.979 0.977 0.974 0.984 0.989 0.989 0.978120 0.293 0.938 0.939 0.945 0.971 0.981 0.981 0.973
Average 0.810 0.974 0.970 0.971 0.984 0.990 0.984 0.966
Likelihood: 38,265 for benchmark and 46,651 for full model.
Peter Carr & Liuren Wu Leverage Effect, Volatility Feedback, & Self-Exciting Disruptions
Relative variance and skew contributions
Θ Estimates Std Error
p 2.8427 0.0074ρ -0.8354 0.0016σJ 5.6355 0.0272vJ+ 0.0000 0.0000vJ− 0.1926 0.0002
Average instantaneous return variance contributions from (Xt , vZt , v
Jt ):
EP[X−2pt ] = 0.0119(10.91%), variance (vol)
EP[vZt ] = 0.0231(15.19%),
EP[(v 2J+ + v 2
J−)v Jt ] = 0.0116(10.79%).
All three types of interactions are strong: Leverage effect (p = 2.8427),volatility feedback (ρ = −0.8354), and self-excitement ( σJ = 5.6355).
Much larger downside jumps than upside jumps (vJ− ≥ vJ+ ).
Peter Carr & Liuren Wu Leverage Effect, Volatility Feedback, & Self-Exciting Disruptions
Different term structure effects
Θ Estimates Std Error
p 2.8427 0.0074κZ 3.0114 0.0127κJ 0.0009 0.0000
Different risk-neutral dynamics lead to different term structureresponses:
vXt : Mean-repelling (drift=p(2p + 1)(vX
t )2dt). Responses toshocks become larger at longer maturities.vZt : Strong mean reversion (κZ = 3.0114). Responses decline
quickly as option maturity increases.v Jt : Slow mean reversion (κJ = 0.0009). Responses do not decline.
⇒ The impacts are vZt are mainly at short term options. Xt and v J
t
extend to long-term options.
Peter Carr & Liuren Wu Leverage Effect, Volatility Feedback, & Self-Exciting Disruptions
The term structure of at-the-money implied volatility
0 1 2 3 4 50.17
0.18
0.19
0.2
0.21
0.22
0.23
0.24
0.25
0.26
0.27Effects of X
t variation
Maturity, years
At−
the
−m
on
ey im
plie
d v
ola
tilit
y, %
0 1 2 3 4 50.12
0.14
0.16
0.18
0.2
0.22
0.24
0.26Effects of vZ
t variation
Maturity, years
At−
the
−m
on
ey im
plie
d v
ola
tilit
y, %
0 1 2 3 4 50.16
0.17
0.18
0.19
0.2
0.21
0.22
0.23
0.24
0.25Effects of vJ
t variation
Maturity, years
At−
the
−m
on
ey im
plie
d v
ola
tilit
y, %
Solid lines: Evaluated at the sample average of (Xt , vZt , v
Jt ).
Dashed lines: Evaluated at 90th-percentile for one state variable.
Dashed lines: Evaluated at 10th-percentile for one state variable.
Peter Carr & Liuren Wu Leverage Effect, Volatility Feedback, & Self-Exciting Disruptions
The capital structure decision
dXt = X 1−pt
(aX − κXX Xt − κXZ vZ
t − κXJv Jt
)dt + X 1−p
t dW Pt
Θ Estimates Std Error
aX 0.0003 0.0000κXX 0.0001 0.0000κXZ 17.5360 0.3087κXJ -0.0774 0.0000
κXX = 0.0001: Capital structure is very persistent.
κXZ = 17.536: High diffusion business risk reduces Xt and henceincreases the financial leverage.
κXJ = −0.0774: High jump business risk increases Xt and hence reducesthe financial leverage.
Peter Carr & Liuren Wu Leverage Effect, Volatility Feedback, & Self-Exciting Disruptions
Market prices of risks
Θ Estimates Std Error
γv -17.4507 0.3048γJ 0.4468 0.0023
(i) Market price of diffusion variance risk (γv ) is highly negative →Negative variance risk premium.
Market price of return jump risk (γv ) is positive:
Risk-neutral return innovation distribution is more negativelyskewed than the statistical distribution:vPJ+ = vJ+/(1− γJvJ+ ) > vJ+ ,
vPJ− = vJ−/(1 + γJvJ−) < vJ− .
(ii) Instantaneous variance contribution from jumps is larger underthe risk-neutral measure than under the statistical measure:(v 2
J+ + v 2J−)v J
t ) > ((vPJ+ )2 + (vP
J−)2)v Jt ).
(iii) Negative risk premium on the jump arrival rate (v Jt ):
σJ(vPJ− − vJ−)v J
t < 0.
(i), (ii), & (iii) ⇒Negative index return variance risk premium.
Peter Carr & Liuren Wu Leverage Effect, Volatility Feedback, & Self-Exciting Disruptions
Time series of the extracted states
97 98 99 00 01 02 03 04 05 06 07 08 09700
800
900
1000
1100
1200
1300
1400
1500
1600Th
e S&P
500 I
ndex
97 98 99 00 01 02 03 04 05 06 07 08 090
0.005
0.01
0.015
0.02
0.025
0.03
0.035
v tX
97 98 99 00 01 02 03 04 05 06 07 08 090
0.02
0.04
0.06
0.08
0.1
0.12
0.14
vZ t
97 98 99 00 01 02 03 04 05 06 07 08 090
0.5
1
1.5
2
2.5
3
3.5
4
4.5
vJ t
The risk contribution from financial leverage (vXt ) reached historical
highs before the burst of the Nasdaq bubble.
The diffusion business risk (vZt ) peaked during the 2003 recession.
The jump risk (v Jt ) reached its highest level during the Asian crises.
Peter Carr & Liuren Wu Leverage Effect, Volatility Feedback, & Self-Exciting Disruptions
Cross-correlogram with the index return
−20 −15 −10 −5 0 5 10 15 20−0.8
−0.6
−0.4
−0.2
0
0.2
Number of lags in weeks
Sa
mp
le c
ross c
orr
ela
tio
n
[SPX return (Lag), ∆ vtX]
−20 −15 −10 −5 0 5 10 15 20−0.8
−0.6
−0.4
−0.2
0
0.2
Number of lags in weeks
Sa
mp
le c
ross c
orr
ela
tio
n
[SPX return (Lag), ∆ vtZ]
−20 −15 −10 −5 0 5 10 15 20−0.8
−0.6
−0.4
−0.2
0
0.2
Number of lags in weeks
Sa
mp
le c
ross c
orr
ela
tio
n
[SPX return (Lag), ∆ vtJ]
Leverage increases in good times.
Diffusion risk and jump risk both increase when market is down.
Some predictions?
Peter Carr & Liuren Wu Leverage Effect, Volatility Feedback, & Self-Exciting Disruptions
Multiple sources of implied volatility skew
Our model embeds 4 mechanisms in generating the implied volatility skew:(1) leverage effect (p > 0), (2) volatility feedback (ρ < 0),(3) crash risk (vJ− � vJ+ ), (4) self-excitement of crash events (σJ).
0 1 2 3 4 52.5
3
3.5
4
4.5
5
5.5
6
6.5
7
Maturity, years
Impli
ed vo
latilit
y ske
w, %
p=2.3p=2.8p=3.3
0 1 2 3 4 53
3.5
4
4.5
5
5.5
6
Maturity, years
Impli
ed vo
latilit
y ske
w, %
ρ=−0.74ρ=−0.84ρ=−0.94
0 1 2 3 4 52
2.5
3
3.5
4
4.5
5
5.5
6
6.5
Maturity, years
Impli
ed vo
latilit
y ske
w, %
vJ
−=0.09
vJ
−=0.19
vJ
−=0.29
0 1 2 3 4 53
3.5
4
4.5
5
5.5
6
Maturity, years
Impli
ed vo
latilit
y ske
w, %
σJ=8.6
σJ=5.6
σJ=2.6
SKt,T = (IVt,T (80%)− IVt,T (120%))/(|dt,T (80%)− dt,T (120%)|)Peter Carr & Liuren Wu Leverage Effect, Volatility Feedback, & Self-Exciting Disruptions
Concluding remarks
Equity return and volatility interact through several distinct channels.
It is helpful to separately model the variations of the financial leverageand the business risk
to bridge the gaps in the literature,to disentangle the different mechanisms of interaction, andto generate good pricing performance on equity options over bothshort and long option maturities.
The approach has potentials in analyzing single name stocks (options).
Peter Carr & Liuren Wu Leverage Effect, Volatility Feedback, & Self-Exciting Disruptions
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