leverage e ect, volatility feedback, and self-exciting...

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.


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.


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.


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 .


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).


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.


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.


The equity index dynamics without index level dependence

An alternative representation:

dFt/Ft = δX−pt dWt +

√vZt dZt

+∫

R0(ex − 1)


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)


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.


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.


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−.


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)


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 .


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).


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.


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.


Standard deviation and autocorrelation

80

90

100

110

120 01

23

45

4

4.5

5

5.5

6

6.5

7


Sta

ndar

d de

viat

ion,

%

80

90

100

110

120 01

23

45

93

94

95

96

97

98

99


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.


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.


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


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


Fa

cto

r lo

ad

ing

on

P3


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


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


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.


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


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.


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)

).


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.


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.


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.


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.


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+ ).


Different term structure effects


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.


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.


The capital structure decision

dXt = X 1−pt

(aX − κXX Xt − κXZ vZ

t − κXJv Jt

)dt + X 1−p

t dW Pt


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.


Market prices of risks


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.


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.


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


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


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


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?


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).


leverage e ect, volatility feedback, and self-exciting...

Documents