volatility and skewness indices using state-preference pricing

Volatility and Skewness Indices UsingState-Preference Pricing

Zhangxin Frank Liu

Finance Theory Module 2March 16th, 2013

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Outline

1 FIX the VIX

2 BEX and BUX

3 SIX is SICK

4 Future Research

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Motivation I

• WHY CARE ABOUT VOLATILITY AT ALL?

“. . . what distinguishes financial economics is thecentral role that uncertainty plays in both financialtheory and its empirical implementation. The start-ing point for every financial model is the uncer-tainty facing investors, . . . Indeed, in the absenceof uncertainty, the problems of financial economicsreduce to exercises in basic microeconomics.”

Campbell, Lo and MacKinlay (1997)

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Volatility Forecasting

• Volatility forecasting has been discussed in following con-texts (Poon and Granger, 2003; Andersen et al. (2005)):

• Historical volatility

• Quick and easy but how far back should one refer to?

• ARCH/GARCH volatility

• ARCH (Engle, 1982): time-varying function of currentobservables.

• GARCH (Bollerslev, 1986; Taylor, 1986):

f (ω1V̄ , ω2V̂t−1, ω3ε; t)

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Volatility Forecasting

• . . .• Implied volatility

• Implied from option prices

• Invert the analytical pricing formula from some option pricingmodels (if exist); or follow some model-free approaches (Du-mas, Fleming and Whaley (1998)).

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VIX History

• A brief history of VIX (Carr and Wu, 2006)

• Old VIX (VXO)

• First introduced by Whaley (1993)

• Based on OEX options (American style)

• An average of the Black-Scholes implied volatilities on eightnear-the-money options at the two nearest maturities

• Artificially induced upward bias from the CBOE trading dayconversion

TV(t ,T ) = ATMV(t ,T )

√NC√NT

≡ ATMV(t ,T )

√NC√

NC− 2× int(NC/7)

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VIX History

• . . .

• New VIX:

• CBOE revised methodology in 2003.

• Based on SPX options (European style)

• Model-free approach in Demeterfi, Derman, Kamal and Zou(DDKZ, 1999)

• Correct artificial upward bias from the previous trading dayconversion

• Trading of VIX futures contracts from May 2004; VIX optionsfrom February 2006

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VIX 101

• VIX formula:

σ2j =

2Tj

∑i

∆Ki

K 2i

erT Q(Ki)−1Tj

[FK0− 1]2

∀ j = 1,2

VIX = 100

√36530

[T1σ

21

NT2 − N30

NT2 − NT1

+ T2σ22

N30 − NT1

NT2 − NT1

]

• Principal of DDKZ (1999): realized volatility can becaptured by the dynamic hedging of a log contract.

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VIX 101

Derivation: Theoretical definition of realized variance for agiven price history is

V =1T

∫ T

0σ2(t , . . .) dt

Think about pricing a variance swap:

F = E(e−rT (V − K ))

For a zero initial value,

Kvar = E(V ) =1T

E

(∫ T

0σ2(t , . . .) dt

)

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VIX 101

DDKZ (1999) only assumes that future underlyer evolution isdiffusive (i.e. no jumps allowed):

dSt

St= µ(t , . . .)dt + σ(t , . . .)dZt

Itô’s lemma⇒ d(ln St ) =

(µ− 1

2σ2)

dt + σdZt

⇒ dSt

St− d(ln St ) =

12σ2dt

or σ2dt = 2(

dSt

St− d(ln St )

)

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VIX 101

Now we have Kvar =1T

E(∫ T

0 σ2(t , . . .) dt)

σ2dt = 2(

dStSt− d(ln St )

)

∴ E(V ) = Kvar =2T

E

(∫ T

0

dSt

St−∫ T

0d(ln St )

)

=2T

E

(∫ T

0

dSt

St

)︸ ︷︷ ︸

A

− 2T

E(

lnST

S0

)︸ ︷︷ ︸

B

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VIX 101

A = E

[∫ T

0(r dt + σ(t , . . .) dZt )

]Zt ∼ N(0, t)

= rT

B = E(

lnST

S0

)= E

(ln

ST

S∗

)︸ ︷︷ ︸Log contract

+ lnS∗S0

where S∗ is some arbitrary number to define the boundary ofOTM calls and puts.

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VIX 101How to value E(ln(ST/S∗))? Suppose we buy a portfolio of op-tions, Π, spanning all strikes K ∈ (0,∞) with expiration T andweighted inversely proportional to K 2, we have

Π =

OTM puts︷ ︸︸ ︷∫ S∗

0

1K 2 max(K − ST ,0) dK +

OTM calls︷ ︸︸ ︷∫ ∞S∗

1K 2 max(ST − K ,0) dK

=

{ ∫ S∗ST

1K 2 (K − ST ) dK , if ST < S∗∫ ST

S∗1

K 2 (ST − K ) dK , if ST ≥ S∗

= −1− ln ST +ST

S∗+ ln S∗

=ST − S∗

S∗− ln

ST

S∗

∴ E(

lnST

S∗

)= E

(ST − S∗

S∗− Π

)13 / 71

VIX 101

Kvar =2T

(rT )− 2T

[E(

ST − S∗S∗

− Π

)+ ln

S∗S0

]=

2T

[rT − E

(ST

S∗− 1)

+ E

(∫ S∗

0

1K 2 max(K − ST ,0) dK +∫ ∞

S∗

1K 2 max(ST − K ,0) dK

)− ln

S∗S0

]

=2T

a︷ ︸︸ ︷

rT −(

S0erT

S∗− 1)− ln

S∗S0

+

b︷ ︸︸ ︷erT∫ S∗

0

P(K )

K 2 dK + erT∫ ∞

S∗

C(K )

K 2 dK

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VIX 101

a =2T

(ln(

erT)−(

S0erT

S∗− 1)− ln(S∗) + ln(S0)

)=

2T

(ln(

S0erT

S∗

)−(

S0erT

S∗− 1))

=2T

(ln(

FS∗

)−(

FS∗− 1))

where F = S0erT

≈ 2T

(

FS∗− 1)− 1

2

(FS∗− 1)2

︸ ︷︷ ︸Taylor expansion of ln(F/S∗)

−(

FS∗− 1)

= − 1T

(FS∗− 1)2

where S∗ ≡ K0

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VIX 101

b =2erT

T

(∫ S∗

0

P(K )

K 2 dK +

∫ ∞S∗

C(K )

K 2 dK

)

≈ 2erT

T

∫ S∗

KL

P(K )

K 2 dK︸ ︷︷ ︸truncation error 0→KL

+

∫ KH

S∗

C(K )

K 2 dK︸ ︷︷ ︸truncation error∞→KH

≈ 2

T

∑i

∆Ki

K 2i

erT Q(Ki)︸ ︷︷ ︸discretization error

Hence we obtain the VIX formula

Kvar = E(V ) ≈ 2T

(∑i

∆Ki

K 2i

erT Q(Ki)

)− 1

T

(FS∗− 1)2

= σ2VIX

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Why the switch?

• SPX options are more popular

• “Model-free approach”

• One can replicate the payoff of VIX futures and options

• VIX futures and options can be traded for volatility hedgingpurposes

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Any Drawbacks?

• Truncation and Discretization errors (Jiang and Tian, 2007)

• Linear interpolation may induce an error, if model-free im-plied variance does not follow a linear function of maturity.

• Mechanically higher weights are allocated to OTM puts i.e.VIX may be manipulable by trading relatively cheaper Deep-OTM put options.

• Why not consider trade volume?

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FIX the VIX

Let’s FIX the VIX.

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FIX the VIX

• A forward-looking volatility index (FIX) as a proxy for marketrealized volatility over the next 30 days.

• State-Preference Pricing Approach

• Arrow (1964) and Debreu (1959)

Pt =S∑

s=1

(Φs,t+1ds,t+1)

• View FIX2 as a financial asset pays you this dollar amount:(ln(

ST + 0.05S0

))2

• How to define state prices?

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FIX the VIX

• State prices (Breeden and Litzenberger, 1978):

Φ(T , . . .) =∂2C(K ,T )

∂K 2 =∂2P(K ,T )

∂K 2

• To see this, construct a butterfly spread to long one call withstrike M −∆M, long one call with strike M + ∆M and shorttwo calls with strike M (Barraclough, 2008).

ST < M − ∆M M − ∆M < ST < M M < ST < M + ∆M M + ∆M < ST

Long 1 call with M − ∆M 0 ST − (M − ∆M) ST − (M − ∆M) ST − (M − ∆M)Short 2 calls with M 0 0 −2(ST − M) −2(ST − M)

Long 1 call with M + ∆M 0 0 0 ST − (M + ∆M)

Total at t = T 0 ∆M + (ST − M) ∆M − (ST − M) 0

• Payoff is $∆M if ST = M at maturity.

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FIX the VIX

• Thus the cost of butterfly spread that produces a paymentof $1 if the future state is ST = M is:

P(M; ∆M) =C(M −∆M,T )− 2C(M,T ) + C(M + ∆M,T )

∆M

• Divide the above by the step size ∆M and in the limit as ∆M → 0yields:

lim∆M→0

P(M; ∆M)

∆M= lim

∆M→0

C(M −∆M,T )− 2C(M,T ) + C(M + ∆M,T )

∆M2

=∂2C(K ,T )

∂K 2

∣∣∣∣K =M

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FIX the VIX

• Thus the price of a security f with payoff d fM at some future

state M is

P f =

∫M

d fM︸︷︷︸

payoff

P(M; dM)︸ ︷︷ ︸state price

=

∫M

d fM︸︷︷︸

payoff

(∂2C(K ,T )

∂K 2

∣∣∣∣K =M

)dM︸ ︷︷ ︸

state price

• As an example, let’s have a look at pricing a European put option.We know the price of the put option can be found as:

P = E(e−rT (K − ST )+) =

∫ ∞0

(K − ST )+︸ ︷︷ ︸payoff

e−rT f (ST ) dST︸ ︷︷ ︸state price

=

∫ K

0(K − ST )e−rT f (ST ) dST

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FIX the VIX• Take the partial derivative w.r.t. K :

∂P∂K

=∂

∂K

{∫ K

0e−rT (K − ST )f (ST ) dST

}

= e−rT

{(K − K )f (K ) +

∫ K

0f (ST )dST

}= e−rT F (K )

where F (·) is the risk-neutral distribution function. Take the par-tial derivative w.r.t K again:

∂2P∂K 2 =

∂

∂K{

e−rT F (K )}

= e−rT f (K )

• That is (note: ∂2P/∂K 2 = ∂2C/∂K 2, as implied in Put-Call Par-ity),

P =

∫ K

0(K−ST )e−rT f (ST )

::::::::dST =

∫ K

0(K−ST )

(∂2P∂K 2

∣∣∣∣K =ST

)::::::::::::

dST

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FIX the VIX

• How to approximate the second derivative?

Model-free approach:∂2C(K ,T )

∂X 2 ≈ Ci−1 − 2Ci + Ci+1

(∆Ki)2

or ≈ −Ci−2 + 16Ci−1 − 30Ci + 16Ci+1 − Ci+2

12(∆Ki)2 (Eberly, 2008)

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FIX the VIX

• It works with simulated data:

∆K True σ CBOE VIX Model-Free State-Price0.1 0.30 0.29999 0.3001955 0.30 0.30002 0.30028

25 0.30 0.30026 0.30240

where S0 = 995, T1 = 17, T2 = 45, K ∈ (100,2000),rf = 3.08% and d = 2%.

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FIX the VIX

• However it fails to deal with real world data:

• Equal option prices for deep OTM options→ zero state price

• Irrational bids in deep OTM options→ negative state price

• Even when prices of OTM options are rational (i.e. increas-ing/decreasing function of its strike price for a put/call), stillpossible to see Pi−1 − 2Pi + Pi+1 < 0.

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FIX the VIX

• Black-Scholes state prices (Breeden and Litzenberger, 1978):

Φ(Ki ,Ki+1) = e−rT (N (d2(Ki))− N (d2(Ki+1)))

where Ki < Ki+1 and

d2(K ) =ln(S0/K ) + (r − d − σ2/2)T

σ√

T

• The key input in N(d2): σ is estimated as the average ofimplied volatilities from 2 ATM calls and puts from 2 maturi-ties that are closer to 30-day (see, Latane and Rendleman(1976), Chiras and Manaster (1978), Beckers (1981), Chris-tensen and Prabhala (1998), Fleming (1998) and Carr andLee (2003, 2009)).

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FIX the VIX

• Data:

• Each trading day from January 4, 1996 to October 29, 2010.

• Daily SPX option quotes from Option Metrics

• Use US 1-month and 3-month T-bill yields (Federal ReserveBulletin), adjusting for the dividends (Option Metrics), as therisk-free interest rates.

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FIX the VIX

• . . .

• Filters:

• SPX options with maturities from 7 to 81 days

• Bid prices less than $0.05 are excluded

• ITM options are excluded

• Apply the put-call parity to exclude any mis-priced options

• Options with implied volatilities > 1 or < 0 are excluded

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FIX the VIX

• States range from

S ∈ (Smin,Smax) = (0.5Slowest 1996 - 2010,1.5Shighest 1996 - 2010)

≈ [300,2400]

with 0.10 increment. This results in 21,001 states per day.

• Modified state payoffs: add 0.05 to get to the center of the10 cents interval. (

ln(

statei + 0.05S0

))2

• FIX is a manipulation-free measure.

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FIX the VIX

• Highly correlated with VIX:

Cor(VIX,FIX) = 99.12%; Cor(∆VIX,∆FIX) = 89.87%

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FIX the VIXMoments FIX VIX VXO RVol30 RVol22Mean 20.44 22.21 23.14 18.71 18.15Median 19.42 21.01 22.25 16.43 15.94Maximum 84.44 80.86 87.24 90.57 87.88Minimum 8.31 9.89 0.00 5.86 5.68Std. Dev. 8.33 8.73 9.57 10.69 10.38Skewness 2.02 1.91 1.70 2.74 2.74Kurtosis 10.42 9.53 8.59 14.24 14.24Auto 0.97 0.98 0.97 0.99 0.99

Moments ∆FIX ∆VIX ∆VXO ∆RVol30 ∆RVol22Mean 0.00 0.00 0.00 0.00 0.00Median 0.00 0.00 0.00 0.00 0.00Maximum 0.54 0.50 0.53 0.66 0.66Minimum -0.55 -0.35 -0.38 -0.39 -0.39Std. Dev. 0.07 0.06 0.07 0.06 0.06Skewness 0.31 0.50 0.47 0.74 0.74Kurtosis 7.63 6.56 6.81 18.43 18.43Auto -0.17 -0.09 -0.14 -0.01 -0.01

RVolt,t+30 = 100×

√√√√√ 365

30

30∑i=1

(ln

(St+i

St+i−1

))2

; RVolt,t+22 = 100×

√√√√√ 252

22

22∑i=1

(ln

(St+i

St+i−1

))2

;

∆FIXt = ln

(FIXt

FIXt−1

)

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FIX the VIX

• Whaley (2009) documents that the change in VIX rises at ahigher absolute rate of change when there is a market fallthan an upswing. How about FIX?

RFIXt = α0 + α1RSPXt + α2RSPX−t + ε1

RVIXt = β0 + β1RSPXt + β2RSPX−t + ε2

RFIXt Coefficient Std. Error t-Statistics Prob. Adj. R2

RSPXt -3.5732 0.1068 -33.4548 0.0000 0.5100RSPX−

t -0.4764 0.1687 -2.8231 0.0048Intercept -0.0014 0.0011 -1.2175 0.2235

RVIXtRSPXt -3.0212 0.0859 -35.1775 0.0000 0.5651RSPX−

t -0.7765 0.1357 -5.7227 0.0000Intercept -0.0028 0.0009 -3.1278 0.0018

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FIX the VIX• Predictability of the market realized volatility over the next

30 days against VIX and VXO:

RVolt ,t+30 = α0 + α1FIXt + ε1RVolt ,t+30 = β0 + β1VIXt + ε2RVolt ,t+30 = γ0 + γ1VXOt + ε3

RVol30 Coefficient Std. Error t-Statistics Prob. Adj. R2 Wald TestFIX 0.9920 0.0655 15.1509 0.0000 0.9032Intercept -1.5834 1.1774 -1.3449 0.1787 0.6010VIX 0.9295 0.0601 15.4727 0.0000 0.2409Intercept -1.9479 1.1761 -1.6562 0.0978 0.5788VXO 0.8556 0.0562 15.2183 0.0000 0.0102Intercept -1.1069 1.1416 -0.9696 0.3323 0.5893

RVol22FIX 0.9626 0.0635 15.1509 0.0000 0.5557Intercept -1.5364 1.1424 -1.3449 0.1787 0.6010VIX 0.9019 0.0583 15.4727 0.0000 0.0926Intercept -1.8900 1.1412 -1.6562 0.0978 0.5788VXO 0.8302 0.0546 15.2183 0.0000 0.0019Intercept -1.0740 1.1077 -0.9696 0.3323 0.5893

The covariance matrix is computed according to Newey and West (1987) with the lag truncation of 8.

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FIX the VIX

• Predictability of the market realized volatility over the next30 days against others:

GARCH(1,1)vol = 100×√

GARCH(1,1)var × 36530

RVolt,t+30 = α0 + α1GARCH(1,1)t + ε1RVolt,t+30 = β0 + β1BAMLt + ε2RVolt,t+30 = γ0 + γ1JPMt + ε3

RVol30 Coefficient Std. Error t-Statistics Prob. Adj. R2 Wald TestFIX 0.9978 0.0660 15.1230 0.0000 0.9728Intercept -1.6155 1.1784 -1.3710 0.1705 0.6043GARCH(1,1) 2.6767 0.2583 10.3626 0.0000 0.0000Intercept 3.4280 1.2960 2.6451 0.0082 0.4522BAML 0.9287 0.0584 15.9078 0.0000 0.2221Intercept -1.5305 1.1138 -1.3741 0.1695 0.5941JPMorgan 0.9478 0.0639 14.8249 0.0000 0.4141Intercept -1.8253 1.2175 -1.4992 0.1339 0.5661

The covariance matrix is computed according to Newey and West (1987) with the lag truncation of 8.

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FIXD in DJIA

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FIXN in NDX

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Motivation II

What’s missing in VIX (Volatility)?

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Motivation II

where µORCL = 41.1% (Estrada, 2006).

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Motivation II

where µORCL = 41.1%, µMSFT = 35.5% and rf = 5% (Estrada, 2006).41 / 71

Motivation II

• FIX hasn’t fixed everything. What’s missing in VIX?

• Volatility takes account of deviations from the mean on bothsides.

• It may be more interested in what the proportion of an upsidepotential versus a downside threat in ∆VIX .

• VIX does not tell how asymmetric the market return will be.

• Market returns are not symmetric

• Campbell, Lo and MacKinlay (1997)

• Bates (2000)

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LPM 101

• Lower Partial Moments Framework

• nth order LPM and UPM for some continuous distribution Fis defined as (Bawa and Lindenberg, 1977):

LPMn(h; F ) ≡∫ h

−∞(h − R)n dF (R)

UPMn(h; F ) ≡∫ ∞

h(R − h)n dF (R)

• h: safety first disaster level return (Roy, 1952)

• Markowitz (1959), semi-variance (LPM degree 2 with h =E(R)):

S := E(min(0,R − E(R))2)

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LPM 101

• . . .

• Mean-Semivariance model (Hogan and Warren, 1974; Bawaand Lindenberg, 1977)

• Psychological studies of Mao (1970a, 1970b), Unser (2000)and Veld and Veld-Merkoulova (2008) support the LPM overvariance as a measure of investor perception of risk.

• Cosemivariance matrix is endogenous and a closed form so-lution does not exist (Cumova and Nawrocki, 2011)

• Asset pricing with LPM (Anthonisz, 2011a, 2011b)

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LPM 101

• . . .

• Andersen and Bondarenko (2009):

• Data: AB use CME option prices of the S&P 500 futures. Weuse implied volatilities from option prices of S&P 500 Index(SPX) itself.

• Methodology: AB apply Positive Convolution Approximation(Bondarenko, 2003) to estimate risk-neutral densities. Weuse state-preference pricing to estimate volatilities.

• Extendibility.

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BEX and BUX

• Decompose FIX into a forward-looking lower partial mo-ment volatility index as a proxy for market downturn, whichwe denote the bear index (BEX); and an upper partial mo-ment counterpart, the bull index (BUX).

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BEX and BUX

• BEX and BUX share the same state price at each statewith modified payoffs:

BEX2t =

∑i

Φi ln(

Si + 0.05SPXt

)2

∀Si ≤ SPXte(rf−d)∗30/365

BUX2t =

∑i

Φi ln(

Si + 0.05SPXt

)2

∀Si > SPXte(rf−d)∗30/365

47 / 71

BEX and BUX

• Predictability of the market realized LPM volatility over thenext 30 days:

RVolLPMt ,t+30 = α0 + α1BEXt + ε1

RVolLPMt ,t+30 = β0 + β1VIXt + ε2

RVolLPM30 Coefficient Std. Error t-Statistics Prob. Adj. R2 Wald Test

BEX 0.8421 0.0713 11.8090 0.0000 0.0269Intercept -0.3446 0.9900 -0.3481 0.7278 0.4296VIX 0.6259 0.0515 12.1632 0.0000 0.0000Intercept -1.0064 1.0190 -0.9876 0.3234 0.4172

RVolLPM22

BEX 0.8171 0.0692 11.8090 0.0000 0.0082Intercept -0.3344 0.9606 -0.3481 0.7278 0.4296VIX 0.6073 0.0499 12.1632 0.0000 0.0000Intercept -0.9765 0.9887 -0.9876 0.3234 0.4172

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BEX and BUX

• Predictability of the market realized UPM volatility over thenext 30 days:

RVolUPMt ,t+30 = γ0 + γ1BUXt + ε3

RVolUPMt ,t+30 = λ0 + λ1VIXt + ε4

RVolUPM30 Coefficient Std. Error t-Statistics Prob. Adj. R2 Wald Test

BUX 1.1765 0.0685 17.1839 0.0000 0.0100Intercept -2.2388 0.7890 -2.8373 0.0046 0.6857VIX 0.6777 0.0397 17.0599 0.0000 0.0000Intercept -1.9310 0.7732 -2.4974 0.0126 0.6567

RVolUPM22

BUX 1.1415 0.0664 17.1839 0.0000 0.0332Intercept -2.1723 0.7656 -2.8373 0.0046 0.6857VIX 0.6576 0.0385 17.0599 0.0000 0.0000Intercept -1.8737 0.7503 -2.4974 0.0126 0.6567

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BEX and BUX

• Daily S&P 500 Index return may be better explained by thecontemporaneous change of BEX and BUX:

RSPXt = α0 + α1RBEXt + α2RBUXt + ε1RSPXt = β0 + β1RVIXt + ε2

RSPXt Coefficient Std.Error t-Statistics Prob. Adj. R2

RBEXt 0.3369 0.0764 4.4112 0.0000RBUXt -0.4903 0.0824 -5.9482 0.0000Intercept 0.0002 0.0001 1.2863 0.1984 0.5255

RSPXtRVIXt -0.1640 0.0067 -24.6618 0.0000Intercept 0.0002 0.0001 1.4417 0.1495 0.5614

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BEX and BUX

• BEX may be a better estimator as “investor fear gauge” thanVIX:

RBEXt = α0 + α1RSPXt + α2RSPX−t + ε1RVIXt = β0 + β1RSPXt + β2RSPX−t + ε2

RBEXt Coefficient Std. Error t-Statistics Prob. Adj. R2

RSPXt -3.6355 0.1096 -33.1633 0.0000RSPX−

t -0.4761 0.1732 -2.7489 0.0060Intercept -0.0013 0.0011 -1.1600 0.2461 0.5051

RVIXtRSPXt -3.0212 0.0859 -35.1775 0.0000RSPX−

t -0.7765 0.1357 -5.7227 0.0000Intercept -0.0028 0.0009 -3.1278 0.0018 0.5651

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BEX and BUX

• If the S&P 500 Index falls by 100 basis points, then VIX willrise by

∆VIXt = −0.0028−3.0212(−0.01)−0.7765(−0.01) = 3.52%

• In contrast, BEX will rise by

∆BEXt = −3.6355(−0.01)− 0.4761(−0.01) = 4.11%

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Motivation III

“Volatility is only a good measure of risk if you feel thatbeing rich then being poor is the same as being poorthen rich.”

Peter Carr

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CBOE SKEW

• A brief history of CBOE SKEW

• Based on Bakshi, Kapadia and Madan (2003): any securitypayoff can be spanned and priced using an explicit position-ing across option strikes.

• Dennis and Mayhew (2002)• Han (2008)• Neumann and Skiadopoulos (2012)• Bali and Murray (2012)• Friesen, Zhang and Zorn (2012)• Buss and Vilkov (2012)• Rehman and Vilkov (2012)• Conrad, Dittmar and Ghysels (2012)

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CBOE SKEW

SKEW := 100− 10× S

S =EQ(R3)− 3EQ(R)EQ(R2) + 2E3

Q(R)

(EQ(R2)− E2Q(R))3/2

=:P3 − 3P1P2 + 2P3

1

(P2 − P21 )3/2

P1 =∑

i

−∆Ki

K 2i

erT Q(Ki )−(

1 + ln(

F0

K0

)−

F0

K0

)︸ ︷︷ ︸

ε1

P2 =∑

i

2∆Ki

K 2i

erT Q(Ki )

(1− ln

(Ki

F0

))

+2 ln(

K0

F0

)(F0

K0− 1)

+12

ln2(

K0

F0

)︸ ︷︷ ︸

ε2

P3 =∑

i

3∆Ki

K 2i

erT Q(Ki )

(2 ln

(Ki

F0

)− ln2

(Ki

F0

))

+3 ln2(

K0

F0

)(13

ln(

K0

F0

)− 1 +

F0

K0

)︸ ︷︷ ︸

ε3

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SIX

• Skewness is hard to measure precisely (Neuberger, 2012)

• A simple solution: the ratio of BUX to BEX forms a marketsymmetric index (SIX).

• Why do we need SIX? If market returns distribution is sym-metric, then we expect

BUXBEX

= 1

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SIX Recap

Pt =S∑

s=1

(Φs,t+1 ds,t+1)

Φs,t+1 =∂2C(K ,T )

∂K 2 ≈ Φ(Ki ,Ki+1) = e−rT (N (d2(Ki ))− N (d2(Ki+1)))

FIX2 : dST =

(ln(

ST + 0.05S0

))2

BEX2 : dST =

(ln(

ST + 0.05S0

))2

IST<S0

BUX2 : dST =

(ln(

ST + 0.05S0

))2

IST>S0

SIX =BEXBUX

TIX : dST =−1

(

√∑Ss=1 Φs

(ln(

Ks+0.05S0

))2)3

(ln(

ST + 0.05S0

))3

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SIX

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SIX

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SIX and VIX

What’s the contemporaneous relationship between the risk-neutralvolatility and skewness?

• Dennis and Mayhew (2002)

• Neuberger (2012)

• Han (2008)

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SIX and VIX

∆SIXt Coefficient Std. Error t-Statistic Prob. Adj. R2

∆VIXt 0.0479 0.0041 11.7741 0.0000 0.3043Intercept 0.0000 0.0000 0.7681 0.4425∆SKEWt

∆VIXt -0.5098 0.0467 -10.9212 0.0000 0.0525Intercept 0.0001 0.0013 0.0882 0.9297

∆SIXt = α0 + α1∆VIXt + εt∆SKEWt = β0 + β1∆VIXt + εt

(1)

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SIX and SPX

VIX responds differently to a decrease in SPX return from anincrease (Whaley, 2009). How about SIX and SKEW?

∆SIXt Coefficient Std. Error t-Statistic Prob. Adj. R2

∆SPXt -0.0840 0.0169 -4.9788 0.0000 0.1517∆ISPX−t 0.0022 0.0003 7.9525 0.0000Intercept -0.0010 0.0001 -6.6267 0.0000∆SKEWt

∆SPXt 2.6165 0.4504 5.8095 0.0000 0.0817∆ISPX−t -0.0106 0.0081 -1.3062 0.1916Intercept 0.0046 0.0041 1.1116 0.2664

∆SIXt = α0 + α1∆SPXt + α2∆ISPX−t + εt∆SKEWt = β0 + β1∆VIXt + β2∆ISPX−t + εt

(2)

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SIX: Return Predictability

Do SIX or SKEW have any return predictability?

• Cremers and Weinbaum (2010)

• Xing, Zhang and Zhao (2010)

• Rehman and Vilkov (2012)

• Conrad, Dittmar and Ghysels (2012)

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SIX: Return Predictability

∆SPXt+2 Coefficient Std. Error t-Statistic Prob. Adj. R2

∆SIXt -0.9370 0.0677 -13.8407 0.0000 0.0739Intercept 0.0004 0.0003 1.1190 0.2632∆SKEWt 0.0256 0.0027 9.4709 0.0000 0.0359Intercept 0.0003 0.0004 0.9746 0.3298

∆SPXt+7∆SIXt -0.8885 0.0760 -11.6838 0.0000 0.0296Intercept 0.0009 0.0008 1.1220 0.2619∆SKEWt 0.0278 0.0032 8.7282 0.0000 0.0188Intercept 0.0009 0.0008 1.0597 0.2893

∆SPXt+30∆SIXt -0.7970 0.1205 -6.6133 0.0000 0.0059Intercept 0.0038 0.0024 1.5985 0.1100∆SKEWt 0.0214 0.0046 4.6375 0.0000 0.0026Intercept 0.0039 0.0024 1.6240 0.1045

∆SPXt+i = α0 + α1∆SIXt + εt∆SPXt+i = β0 + β1∆SKEWt + εt

(3)

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SIX and Realized Skewness

Can SIX or SKEW forecast the physical skewness?

RSkew30 Coefficient Std. Error t-Statistic Prob. Adj. R2

SIX -1.0539 0.3088 -3.4135 0.0006 0.0214Intercept 1.2536 0.3762 3.3327 0.0009SKEW -0.0136 0.0394 -0.3457 0.7296 -0.0001Intercept 0.0156 0.0716 0.2176 0.8278

RSkewt,t+30 = α0 + α1SIXt + εtRSkewt,t+30 = γ0 + γ1SKEWt + εt

(4)

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SIXD in DJIA

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SIXN in NDX

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something interesting (perhaps)

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Future Research

Way too many ...

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Future Research

• Examine the intra-day data on S&P 500 Index options tosee if these results still hold.

• Extend this study to other markets: ASX 200, etc.

• Other Applications

• Investigation on the impact of investor fear levels (measuredby VIX) on the earnings management behaviour of US com-panies (Gassen and Markarian, 2009).

• Relationship between changes in expected market volatility(measured by VIX) and net equity fund flows to US equitymutual funds (Ederington and Golubeva, 2009).

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Thank you for your time and questions!

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