Volatility and Skewness Indices Using State-Preference Pricing - - PowerPoint PPT Presentation

Volatility and Skewness Indices Using State-Preference Pricing

Zhangxin Frank Liu Finance Theory Module 2 March 16th, 2013

1 / 71

Outline

1

FIX the VIX

2

BEX and BUX

3

SIX is SICK

4

Future Research

2 / 71

Motivation I

• WHY CARE ABOUT VOLATILITY AT ALL?

“. . . what distinguishes financial economics is the central role that uncertainty plays in both financial theory and its empirical implementation. The start- ing point for every financial model is the uncer- tainty facing investors, . . . Indeed, in the absence

• f uncertainty, the problems of financial economics

reduce to exercises in basic microeconomics.” Campbell, Lo and MacKinlay (1997)

3 / 71

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 current
• bservables.
• GARCH (Bollerslev, 1986; Taylor, 1986):

f(ω1 ¯ V, ω2 ˆ Vt−1, ω3ǫ; t)

4 / 71

Volatility Forecasting

• . . .
• Implied volatility
• Implied from option prices
• Invert the analytical pricing formula from some option pricing

models (if exist); or follow some model-free approaches (Du- mas, Fleming and Whaley (1998)).

5 / 71

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 eight

near-the-money options at the two nearest maturities

• Artificially induced upward bias from the CBOE trading day

conversion TV(t, T) = ATMV(t, T) √ NC √ NT ≡ ATMV(t, T) √ NC

• NC − 2 × int(NC/7)

6 / 71

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 day

conversion

• Trading of VIX futures contracts from May 2004; VIX options

from February 2006

7 / 71

VIX 101

• VIX formula:

σ2

j = 2

Tj

• i

∆Ki K 2

i

erTQ(Ki) − 1 Tj F K0 − 1 2 ∀ j = 1, 2 VIX = 100

• 365

30

• T1σ2

1

NT2 − N30 NT2 − NT1 + T2σ2

2

N30 − NT1 NT2 − NT1

• Principal of DDKZ (1999): realized volatility can be

captured by the dynamic hedging of a log contract.

8 / 71

VIX 101

Derivation: Theoretical definition of realized variance for a given price history is V = 1 T T σ2(t, . . .) dt Think about pricing a variance swap: F = E(e−rT(V − K)) For a zero initial value, Kvar = E(V) = 1 T E T σ2(t, . . .) dt

• 9 / 71

VIX 101

DDKZ (1999) only assumes that future underlyer evolution is diffusive (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) = 1 2σ2dt

• r

σ2dt = 2 dSt St − d(ln St)

• 10 / 71

VIX 101

Now we have    Kvar = 1 T E T

0 σ2(t, . . .) dt

• σ2dt

= 2

• dSt

St − d(ln St)

• ∴ E(V) = Kvar = 2

T E T dSt St − T d(ln St)

• = 2

T E T dSt St

• A

− 2 T E

• ln ST

S0

• B

11 / 71

VIX 101

A = E T (r dt + σ(t, . . .) dZt)

• Zt ∼ N(0, t)

= rT B = E

• ln ST

S0

• = E
• ln ST

S∗

• Log contract

+ ln S∗ S0 where S∗ is some arbitrary number to define the boundary of OTM calls and puts.

12 / 71

VIX 101

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

OTM puts

• S∗

1 K 2 max(K − ST, 0) dK +

OTM calls

• ∞

S∗

1 K 2 max(ST − K, 0) dK =

• S∗

ST 1 K 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

• ln ST

S∗

• = E

ST − S∗ S∗ − Π

• 13 / 71

VIX 101

Kvar = 2 T (rT) − 2 T

• E

ST − S∗ S∗ − Π

• + ln S∗

S0

• = 2

T

• rT − E

ST S∗ − 1

• + E

S∗ 1 K 2 max(K − ST, 0) dK+ ∞

S∗

1 K 2 max(ST − K, 0) dK

• − ln S∗

S0

• = 2

T     

a

• rT −

S0erT S∗ − 1

• − ln S∗

S0 +

b

• erT

S∗ P(K) K 2 dK + erT ∞

S∗

C(K) K 2 dK     

14 / 71

VIX 101

a = 2 T

• ln
• erT

− S0erT S∗ − 1

• − ln(S∗) + ln(S0)
• = 2

T

• ln

S0erT S∗

• −

S0erT S∗ − 1

• = 2

T

• ln

F S∗

• −

F S∗ − 1

• where F = S0erT

≈ 2 T      F S∗ − 1

• − 1

2 F S∗ − 1 2

• Taylor expansion of ln(F/S∗)

− F S∗ − 1

• 

    = − 1 T F S∗ − 1 2 where S∗ ≡ K0

15 / 71

VIX 101

b = 2erT T S∗ 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 2

i

erTQ(Ki)

• discretization error

Hence we obtain the VIX formula Kvar = E(V) ≈ 2 T

• i

∆Ki K 2

i

erTQ(Ki)

• − 1

T F S∗ − 1 2 = σ2

VIX

16 / 71

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 hedging

purposes

17 / 71

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?

18 / 71

FIX the VIX Let’s FIX the VIX.

19 / 71

FIX the VIX

• A forward-looking volatility index (FIX) as a proxy for market

realized 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 + 0.05 S0 2

• How to define state prices?

20 / 71

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 with

strike M − ∆M, long one call with strike M + ∆M and short two 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 ST − (M − ∆M) ST − (M − ∆M) ST − (M − ∆M) Short 2 calls with M −2(ST − M) −2(ST − M) Long 1 call with M + ∆M ST − (M + ∆M) Total at t = T ∆M + (ST − M) ∆M − (ST − M)

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

21 / 71

FIX the VIX

• Thus the cost of butterfly spread that produces a payment
• f $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 → 0

yields: 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

22 / 71

FIX the VIX

• Thus the price of a security f with payoff df

M at some future

state M is

Pf =

• M

df

M

• payoff

P(M; dM)

• state price

=

• M

df

M

• 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)+) = ∞ (K − ST)+

• payoff

e−rTf(ST) dST

• state price

= K (K − ST)e−rTf(ST) dST

23 / 71

FIX the VIX

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

∂P ∂K = ∂ ∂K K e−rT(K − ST)f(ST) dST

• = e−rT
• (K − K)f(K) +

K f(ST)dST

• = e−rTF(K)

where F(·) is the risk-neutral distribution function. Take the par- tial derivative w.r.t K again: ∂2P ∂K 2 = ∂ ∂K

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

ity), P = K (K −ST)e−rTf(ST)

✿✿✿✿✿✿✿✿ dST =

K (K −ST)

• ∂2P

∂K 2

• K=ST
• ✿✿✿✿✿✿✿✿✿✿✿✿

dST

24 / 71

FIX the VIX

• How to approximate the second derivative?

Model-free approach: ∂2C(K, T) ∂X 2 ≈ Ci−1 − 2Ci + Ci+1 (∆Ki)2

• r ≈ −Ci−2 + 16Ci−1 − 30Ci + 16Ci+1 − Ci+2

12(∆Ki)2 (Eberly, 2008)

25 / 71

FIX the VIX

• It works with simulated data:

∆K True σ CBOE VIX Model-Free State-Price 0.1 0.30 0.29999 0.300195 5 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%.

26 / 71

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), still possible to see Pi−1 − 2Pi + Pi+1 < 0.

27 / 71

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 of

implied 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 and Lee (2003, 2009)).

28 / 71

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 Reserve

Bulletin), adjusting for the dividends (Option Metrics), as the risk-free interest rates.

29 / 71

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

30 / 71

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 the

10 cents interval.

• ln

statei + 0.05 S0 2

• FIX is a manipulation-free measure.

31 / 71

FIX the VIX

• Highly correlated with VIX:

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

32 / 71

FIX the VIX

Moments FIX VIX VXO RVol30 RVol22 Mean 20.44 22.21 23.14 18.71 18.15 Median 19.42 21.01 22.25 16.43 15.94 Maximum 84.44 80.86 87.24 90.57 87.88 Minimum 8.31 9.89 0.00 5.86 5.68

• Std. Dev.

8.33 8.73 9.57 10.69 10.38 Skewness 2.02 1.91 1.70 2.74 2.74 Kurtosis 10.42 9.53 8.59 14.24 14.24 Auto 0.97 0.98 0.97 0.99 0.99 Moments ∆FIX ∆VIX ∆VXO ∆RVol30 ∆RVol22 Mean 0.00 0.00 0.00 0.00 0.00 Median 0.00 0.00 0.00 0.00 0.00 Maximum 0.54 0.50 0.53 0.66 0.66 Minimum

• 0.55
• 0.35
• 0.38
• 0.39
• 0.39
• Std. Dev.

0.07 0.06 0.07 0.06 0.06 Skewness 0.31 0.50 0.47 0.74 0.74 Kurtosis 7.63 6.56 6.81 18.43 18.43 Auto

• 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

• 33 / 71

FIX the VIX

• Whaley (2009) documents that the change in VIX rises at a

higher absolute rate of change when there is a market fall than an upswing. How about FIX? RFIXt = α0 + α1RSPXt + α2RSPX−

t + ǫ1

RVIXt = β0 + β1RSPXt + β2RSPX−

t + ǫ2

RFIXt Coefficient

• Std. Error

t-Statistics Prob.

• Adj. R2

RSPXt

• 3.5732

0.1068

• 33.4548

0.0000 0.5100 RSPX−

t

• 0.4764

0.1687

• 2.8231

0.0048 Intercept

• 0.0014

0.0011

• 1.2175

0.2235 RVIXt RSPXt

• 3.0212

0.0859

• 35.1775

0.0000 0.5651 RSPX−

t

• 0.7765

0.1357

• 5.7227

0.0000 Intercept

• 0.0028

0.0009

• 3.1278

0.0018 34 / 71

FIX the VIX

• Predictability of the market realized volatility over the next

30 days against VIX and VXO: RVolt,t+30 = α0 + α1FIXt + ǫ1 RVolt,t+30 = β0 + β1VIXt + ǫ2 RVolt,t+30 = γ0 + γ1VXOt + ǫ3

RVol30 Coefficient

• Std. Error

t-Statistics Prob.

• Adj. R2

Wald Test FIX 0.9920 0.0655 15.1509 0.0000 0.9032 Intercept

• 1.5834

1.1774

• 1.3449

0.1787 0.6010 VIX 0.9295 0.0601 15.4727 0.0000 0.2409 Intercept

• 1.9479

1.1761

• 1.6562

0.0978 0.5788 VXO 0.8556 0.0562 15.2183 0.0000 0.0102 Intercept

• 1.1069

1.1416

• 0.9696

0.3323 0.5893 RVol22 FIX 0.9626 0.0635 15.1509 0.0000 0.5557 Intercept

• 1.5364

1.1424

• 1.3449

0.1787 0.6010 VIX 0.9019 0.0583 15.4727 0.0000 0.0926 Intercept

• 1.8900

1.1412

• 1.6562

0.0978 0.5788 VXO 0.8302 0.0546 15.2183 0.0000 0.0019 Intercept

• 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. 35 / 71

FIX the VIX

• Predictability of the market realized volatility over the next

30 days against others:

GARCH(1,1)vol = 100 ×

• GARCH(1,1)var × 365

30

RVolt,t+30 = α0 + α1GARCH(1,1)t + ǫ1 RVolt,t+30 = β0 + β1BAMLt + ǫ2 RVolt,t+30 = γ0 + γ1JPMt + ǫ3

RVol30 Coefficient

• Std. Error

t-Statistics Prob.

• Adj. R2

Wald Test FIX 0.9978 0.0660 15.1230 0.0000 0.9728 Intercept

• 1.6155

1.1784

• 1.3710

0.1705 0.6043 GARCH(1,1) 2.6767 0.2583 10.3626 0.0000 0.0000 Intercept 3.4280 1.2960 2.6451 0.0082 0.4522 BAML 0.9287 0.0584 15.9078 0.0000 0.2221 Intercept

• 1.5305

1.1138

• 1.3741

0.1695 0.5941 JPMorgan 0.9478 0.0639 14.8249 0.0000 0.4141 Intercept

• 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. 36 / 71

FIXD in DJIA

37 / 71

FIXN in NDX

38 / 71

Motivation II

What’s missing in VIX (Volatility)?

39 / 71

Motivation II

where µORCL = 41.1% (Estrada, 2006).

40 / 71

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 both

sides.

• It may be more interested in what the proportion of an upside

potential 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)

42 / 71

LPM 101

• Lower Partial Moments Framework
• nth order LPM and UPM for some continuous distribution F

is 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)

43 / 71

LPM 101

• . . .
• Mean-Semivariance model (Hogan and Warren, 1974; Bawa

and Lindenberg, 1977)

• Psychological studies of Mao (1970a, 1970b), Unser (2000)

and Veld and Veld-Merkoulova (2008) support the LPM over variance 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)

44 / 71

LPM 101

• . . .
• Andersen and Bondarenko (2009):
• Data: AB use CME option prices of the S&P 500 futures. We

use 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. We use state-preference pricing to estimate volatilities.

• Extendibility.

45 / 71

BEX and BUX

• Decompose FIX into a forward-looking lower partial mo-

ment volatility index as a proxy for market downturn, which we denote the bear index (BEX); and an upper partial mo- ment counterpart, the bull index (BUX).

46 / 71

BEX and BUX

• BEX and BUX share the same state price at each state

with modified payoffs: BEX2

t =

• i

Φi ln Si + 0.05 SPXt 2 ∀Si ≤ SPXte(rf −d)∗30/365 BUX2

t =

• i

Φi ln Si + 0.05 SPXt 2 ∀Si > SPXte(rf −d)∗30/365

47 / 71

BEX and BUX

• Predictability of the market realized LPM volatility over the

next 30 days: RVolLPM

t,t+30

= α0 + α1BEXt + ǫ1 RVolLPM

t,t+30

= β0 + β1VIXt + ǫ2

RVolLPM

30

Coefficient

• Std. Error

t-Statistics Prob.

• Adj. R2

Wald Test BEX 0.8421 0.0713 11.8090 0.0000 0.0269 Intercept

• 0.3446

0.9900

• 0.3481

0.7278 0.4296 VIX 0.6259 0.0515 12.1632 0.0000 0.0000 Intercept

• 1.0064

1.0190

• 0.9876

0.3234 0.4172 RVolLPM

22

BEX 0.8171 0.0692 11.8090 0.0000 0.0082 Intercept

• 0.3344

0.9606

• 0.3481

0.7278 0.4296 VIX 0.6073 0.0499 12.1632 0.0000 0.0000 Intercept

• 0.9765

0.9887

• 0.9876

0.3234 0.4172 48 / 71

BEX and BUX

• Predictability of the market realized UPM volatility over the

next 30 days: RVolUPM

t,t+30

= γ0 + γ1BUXt + ǫ3 RVolUPM

t,t+30

= λ0 + λ1VIXt + ǫ4

RVolUPM

30

Coefficient

• Std. Error

t-Statistics Prob.

• Adj. R2

Wald Test BUX 1.1765 0.0685 17.1839 0.0000 0.0100 Intercept

• 2.2388

0.7890

• 2.8373

0.0046 0.6857 VIX 0.6777 0.0397 17.0599 0.0000 0.0000 Intercept

• 1.9310

0.7732

• 2.4974

0.0126 0.6567 RVolUPM

22

BUX 1.1415 0.0664 17.1839 0.0000 0.0332 Intercept

• 2.1723

0.7656

• 2.8373

0.0046 0.6857 VIX 0.6576 0.0385 17.0599 0.0000 0.0000 Intercept

• 1.8737

0.7503

• 2.4974

0.0126 0.6567 49 / 71

BEX and BUX

• Daily S&P 500 Index return may be better explained by the

contemporaneous change of BEX and BUX: RSPXt = α0 + α1RBEXt + α2RBUXt + ǫ1 RSPXt = β0 + β1RVIXt + ǫ2

RSPXt Coefficient Std.Error t-Statistics Prob.

• Adj. R2

RBEXt 0.3369 0.0764 4.4112 0.0000 RBUXt

• 0.4903

0.0824

• 5.9482

0.0000 Intercept 0.0002 0.0001 1.2863 0.1984 0.5255 RSPXt RVIXt

• 0.1640

0.0067

• 24.6618

0.0000 Intercept 0.0002 0.0001 1.4417 0.1495 0.5614 50 / 71

BEX and BUX

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

VIX: RBEXt = α0 + α1RSPXt + α2RSPX−

t + ǫ1

RVIXt = β0 + β1RSPXt + β2RSPX−

t + ǫ2

RBEXt Coefficient

• Std. Error

t-Statistics Prob.

• Adj. R2

RSPXt

• 3.6355

0.1096

• 33.1633

0.0000 RSPX−

t

• 0.4761

0.1732

• 2.7489

0.0060 Intercept

• 0.0013

0.0011

• 1.1600

0.2461 0.5051 RVIXt RSPXt

• 3.0212

0.0859

• 35.1775

0.0000 RSPX−

t

• 0.7765

0.1357

• 5.7227

0.0000 Intercept

• 0.0028

0.0009

• 3.1278

0.0018 0.5651 51 / 71

BEX and BUX

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

rise 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%

52 / 71

Motivation III

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

53 / 71

CBOE SKEW

• A brief history of CBOE SKEW
• Based on Bakshi, Kapadia and Madan (2003): any security

payoff 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)

54 / 71

CBOE SKEW

SKEW := 100 − 10 × S S = EQ(R3) − 3EQ(R)EQ(R2) + 2E3

Q(R)

(EQ(R2) − E2

Q(R))3/2

=: P3 − 3P1P2 + 2P3

1

(P2 − P2

1)3/2

P1 =

• i

−∆Ki K 2

i

erT Q(Ki) −

• 1 + ln

F0 K0

• − F0

K0

• ε1

P2 =

• i

2∆Ki K 2

i

erT Q(Ki)

• 1 − ln

Ki F0

• +2 ln

K0 F0 F0 K0 − 1

• + 1

2 ln2 K0 F0

• ε2

P3 =

• i

3∆Ki K 2

i

erT Q(Ki)

• 2 ln

Ki F0

• − ln2

Ki F0

• +3 ln2

K0 F0 1 3 ln K0 F0

• − 1 + F0

K0

• ε3

55 / 71

SIX

• Skewness is hard to measure precisely (Neuberger, 2012)
• A simple solution: the ratio of BUX to BEX forms a market

symmetric index (SIX).

• Why do we need SIX? If market returns distribution is sym-

metric, then we expect BUX BEX = 1

56 / 71

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 + 0.05 S0 2 BEX2 : dST =

• ln

ST + 0.05 S0 2 IST <S0 BUX2 : dST =

• ln

ST + 0.05 S0 2 IST >S0 SIX = BEX BUX TIX : dST = −1 (

• S

s=1 Φs

• ln
• Ks+0.05

S0

2 )3

• ln

ST + 0.05 S0 3

57 / 71

SIX

58 / 71

SIX

59 / 71

SIX and VIX

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

• Dennis and Mayhew (2002)
• Neuberger (2012)
• Han (2008)

60 / 71

SIX and VIX

∆SIXt Coefficient

• Std. Error

t-Statistic Prob.

• Adj. R2

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

• 0.5098

0.0467

• 10.9212

0.0000 0.0525 Intercept 0.0001 0.0013 0.0882 0.9297 ∆SIXt = α0 + α1∆VIXt + ǫt ∆SKEWt = β0 + β1∆VIXt + εt (1)

61 / 71

SIX and SPX

VIX responds differently to a decrease in SPX return from an increase (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 0.0000 Intercept

• 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

0.1916 Intercept 0.0046 0.0041 1.1116 0.2664 ∆SIXt = α0 + α1∆SPXt + α2∆ISPX−

t + ǫt

∆SKEWt = β0 + β1∆VIXt + β2∆ISPX−

t + εt

(2)

62 / 71

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)

63 / 71

SIX: Return Predictability

∆SPXt+2 Coefficient

• Std. Error

t-Statistic Prob.

• Adj. R2

∆SIXt

• 0.9370

0.0677

• 13.8407

0.0000 0.0739 Intercept 0.0004 0.0003 1.1190 0.2632 ∆SKEWt 0.0256 0.0027 9.4709 0.0000 0.0359 Intercept 0.0003 0.0004 0.9746 0.3298 ∆SPXt+7 ∆SIXt

• 0.8885

0.0760

• 11.6838

0.0000 0.0296 Intercept 0.0009 0.0008 1.1220 0.2619 ∆SKEWt 0.0278 0.0032 8.7282 0.0000 0.0188 Intercept 0.0009 0.0008 1.0597 0.2893 ∆SPXt+30 ∆SIXt

• 0.7970

0.1205

• 6.6133

0.0000 0.0059 Intercept 0.0038 0.0024 1.5985 0.1100 ∆SKEWt 0.0214 0.0046 4.6375 0.0000 0.0026 Intercept 0.0039 0.0024 1.6240 0.1045 ∆SPXt+i = α0 + α1∆SIXt + ǫt ∆SPXt+i = β0 + β1∆SKEWt + εt (3) 64 / 71

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 0.0214 Intercept 1.2536 0.3762 3.3327 0.0009 SKEW

• 0.0136

0.0394

• 0.3457

0.7296

• 0.0001

Intercept 0.0156 0.0716 0.2176 0.8278 RSkewt,t+30 = α0 + α1SIXt + ǫt RSkewt,t+30 = γ0 + γ1SKEWt + εt (4)

65 / 71

SIXD in DJIA

66 / 71

SIXN in NDX

67 / 71

something interesting (perhaps)

68 / 71

Future Research

Way too many ...

69 / 71

Future Research

• Examine the intra-day data on S&P 500 Index options to

see if these results still hold.

• Extend this study to other markets: ASX 200, etc.
• Other Applications
• Investigation on the impact of investor fear levels (measured

by 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 equity mutual funds (Ederington and Golubeva, 2009).

70 / 71

Thank you for your time and questions!

71 / 71