SPX and VIX: A Copula Approach

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SPX and VIX A Copula Approach

• Kuan-Pin Lin
• Copula Seminar, February 23, 2006

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SPX and VIX

• SPX Standard And Pools 500 Stock Index
• VIX Implied Volatility Index
• Daily Time Series 1990.1.2 to 2004.12.31
• Questions
• Are They Correlated? How?
• Is There Tail Dependence?
• Does the Dependence Time-Varying?

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Notations

• SPX ? ln(SPX) ? R ? y ? v ? 0,1
• VIX ? ln(VIX) ? V ? x ? u ? 0,1

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SPX

• SPX is the Standard and Pools 500 Index. It
  represents 70 of U.S. publicly traded companies
  listed on NYSE. The index is an average of the
  share prices weighted by the companys market
  capitalization.

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VIX

• VIX is the Chicago Board of Trades index for
  implied volatility. It is calculated based on the
  daily at-the-money prices in both current and
  future contract periods. VIX or implied
  volatility is not about stock price swings, but
  about the associated option swings.

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ln(SPX) and R 100ln(SPX/SPX-1)

• ln(SPX) is non-stationary.
• R 100ln(SPX/SPX-1), the rate of returns.
• R is stationary.

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ln(SPX) and R 100ln(SPX/SPX-1)
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ln(VIX) and V 100ln(VIX/VIX-1)

• ln(VIX) is non-stationary (although VIX is
  stationary).
• V 100ln(VIX/VIX-1), the rate of volatility
  change.
• V is stationary.

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ln(VIX) and V 100ln(VIX/VIX-1)
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R and V are Stationary but Non-Normal
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The Univariate Time Series Model
Let Y be either R or V.   Conditional Mean
Equation--AR(p) Yt b0 b Zt r1Yt-1
rpYt-p et et stut, where ut nid(0,1)
Conditional Variance EquationAsymmetric
GARCH(1,1) st2 g0 g Zt d1st-12 (de da
Dt-1) et-12 Dt asymmetry 1 if etlt0 0
otherwise. Zt is the exogenous quantitative or
qualitative variable(s). In this study, Zt
Zt1, Zt2 where Zt1 shift 1 if t gt
1997.6.30 0 otherwise. Zt2 trend
(t-1997.6.30)/(2004.12.31-1997.7.1) if t gt
1997.6.30 0 otherwise. 
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Quasi-Maximum-Likelihood Estimation
Define ut et/st. Log-likelihood function
?t1,2,,N lnf(ut)/st   ll(q) ½ N ln(2p)
- ½ ?t1,2,,N ln(st2) ½ ?t1,2,,N
(et2/st2) Quasi-Maximum-Likelihood Estimation is
robust with respect to the potential model
misspecification due to non-normality in the
data q (b,r,g,d) maxarg q ll(q)
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QML Estimates of Model Parameters
Note Numbers in parentheses are estimated
standard errors.
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Summary of Univariate Analysis I

• There is 2-3 week or 12-day autocorrelation in
  the mean, for both R and V.
• V has shifted down in the mean since 1997.7.
• The mean of R has not changed.

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Summary of Univariate Analysis II

• There are persistence and asymmetry in the
  variance, for both R and V.
• V has negative asymmetry in the variance, while R
  has positive asymmetry.
• Both variances have shifted since 1997.7. For R,
  it has shifted upward, while the variance of V
  has shifted down.
• Negative trend since 1997.7 in the variance is
  found for both R and V.

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QML Residual Analysis
Note First 12 observations are deleted due to 12
lags used in the model estimation.
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Standardized Residuals (Shocks)
yt QML estimates of (et/st) for R (top) xt
QML estimates of (et/st) for V (bottom)
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y and x
Numbers in parentheses are p-values. K-S test
critical values (N3771) 10, 0.019892651 5,
0.022070442 1 0.026458526
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Summary of Univariate Analysis II

• Standardized residuals y and x are i.i.d.
• They are non-normal.
• They are uniformly distributed.
• Ready for copula-based correlation analysis.

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Scatter Plot of (x,y) x on horizontal and y on
vertical axis
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Scatter Plot of (u,v) uFX(x) on horizontal and
vFY(y) on vertical axis
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Unconditional Correlations of (y,x)

• Pearson c
• Kendall t
• Spearman r

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Unconditional Correlations of (y,x)

• y (SPX shocks) and x (VIX shocks) are negatively
  correlated.
• Is there a tail dependency?

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The Bivariate Copula Model

• Sklars TheoremF(x,y) C(u,v)u FX(x) in
  0,1v FY(y) in 0,1
• F(x,ya)C(FX(x),FY(y)a)F(FX-1(u),FY
  -1(v)a)where the estimated or empirical margins
  areu FX(x), v FY(y). a is the parameter(s)
  in the copula C.

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Likelihood Function

• Since most members of Archimedian copula family
  assume positive dependence, we re-defineyt
  estimates of (et/st) (shocks of returns based on
  SPX)xt estimates of (-et/st) (shocks of
  negative change in VIX)
• Assuming independent sample observations of
  (x,y), the likelihood function is L ?(x,y)
  f(x,ya)with f(x,ya) ?2F(x,ya)/?x?y
  (?2C/?u?v)(?FX/?x)(?FY/?y) c(FX(x),FY(y)a)
  fX(x) fY(y)where fX, fY, and c(u,va) are the
  density functions.

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Maximum Likelihood Estimation

• The log-likelihood function isln L(aXx,Yy)
  ln c(FX(x),FY(y)a) ln fX(x) ln fY(y)
• Since fX and fY are the estimated density
  functions of X and Y from the first stage of
  univariate model estimation, they are independent
  of the unknown association parameter a in this
  stage. We have,a arg max ?(x,y) ln
  L(aXx,Yy) arg max ?(x,y) ln
  c(FX(x),FY(y)a)
• Because of tail dependence both in the left and
  right, we assume the copula takes the mixture of
  Gumbel and Gumbel Survival copulas.

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Gumbel Copula

• CG(u,va) exp-(-ln u)a (-ln v)a1/a, where
  agt1. If a 1, u and v are independent.
• cG(u,va) (1a) uv-1-a u-a v-a - 1 -21/a
• ln cG(u,va) ln(1a) (a1)(ln u ln v)
  (1/a2) lnu-a v-a - 1
• Kendalls t and Tail Dependency tU 2-21/a, tL
  0a 1/(1-t) ln(2)/ln(2-tL)

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Gumbel Survival Copula

• CGS(u,vb) u v 1 exp-(-ln(1-u))b
  (-ln(1-v))b1/b, where bgt1. If b 1, u and v
  are independent.
• cGS(u,vb) cG(1-u,1-vb)
• Kendalls t and Tail DependencytL 2-21/b, tU
  0b 1/(1-t) ln(2)/ln(2-lU)

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Gumbel Mixture Copula

• CGM(u,va,b,w) w CG(u,va) (1-w) CGS(u,vb)
  where 0 ? w ? 1
• Kendalls t w(1-1/a)(1-w)(1-1/b)
• Tail Dependency tU w(2-21/a)tL
  (1-w)(2-21/b)

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ML Estimates of Copula Parameters
Note Numbers in parentheses are estimated
standard errors.
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Summary of Copula Analysis

• Gumbel mixture copula fits the data better than
  either Gumbel or Gumbel survival.
• There is negative dependence (0.476) between y
  (SPX shocks) and x (VIX shocks).
• There is negative tail dependence more on the
  lower-left (0.47) than on the upper-right corner
  (0.09).

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Conditional Correlations

• Does the correlation change over time?
• Can we forecast correlation based on the
  historical dependence relationship?
• Study of time-varying correlation is important
  for portfolio diversification strategy and for
  risk management.

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100-Day Rolling Window Correlations
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Discrete Statistics of Rolling Correlations
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Conditional Correlation Model

• To allow for time-varying dependence between x
  and y, the autoregressive conditional correlation
  can be formulated similar to the structure of
  conditional variance as follows (Patton
  2006)tt w0w1 tt-1w2? s1,,12 ut-s
  vt-s/12
• The last term is the forcing variable computed
  from the 12-day average of probability
  differentials. This is because the 12-lag AR
  process was used in constructing the data (u,v).

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Asymmetric Conditional Correlation Model

• Further, to allow for asymmetry in the
  conditional correlation equationtt w0 w1
  tt-1 (w2 wd Dt-1) ? s1,,12 ut-s
  vt-s/12 where Dt 1 if utlt0.5 and vtlt0.5 0
  otherwise.

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Future Research

• Conditional Copula.
• Copula-based Asymmetric Conditional Correlations.
  (To be continued)
• Time-Varying Multivariate Dependence is a more
  realistic and practical application.

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References

• A. J. Patton, Modelling Asymmetric Exchange Rate
  Dependence, forthcoming in the Journal of
  International Economics, 2006.
• A. J. Patton, Estimation of Multivariate Models
  for Time Series of Possibly Different Lengths,
  forthcoming in the Journal of Applied
  Econometrics, 2006.
• G. Tsafack, Dependence Structural and Extreme
  Comovements in International Equity and Bonds
  Markets, Universite de Montreal, CIRANO and
  CIREQ, January 2006.