Applications of Local Gaussian Correlation in Finance

Transcription

1 Applications of Local Gaussian Correlation in Finance and towards a parametric version of the LGC Bård Støve Universit of Bergen, Norwa Department of Mathematics Joint work in progress with Dag Tjøstheim, Håkon Otneim, Geir D. Berentsen, Karl O. Hufthammer and Jan Bulla BBB - 17

2 Financial returns Figure : Non-standardised dail returns Jul 1987-Jul 11 for the market indices for German, the United Kingdom and the United States (in this order, from top to bottom). LGC in Finance B. Støve Universit of Bergen /

3 Descriptive statistics Statistic S&P 5 FTSE 1 DAX 3 Mean..1.3 Variance Skewness Kurtosis Maimum Minimum Table : Summar inde statistics for dail data returns from Jul 1987 to Jul 11 - in total 675 observations. LGC in Finance B. Støve Universit of Bergen 3 /

4 Local Gaussian approimation The central idea: Approimate a bivariate densit f of (X 1, X ) at a point = ( 1, ) b a bivariate Gaussian densit ψ(v, µ(), Σ()). We take the correlation ρ() parameter of that Gaussian densit as our measure of local dependence, and we call it the local Gaussian correlation. As we move to another point = ( 1, ) of f, another Gaussian ψ(v, µ( ), Σ( )) is required to approimate f in a neighbourhood A of. In this wa the dependence in f is described b the famil of Gaussian distributions {ψ(v, µ(), Σ())} and the associated correlations {ρ()}. Note: µ() = (µ 1 (), µ ()) T is the local mean vector and Σ() = (σ ij ()) is the local covariance matri. The local correlation at the point is ρ() =. Cfr. Tjøstheim & Hufthammer (13). σ 1() σ 11 ()σ 1 () LGC in Finance B. Støve Universit of Bergen /

5 LGC: S&P 5 & FTSE 1. Dail log returns Jul Jul 11. Figure : Local Gaussian correlation map for FTSE 1 against S&P 5 returns. LGC in Finance B. Støve Universit of Bergen 5 /

6 Diagonal LGC Figure : Diagonal local Gaussian correlation values for FTSE 1 against S&P 5 returns. LGC in Finance B. Støve Universit of Bergen 6 /

7 Diagonal LGC for the three markets using standardised returns Figure : Local Gaussian correlation curves with approimate 9% confidence intervals based on 1 bootstrap replications. LGC in Finance B. Støve Universit of Bergen 7 /

8 Background The degree of comovement of financial returns is necessar to know/estimate for constructing a well diversified portfolio - and for evaluating risk. It is well documented that there are asmmetries in the distribution of financial returns, e.g. Okimoto (8), Longin & Solnik (1), Campbell et al (), Chollete et al (9). In particular, there are often stronger dependence between returns of financial objects when the market is going down than when it is stable or going up. It does not seem to be consensus, however, as to how the asmmetries should be measured, quantitativel interpreted and tested for. The LGC is perhaps a viable alternative (Støve & Tjøstheim (1)). LGC in Finance B. Støve Universit of Bergen 8 /

9 Financial contagion LGC in Finance B. Støve Universit of Bergen 9 /

10 Financial contagion Contagion: Do financial markets become more interdependent during financial crisis? Contagion is defined as in Forbes and Rigobon (): a significant increase in cross-market linkages after a shock to one countr (or a group of countries). We stud the economic crisis of 7 9, and whether contagion has occurred from the US to other countries. Dail log returns of stock indices from to 9-7-8, i.e. 1 observations. The stable period: from to (678 observations) The turmoil period: from to (5 observations) LGC in Finance B. Støve Universit of Bergen 1 /

11 Market indices LGC in Finance B. Støve Universit of Bergen 11 /

12 Returns LGC in Finance B. Støve Universit of Bergen 1 /

13 Local Gaussian correlations US-UK US-German US-Norwa US-France LGC in Finance B. Støve Universit of Bergen 13 /

14 Contagion bootstrap test Test: H : ρ NC = ρ C against H 1 : ρ NC < ρ C. Observations d t = (X t, Y t ), i.e. {d 1,..., d T } (filtered returns data). 1 Draw at random & with replacement resamples {d 1,..., d T } b, b = 1,..., B. Divide into time periods NC and C and compute ˆρ NC ( i, i ) and ˆρ C ( i, i ) on a (diagonal) grid i = 1,..., n b. 3 Calculate test variable D 1 = 1 n n i=1 [ˆρ C ( i, i ) ˆρ NC ( i, i )]w( i, i ) b (w i = weight function) and find the distribution of D 1 (distribution under H ). For the observations {d 1,..., d T } calculate ˆρ NC ( i, i ), ˆρ C ( i, i ) and the test statistic D 1 = 1 n n i=1 [ˆρ C( i, i ) ˆρ NC ( i, i )]w( i, i ). The p-value in terms of the D 1 distribution is found. LGC in Finance B. Støve Universit of Bergen 1 /

15 Contagion during 7-9 crisis The plots provide some evidence that the dependence between the US and the European markets have increased during the 7-9 crisis compared to the stable period from Januar 5 to August 7. Test: H : ρ NC = ρ C against H 1 : ρ NC < ρ C. The bootstrap test gives (for B=1, and w i = 1 if i [.5,.5] and elsewhere): UK: p-value =.53 FR: p-value =.9 GE: p-value =.15 NO: p-value =.31 Reject H. Evidence of contagion. For more, see Støve, Tjøstheim and Hufthammer (1). LGC in Finance B. Støve Universit of Bergen 15 /

16 Time dnamics of the dependence structure for financial returns Figure : Diagonal local Gaussian correlations (with approimate 9% confidence intervals) between standardised returns from S&P 5 and FTSE 1, divided into five-ear long intervals from 1987 to 11. LGC in Finance B. Støve Universit of Bergen 16 /

17 Background Other studies have documented increasing trends in the dependence of international stock markets, see e.g. Okimoto (1) and Christoffersen et al (1). Using regime-switching models (to cater for the time dnamics) combined with copulas (to cater for the nonlinearities in the dependence structure). We aim for something similar; replacing copulas with the LGC. But; in order to use the framework of RS-models - we need a parametric version of the LGC (to the best of our knowledge). LGC in Finance B. Støve Universit of Bergen 17 /

18 Parametric LGC - work in progress Proposal for a smmetric LGC: Let ρ(, ) = A + φ( + ) φ( ). Then ρ(, ) = A, ρ(, ) = A + φ() and ρ(, ) = A φ(). The function φ can contain a number of parameters, for eample φ(z) = Furthermore, D (ep(γz + ep( γz)) + ep(γz) + ep( γz) φ() = D, and φ( ) = φ( ) = D. ρ(, ) = ρ(, ) = A + D D = A + D ρ(, ) = ρ(, ) = A + D D = A D. LGC in Finance B. Støve Universit of Bergen 18 /

19 Eample The colour scale goes from 1 (dark blue) to +1 (dark red). White is zero, but also values outside of [ 1, 1], because the functions, as defined above, can take values outside of this interval. This is perhaps also a restriction to possible parameter values that we should take in to consideration when optimizing. A =.5, D =.5, gamma = 1 A =, D =.8, gamma =.5 A =.5, D =.33, gamma = 1 Figure : Case 1 LGC in Finance B. Støve Universit of Bergen 19 /

20 Generalizations Different strengths of smmetr in = and = directions. 1 ρ(, ) = A + φ( + ) Bφ( ), where B > (but B = gives just one ridge and no valles). This can be combined with different growth rates: ρ(, ) = A + φ 1 ( + ) Bφ ( ), where φ 1 (z) = D(ep(γ 1z) + ep( γ 1 z)) + ep(γ 1 z) + ep( γ 1 z) φ (z) = D(ep(γ z) + ep( γ z)) + ep(γ z) + ep( γ z). LGC in Finance B. Støve Universit of Bergen /

21 Figure : Case 1-, with different strengths of smmetr in the - and the -directions as well as different growth rates. LGC in Finance B. Støve Universit of Bergen 1 / Eample A =, B =, D =.8, gamma =.5 A =, B =.5, D =.8, gamma =.5 A =.5, B =, D =.8, gamma =.5 Figure : Case 1-1, with different strengths of smmetr in the - and the -directions. A =, B = 1, D =.8, gamma1 =., gamma = 1 A =, B =.5, D = 1, gamma1 =.5, gamma =.5 A =, B = 1, D =.6, gamma1 =.8, gamma =.

22 Other suggestions 3. Instead of (1), one can have with ρ(, ) = A + φ 1 ( + ) φ ( ), φ ( z) = D i(ep(γz) + ep( γz)) + ep(γz) + ep( γz). This can of course be combined with ().. Saddle point not at (, ). This can be obtained with where for eample ρ(, ) = A + φ( + β 1 ) φ( β ), φ(z β) = D(ep(γ(z β)) + ep(γ(z β))) + ep(γ(z β)) + ep( γ(z β)). LGC in Finance B. Støve Universit of Bergen /

23 Eample A =, D1 =.8, D =., gamma = 1 A =.5, D1 = 1, D =.5, gamma =.5 A =, D1 =.6, D =.8, gamma =. Figure : Case 1-3, with different Ds. A =, D =.8, gamma =.5, beta1 = 1, beta = A =, D =.8, gamma =.5, beta1 =, beta = A =, D =.8, gamma =.5, beta1 = 1, beta = Figure : Case 1-, moving the saddle point. LGC in Finance B. Støve Universit of Bergen 3 /

24 Further suggestions 5. Changing the direction of the ridges and valles. In the original model, the ridge runs along =, and the valle along =. This can of course be changed b replacing + with α 1 + and = b α, which again can be combined with (). 6. Asmmetric behaviour along the ridges and valles. This can be obtained b letting ρ be asmmetric, for eample b letting φ(z) = D(ep(γ 1z) + ep(γ z)) + ep(γ 1 z) + ep( γ z), and again, this can be combined with all the other possibilities. LGC in Finance B. Støve Universit of Bergen /

25 Eample A =, D =.8, gamma =.5, alpha1 = 1, alpha = A =, D =.8, gamma =.5, alpha1 = 1, alpha = A =, D =.8, gamma =.5, alpha1 = 1, alpha = Figure : Case 1-5, changing the direction of the valles and ridges A =, D =.5, gamma1 =.3, gamma =.9 A =, D =.8, gamma1 =.5, gamma =.1 A =, D =.33, gamma1 =.6, gamma =.3 Figure : Case 1-6, asmmetric behaviour along ridges and valles LGC in Finance B. Støve Universit of Bergen 5 /

26 In general; Most of the changes outlined in (1)-(6) can be accomplished b taking ρ(, ) = A + D [ 1 ep(γ1 (α 1 + β 1 )) + ep( γ (α 1 + β 1 )) ] + ep(γ 1 (α 1 + β 1 )) + ep( γ (α 1 + β 1 )) D [ ep(γ3 (α + β )) + ep( γ (α + β )) ] + ep(γ 3 (α + β )) + ep( γ (α + β )) LGC in Finance B. Støve Universit of Bergen 6 /

27 Asmmetric - as in man financial data set Need a function ψ like ψ 1 (z) = D ep( γ(z β)) 1 + ep( γ(z β)), γ > and ψ (z) = Since there seems to be a ridge along =, a function could be with φ as before. ρ(, ) = A + ψ( + ) φ( ), D ep(γ(z β)) 1 + ep(γ(z β)), LGC in Finance B. Støve Universit of Bergen 7 /

28 Eample A =.33, D =.5, gamma =.5, beta =.3 A =, D =.6, gamma =.5, beta = A =.5, D =.8, gamma = 1, beta = 1 Figure : Case, asmmetric behaviour as in financial data LGC in Finance B. Støve Universit of Bergen 8 /

29 Estimation of PLGC - work in progress Work with the Gaussian pseudo-observations Z = (Z 1, Z ) = ( φ 1 (F 1,n (X 1 )), φ 1 (F,n (X ))), (1) X 1 and X are the original observations, φ is the standard normal distribution function, F i,n ( ) is the empirical distribution function. Define the function φ θ b 1 φ θ (z) = φ θ (z 1, z ) = π(1 ρ θ (z 1, z ) ) ( 1 ( = ep z (1 ρ θ (z 1, z ) ) 1 + z ) ) ρ θ (z 1, z )z 1 z where ρ θ () = ρ θ (z 1, z ) is one of the parametrisation described. LGC in Finance B. Støve Universit of Bergen 9 /

30 Estimation of PLGC II It ma be tempting to treat φ θ as an ordinar densit function and proceed with the standard likelihood approach b maimizing L (θ) = n 1 n log (φ θ ( i )) () i=1 However, for the parametrisations described and their accompaning parameter spaces ρ θ () frequentl violates the equalit I = φ θ (t)dt = 1 (3) LGC in Finance B. Støve Universit of Bergen 3 /

31 Estimation of PLGC III There are at least two approaches to this problem: 1 Optimize L under the constraint I = φ θ (t)dt (1 ɛ, 1 + ɛ) for some ɛ > () Adjust the likelihood function For the latter approach some candidates are: L (θ) = n 1 L 3 (θ) = n 1 n i=1 n i=1 ( log (φ θ (z i )) ( log (φ θ (z i )) log φ θ (t)dt 1) (5) ) φ θ (t)dt (6) LGC in Finance B. Støve Universit of Bergen 31 /

32 Estimation of PLGC IV The adjusted likelihood L 3 amounts to fitting the densit φ θ (z) = φ θ(z) φθ (t)dt (7) Some simulation results indicate that approach 1 (constrained optimization) does not work well. Focus on approach, and the candidate L 3. LGC in Finance B. Støve Universit of Bergen 3 /

33 Eample 1: Gumbel copula Simulate n = 1 data from F(z) = C gumbel (φ(z 1 ), φ(z )) (here φ is the standard normal!) Use the general parameterization and let α 1 = α = 1, β 1 = β = 1, γ 1 = γ 3 and γ = γ. The parameter of the gumbel copula corresponds to a global correlation of.7. LGC in Finance B. Støve Universit of Bergen 33 /

34 Eample 1: Scatterplot and levelplot z1 z A =.3, D1 = 3.9, D = 1.17, alpha1 = 1, alpha = 1, beta1 =, beta =, gamma1 =.7, gamma =.5, gamma3 =.7, gamma =.5, I = Figure : Estimated PLGC for the gumbel model using L 3 LGC in Finance B. Støve Universit of Bergen 3 /

35 Eample 1: Diagonal plot of the estimated PLGC rho_diag z LGC in Finance B. Støve Universit of Bergen 35 /

36 Eample : Claton copula Simulate n = 1 data from F(z) = C claton (φ(z 1 ), φ(z )). Same parameterization as above. The parameter of the claton copula corresponds to a global correlation of.63. For comparison we have that ρˆθ( 5, 5) =.93 and ρˆθ(5, 5) =.6, which seems reasonable. LGC in Finance B. Støve Universit of Bergen 36 /

37 Eample : Scatterplot and levelplot z1 z A = 3.9, D1 =.33, D = 1.1, alpha1 = 1, alpha = 1, beta1 =, beta =, gamma1 =.9, gamma = 1.58, gamma3 =.9, gamma = 1.58, I = Figure : Estimated PLGC for the claton model using L 3 LGC in Finance B. Støve Universit of Bergen 37 /

38 Eample : Diagonal plot of the estimated PLGC rho_diag z LGC in Finance B. Støve Universit of Bergen 38 /

39 Concluding remarks Used the LGC to stud that financial returns ehibit asmmetric dependence, such as increased dependence during bear markets LGC and financial contagion Proposed a parametric version of the LGC - work in progress LGC in Finance B. Støve Universit of Bergen 39 /

40 Some references Tjøstheim & Hufthammer (13). Local Gaussian correlation: A new measure of dependence. Journal of Econometrics Støve & Tjøstheim (1). Asmmetric dependence patterns in financial returns: An empirical investigation using local Gaussian correlation. Nonlinear Time Series Econometrics, OUP. Støve, Tjøstheim & Hufthammer (1). Using local Gaussian correlation in a nonlinear re-eamination of financial contagion. Journal of Empirical Finance. Okimoto (8). New evidence of asmmetric dependence structures in international equit markets. Journal of Financial and Quantitative Analsis. Okimoto (1). Asmmetric increasing trends in dependence in international equit markets. Journal of Banking and Finance. Christoffersen et al (1). Is the potential for international diversification disappearing? A dnamic copula approach. Review of Financial Studies. LGC in Finance B. Støve Universit of Bergen /