Properties of a Diversified World Stock Index

Transcription

1 Properties of a Diversified World Stock Index Eckhard Platen School of Finance and Economics and School of Mathematical Sciences University of Technology, Sydney Platen, E. & Heath, D.: A Benchmark Approach to Quantitative Finance Springer Finance, 700 pp., 199 illus., Hardcover, ISBN (2006). Le, T. & Platen. E.: Approximating the growth optimal portfolio with a diversified world stock index. J. Risk Finance 7(5), (2006). Platen, E. & Sidorowicz, R.:Empirical evidence on Student-t log-returns of diversified world stock indices. University of Technology, Sydney. QFRC Research Paper 194 (2007).

2 Springer Finance S F Springer Finance E. Platen D. Heath The benchmark approach provides a general framework for financial market modeling, which extends beyond the standard risk neutral pricing theory. It allows for a unified treatment of portfolio optimization, derivative pricing, integrated risk management and insurance risk modeling. The existence of an equivalent risk neutral pricing measure is not required. Instead, it leads to pricing formulae with respect to the real world probability measure. This yields important modeling freedom which turns out to be necessary for the derivation of realistic, parsimonious market models. The first part of the book describes the necessary tools from probability theory, statistics, stochastic calculus and the theory of stochastic differential equations with jumps. The second part is devoted to financial modeling under the benchmark approach. Various quantitative methods for the fair pricing and hedging of derivatives are explained. The general framework is used to provide an understanding of the nature of stochastic volatility. The book is intended for a wide audience that includes quantitative analysts, postgraduate students and practitioners in finance, economics and insurance. It aims to be a self-contained, accessible but mathematically rigorous introduction to quantitative finance for readers that have a reasonable mathematical or quantitative background. Finally, the book should stimulate interest in the benchmark approach by describing some of its power and wide applicability. ISBN springer.com Platen Heath 1 A Benchmark Approach to Quantitative Finance A Benchmark Approach to Quantitative Finance Eckhard Platen David Heath 1 23

3 Benchmark Approach Pl. & Heath (2006) best performing strictly positive portfolio as benchmark growth optimal portfolio (GOP) benchmark in portfolio optimization numeraire in derivative pricing approximate GOPs Diversification Theorem Eckhard Platen AMAMEF07, Bedelow 1

4 log-return density for diversified stock indices Markowitz & Usmen (1996a, 1996b): S&P500 log-returns Student t (4.5) Hurst & Pl. (1997): regional stock market indices symmetric generalized hyperbolic distribution Student t (3.0) (4.5) Eckhard Platen AMAMEF07, Bedelow 2

5 Fergusson & Pl. (2006): maximum likelihood ratio test Student t (4) McNeil, Frey & Embrechts (2005): Student t type log-returns Pl. & Sidorowicz (2007): EWI104s Student t (4) 99.9% significance Eckhard Platen AMAMEF07, Bedelow 3

6 benchmark approach Pl. & Heath (2006) growth optimal portfolio (GOP) Kelly (1956) diversified portfolios (DPs) diversification theorem Pl. (2005) equally weighted index (EWI) EWI104s Eckhard Platen AMAMEF07, Bedelow 4

7 Index Construction market capitalization weighted indices (MCIs) diversity weighted indices (DWIs) Fernholz (2002) equally weighted indices (EWIs) world stock indices (WSIs) Le & Pl. (2006) Eckhard Platen AMAMEF07, Bedelow 5

8 portfolio generating function given any fractions π δ,t = (π 1 δ,t, π2 δ,t,..., πd δ,t ) forms vector of nonnegative fractions π δ,t = ( π 1 δ,t, π2 δ,t,..., πd δ,t ) = A(π δ,t ) [0, 1] d d j=1 π j δ,t = 1 Eckhard Platen AMAMEF07, Bedelow 6

9 Market Capitalization Weighted Indices MCI π j δ MCI,t = δj ts j t d i=1 δi ts i t δ j t number of units of jth constituent Eckhard Platen AMAMEF07, Bedelow 7

10 Diversity Weighted Index DWI Fernholz (2002) π j δ,t = (πj δ MCI,t )p d l=1 (πl δ MCI,t )p p [0, 1] p = 0.5 Eckhard Platen AMAMEF07, Bedelow 8

11 Equally Weighted Index EWI π j δ EWI,t = 1 d j {1, 2,..., d} Eckhard Platen AMAMEF07, Bedelow 9

12 world stock index WSI π j δ,t = (πj δ,t + µ t) p d l=1 (πl δ,t + µ t) p fractions of GOP π δ,t = Σ 1 t (a t r t 1) µ t = inf j π j δ,t + µ Eckhard Platen AMAMEF07, Bedelow 10

13 WSI EWI DWI MCI 28/08/76 18/02/82 11/08/87 31/01/93 24/07/98 14/01/04 Figure 1: Indices constructed from regional stock market indices. Eckhard Platen AMAMEF07, Bedelow 11

14 WSI35s EWI35s DWI35s MCI35s 28/08/76 18/02/82 11/08/87 31/01/93 24/07/98 14/01/04 Figure 2: Indices constructed from sector indices based on 35 industries. Eckhard Platen AMAMEF07, Bedelow 12

15 WSI104s EWI104s DWI104s MCI104s 28/08/76 18/02/82 11/08/87 31/01/93 24/07/98 14/01/04 Figure 3: Indices constructed from sector indices based on 104 industries. Eckhard Platen AMAMEF07, Bedelow 13

16 10 4 EWI 10 3 EWI104s /08/76 18/02/82 11/08/87 31/01/93 24/07/98 14/01/04 Figure 4: The regional EWI and sector EWI104s indices in log-scale. Eckhard Platen AMAMEF07, Bedelow 14

17 Log-return Distributions Barndorff-Nielsen (1978), Hurst & Pl. (1997) McNeil, Frey & Embrechts (2005) normal mean-variance mixture distribution Z N(0, 1) X = m(w) + WσZ W 0 is nonnegative random variable independent of Z symmetric case = normal variance-mixture distribution X = WσZ Eckhard Platen AMAMEF07, Bedelow 15

18 Generalized Hyperbolic Distributions mixing density generalized inverse Gaussian W GIG(λ, χ, ψ) X GH(λ, χ, ψ, µ, σ, γ) f X (x) = ψλ (ψ + γβ) 1 2 λ ( χψ) λ 2πσKλ ( χψ) K λ 1 2 ( (χ + Q)(ψ + γβ) ) ( eξβ (χ )1 2 λ + Q)(ψ + γβ) ξ = x µ, β = γσ 2, Q = (x µ) 2 σ 2 K λ ( ) modified Bessel function of the third kind Eckhard Platen AMAMEF07, Bedelow 16

19 symmetric generalized hyperbolic density f X (x) = 1 δσk λ (ᾱ) ᾱ 2π ( 1+ x2 (δσ) 2 )1 2 (λ 1 2 ) ( ) K λ 1 ᾱ 1 + x2 2 (δσ) 2 λ R, α, δ 0, α 0 if λ 0, δ 0 if λ 0 ᾱ = αδ unique scale parameter c 2 = (δσ) 2 2(λ+1) if α = 0 for λ < 0 and ᾱ = 0, 2λσ 2, α 2 if δ = 0 for λ > 0 and ᾱ = 0, (δσ) 2 K λ+1 (ᾱ) ᾱk λ (ᾱ) otherwise Eckhard Platen AMAMEF07, Bedelow 17

20 Special Cases of the SGH Distribution Variance Gamma: ᾱ = 0 and λ > 0 Madan & Seneta (1990) Student t: ᾱ = 0 and λ < 0 Praetz (1972) Hyperbolic: λ = 1 Eberlein & Keller (1995) Normal Inverse Gaussian: λ = 0.5 Barndorff-Nielsen (1995) Eckhard Platen AMAMEF07, Bedelow 18

21 Variance Gamma Density ᾱ = 0, α = 2λ, δ = 0 gamma distribution mixing f X (x) = λ πσ2 λ 1 Γ(λ) ( 2λ x σ ) λ 1 2 K λ 1 2 ( ) 2λ x σ Madan & Seneta (1990) Eckhard Platen AMAMEF07, Bedelow 19

22 Student t Density Praetz (1972), Blattberg & Gonedes (1974) inverse gamma distribution mixing degrees of freedom ν = 2 λ 2 f X (x) = Γ ( ν ν 2 ( ) 1 + Q πνσ ν ( ) ν+1 (ν ) 2 K ν+1 + Q)γβ 2 ( (ν ) ν Q)γβ e ξβ Eckhard Platen AMAMEF07, Bedelow 20

23 Likelihood Ratio Test likelihood ratio Λ = L model L nesting model L model maximized likelihood function test statistic L n = 2 ln(λ) Eckhard Platen AMAMEF07, Bedelow 21

24 P(L n < χ 2 1 α,1 ) F χ 2 (1)(χ 2 1 α,1 ) = 1 α L n < χ , L n < χ , not rejected at the 99.9% level Eckhard Platen AMAMEF07, Bedelow 22

25 Fitted Log-return Distributions daily log-returns EWI104s denominated in 27 currencies > observations Eckhard Platen AMAMEF07, Bedelow 23

26 Figure 5: Log-histogram of the EWI104s log-returns and Student t density with four degrees of freedom. Eckhard Platen AMAMEF07, Bedelow 24

27 Estimated LLF Estimated λ λ α Figure 6: Log-likelihood function based on the EWI104s. Eckhard Platen AMAMEF07, Bedelow 25

28 1 MCI 1 DWI λ 2 λ α EWI α WSI λ 2 λ α α Figure 7: (ᾱ, λ)-plot for log-returns of indices in different currencies constructed from regional stock market indices as constituents. Eckhard Platen AMAMEF07, Bedelow 26

29 1 MCI35s 1 DWI35s λ 2 λ α EWI35s α WSI35s λ 2 λ α α Figure 8: (ᾱ, λ)-plot for log-returns of indices in different currencies constructed from 35 sector indices as constituents. Eckhard Platen AMAMEF07, Bedelow 27

30 1 MCI104s 1 DWI104s λ 2 λ α EWI104s α WSI104s λ 2 λ α α Figure 9: (ᾱ, λ)-plot for log-returns of indices in different currencies constructed from 104 sector indices as constituents. Eckhard Platen AMAMEF07, Bedelow 28

31 SGH Student t NIG Hyperbolic VG σ ᾱ λ ν ln(l ) L n Table 1: Results for log-returns of the EWI104s Eckhard Platen AMAMEF07, Bedelow 29

32 Country Student-t NIG Hyperbolic VG ν Australia Austria Belgium Brazil Canada Denmark Finland France Germany Greece Hong.Kong India Ireland Italy Japan Korea.S Eckhard Platen AMAMEF07, Bedelow 30

33 Malaysia Netherlands Norway Portugal Singapore Spain Sweden Taiwan Thailand UK USA Table 2: L n test statistic of the EWI104s for different currency denominations Eckhard Platen AMAMEF07, Bedelow 31

34 Stochastic Volatility Model mixing density for returns is inverse gamma squared volatility dσ 2 t = 1 4 γ2 (ν ξ) σ 4(ξ 1) t ( σ 2 σ 2 t ) dt + γ σ 2ξ d W t stationary density is inverse gamma Heath, Hurst & Pl. (2001) d dt [ ] ln(σ 2 ) t = γ 2 σ 2(ξ 1) t γ 2 = ξ = 1 Eckhard Platen AMAMEF07, Bedelow 32

35 Figure 10: Returns of industry index. Eckhard Platen AMAMEF07, Bedelow 33

36 DF= Figure 11: Histogram of returns. Eckhard Platen AMAMEF07, Bedelow 34

37 Figure 12: Squared volatility. Eckhard Platen AMAMEF07, Bedelow 35

38 DF= e e e e Figure 13: Histogram of inverse squared volatility. Eckhard Platen AMAMEF07, Bedelow 36

39 Figure 14: Quadratic variation of log-squared volatility. Eckhard Platen AMAMEF07, Bedelow 37

40 Financial Market Model Wiener processes B k = {B k t, t R +} for k {1, 2,..., m} compensated normalized jump martingales trading uncertainties dq k t = (hk t ) 1 2 (dp k t hk t dt) W = {W t = (W 1 t,..., W m t, W m+1 t,..., W d t ), t R + } W 1 t = B1 t,..., W m t = B m t W m+1 t = q m+1 t,..., Wt d = qd t Eckhard Platen AMAMEF07, Bedelow 38

41 primary security accounts savings account S 0 t = exp { t 0 } r s ds < jth risky asset ds j t = S j t ( a j tdt + d b j,k t dw k t k=1 ) volatility matrix invertible assume b j,k t h k t Eckhard Platen AMAMEF07, Bedelow 39

42 market price of risk θ t = (θ 1 t,..., θd t ) = b 1 t (a t r t 1) assume θ k t < h k t Eckhard Platen AMAMEF07, Bedelow 40

43 portfolio d S δ t = δ j t S j t j=0 fraction ds δ t = Sδ t π j δ,t = δj t S j t S δ t ( ) r t dt + π δ,t b t (θ t dt + dw t ) assume π j δ,t 0 Eckhard Platen AMAMEF07, Bedelow 41

44 Growth Optimal Portfolio ds δ t = S δ t ( + r t dt + d k=m+1 m θ k t (θk t dt + dw k t ) k=1 θ k t 1 θ k t (h k t ) 1 2 ( θ k t dt + dw k t ) lim sup T ( 1 S δ T ln T S0 δ ) lim sup T ( 1 S δ T ln T S δ 0 ) Eckhard Platen AMAMEF07, Bedelow 42

45 sequence of diversified portfolios (DPs) π j δ,t K 2 d 1 2 +K 1 assume sequence of markets regular : k {1, 2,..., d} E ( (ˆσ k (d) (t))2) K Eckhard Platen AMAMEF07, Bedelow 43

46 tracking rate d R δ (d) (t) = d π j δ,t σj,k (d) (t) k=1 j=0 2 R δ (d) (t) = 0 Diversification Theorem For any DP lim d Rδ (d) (t) P = 0 for all t R + model independent Eckhard Platen AMAMEF07, Bedelow 44

47 References Barndorff-Nielsen, O. (1978). Hyperbolic distributions and distributions on hyperbolae. Scand. J. Statist. 5, Barndorff-Nielsen, O. (1995). Normal-Inverse Gaussian processes and the modelling of stock returns. Technical report, University of Aarhus Blattberg, R. C. & N. Gonedes (1974). A comparison of the stable and Student distributions as statistical models for stock prices. J. Business 47, Eberlein, E. & U. Keller (1995). Hyperbolic distributions in finance. Bernoulli 1, Fergusson, K. & E. Platen (2006). On the distributional characterization of log-returns of a world stock index. Appl. Math. Finance 13(1), Fernholz, E. R. (2002). Stochastic Portfolio Theory, Volume 48 of Appl. Math. Springer. Heath, D., S. R. Hurst, & E. Platen (2001). Modelling the stochastic dynamics of volatility for equity indices. Asia-Pacific Financial Markets 8, Hurst, S. R. & E. Platen (1997). The marginal distributions of returns and volatility. In Y. Dodge (Ed.), L 1 -Statistical Procedures and Related Topics, Volume 31 of IMS Lecture Notes - Monograph Series, pp Institute of Mathematical Statistics Hayward, California. Eckhard Platen AMAMEF07, Bedelow 45

48 Kelly, J. R. (1956). A new interpretation of information rate. Bell Syst. Techn. J. 35, Le, T. & E. Platen (2006). Approximating the growth optimal portfolio with a diversified world stock index. J. Risk Finance 7(5), Madan, D. & E. Seneta (1990). The variance gamma (V.G.) model for share market returns. J. Business 63, Markowitz, H. & N. Usmen (1996a). The likelihood of various stock market return distributions, Part 1: Principles of inference. J. Risk & Uncertainty 13(3), Markowitz, H. & N. Usmen (1996b). The likelihood of various stock market return distributions, Part 2: Empirical results. J. Risk & Uncertainty 13(3), McNeil, A., R. Frey, & P. Embrechts (2005). Quantitative Risk Management. Princeton University Press. Platen, E. (2005). Diversified portfolios with jumps in a benchmark framework. Asia-Pacific Financial Markets 11(1), Platen, E. & D. Heath (2006). A Benchmark Approach to Quantitative Finance. Springer Finance. Springer. Platen, E. & Sidorowicz (2007). Empirical evidence on Student-t log-returns of diversified world stock indices. Technical report, University of Technology, Sydney. QFRC Research Paper 194. Eckhard Platen AMAMEF07, Bedelow 46

49 Praetz, P. D. (1972). The distribution of share price changes. J. Business 45, Eckhard Platen AMAMEF07, Bedelow 47