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-10 3-540-26212-1 (2006). Le, T. & Platen. E.: Approximating the growth optimal portfolio with a diversified world stock index. J. Risk Finance 7(5), 559 574 (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).
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 3-540-26212-1 springer.com Platen Heath 1 A Benchmark Approach to Quantitative Finance A Benchmark Approach to Quantitative Finance Eckhard Platen David Heath 1 23
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
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
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
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
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
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
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
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
Equally Weighted Index EWI π j δ EWI,t = 1 d j {1, 2,..., d} Eckhard Platen AMAMEF07, Bedelow 9
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
10000 9000 8000 7000 6000 5000 WSI EWI 4000 3000 DWI 2000 1000 0 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
10000 9000 8000 7000 6000 WSI35s 5000 4000 EWI35s 3000 2000 1000 0 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
10000 9000 8000 7000 6000 5000 4000 WSI104s EWI104s 3000 2000 1000 0 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
10 4 EWI 10 3 EWI104s 10 2 10 1 28/08/76 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
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
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
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
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
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
Student t Density Praetz (1972), Blattberg & Gonedes (1974) inverse gamma distribution mixing degrees of freedom ν = 2 λ 2 f X (x) = Γ ( ν 2 2 1 ν 2 ( ) 1 + Q πνσ ν ( ) ν+1 (ν ) 2 K ν+1 + Q)γβ 2 ( (ν ) ν+1 2 + Q)γβ e ξβ Eckhard Platen AMAMEF07, Bedelow 20
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
P(L n < χ 2 1 α,1 ) F χ 2 (1)(χ 2 1 α,1 ) = 1 α L n < χ 2 0.01,1 0.000157 L n < χ 2 0.001,1 0.000002 not rejected at the 99.9% level Eckhard Platen AMAMEF07, Bedelow 22
Fitted Log-return Distributions daily log-returns 1973 2006 EWI104s denominated in 27 currencies > 200.000 observations Eckhard Platen AMAMEF07, Bedelow 23
10 1 10 2 10 3 10 4 10 5 0 5 10 Figure 5: Log-histogram of the EWI104s log-returns and Student t density with four degrees of freedom. Eckhard Platen AMAMEF07, Bedelow 24
Estimated LLF 10 5 2.84 2.86 2.88 2.9 2 2.92 2.94 1.5 5 4 3 2.15 1 Estimated λ 1 0 1 λ 2 3 4 5 0 0.5 α Figure 6: Log-likelihood function based on the EWI104s. Eckhard Platen AMAMEF07, Bedelow 25
1 MCI 1 DWI 1.5 1.5 λ 2 λ 2 2.5 2.5 3 0 0.2 0.4 0.6 1 α EWI 3 0 0.2 0.4 0.6 1 α WSI 1.5 1.5 λ 2 λ 2 2.5 2.5 3 0 0.2 0.4 0.6 α 3 0 0.2 0.4 0.6 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
1 MCI35s 1 DWI35s 1.5 1.5 λ 2 λ 2 2.5 2.5 3 0 0.2 0.4 0.6 1 α EWI35s 3 0 0.2 0.4 0.6 1 α WSI35s 1.5 1.5 λ 2 λ 2 2.5 2.5 3 0 0.2 0.4 0.6 α 3 0 0.2 0.4 0.6 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
1 MCI104s 1 DWI104s 1.5 1.5 λ 2 λ 2 2.5 2.5 3 0 0.2 0.4 0.6 1 α EWI104s 3 0 0.2 0.4 0.6 1 α WSI104s 1.5 1.5 λ 2 λ 2 2.5 2.5 3 0 0.2 0.4 0.6 α 3 0 0.2 0.4 0.6 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
SGH Student t NIG Hyperbolic VG σ 0.9807068 0.7191163 0.9697258 0.9584118 0.9593693 ᾱ 0.0000000 0.9694605 0.7171357 λ -2.1629649 1.4912414 ν 4.3259646 ln(l ) -285796.3865295-285796.3865297-286448.9371892-287152.0787956-287499.8259143 L n 0.0000004 1305.1013194 2711.3845322 3406.8787696 Table 1: Results for log-returns of the EWI104s Eckhard Platen AMAMEF07, Bedelow 29
Country Student-t NIG Hyperbolic VG ν Australia 0.000000 76.770817 150.202282 181.632971 4.281222 Austria 0.000000 39.289103 77.505683 102.979330 4.725907 Belgium 0.000000 31.581622 60.867570 83.648470 4.989912 Brazil 2.617693 5.687078 63.800349 60.078395 2.713036 Canada 0.000000 47.506215 79.917741 104.297607 5.316154 Denmark 0.000000 41.509921 87.199686 114.853658 4.512101 Finland 0.000000 28.852844 68.677271 88.553080 4.305638 France 0.000000 26.303544 57.639325 80.567283 4.722787 Germany 0.000000 27.290205 52.667918 71.120798 5.005856 Greece 0.000000 60.432172 104.789463 125.601499 4.674626 Hong.Kong 0.000000 42.066531 100.834255 122.965326 3.930473 India 0.000000 74.773701 163.594078 198.002956 3.998713 Ireland 0.000000 77.727856 136.505582 170.013644 4.761519 Italy 0.000000 25.196598 55.185625 75.481897 4.668983 Japan 0.000000 37.630363 77.163656 102.967380 4.649745 Korea.S. 0.000000 120.904983 304.829431 329.854620 3.289204 Eckhard Platen AMAMEF07, Bedelow 30
Malaysia 0.000000 79.714054 186.013963 221.061290 3.785195 Netherlands 0.000000 26.832761 51.625813 71.541627 5.084056 Norway 0.000000 42.243851 89.012090 115.059003 4.472349 Portugal 0.000000 61.177624 137.681039 165.689683 3.984860 Singapore 0.000000 36.379685 77.600590 98.124375 4.251472 Spain 0.000000 56.694545 109.533768 138.259224 4.517153 Sweden 0.000000 77.618384 143.420049 178.983373 4.546640 Taiwan 0.000000 41.162560 96.283628 115.186585 3.914719 Thailand 0.000000 78.250621 254.590254 267.508143 3.032038 UK 0.000000 26.693076 55.937248 80.678494 4.952843 USA 0.000000 40.678242 79.617362 100.901197 4.636661 Table 2: L n test statistic of the EWI104s for different currency denominations Eckhard Platen AMAMEF07, Bedelow 31
Stochastic Volatility Model mixing density for returns is inverse gamma squared volatility dσ 2 t = 1 4 γ2 (ν + 2 4 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
0.10 0.05 0.00 0.05 0 2000 4000 6000 8000 Figure 10: Returns of industry index. Eckhard Platen AMAMEF07, Bedelow 33
DF= 4.4679 0.0 0.1 0.2 0.3 0.4 4 2 0 2 4 Figure 11: Histogram of returns. Eckhard Platen AMAMEF07, Bedelow 34
0.0000 0.0005 0.0010 0.0015 0.0020 0 2000 4000 6000 8000 Figure 12: Squared volatility. Eckhard Platen AMAMEF07, Bedelow 35
DF= 4.4554 0.0 e+00 1.0 e 05 2.0 e 05 3.0 e 05 0 50000 100000 150000 200000 Figure 13: Histogram of inverse squared volatility. Eckhard Platen AMAMEF07, Bedelow 36
0 50 100 150 200 250 300 350 0 2000 4000 6000 8000 Figure 14: Quadratic variation of log-squared volatility. Eckhard Platen AMAMEF07, Bedelow 37
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
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
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
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
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
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
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
References Barndorff-Nielsen, O. (1978). Hyperbolic distributions and distributions on hyperbolae. Scand. J. Statist. 5, 151 157. Barndorff-Nielsen, O. (1995). Normal-Inverse Gaussian processes and the modelling of stock returns. Technical report, University of Aarhus. 300. Blattberg, R. C. & N. Gonedes (1974). A comparison of the stable and Student distributions as statistical models for stock prices. J. Business 47, 244 280. Eberlein, E. & U. Keller (1995). Hyperbolic distributions in finance. Bernoulli 1, 281 299. Fergusson, K. & E. Platen (2006). On the distributional characterization of log-returns of a world stock index. Appl. Math. Finance 13(1), 19 38. 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, 179 195. 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. 301 314. Institute of Mathematical Statistics Hayward, California. Eckhard Platen AMAMEF07, Bedelow 45
Kelly, J. R. (1956). A new interpretation of information rate. Bell Syst. Techn. J. 35, 917 926. Le, T. & E. Platen (2006). Approximating the growth optimal portfolio with a diversified world stock index. J. Risk Finance 7(5), 559 574. Madan, D. & E. Seneta (1990). The variance gamma (V.G.) model for share market returns. J. Business 63, 511 524. Markowitz, H. & N. Usmen (1996a). The likelihood of various stock market return distributions, Part 1: Principles of inference. J. Risk & Uncertainty 13(3), 207 219. Markowitz, H. & N. Usmen (1996b). The likelihood of various stock market return distributions, Part 2: Empirical results. J. Risk & Uncertainty 13(3), 221 247. 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), 1 22. 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
Praetz, P. D. (1972). The distribution of share price changes. J. Business 45, 49 55. Eckhard Platen AMAMEF07, Bedelow 47