Properties of a Diversiﬁed World Stock Index · Springer Finance A Benchmark Approach to Quantitative Finance 1 A Benchmark Approach to Quantitative Finance S F Platen · Heath

Properties of a Diversified World Stock Index

Eckhard PlatenSchool 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 Finance7(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).

1 23

Springer Finance

A Benchmark Approach to Q

uantitative Finance

1 A Benchmark Approach to

Quantitative Finance

S F

Platen · Heath

Eckhard Platen David Heath

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Springer Finance

E. Platen · D. HeathThe 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 bench-

mark approach. Various quantitative methods for the fair pricing and hedging

of derivatives are explained. The general framework is used to provide an under-

standing 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 quanti-

tative 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

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

Studentt (4.5)

Hurst & Pl. (1997):

regional stock market indices

symmetric generalized hyperbolic distribution

Studentt (3.0)–(4.5)


Fergusson & Pl. (2006):

maximum likelihood ratio test

Studentt (4)

McNeil, Frey & Embrechts (2005):

Studentt type log-returns

Pl. & Sidorowicz (2007):

EWI104s

Studentt (4)

99.9% significance


• benchmark approach

Pl. & Heath (2006)

• growth optimal portfolio (GOP)

Kelly (1956)

• diversified portfolios (DPs)

diversification theorem

Pl. (2005)

equally weighted index (EWI)

EWI104s


Index Construction

• market capitalization weighted indices (MCIs)

• diversity weighted indices (DWIs)

Fernholz (2002)

• equally weighted indices (EWIs)

• world stock indices (WSIs)

Le & Pl. (2006)


• portfolio generating function

given any fractions

πδ,t = (π1δ,t, π

2δ,t, . . . , π

dδ,t)

⊤

forms vector of nonnegative fractions

πδ,t = (π1δ,t, π

2δ,t, . . . , π

dδ,t)

⊤ = A(πδ,t) ∈ [0, 1]d

d∑

j=1

πjδ,t = 1


Market Capitalization Weighted Indices

MCI

πjδMCI ,t

=δ

jtS

jt

∑d

i=1 δitS

it

δjt number of units ofjth constituent


Diversity Weighted Index

DWI

Fernholz (2002)

πjδ,t =

(πjδMCI ,t

)p

∑d

l=1(πlδMCI ,t

)p

p ∈ [0, 1]

p = 0.5


Equally Weighted Index

EWI

πjδEWI,t

=1

d

j ∈ {1, 2, . . . , d}


world stock index

WSI

πjδ,t =

(πjδ,t + µt)

p

∑d

l=1(πlδ,t + µt)p

fractions of GOP

πδ∗,t = Σ−1t (at − rt1)

µt =∣

∣ infjπ

jδ,t

∣

∣+ µ


28/08/76 18/02/82 11/08/87 31/01/93 24/07/98 14/01/04 0

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

WSI

EWI

DWI

MCI

Figure 1: Indices constructed from regional stock market indices.


28/08/76 18/02/82 11/08/87 31/01/93 24/07/98 14/01/04 0

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

WSI35s

EWI35s

DWI35s

MCI35s

Figure 2: Indices constructed from sector indices based on 35 industries.


28/08/76 18/02/82 11/08/87 31/01/93 24/07/98 14/01/04 0

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

EWI104s

WSI104s

DWI104s MCI104s

Figure 3: Indices constructed from sector indices based on 104 industries.


28/08/76 18/02/82 11/08/87 31/01/93 24/07/98 14/01/04 10

1

102

103

104

EWI

EWI104s

Figure 4: The regional EWI and sector EWI104s indices in log-scale.


Log-return Distributions

Barndorff-Nielsen (1978), Hurst & Pl. (1997)McNeil, Frey & Embrechts (2005)

• normal mean-variance mixture distribution

X = m(W ) +√WσZ

Z ∼ N(0, 1)

W ≥ 0 is nonnegative random variable

independent ofZ

• symmetric case =⇒ normal variance-mixture distribution

X =√WσZ


Generalized Hyperbolic Distributions

mixing density

generalized inverse Gaussian

W ∼ GIG(λ, χ, ψ)

X ∼ GH(λ, χ, ψ, µ, σ, γ)

fX(x) =ψλ(ψ + γβ)

12−λ(

√χψ)−λ

√2πσKλ(

√χψ)

Kλ−12

(

√

(χ+Q)(ψ + γβ))

(

√

(χ+Q)(ψ + γβ))

12−λ

eξβ

ξ = x− µ, β = γσ−2, Q = (x− µ)2σ−2

Kλ(·) modified Bessel function of the third kind


• symmetric generalized hyperbolic density

fX(x) =1

δσKλ(α)

√

α

2π

(

1+x2

(δσ)2

)12 (λ−

12 )

Kλ−12

(

α

√

1 +x2

(δσ)2

)

λ ∈ ℜ,α, δ ≥ 0, α 6= 0 if λ ≥ 0, δ 6= 0 if λ ≤ 0

α = αδ

unique scale parameter

c2 =

(δσ)2

−2(λ+1)if α = 0 for λ < 0 andα = 0,

2λσ2

α2 , if δ = 0 for λ > 0 andα = 0,(δσ)2Kλ+1(α)

αKλ(α)otherwise


Special Cases of the SGH Distribution

• Variance Gamma:α = 0 and λ > 0

Madan & Seneta (1990)

• Studentt: α = 0 andλ < 0

Praetz (1972)

• Hyperbolic: λ = 1

Eberlein & Keller (1995)

• Normal Inverse Gaussian:λ = −0.5

Barndorff-Nielsen (1995)


Variance Gamma Density

α = 0, α =√

2λ, δ = 0

gamma distribution mixing

fX(x) =

√λ

√πσ2λ−1Γ(λ)

(√2λ|x|σ

)λ−12

Kλ−12

(√2λ|x|σ

)

Madan & Seneta (1990)


Student t Density

Praetz (1972), Blattberg & Gonedes (1974)

inverse gamma distribution mixing

degrees of freedomν = −2λ ≥ 2

fX(x) =2

1−ν

2

Γ(

ν2

)√πνσ

(

1 +Q

ν

)−ν+12 K ν+1

2

(

√

(ν +Q)γβ)

(

√

(ν +Q)γβ)−

ν+12

eξβ


Likelihood Ratio Test

• likelihood ratio

Λ =L∗

model

L∗nesting model

L∗model maximized likelihood function

• test statistic

Ln = −2 ln(Λ)


P (Ln < χ21−α,1) ≈ Fχ2(1)(χ

21−α,1) = 1 − α

Ln < χ20.01,1 ≈ 0.000157

Ln < χ20.001,1 ≈ 0.000002

not rejected at the99.9% level


Fitted Log-return Distributions

daily log-returns 1973 – 2006

EWI104s

denominated in 27 currencies

> 200.000 observations


−10 −5 0 5 10

10−4

10−3

10−2

10−1

Figure 5: Log-histogram of the EWI104s log-returns and Student t density

with four degrees of freedom.Eckhard Platen AMAMEF07, Bedelow 24

0

0.5

1

1.5

2

−5−4

−3−2.15

−10

1 2

34

5

−2.94

−2.92

−2.9

−2.88

−2.86

−2.84

αλ

Estimated λ

Estimated LLF

× 105

Figure 6: Log-likelihood function based on the EWI104s.


0 0.2 0.4 0.6−3

−2.5

−2

−1.5

−1MCI

α

λ

0 0.2 0.4 0.6−3

−2.5

−2

−1.5

−1DWI

α

λ

0 0.2 0.4 0.6−3

−2.5

−2

−1.5

−1EWI

α

λ

0 0.2 0.4 0.6−3

−2.5

−2

−1.5

−1WSI

α

λ

Figure 7:(α, λ)-plot for log-returns of indices in different currencies con-

structed from regional stock market indices as constituents.Eckhard Platen AMAMEF07, Bedelow 26

0 0.2 0.4 0.6−3

−2.5

−2

−1.5

−1MCI35s

α

λ

0 0.2 0.4 0.6−3

−2.5

−2

−1.5

−1DWI35s

α

λ

0 0.2 0.4 0.6−3

−2.5

−2

−1.5

−1EWI35s

α

λ

0 0.2 0.4 0.6−3

−2.5

−2

−1.5

−1WSI35s

α

λ


structed from 35 sector indices as constituents.Eckhard Platen AMAMEF07, Bedelow 27

0 0.2 0.4 0.6−3

−2.5

−2

−1.5

−1MCI104s

α

λ

0 0.2 0.4 0.6−3

−2.5

−2

−1.5

−1DWI104s

α

λ

0 0.2 0.4 0.6−3

−2.5

−2

−1.5

−1EWI104s

α

λ

0 0.2 0.4 0.6−3

−2.5

−2

−1.5

−1WSI104s

α

λ


structed from 104 sector indices as constituents.Eckhard Platen AMAMEF07, Bedelow 28

SGH Studentt NIG Hyperbolic VG

σ 0.9807068 0.7191163 0.9697258 0.9584118 0.9593693

α 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

Ln 0.0000004 1305.1013194 2711.3845322 3406.8787696

Table 1: Results for log-returns of the EWI104s


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


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:Ln test statistic of the EWI104s for different currency denominations


Stochastic Volatility Model

mixing density for returns is inverse gamma

• squared volatility

dσ2t =

1

4γ2(ν + 2 − 4 ξ)σ

4(ξ−1)t

(

σ2 − σ2t

)

dt+ γ σ2ξ dWt

stationary density is inverse gamma

Heath, Hurst & Pl. (2001)

d

dt

[

ln(σ2)]

t= γ2 σ

2(ξ−1)t ≈ γ2 =⇒ ξ = 1


0 2000 4000 6000 8000

−0.10

−0.05

0.00

0.05

Figure 10: Returns of industry index.


DF= 4.4679

−4 −2 0 2 4

0.00.1

0.20.3

0.4

Figure 11: Histogram of returns.


0 2000 4000 6000 8000

0.000

00.0

005

0.001

00.0

015

0.002

0

Figure 12: Squared volatility.


DF= 4.4554

0 50000 100000 150000 200000

0.0 e+

001.0

e−05

2.0 e−

053.0

e−05

Figure 13: Histogram of inverse squared volatility.


0 2000 4000 6000 8000

050

100

150

200

250

300

350

Figure 14: Quadratic variation of log-squared volatility.


Financial Market Model

• Wiener processesBk = {Bkt , t ∈ R+} for k ∈ {1, 2, . . . ,m}

• compensated normalized jump martingales

dqkt = (hk

t )−

12 (dpk

t − hkt dt)

• trading uncertainties

W = {Wt = (W 1t , . . . ,W

mt ,W

m+1t , . . . ,W d

t )⊤, t ∈ R+}

W 1t = B1

t , . . . ,Wmt = Bm

t

Wm+1t = q

m+1t , . . . ,W d

t = qdt


• primary security accounts

savings account

S0t = exp

{∫ t

0

rsds

}

< ∞

jth risky asset

dSjt = S

jt−

(

ajtdt+

d∑

k=1

bj,kt dW k

t

)

volatility matrix invertible

assume

bj,kt ≥ −

√

hkt


• market price of risk

θt = (θ1t , . . . , θ

dt )⊤ = b

−1t (at − rt 1)

assume

θkt <

√

hkt


• portfolio

Sδt =

d∑

j=0

δjt S

jt

• fraction

πjδ,t = δ

jt

Sjt

Sδt

dSδt = Sδ

t−

(

rt dt+ π⊤

δ,tbt (θt dt+ dWt))

assume

πjδ,t ≥ 0


Growth Optimal Portfolio

dSδ∗

t = Sδ∗

t−

(

rt dt+

m∑

k=1

θkt (θk

t dt+ dW kt )

+

d∑

k=m+1

θkt

1 − θkt (hk

t )−

12

(

θkt dt+ dW k

t

)

lim supT →∞

1

Tln

(

SδT

Sδ0

)

≤ lim supT →∞

1

Tln

(

Sδ∗

T

Sδ∗

0

)


• sequence of diversified portfolios (DPs)

|πjδ,t| ≤ K2

d12+K1

• assume sequence of markets

regular :

E(

(σk(d)(t))

2)

≤ K

k ∈ {1, 2, . . . , d}


• tracking rate

Rδ(d)(t) =

d∑

k=1

d∑

j=0

πjδ,t σ

j,k

(d)(t)

2

Rδ∗

(d)(t) = 0

• Diversification Theorem

For any DP

limd→∞

Rδ(d)(t)

P= 0

for all t ∈ R+

model independent


ReferencesBarndorff-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 ofstock returns. Technical report, University of Aarhus. 300.

Blattberg, R. C. & N. Gonedes (1974). A comparison of the stable and Student distributionsas statistical models for stock prices.J. Business47, 244–280.

Eberlein, E. & U. Keller (1995). Hyperbolic distributions in finance.Bernoulli1, 281–299.

Fergusson, K. & E. Platen (2006). On the distributional characterization of log-returns of aworld stock index.Appl. Math. Finance13(1), 19–38.

Fernholz, E. R. (2002).Stochastic Portfolio Theory, Volume 48 ofAppl. Math.Springer.

Heath, D., S. R. Hurst, & E. Platen (2001). Modelling the stochastic dynamics of volatilityfor equity indices.Asia-Pacific Financial Markets8, 179–195.

Hurst, S. R. & E. Platen (1997). The marginal distributions of returns and volatility. InY. Dodge (Ed.),L1-Statistical Procedures and Related Topics, Volume 31 ofIMS Lec-ture Notes - Monograph Series, pp. 301–314. Institute of Mathematical Statistics Hay-ward, California.


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 diversifiedworld stock index.J. Risk Finance7(5), 559–574.

Madan, D. & E. Seneta (1990). The variance gamma (V.G.) modelfor share market returns.J. Business63, 511–524.

Markowitz, H. & N. Usmen (1996a). The likelihood of various stock market return distribu-tions, Part 1: Principles of inference.J. Risk& Uncertainty13(3), 207–219.

Markowitz, H. & N. Usmen (1996b). The likelihood of various stock market return distribu-tions, Part 2: Empirical results.J. Risk& Uncertainty13(3), 221–247.

McNeil, A., R. Frey, & P. Embrechts (2005).Quantitative Risk Management. Princeton Uni-versity Press.

Platen, E. (2005). Diversified portfolios with jumps in a benchmark framework.Asia-PacificFinancial Markets11(1), 1–22.

Platen, E. & D. Heath (2006).A Benchmark Approach to Quantitative Finance. SpringerFinance. Springer.

Platen, E. & Sidorowicz (2007). Empirical evidence on Student-t log-returns of diversifiedworld stock indices. Technical report, University of Technology, Sydney. QFRC Re-search Paper 194.


Praetz, P. D. (1972). The distribution of share price changes.J. Business45, 49–55.


Documents

Properties of a Diversiﬁed World Stock Index · Springer Finance A Benchmark Approach to Quantitative Finance 1 A Benchmark Approach to Quantitative Finance S F Platen · Heath