THE SHANGHAI STOCK EXCHANGE: STATISTICALPROPERTIES AND SIMULATION

by

Xiongzhi Chen

Master of Science in MathematicsSichuan University, China

Submitted to the Department of Mathematicsin Partial Ful�llment of the Degree of

Master of Arts in Mathematics

at the

University of HawaiiApril, 2009

THESIS COMMITTEE:

Thomas Ramsey

George Wilkens

ii

We certify that we have read this document and that, in our opinion, it is satisfactoryin scope and quality as a thesis for the degree of Master of Arts in Mathematics.

THESIS COMMITTEE

� � � � � � � � � � � � � �

� � � � � � � � � � � � � � -

� � � � � � � � � � � � � � �

iii

THE SHANGHAI STOCK EXCHANGE: STATISTICALPROPERTIES AND SIMULATION BY GENERALIZED HYPERBOLIC

DIFFUSION

by

Xiongzhi Chen

Submitted to the Department of Mathematics on April 21, 2009

in partial ful�llment of the requirements for the degree of

Master of Arts in Mathematics

Abstract

Empirical data of asset returns, including that of stock market index series, always ex-hibit non-stationarity, heavy tails, correlations, and other stylized features as well. Amongthese features the tail indices have wide applications in assessing the distribution of outliers.Also the Hurst index reveals roughly how strongly and how long the correlation structurepersists.While estimating these statistics is usually hard, simulating �nancial markets by contin-

uous-time models is even more challenging since the proposed model should be able to atleast generate the empirical distribution, capture the intrinsic dependence structure, andsimulate the volatilities of the �nancial data.This thesis contains two independent papers that summarize the author�s study of the

statistical properties of the return series of the daily price index series of the ShanghaiStock Exchange (SSE) 1 for trading days between 01/01/1991 and 11/14/2007, and theauthor�s simulation of the linearly detrended logarithmic SSE index series by an ergodic,generalized hyperbolic (GH) di¤usion whose invariant density is a generalized hyperbolic(GH) density.More speci�cally, the �rst paper herein gives estimates of the right tail index, the left tail

index, and the Hurst index for the return series 2 of the SSE index. All of these estimatesare consistent with what most researchers have obtained for stock market index series. Thenon-trivial quality of the Hurst and tail indices imply that extreme care should be takenwhen making assumptions about the underlying stochastic process that generates the stockprice index series. These estimates are obtained by the simplest method � the traditionalordinary least squares (OLS) method, to avoid unnecessary or even wrong assumptionsabout stationarity or dependence structures of the index series, but some comparativeestimates via di¤erent methods are also provided.

1The "SSE index" will speci�cally denote the "daily price index series of the SSE".2The "return series" denotes speci�cally the "return series of the daily price index series of the SSE ".

iv

The second paper is devoted to testing whether an ergodic GH di¤usion process cansimulate well the linearly detrended logarithmic SSE index 3. The ergodic GH di¤usionmodel was proposed by several researchers in the mid 1990s and has successfully simulatedsome stock price series. Its main advantages are: its capability to generate the generalizedautoregressive conditional heteroskedastic (GARCH) e¤ect that is compatible with thecorrelation structure of most �nancial time series, and its capability to approximate theemprical densities of the �nancial time series by the model�s invariant density which is ageneralized hyperbolic (GH) density.This second paper describes for the detrended index an ergodic GH di¤usion whose in-

variant density is proportional to the GH density and whose parameters were estimated byMarkov chain Monte Carlo (MCMC) methods. Six possible models were found since sixindependent, convergent Markov chains are obtained by the Monte Carlo standard error(MCSE) convergence diagnostic. Unfortunately the Gelman-Rubin convergence diagnosticfailed to converge to one even after 350; 000 iterations of the Metroplolis-Hastings algo-rithm, since each of the six chains converged to its own stationary distribution at around50; 000 iterations (which might be the consequence of trapped iterations for the chains inthe simulation).After six Monte Carlo (MC) estimates of the unknown parameters were obtained, we

simulated the detrended index by the six competing models. However, none of themsuccessfully generated paths that match the data well. Also, a uniform residual test wascarried out for only four of the six possible models due to the computational capabilityof Mathematica and Matlab. The test results statistically reject all four selected models,thus rejecting the null hypothesis that the underlying stochastic process for the detrendedindex is an ergodic GH di¤usion with GH invariant density.One reason for such a failure might be that the invariant GH density is inappropriate

for the empirical density of the detrended index, since its empirical density is tri-modalwhereas the GH density is unimodal.

3The "detrended index" will speci�cally denote the "linearly detrended logarithmic SSE index".

v

Acknowledgements

I am deeply indebted to my adviser, Professor Thomas Ramsey, whose constant guidance,inspiring suggestions and supports motivated my study of the statistical properties of�nancial markets and of the simulation of �nancial time series by continuous-time models.Without his help, this thesis would not have been possible.

I would like to thank Professor George Wilkens for his help with my study of the theoryof stochastic control, and, Professor James B. Nation, Professor Wayne Smith, ProfessorWilliam Lampe for their decisive support to my pursuit of further academic opportunitiesin the near future.

I am grateful to other faculty members and stu¤, particularly, the departmental secre-taries, for their help that made my life and study here at Hawaii much easier and moreenjoyable. I am also grateful to fellow graduate students in the department, particularly,to Shashi Bhushan Singh and Gretel Sia for their kind help.

Finally, I would like to express my appreciation to my family, especially my wife, mybrother, and to my friends, especially Jin Li, whose love and contribution paved the wayfor my dream in mathematics.

Please notice that the two papers incorporated into this thesis have their own pagingschemes. The next page will be the starting page of the �rst paper, and the second startsright after the ending page of the �rst.Please go to the next page for the start of the �rst paper and then to the second paper

after around 23 pages .......

THE SHANGHAI STOCK EXCHANGE I: STATISTICALPROPERTIES

XIONGZHI CHEN

Abstract. This paper studies some of the statistical properties of thedaily return series of the SSE index (hereunder abbreviated as the "re-turn series") for trading days between 01/01/1991 and 11/14/2007, andgives their corresponding statistical estimates.For this return series, the estimated right tail index is around 4:6,

that of the left tail index around 2:19 and the Hurst index around0:61. These statistics are obtained by linear regression via ordinaryleast squares (OLS), to avoid unnecessary or even wrong assumptionsabout the stationarity or dependence structures of the return series.Results provided herein comply with what most researchers have

found for stock market indices (see Cont01 [5] and Mills99 [21]). The tailindices show that statistically the underlying stochastic process that gen-erates the return series can be assumed to have �nite variance, whereasthe Hurst index shows that the underlying process is not a random walkbut has long-range dependence.

Key words and phrases. Hill�s Estimator, Hurst Index, Lo�s Modi�ed R/S Statistic,Classic R/S Statistic, R/S Analysis, Short/Long-range Dependence, Skewness, Tail Index.

1

2 XIONGZHI CHEN

Contents

List of Figures 31. Introduction 42. Tail Index 52.1. Hill�s Estimator 82.2. Linear Regressor 102.3. Mandelbrot�s Visual Check 123. Kurtosis and Skewness 144. Dependence 154.1. Hurst�s R/S Statistic 164.2. Mandelbrot�s R/S Analysis 184.3. Direct Regression 225. Conclusions 22References 23

SSE DAILY RETURNS: TAIL INDICES, SKEWNESS, HURST INDEX 3

List of Figures

1 Daily SSE Price Index 6

2 Returns of SSE Index 6

3 EPDF of Returns 10

4 Log Left Tail of Returns 11

5 Log Right Tail of Returns 11

6 Log Left Tail with Linear Regressor 12

7 Log Right Tail with Linear Regressor 12

8 Sample Variance 13

9 4th Sample Moment 13

10 5th Absolute Sample Moment 14

11 Skewness 15

12 Zoom-in EPDF 15

13 Log-log Plot of Classic R/S Statistic 17

14 Classic R/S Statistic with Linear Regressor I 17

15 Classic R/S Statistic with Linear Regressor II 18

16 Pox Plot of R/S Statistic 19

17 ACFs with Max Lag Order 1000 20

18 Lo�s Modi�ed R/S Statistic 21

19 Hurst Index via Direct Regression 22

4 XIONGZHI CHEN

1. Introduction

It�s widely acknowledged that most asset returns1 have the following mainstylized features revealed by empirical studies: leptokurtosis 2, skewness, andpower-decaying correlations for absolute returns3 (see Cont01 [5], Mandel-brot01a [18]).As a reference for interested readers, the eleven empirical features of asset

returns given in Cont01 [5] are listed below:

F1: Absence of autocorrelations: (linear) autocorrelations of assetreturns are often insigni�cant, except for very small intraday timescales (about 20 minutes) for which microstructure e¤ects come intoplay.

F2: Heavy tails: the (unconditional) distribution of returns seems todisplay a power-law or Pareto-like tail, with a tail index which is�nite, higher than two but less than �ve for most data sets studied.In particular this excludes stable laws with in�nite variance and thenormal distribution. However the precise form of the tails is di¢ cultto determine.

F3: Gain/loss asymmetry: one observes large drawdowns in stockprices and stock index values but not equally large upward move-ments.

F4: Aggregational Gaussianity: as one increases the time scale�t over which returns are calculated, their distribution looks moreand more like a normal distribution. In particular, the shape of thedistribution is not the same at di¤erent time scales.

F5: Intermittency: returns display, at any time scale, a high degreeof variability. This is quanti�ed by the presence of irregular burstsin time series of a wide variety of volatility estimators.

F6: Volatility clustering: di¤erent measures of volatility display apositive autocorrelation over several days, which quanti�es the factthat high-volatility events tend to cluster in time.

F7: Conditional heavy tails: even after correcting returns for volati-lity clustering (e.g., via GARCH-type models), the residual timeseries exhibit heavy tails. However, the tails are less heavy than inthe unconditional distribution of returns.

F8: Slow decay of autocorrelation in absolute returns: the au-tocorrelation function of absolute returns decays slowly as a func-tion of the time lag, roughly as a power law with an exponent� 2 [0:2; 0:4] : This is sometimes interpreted as a sign of long-rangedependence.

1Asset returns or returns will be used to also denote the series of the returns of someasset.

2A leptokurtic distribution is one that is peaked near the center and has fat tails.3An absolute return is the absolute value of a return.

SSE DAILY RETURNS: TAIL INDICES, SKEWNESS, HURST INDEX 5

F9: Leverage e¤ect: most measures of volatility of an asset are neg-atively correlated with the returns of that asset.

F10: Volume/volatility correlation: trading volume is correlatedwith all measures of volatility.

F11: Asymmetry in time scales: coarse-grained measures of volatil-ity predict �ne-scale volatility better than the other way.

These characteristics are extremely helpful for researchers who seek to un-derstand asset returns statistically and simulate them e¢ ciently, but theyseem to have frustrated any attempts to build mathematically and computa-tionally feasible continuous-time models that are able to incorporate almostall of them.To verify and better understand these empirical features and construct

good models for stock market index series, we take the daily return series ofthe SSE index 4 and study its statistical properties, such as skewness, kurto-sis, tail index, and long-term dependence structure. These statistics providestatistical support for simulating this return series by a generalized hyperbolic(GH) di¤usion model 5. In order to avoid unnecessary, even wrong assump-tions about this return series, all statistical estimates herein are obtainednonparametrically by linear regression with ordinary least squares (OLS).We also provide comparisons with other popular estimation techniques ofthese statistics that would be valid under the independent and identicallydistributed (i.i.d.) assumption. However no non-parametric check on the sta-tionarity of this return series is done due to the lack of plausible assumptionsand the reluctance of the author to impose i.i.d or a normality assumptionon the background noise of this series as proposed by most researchers6.The statistical properties estimated in this paper comply with their corre-

sponding stylized empirical features and tend to argue against the commontradition of assuming stationarity for return series of stock price indices andagainst the possibly inappropriate stable-distribution assumption for �nan-cial time series.

2. Tail Index

Figure 1 and Figure 2 depict the SSE index ~P = fPt : 1 � t � 4129g andthe corresponding return series ~R = fRt = lnPt+1�lnPt : 1 � t � 4128g fortrading days between 01=01=1990 and 11=24=2007, where the horizontal axisdenotes the number of trading days from Jan 1, 1991 and the vertical axisthe price index series and the corresponding returns, respectively. It is clearthat the original stock index series has an increasing trend for most part ofthe period considered, while the return series looks more like trendless noise

4Unless otherwise speci�ed, notationally we will not distinguish a random process X =fXt; t 2 Ig from an observation X = fXt; t 2 I0g ; I0 � I; of X; where I is a nonemptyindex set and I0 is usually nonempty, that is, fXt; t 2 I0g also denotes the observation X.

5For a hyperbolic di¤usion model for stock prices, see Bibby97 [2] and Rydberg99 [27].6It is widely agreed that most �nancial time series are not strictly stationary or even

covariance stationary (see, Mills95 [20], Mills99 [21], and Loretan94 [11] ) .

6 XIONGZHI CHEN

0 500 1000 1500 2000 2500 3000 3500 4000 45000

1000

2000

3000

4000

5000

6000

7000

Number of trading days from 01/01/1991

Pric

e In

dex

Series of Daily Price Index of Shanghai Stock Exchange

Figure 1. Daily SSE Price Index

0 500 1000 1500 2000 2500 3000 3500 4000 4500­0.4

­0.3

­0.2

­0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Number of Trading Days from 01/01/1991

Ret

urns

Return Series of SSE Index

Figure 2. Returns of SSE Index

but displays more volatility during the period roughly from Jan 1991 to Aug2006.Because under most circumstances only observations of a stochastic pro-

cess are available, it is reasonable to start from the empirical cumulativedistribution function (ecdf) of a process to obtain some of the required sta-tistics. Under the traditional assumption that all random variables of theunderlying stochastic process that generates the observations are identicallydistributed, the ecdf is even more crucial to statistical inference about theprocess. Studies show that the ecdf�s of most �nancial time series displaypower-decaying or semi-heavy tails (see, Cont01 [5], Mantegal95 [22], andLoretan [11]) with tail index � 2 (0; 5) (see, Mills99 [21]).

SSE DAILY RETURNS: TAIL INDICES, SKEWNESS, HURST INDEX 7

Given a stochastic process 7 X = fXt; t 2 Tg de�ned on some underlyingprobability space (;F ; P ) such that all Xt; t 2 T are identically distributed(i.d.) 8, the tail index � of X is usually de�ned as (see, Samorodnistky94[28])

� = supn� > 0 : EP

�jXtj�

�<1

owhere EP denotes the mathematical expectation with respect to the prob-ability measure P . Tail index is extremely important in risk managementbecause it helps measure the probability of outliers.Several methods for tail index estimation have been proposed (see, Hill75

[8], Loretan94 [11], Samorodnitsky94 [28], and Taqqu97 [30]), and theirproperties and performances have been compared (see, Taqqu97 [30] andMcCulloch [19]). But it seems that the classical Hill�s estimator, the linearregression method based on ordinary least squares (OLS) and the maximumlikelihood estimation (MLE) perform better in an overall fashion than theothers 9. Therefore, we base our estimation of the tail index � on linearregression by OLS but also provide the estimate obtained by the modi�edHill�s estimator introduced in Loretan94 [11].Before estimating the tail index for the ecdf of return series ~R = fRt : 1 �

t � 4128g, we provide a slightly di¤erent version (compared to Loretan94[11]) of the de�nition for the generalized tail index :

De�nition 1. Let X = fXt : t 2 Tg 10 be a stochastic process such thatXt; t 2 T are identically distributed (i.d.) with F (x) = P (Xt0 < x) wheret0 2 T is �xed and x 2 R. If

limx!1

P (Xt0 > x) = Cx�� (1 +BR (x)) ; x > 0 (2.1)

holds for some constants � > 0; C and for some BR (x) with

limx!1

BR (x) = 0

Then � is called the right tail index of X:

The left tail index � is de�ned similarly. The above de�nition actuallyrelaxes the i.i.d. assumption introduced in Loretan94 [11] and Mills95 [20],and such relaxation of assumptions may be important for analyzing returnseries, in that for most �nancial stochastic processes X = fXt : t 2 Tg therandom variables Xt; t 2 T may not be i.i.d or even i.d.. Thus in practicewe would have to content ourselves with the following interpretation of what

7For more details on the de�nition of stochastic process, see Karatzas98 [10].8For most �nancial time series, the random variables of the underlying stochastic

process that generates the observation are not identically distributed. For evidence onthis, see Mills99 [21].

9It should be noted that for stable distributions Hill�s estimator behaves somewhatbadly, according to Taqqu97 [30] and McCulloch [19].

10T � R is some nonempty index set and hereunder all index sets will not be explicitlydescribed unless confusion is possible.

8 XIONGZHI CHEN

is meant by saying " the empirical distributions of a stochastic process hasheavy tails ":

De�nition 2. Let X = fXt : t 2 Tg be a stochastic process with underlyingprobability space (;F ; P ) and let

ON = fXti (!i) : ti 2 T; !i 2 ; 1 � i � Ng ; N 2 Nbe a discrete observation of X: We say that the epdf of X has right tail index� if and only if

limx!1

�limN!1

~PN

�[N

i=1fXti (!i) > xg

��= Cx�� (1 +BR (x)) ; x > 0

(2.2)holds for any such observation, for some constants � > 0; C 2 R and forsome BR (x) 2 R with

limx!1

BR (x) = 0

where ~PN denotes the empirical density11 obtained from the observation ON :

The above de�nition describes more realistically what we see and do whenplotting the ecdf of samples by smoothing techniques, and it is almost equiv-alent to the assumption that the process X is ergodic.

2.1. Hill�s Estimator. In econometrics, the most popular estimators of theconstants C and � (see, Hill75 [8]) in (2.1) are obtained by using rank statis-tics of the observation ~X = fXi : 1 � i � Ng, that is, if

�X(i); 1 � i � N

are the order statistics of ~X such that X(1) � X(2) � � � � � X(N) then

� =

0@s�1 sXj=1

logX(N�j+1) � logX(N�s)

1A�1

(2.3)

is a right tail index estimator for fXt : t 2 Tg (assuming that all Xt; t 2 Tare identically distributed). Further, the estimator for C is given by

C =s

NX �(N�s) (2.4)

Empirical study shows that the tails of the ecdf�s of asset returns arenot asymptotically so heavy as those of the stable distributions but mostlyare power-law decaying or semi-heavy. McCulloch97 [19] has pointed thatthe Hill�s estimator behaves badly for stable distributions with tail index� close to 2 since it is usually up-biased, and that a tail index estimate� > 2 obtained by Hill�s estimator only rejects the joint hypothesis that theunderlying innovations 12 are i.i.d. and have stable distributions.

11By the law of large numbers, the empirical density converges to the true density asthe sample size tends to in�nity.

12"Innovations" is used by several authors to mean "the underlying noise that conta-minates the observations".

SSE DAILY RETURNS: TAIL INDICES, SKEWNESS, HURST INDEX 9

Due to the appeal of the Hill�s estimator, we still estimated � by using(2.3) � (2.4). We followed the suggestions from Philips96 [24] by setting

s (N) =h�N2=3

i13 (2.5)

and estimating � by

� =

���� �1p2Ns2 (�1 � �2)����2=3 (2.6)

where �1; �2 are preliminary estimates of � based on (2.3) using truncatess1 = [N

�] ; s2 = [N� ] with 0 < � < 2=3 < � < 1; � = 0:6 and � = 0:9 in the

actual implementation. Note that these estimates are for the right or theleft tail of the distribution respectively.The most devastating shortcoming of the above method is that it is based

on the i.i.d. assumption about the innovations of the sample. Since su¢ cientevidence show that most return series are not i.i.d., we can assume that thisis also true for ~R and consequently expect that the Hill�s estimator � forthe right tail index will be bigger than 2 .Hereunder we provide the detailed estimation procedure for the Hill�s tail

index estimator. By setting s1 =�41280:6

�= 147, s2 =

�41280:9

�= 1795,

we have

�1 =

240@s�11 s1Xj=1

logX(N�j+1)

1A� logX�1(N�s1)

35�1

=

�1237:917

147� 8:1118

��1= 3:2320

and

�2 =

0@s�12 s2Xj=1

log�X(N�j+1) � logX(N�s1)

�1A�1

=

�13473:64

1795� 7:1903

��1= 3:1655

which gives

� =

����3:2320p2

4128

1795(3:2320� 3:1655)

����2=3 = 0:4963and

s (N) =h0:4963 � (4128)2=3

i= [1366:1] = 1366

Thus

� =

0@s�1 sXj=1

logX(N�j+1) � logX(N�s)

1A�1

= 3:4100

13[x] ; x 2 R denotes the maximal interger not exceeding x.

10 XIONGZHI CHEN

andC =

s

NX �(N�s) =

1366

4128(7:2958)3:4100 = 290:19

Consequently, assuming all Rt; 1 � t � 4128 in the returns series ~R areidentically distributed (which might be false) and that the sample size 4128for ~R is large enough to stabilize the right tail index estimate (see, Mills99[21]), we have asymptotically

~PN (R1 > x) � 290:193:4100x�3:4100 (1 + �R (x))for su¢ ciently large x > 0, where limx!+1 �R (x) = 0; i.e.,

log ~PN (X > x) � 19:337� 3:41 log xThus the Hill�s estimator of the right tail index is 3:41.

2.2. Linear Regressor. Now we will use visual examination and linearregression to estimate the tail index � without assumptions about the un-derlying innovations for the return series ~R. Figure 3 shows the comparison

­0.4 ­0.2 0 0.2 0.4 0.6 0.80

3

6

9

12

15

18

21

24

27

29

Returns

Pro

babi

lity

Empirical Pdf against Nomal Density

Empirical PDFN(0.00091,0.0289)

Figure 3. EPDF of Returns

between the empirical probability density function (epdf) of ~R obtained bya kernel estimator and the normal density whose mean and standard de-viation are the same as those of this epdf. Figure 3 clearly suggests thatthe epdf has heavier tail and bigger kurtosis than those of a normal densityN (0:00091; 0:00289).We also provide in Figure 4 and Figure 5 the log-log plots of both right

and left tails. A visual check of these two �gures reveals that there aresome points for x right beyond which logFN (�x) and log (1� FN (x)) arestrongly linear against log x, respectively, where FN (�) is the epdf of a ran-dom variable with sample size N . These points are x � 0:024 for the lefttail and x � 0:03 for the right tail of ~R.Since all we want are the asymptotic properties (assuming that a sample

sizeN = 4128 is su¢ ciently large for the estimation of asymptotic statistics),

SSE DAILY RETURNS: TAIL INDICES, SKEWNESS, HURST INDEX 11

­12 ­10 ­8 ­6 ­4 ­2 0­8

­7

­6

­5

­4

­3

­2

­1

0

log(x), x > 0

log(

F N(­x

))

Log Left  Tail of Returns(SSE: 01/01/1991~11/14/2007)

Figure 4. Log Left Tail of Returns

­14 ­12 ­10 ­8 ­6 ­4 ­2 0­8

­7

­6

­5

­4

­3

­2

­1

0

log(x), x > 0

log(

1­F N

(x))

Log Right Tail of Returns(SSE:01/01/1991~11/14/2007)

Figure 5. Log Right Tail of Returns

we can safely choose x � 0:024 and x � 0:03 respectively and then uselinear regression by OLS to estimate the right and left tail index. Figure 6and Figure 7 display where the log-linearity seems valid as compared withreference linear regressors14.The regression equation for the right tail is

ln (1� FN (x)) = �0:9766� 4:670 lnx; x � 0:030061and that of left tail is

lnFN (�x) = �10:370� 2:192 lnx; x � 0:024045Consequently, the OLS estimate for the right tail index is

�R = 4:670

14Figure 6 and Figure 7 show that both ends of the two tails diverge a lot from theirlinear regressors. This is the consequence of a few outliers in the sample ~R due to relativelylarge gains or losses that happened approximately in Oct 2007

12 XIONGZHI CHEN

­4 ­3.8 ­3.6 ­3.4 ­3.2 ­3 ­2.8 ­2.6 ­2.4 ­2.2 ­2 ­1.8 ­1.6 ­1.4 ­1.2 ­1­10

­9.2

­8.4

­7.6

­6.8

­6

­5.2

­4.4

­3.6

­2.8

­2

log(x),x>0.024

log(

F N(­x

))

Log­log Plot of Left Tail versus Linear Regressor

Log­log Plot of Left TailLine of Regressiony=­10.37­2.19*log(x)

Figure 6. Log Left Tail with Linear Regressor

­1.6 ­1.52 ­1.44 ­1.36 ­1.28 ­1.2 ­1.12 ­1.04 ­0.96 ­0.88 ­0.8 ­0.72 ­0.64 ­0.56 ­0.48 ­0.4 ­0.32 ­0.24 ­0.2­9

­8.5

­8

­7.5

­7

­6.5

­6

­5.5

­5

­4.5

­4

­3.5

­3

­2.5

­2Log­Log Plot of Right Tail versus Linear Regressor

log(x), x >= 0.03

log(

1­ F

N(x

))

Log­log Plot of Right TailLine of Regressiony=­9.76 ­ 4.67*log(x)

Figure 7. Log Right Tail with Linear Regressor

and that for left tail index is

�L = 2:192

These indices are consistent with the usual observation that the ecdf�sof stock return series are not stable distributions but Paretian with power-decaying tails. They are further evidence against assuming in�nite vari-ance for stock return series whose innovations are identically distributed(see, Mills99 [21]). They are also in accordance with the tail range most re-searchers have found empirically and the observation that such ecdf�s usuallyhave relatively heavier left tails than right tails (see Mills99 [21]).

2.3. Mandelbrot�s Visual Check. Supplementary to the tail estimationconducted above, we plot the sample moments computed by

�p (n) =1

n

nXi=1

0@Ri � 1

n

nXj=1

Rj

1A

SSE DAILY RETURNS: TAIL INDICES, SKEWNESS, HURST INDEX 13

of order p = 2; 4 for ~R for 1000 � n � 4128 in Figure 8 and Figure 9

1000 1500 2000 2500 3000 3500 40000

0.5

1

1.5

2

2.5x 10­6

sample size

sam

ple 

varia

nce

Sample Variance against Sample Size

Figure 8. Sample Variance

1000 1500 2000 2500 3000 3500 4000 45000

0.5

1

1.5

2

2.5

3

3.5x 10­13

Sample size

Sam

ple

 4th

 mom

ent

Sample 4th moment versus sample size

Figure 9. 4th Sample Moment

These graphs seem to show that as the sample size increases to in�nity, thesample variance and 4th order sample moment stabilize themselves aroundcertain values respectively. Thus by the idea of Mandelbrot63 [15], we seemto be able to infer that ~R does have �nite variance and 4th order moment 15

(if ~R has a unique law of probability 16). This is consistent with the estimatesof the tail indices herein. Actually we also plotted the 5th absolute samplemoment for ~R1 = fRt; 4128 � t � 1500g in Figure 10 to check whether theright-tail index is low-biased or not (since if it is low-biased then the 5thabsolute sample moment might diverge eventually) and it seems that the5th absolute moment is also �nite by Mandelbrot�s idea.

15We say that "a stochastic process X = fXt; t 2 Tg has some property" if (it isassumed that) all Xt; t 2 T have the same property.

16A stochastic process X = fXt; t 2 Tg is said to have a unique law of probability ifall Xt; t 2 T have the same law of probability.

14 XIONGZHI CHEN

1500 2000 2500 3000 3500 4000 45000

0.5

1

1.5

2

2.5

x 10­17

sample size

 5th

 absolu

te s

am

ple

 mom

ent

5th absolute sample moment against sample size

Figure 10. 5th Absolute Sample Moment

It should be pointed out that this visually checked convergence might befalse due to insu¢ cient sample size. But proof or disproof of this seemsto be impossible in this paper since we are not able estimate the exact as-ymptotic tail properties of the ecdf without assuming much more about theunderlying noise for ~R. Also this visual check can be misleading particularlywhen the sample is from a distribution with tail index 2; as does sometimeshappen when sampling a Cauchy distribution. However under very strin-gent assumptions about the mixing properties and the order of moments ofthe innovations, the �niteness of the 4th moment can be tested as done inLoretan94 [11].

3. Kurtosis and Skewness

Kurtosis is easily recognized from the �gure of epdf (Figure 3). To checkthe skewness, the easiest way is plot X(N�p) � Xmed against Xmed � X(p)since for symmetric epdf�s

(X(N�p) �Xmed; Xmed �X(p))will be points one a line of slope �1; where X(i); i = 1; :::; N are the rankstatistics of the sample ~X = fXi : 1 � i � Ng, Xmed is the median and1 � p � [N=2] + 1. Empirical evidence show that within a fairly wide rangearound its mean the epdf of a �nancial time series is symmetric and skewnessappears in the tails (see, Mills99 [21] and Mantega95 [22]). Moreover, mostsuch epdf�s are skewed to the left (see, Mills95 [20] and Badrinath98 [1]).Figure 11 plots R(T�p) � Rmed against Rmed � R(p) for ~R = fRt : 1 � t

� 4128g. An OLS linear regression estimate of R(p) against R(T�p) gives

R(N�p) = �1:0629 �R(p) + 0:0004; 1 � p � [N=2] + 1Surprisingly it shows that the epdf is skewed more to the right than to the

left (see the "zoom in" Figure 12 of a kernel density estimate of the epdf),

SSE DAILY RETURNS: TAIL INDICES, SKEWNESS, HURST INDEX 15

­0.35 ­0.3 ­0.25 ­0.2 ­0.15 ­0.1 ­0.05 0 0.05­0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Lower quantiles of returns

Hig

her q

uant

iles 

of re

turn

s

Higher quantiles agains lower ones

dataHypothetical line of slope ­1linear regressor

Figure 11. Skewness

­0.1 ­0.05 0 0.05 0.1 0.150

5

10

15

20

25

30

Returns

Pro

ba

bility

Kernel density estimate of the epdf of returns

Figure 12. Zoom-in EPDF

although the epdf of the returns ~R is symmetric in a relatively wide rangearound its mean. This strange observation complies with the fact that formost of the trading period from July 1994 to Oct 2007 the SSE index wasincreasing, and that the probability of a large decrease in this return seriesis relatively lower than that of a large increase.

4. Dependence

Dependence, long-term or short-term, exists in time series observed froma lot of natural phenomena (see, Cont01 [5], Mandelbrot62, [14] and Man-delbrot01a [18]) and has been paid a lot of attention in recent years. Fordiscrete time models that incorporate such dependence, the FIGARCH (i.e.,fractional integrated GARCH) model has been introduced, while for continu-ous time model, the fractional Brownian motion (fBm) has been constructed(see, Mandelbrot62 [14] and Mandelbrot01a [18]), both of which have wideapplications.

16 XIONGZHI CHEN

For a discretely sampled stochastic process, the Hurst indexH can be usedto measure the strength and duration of its dependence structure. Hereunderwe describe three methods to estimate this index and provide the estimatedHurst index for ~R.

4.1. Hurst�s R/S Statistic. Hurst (see, Hurst51 [9]) found out empiricallyor how the Hurst index is de�ned is that

E (R=S(n)) / CnH (4.1)

holds for su¢ ciently large sample size n for some constants C > 0 and someH 2 (1=2; 1) : Here E is the expectation with respect to the probabilitymeasure P of the underlying probability space (;F ; P ) on which the ran-dom process that generates the sample fxi; i = 1; 2; :::; ng is de�ned. Thestatistic

R=S (n) =R (n)

S (n)(4.2)

is called the classic rescaled range statistic or R=S statistic with

� =1

n

nXi=1

xi; �2 =

1

n

nXi=1

(xi � �)2 ; S (n) =

vuut 1

n

nXi=1

(xi � �)2

and

R (n) = max1�j�n

(jXi=1

xi � j�)� min1�j�n

(jXi=1

xi � j�)

One major disadvantage of the classic Hurst R=S estimator is its excessivesensitivity to short-range dependence 17. But its key advantage is that, ifwe de�ne a long-range dependence parameter d by

d = H � �with � = 1=2 for �nite variance processes or 1=� for in�nite varianceprocesses with tail index 0 < � < 2; then the R=S statistic consistently

provides an estimate of d +1

2independent of the value of �. This last is

true becauseR (n) / nH = nd+1=�

andS (n) / n1=��1=2

asymptotically (see, Taqqu98 [31] and Mandelbrot75 [17]).Figure 13 is a log-log plot of the classic R=S (n) against n for

~Rn :=18 fRt : t = 1; ::; ng ; 1 � n � 4025

As can been seen from the log-log plot Figure 13, linearity is very strongfor two sections in terms of the sample size n : exp (0:8) � n � exp (6) and

17See Mandelbrot01a [18] for details on "short/long-range dependence".18" := " means " de�ned as ".

SSE DAILY RETURNS: TAIL INDICES, SKEWNESS, HURST INDEX 17

0 1 2 3 4 5 6 7 8 90

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

log(n)

log

((R

/S)(

n))

Log­log Plot of R/S Statistic against Sample Size

Figure 13. Log-log Plot of Classic R/S Statistic

exp(8) � n � exp(6): Consequently it�s reasonable to estimate H for eachsection. An OLS regressor for the �rst section is given as

L1 (n) = 0:3366 � lnn+ 1:6223; 4 � n � 410and that for the second section

L2 (n) = 0:4265 � lnn+ 0:9152; 410 � n � 4025which are combined into

logR=S (n) =

�0:3366 � lnn+ 1:6223 if 4 � n � 4100:4265 � lnn+ 0:9152 if 410 � n � 4025

The corresponding plots of logR=S (n) against log n with linear regressorsare provided in Figure 14 and Figure 15.

1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 60

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Sample size in log scale

Lo

g c

lassic

 R/S

 sta

tistic

Classic R/S statistic against sample size in log scalewith linear regressor

ln(R/S) vs ln(n)regressor

Figure 14. Classic R/S Statistic with Linear Regressor I

Thus focusing on large sample properties, we can just take 0:4265 to bethe estimate of d + 1=2 since we showed that ~R has �nite variance by its

18 XIONGZHI CHEN

6 6.5 7 7.5 8 8.53.4

3.6

3.8

4

4.2

4.4

4.6

4.8

5

Sample size n>=exp(6)

Lo

g c

lassic

 R/S

 Sta

tistic

Classic R/S Statistic against sample size in log scalewith linear regressor

Figure 15. Classic R/S Statistic with Linear Regressor II

tail index estimate (by explicitly assuming that ~R has a unique probabilitylaw). This suggests that the estimated d = �0:0735 and that ~R is a processwith short-range dependence.However, Mandelbrot�s R=S analysis (abbreviated as the "R=S analysis")

and the Lo�s modi�ed R=S statistic (abbreviated as the "Lo�s statistic") inthe following subsections both show the existence of long-range dependencein ~R, contradicting the short-range dependence revealed by classic R=S sta-tistics. This might be a consequence of the excessive sensitivity of the clas-sic R=S statistic to short-range dependences in various durations where thesample ~R is taken. We present the R=S analysis next.

4.2. Mandelbrot�s R/S Analysis. The actual and more realistic imple-mentation of the classic R=S statistic is through the so called "R=S analysis"(see, Mandelbrot69 [16]). It is described brie�y as follows (see, Rachev00[26]): let N be the sample size; divide the sample into K blocks each of

size N=K and de�ne km = 1 + (m� 1) NKwith m = 1; � � � ;K; then esti-

mate Rn (km) and Sn (km) for each lag n such that km + n < N ; �nally,

plot logRn (Km)

Sn (km)against log n for each m = 1; � � � ;K in one �gure. The

resulting graph is called " rescaled adjusted range plot " or " pox plot ofthe (classic) R=S statistic (see, Rachev00 [26]) ".To allow �exibility in grouping, we only selected the �rst 4000 entries

from ~R with group size l = 400 and provide the pox plot of R=S statistic inFigure 16.Note that in Figure 16 the red line has slope 0:62 and the blue one has

slope 0:5, the latter of which corresponds to the Hurst index of a standardBrownian motion. It can be seen clearly from the plot that for sample size

SSE DAILY RETURNS: TAIL INDICES, SKEWNESS, HURST INDEX 19

0 1 2 3 4 5 6 7 8 90

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

log(n)

(RS)

(n)

Pox Plot of R/S with Reference Lines

Red line:slope 0.62

Green LineSlope 0.5

Figure 16. Pox Plot of R/S Statistic

n � exp (5) ; all Rn (Km)Sn (km)

; m = 1; � � � ;K stay within the cone correspond-

ingly formed by the graphs of L1 = 0:62 � log n and L1 = 0:5 � log n (whichis relatively su¢ cient for asymptotic property). The most striking featureis that the line

L = 0:57 � log n; 4000 � n � [exp (5)]

gives a good least squares �t to these scattered R=S statistics, which suggeststhe estimated Hurst index is

H = 0:57 (4.3)

This is consistent with what most researchers have found as the Hurst indexestimates for �nancial time series (see, Cont01 [5]).Thus we can reject the hypothesis that the return series of the SSE in-

dex follows a standard Brownian motion. Moreover, if the process ~R has�nite variance we then can conclude that the return series ~R is a long-rangedependence process with long-range dependence parameter

d = 0:07 (4.4)

Note that this is also supported by the autocorrelations of ~R till maximalorder of lag L = 1000 in Figure 17.To reduce the excessive sensitivity of the R=S analysis to short-range

dependences (see Figure 13 ), Lo91 ([12]) introduced the so called " Lo�smodi�ed R=S statistic " to detect overall long-range dependence withoutgiving any estimation of the Hurst index H. Lo�s modi�ed R=S statistic isde�ned as

VN (q) =1pN

RNSN (q)

(4.5)

20 XIONGZHI CHEN

100 200 300 400 500 600 700 800 900 1000­0.2

­0.15

­0.1

­0.05

0

0.05

0.1

0.15

0.2

Lag Order

Sa

mp

le A

uto

co

rre

latio

n

Sample Autocorrelation Function (ACF)

Figure 17. ACFs with Max Lag Order 1000

where

� =1

N

NXi=1

Xi; !j (q) = 1�j

q + 1; q < N (4.6)

and

RN = max1�q�N

0@ qXj=1

Xj � q�

1A� min1�q�N

0@ qXj=1

Xj � q�

1A (4.7)

and

S2N (q) =1

N

NXi=1

(Xi � �)2 + (4.8)

2

N

qXj=1

!j (q)

0@ NXi=j+1

(Xi � �) (Xi�j � �)

1ATo use the Lo�s modi�edR=S statistic, we cite from Lo91 [12] the following

theorem.

Theorem 1 (Lo91 [12]). Suppose the random process " = f"t : t � 0g sat-is�es the following conditions

a1): E ("t) = 0a2): supt�0E

�j"tj2�

�<1 for some � > 2

a3): 0 < �2 = limn!1E�1

n

Pnj=1 "

2j

�<1

a4): f"tg is strongly mixing with mixing coe¢ cients �k that satisfy1Xj=

�1�2=�j <1

SSE DAILY RETURNS: TAIL INDICES, SKEWNESS, HURST INDEX 21

a5): q � o�n1=4

�as both n and q approach to +1

Then

VN (q)!d W1 = maxW0 (t)�minW0 (t) (4.9)

if f"tg has no long-range dependence, where W0 (t) is the standard Brownianbridge and

P (W1 2 [0:809; 1:862]) = 0:95

Note that in the above theorem, at least uniformly the fourth momentis required to be �nite for all "t; t � 0 in order to get the asymptoticdistribution. Assumption a3) is a weak condition about the ergodicity of thevariance and assumption a4) requires that the random variables "t; t � 0can not be so dependent.From the estimates already obtained, we know that ~R = fRt : 1 � t

� 4128g does not satisfy assumption a2) even though it might satisfy as-sumption a4) since the daily return series have very weak dependence (see,Cont01 [5]). Based on the previous tail index estimates and running therisk that ~R might violate the fourth moment requirement a4), we stillplotted the Lo�s R=S statistic VN (q) in Figure 18 for the detrended �R =

0 500 1000 1500 2000 2500 3000 3500 4000 45000.5

1

1.5

2

2.5

3

3.5

Modified lags q

VN

(q)

Lo's RS Statistic with Confidence Interval

Figure 18. Lo�s Modi�ed R/S Statistic

fRt � � : 1 � t � 4128g where � is the sample mean of ~R:Note in Figure 18 the con�dence interval of the Lo�s modi�ed R=S statistic

is given by the red and green horizontal lines (under the assumption of theprevious theorem). Obviously, approximately for q � 1500; this statisticstays outside the 95% con�dence interval and thus suggests the existence oflong-range dependence. (It should be noted that Lo�s modi�ed R=S statisticis widely used even though not all assumptions in the above theorem aresatis�ed. See Rachev00 [26].)

22 XIONGZHI CHEN

4.3. Direct Regression. As Rachev00 [26] has pointed out that the R=Sanalysis is still sensitive to the presence of short-range dependence, we adoptthe method of direct regression therein to estimate the Hurst index, thatis, for a discrete observation ~X = fXi : 1 � i � Ng of a random processX = fXt : t 2 Tg we can regress

�k = log E (jXi+k �Xij) 19; 1 � i � N � kagainst log k to estimate the Hurst index H; where E is the sample meanand 1 � k � N � 1.For the returns ~R = fRt : 1 � t � 4128g, the results are presented in

Figure 19 in which the linear regressor is given by

0 1 2 3 4 5 6 7 8 9­5

­4

­3

­2

­1

0

1

2

Log Lag order

Lo

g A

bso

lute

 Ave

rag

e R

etu

rn

L­L Plot of Absolute Average Returns  againstLag Order with Linear Regressor

log absolute average returnsLinear Regressor

Figure 19. Hurst Index via Direct Regression

�k = log E (jRt+k �Rtj) = 0:6133 log k � 4:6021; 1 � t � 4128� kwhere 1 � k � 4127:Thus the estimated Hurst index for the return series ~R is

H� = 0:6133

This result con�rms the existence of long-range dependence in ~R and isconsistent with similar results obtained in Oh06 [23].

5. Conclusions

We have mainly estimated the tail indices and Hurst index of the returnseries of the SSE index, by non-parametric linear regression. These statisticsare consistent with the empirical estimates obtained by most researchers,except that the corresponding return series skews more to the right. Animperfection of this paper is that no statistical tests have been carried outfor the estimates, since we do not pre-assume any speci�c properties ofthe stochastic process that generates this return series. However, the results

19If fXt; t = 1; 2; ::; Ng is the logarithmic SSE index, then E (jXt+k �Xtj) is called theaverage absolute return over k quotes. (See, Rachev00 [26]) .

SSE DAILY RETURNS: TAIL INDICES, SKEWNESS, HURST INDEX 23

will de�nitely help to measure the appropriateness of the ergodic generalizedhyperbolic di¤usion as a model for the linearly detrended logarithmic SSEindex in the next paper.

Please continue to the second part of the thesis after the references .......

References

[1] S. G. Badrinath and Sangit Chatterjee (1998): On Measuring Skewness and Elon-gation in Common Stock Return Distributions: The Case of the Market Index. TheJournal of Business, Vol. 61, No. 4 (Oct., 1988), pp. 451-472

[2] Bibby, B. M. & Sorensen, M. (1997): A Hyperbolic Di¤usion Model for Stock Prices.Finance and Stochastics 1(1), 25�41.

[3] Chang Xuejiang, et al. (1993): Time series analysis. China Higher Education Press[4] Rama Cont, Peter Tankov (2004): Financial modelling with jump processes. Chapman

& Hall[5] Rama Cont (2001), Empirical properties of asset returns: stylized facts and statistical

issues. Quantitative Finance, Volume 1 (2001), 223-236[6] Peter Hall and A. H. Welsh (1985): Adaptive Estimates of Parameters of Regular

Variation. The Annals of Statistics, Vol. 13, No. 1 (Mar., 1985), pp. 331-341[7] J.D. Hamilton, R. Susmel (1994): Autoregressive conditional heteroskedasticity and

changes in regime. Journal of econometrics, 64, 307-33[8] B. M Hill (1975): A simple general approach to inference about the tail of a distribu-

tion. The Annals of statistics, 3, 163-74[9] H. Hurst (1951): Long term storage capacity of reservoirs. Transactions of the Amer-

ican Society of Civil Engineering, 116, 770-99[10] I. Karatzas, S.E. Shreve 1998, Brownian Motion and Stochastic Calculus. Springe-

Verlag[11] Mico Loretan, Peter C.B. Philips (1994): Testing the covariance stationarity of heavy-

tailed time series. Journal of empirical �nance 1 (1994) 211-248 North-Holland[12] Andrew W. Lo (1991), Long-term memory in stock market prices. Econometrica, Vol.

59, No. 5 (September, 1991), 1279-1313[13] Andrew W. Lo, A. Craig MacKinlay (1999): A non-random walk down wall street.

Princeton University Press[14] Benoit B. Mandelbrot (1962): New methods in statistic economics. Journal of political

economy, 71, 421-40[15] Benoit B. Mandelbrot (1963): The variation of certain speculative prices. Journal of

Business XXXVI, 392-417.[16] BB Mandelbrot, JM Wallis (1969), Robustness of the rescaled range R/S in the mea-

surement of noncyclic long run statistical dependence, Water Resources Research,5:967-988.

[17] Benoit B. Mandelbrot(1975): Limit theorems on self-normalized range for weaklyand strongly dependent processes. Zeitschrift fur Wahrscheinlichkeitstheorie und ver-wandte Gebieter, 31:271-285

[18] Benoit .B. Mandelbrot (2001): Scaling in �nancial prices I: tails and dependence.Quantitative �nance, Volume 1 (2001) 113-123

[19] J. Huston McCulloch (1997): Measuring Tail Thickness to Estimate the Stable Index�: A Critique. Journal of Business & Economic Statistics, Vol. 15, No. 1 (Jan., 1997),pp. 74-81

[20] Terence C. Mills (1995): Modelling Skewness and Kurtosis in the London Stock Ex-change FT-SE Index Return Distributions. The Statistician, Vol. 44, No. 3 (1995),pp. 323-332

24 XIONGZHI CHEN

[21] Terence C Mills (1999): The econometric modelling of �nancial time series. Cam-bridge University Press

[22] Rosario N. Mantenga, H. Eugene Stanley (1995): Scaling behavior in the dynamicsof an economic index. Nature, Vol 376, July 1995

[23] Gabjin Oh, Seunghwan Kim (2006): Statistical properties of the returns of stock pricesof international markets. Journal of the Korean Physical Society, Vol 48, February2006, pp S197-S201.

[24] Peter C. B. Phillips, James W. McFarland, Patrick C. McMahon (1996): Robust Testsof Forward Exchange Market E¢ ciency with Empirical Evidence From the 1920s.Journal of Applied Econometrics, Vol. 11, No. 1 (Jan. - Feb., 1996), pp. 1-22

[25] C.-K. Peng, et al (1994): Mosaic organization of DNA nucleotides. Phys. Rev. E 49,1685 (1994).

[26] Svetlozar T. Rachev, Stefan Mittnik (2000): Stable Paretian Models in Finance. Wiley2000

[27] T. H. Rydberg (1999): Generalized hyperbolic di¤usion processes with applications in�nance. Mathematical Finance, Vol 9, 183-2001

[28] Gennady Samorodnitsky, Murad S. Taqqu (1994): Stable non-Gaussion randomprocess. Chapman & Hall

[29] Michael F. Shlesinger (1994): Comment on "stochastic process with ultraslow conver-gence to a Gaussian: the truncated Levy �ight". Physical Review Letters, Volume 74Number 24

[30] Murad Taqqu, Estimators for long-range dependence: an empirical study, workingpaper

[31] Murad S. Taqqu, Vadim Teverovsky (1998): On Estimating the Intensity of Long-Range Dependence in Finite and In�nite Variance Time Series. Unpublished work.

[32] Xiao-tian Wang, et al. (2006): Fractional poisson process. Chaos, solitons and fractals28 (2006) 143-147

SHANGHAI STOCK EXCHANGE II: NOT AN ERGODICGENERALIZED HYPERBOLIC DIFFUSION

XIONGZHI CHEN

Abstract. Motivated by recent successes of modelling stock prices byhyperbolic distributions, (generalized) hyperbolic di¤usions (Bibby97[5], Elerian98 [18] and Rydberg99 [37]), this paper attempts to simulatethe linearly detrended logarithmic SSE index (abbreviated hereunder as"detrended index" ) by an ergodic generalized hyperbolic (GH) di¤usion.The goal is a simulation whose tail index and correlation structure matchthe estimates obtained in part one of this thesis and whose invariantdensity is proportional to some generalized hyperbolic (GH) density.We adopt the methodologies from Bibby97 [5], Bibby04 [7], and

Sorensen02 [40] to build such a model. We estimate the six parametersinvolved in the model by Markov chain Monte Carlo (MCMC) methodswhose convergence criteria are the Monte Carlo standard error (MCSE)(Roberts96 [35]) and the Gelman-Rubin (GR) convergence diagnostic(Gelman92 [20], Brooks98 [10] and Brooks00 [11]), while retaining agood acceptance rate (Chib95 [12], Johannes03 [25]).We run 6 independent chains in the MCMC simulation whose ini-

tial values are widely dispersed but centered around the nonlinear leastsquares estimate of the parameters of a GH density based on the de-trended index.MCSEs suggest that all six independent chains have converged to

their own stable distributions at around 50; 000 iterations and the MC-SEs remain at around 10�3 afterwards. Unfortunately the GR statisticdoes not converge to 1 even after 350; 000 iterations. Because the ac-ceptance rates became too low for all six chains after around 50; 000iterations, the MCMC simulation has to be terminated. Further itera-tions will not signi�cantly move the chains.We take the last 10% of the �rst 50; 000 samples from each Markov

chain in the MCMC simulation to obtain 6 Monte Carlo estimates forthe parameter vector. Due to the computational inability of both Math-ematica and Matlab resulted from the 2 badly mixed chains, only 4chains are selected for the uniform residual test (Bibby97 [5]).Test results show that all 4 possible GH models are rejected statisti-

cally. Thus under the scheme of approximating the unknown posteriorfor the parameter vector by the likelihood function obtained from Mil-stein�s discretization, we reject the hypothesis that " the underlyingstochastic process for the detrended index in the trading period from01/01/1991 to 11/14/2007 is an ergodic, GH di¤usion", i.e., the de-trended index is not an ergodic GH di¤usion.

Key words and phrases. Generalized Hyperbolic Distribution, Ergodicity, InvariantMeasure, Markov chain Monte Carlo Methods, Metropolis-Hastings Algorithm, Gelman-Rubin Convergence Diagnostic/Statistic, Uniformal Residual Test.

1

2

Contents

List of Tables 3List of Figures 41. INTRODUCTION 52. CONSTRUCTION OF AN ERGODIC, GH DIFFUSION 62.1. General Theory of Solutions of One Dimensional SDEs 72.2. Ergodic Di¤usion with GH Invariant Density 83. MCMC ESTIMATION OF PARAMETERS 113.1. Milstein Discretization Scheme 113.2. MCMC Methods and Metropolis-Hastings Algorithm 163.3. Convergence Checking 183.4. Implementation and Estimates 213.5. Test of Model 234. CONCLUSION AND DISCUSSION 25References 265. APPENDIX 285.1. Numerical Results 28

SSE: NON-GENERALIZED HYPERBOLIC DIFFUSION 3

List of Tables

1 Monte Carlo Estimates for Parameter Vector for 6 Chains 23

2 MCSEs at the 100 th Group 24

3 Uniform Residual Test Statistics for 4 Models 24

4 XIONGZHI CHEN

List of Figures

1 Detrended Index 12

2 EPDF of Detrended Index 21

3 Squared Di¤usion Function in Model 2 22

4 Max MCSEs for 6 Chains 22

5 Acceptance Rates for 6 Chains 23

6 QQ Plot of Residuals for Model 2 25

7 E¤ective Sample Paths in 6 Chains for � 28

8 E¤ective Sample Paths in 6 Chains for � 29

9 E¤ective Sample Paths in 6 Chains for � 30

10 E¤ective Sample Paths in 6 Chains for � 31

11 E¤ective Sample Paths in 6 Chains for � 31

12 E¤ective Sample Paths in 6 Chains for � 32

13 Simulated Squared Volatilities for 6 Models 33

14 Simualted Detrended Index for Models 1~3 34

15 Simualted Detrended Index for Models 4~6 35

16 QQ Plot of Residuals of 4 Tested Models 36

SSE: NON-GENERALIZED HYPERBOLIC DIFFUSION 5

1. INTRODUCTION

In view of recent successful applications of generalized hyperbolic (GH)distributions in modelling asset returns, stock prices and option prices (seeBibby97 [5], Karsten99 [27], and Eberlein95 [16]), we tentatively assumethat the underlying stochastic process X = fXt : t � 0g for the linearlydetrended logarithmic daily SSE price index 1 is an ergodic GH di¤usion 2

whose invariant density is some GH density and which for the trading daysconsidered herein induces a daily return series that has the tail indices anddependence structures as previously estimated in part one of this thesis. Tobe more speci�c, we try to solve following simulation

Problem 1. Construct an ergodic, weak solution X = fXt : t � 0g to theparametric stochastic di¤erential equation (SDE)

dXt = b (Xt; �) dt+ v (Xt; �) dBt3 (1.1)

such that(1) X has the generalized hyperbolic (GH) density

fgh (x;�; �; �; �; �) (1.2)

=

��2 � �2

��=2K��1=2

��q�2 + (x� �)2

�p2����1=2��K�

��p�2 � �2

���2 + (x� �)2

�(��1=2)=2exp (� (x� �))

as its invariant or initial or asymptotic density, where the domainof variation of the parameters is � 2 R and8<: � � 0; j�j < � if � > 0

� > 0; j�j < � if � = 0� > 0; j�j � � if � < 0

(1.3)

and the modi�ed Bessel function K� of the second kind is given by

K� (x) =1

2

Z 1

0y��1 exp

��x2

�y + y�1

��dy; x > 0 (1.4)

(See Karsten99 [27] for more details on the GH density)(2) and discretized on the trading days considered in part one of this

thesis, X induces a daily return series that has tail indices and de-pendence structures as estimated in part one of this thesis.

1"Detrended index" will denote " the linearly detrended logarithmic daily SSE indexseries", and the exact linear detrending will be explained in later sections.

2For a de�nition of "di¤usion", please see Shreve98 [38]. For a de�nition of "generalizedhyperbolic di¤usion", please see Rydberg99 [37].

3Here in (1.1), b (�; �) and v (�; �) are appropriately chosen drift and di¤usion functionsparametrised by �, respectively. The parameter � lies in some bounded, open subset ofRm for some m 2 N; and B = fBt; t � 0g is a standard Brownian motion.

6 XIONGZHI CHEN

We adopt the traditional notations and rewrite fgh compactly as

fgh (x; �) = a (�; �; �; �)K��1=2

��

q�2 + (x� �)2

�(1.5)

���2 + (x� �)2

�(��1=2)=2exp (� (x� �))

with parameter vector � = (�; �; �; �; �) by letting

a (�; �; �; �) =

��2 � �2

��=2p2����1=2��K�

��p�2 � �2

� (1.6)

Further, we simply denote the unnormalized GH density fgh (x; �) by

f (x; �) (1.7)

= K��1=2

��

q�2 + (x� �)2

���2 + (x� �)2

�(��1=2)=2exp (� (x� �))

We adopt the method in Bibby95 [4] and Bibby97 [5] to build such anergodic GH di¤usion for the detrended index. As for parameter estima-tion we use Markov chain Monte Carlo (MCMC) methods driven by theMetropolis-Hastings (MH) algorithm to sample from the unknown posteriorof the parameter vector (see Johannes03 [25] and Tes04 [42] 4).It is unfortunate that for the detrended index, due to the intense com-

putational complexity, the method of martingale estimating function (seeBibby95 [4] and Bibby97 [5]) has not been successfully implemented, andthat due to lack of time, estimation methods introduced in Ait-Sahalia02 [3]5 have not been applied.

2. CONSTRUCTION OF AN ERGODIC, GH DIFFUSION

There are many references to building di¤usion-type models with givenmarginal distributions and (exponentially-decreasing) autocorrelation func-tions (see Bibby97 [5], Bibby04 [7], Madan02 [30] and Ait-Sahalia02 [3]). Allmodels constructed fall into three categories: (i)X is a di¤usion with a giveninitial density; (ii) X is an ergodic di¤usion whose asymptotic density or in-variant density is the given density; (iii) X is a di¤usion with a given familyof marginal densities. Herein, we only concentrate on the construction of adriftless, ergodic GH di¤usion with a GH invariant density.

4Note however that in Tse04 [42] at least two mistakes have been found by the authorof this thesis.

5Ait-Sahalia�s method is the only one that gives explicit approximation to the transitiondensities of su¢ ciently good di¤usions and estimates the unknown parameters only after agood approximation of the transition densities is obtained, thus distinguishing itself fromall other methods that estimate the unknown parameters with a pre-speci�ed model. Fora comparison of estimation methods on discretely sampled di¤usions, please see Jessen99[24].

SSE: NON-GENERALIZED HYPERBOLIC DIFFUSION 7

2.1. General Theory of Solutions of One Dimensional SDEs. Firstof all, we would like to cite some powerful theorems about the existenceand uniqueness of weak solutions to one-dimensional stochastic di¤erentialequations (SDEs). These theorems give the su¢ cient or necessary and suf-�cient conditions for the existence, uniqueness, recurrence and ergodicity ofsolutions to such SDE�s.From Kutoyants04 [26] comes the following su¢ cient condition for the

existence and ergodicity of a weak solution as a di¤usion.

Theorem 1 (Kutoyants04[26]). Suppose b (�) is locally bounded, v (�) is con-tinuous and positive, and

xb (x) + v2 (x) � A�1 + x2

�for some constant A > 0: Then the stochastic di¤erential equation

dXt = b (Xs) dt+ v (Xt) dWt

has a unique weak solution.Moreover, if

V (b; x) =

Z x

0exp

��2Z y

0

b (s)

v2 (s)ds

�dy ! �1; as x! �1

and

G (b) =

Z +1

�1v�2 (x) exp

�2

Z x

0

b (s)

v2 (s)ds

�dx <1

then the solution is ergodic with invariant density

fb (x) = G�1 (b) v�2 (x) exp

�2

Z x

0

b (s)

v2 (s)ds

�Remark 1. The conditions for the existence of weak solutions in the abovetheorem are relatively strong but necessary for statistical inference.

From Shreve98 ([38]), we cite the elegant theorem proved by H. J. Engel-bert and W. Schmidt (1984).

Theorem 2 (Engelbert and Schmidt 1984). Equation

dXt = v (Xt) dWt

has a non-exploding weak solution unique in probability law for every initialdistribution �0 if and only if for all real x

v (x) = 0, (8" > 0)Z "

�"

dy

[v (x+ y)]2=1

Finally, we cite from Skorokhod89 [39] this powerful theorem about er-godicity:

8 XIONGZHI CHEN

Theorem 3 (Skorokhod89). If r1 = �1; r2 = +1; andR +1�1 v�2 (x) dx <

1; then dXt = v (Xt) dWt is ergodic with ergodic distribution

� (A) =

RA v

�2 (z) dzRR v

�2 (z) dz

(where A is any Lebesgue measurable subset of R)

Remark 2. In Theorem 3, r1 = limx!�1 p (x) ; r2 = limx!1 p (x) and forthe SDE dXt = b (Xt) dt+ v (Xt) dWt;

p (x) =

Z x

0exp

��2Z y

0

b (s)

v2 (s)ds

�dy

is called the scale function. For an ergodic di¤usion, its ergodic distributioncorresponds to its invariant distribution. For more details on this, please seeSkorokhod89 [39]. Here we brie�y cite from Skorokhod89 [39] a de�nitionthat, a �-�nite measure � is said to be invariant if �Tt = � for all t �0; where �Tt (A) =

RP (t; x; A)� (dx) = E� (IA (x (t))) and P (t; x; A) is

the underlying transition density. The relationship between the di¤usioncoe¢ cient and the invariant density can be also found in Bibby97 [5], wherethe above theorem is used.

2.2. Ergodic Di¤usion with GH Invariant Density. With these the-orems at hand, we are now ready to construct the proposed ergodic GHdi¤usion. We borrow the model from Bibby97 [5] and Rydberg99 [37], andassume that the underlying stochastic process S for the SSE index satis�es

St = exp (Xt + �t) ; t � 0 (2.1)

where � is some constant to be determined and

Xt = X0 +

Z t

0v (Xs) dWs (2.2)

for some v (�) satisfying the conditions which give existence and uniquenessin probability law of a weak solution to (2.2).

2.2.1. Model I. With the speci�cation in (2.1) � (2.2), we have two choices.One choice is to estimate � by the ordinary least squares (OLS) linear regres-sion to obtain � based on the observed SSE index 6 ~P = fPt; t = 1; :::; 4129g :Thus

lnPt � �t; t = 1; :::; 4129; (2.3)

denoted by X, forms an observation of the ergodic di¤usion

Xt = X0 +

Z t

0v (Xs) dWs; t � 0 (2.4)

which is called the "underlying detrended index". The invariant density of(2.4) is assumed to be proportional to some unnormalized GH density f (�; �)

6Please see the �rst part of the thesis for the observed SSE index.

SSE: NON-GENERALIZED HYPERBOLIC DIFFUSION 9

as de�ned in (1.7). One then estimates the unknown parameter � based on(2.3) and (2.4) . The second choice is to assume that

lnSt = Xt + �t = X0 +

Z t

0v (Xs) dWs +

Z t

0�ds (2.5)

is itself an ergodic di¤usion whose invariant density is some GH densityde�ned in (1.2), then estimate the involved unknown parameter vector �� =(�; �; �; �; �; �; �) based on ~P .Hereunder we call (2.3) � (2.4) the simpli�ed Model I. Setting the di¤u-

sion function v (�) to be

v�x; ~�

�=p�2=f (x; �) (2.6)

with the augmented parameter vector ~� = (�; �) yields

Xt = X0 +

Z t

0

r�2=f

�x; ~�

�dWs (2.7)

Note that v�x; ~�

�> 0 for x 2 R; � in (1.5) and � 6= 0: Also f has �nite

second order moment because the moment generating function for the GHdensity is

M (t) = et��

�2 � �2

�2 � (� + t)2

��=2 K���q�2 � (� + t)2�K�

��p�2 � �2

� ; j� + tj < �

(2.8)(see Karsten99 [27]). Therefore the conditions of Theorem 2 and Theorem 3are satis�ed and X is weakly uniquely de�ned by (2.7) with ergodic densityproportional to f (�; �).

2.2.2. Model II. However, if we take (2.5) � (2.6) as Model II and wantto ensure that the invariant density is the GH density, then according toTheorem 1 we have to �nd v (�; �) such that

V (x; �) =

Z x

0exp

��2Z y

0

�

v (s; �)2ds

�dy ! �1; as x! �1 (2.9)

and

G =

Z +1

�1v (x)�2 exp

�2

Z x

0

�

v (s; �)2ds

�dx <1 (2.10)

so that the invariant density is

f (x; �) = G�1v (x; �)�2 exp

�2

Z x

0

�

v (s; �)2ds

�(2.11)

Remark 3. If the initial X0 has the density f (�; �) ; then X is stationary.

10 XIONGZHI CHEN

More speci�cally, we have to solve for v (�; �) the following equations8>><>>:fgh (x; �) = G

�1v (x; �)�2 exp

�2R x0

�

v (s; �)2ds

�R x0 exp

��2R y0

�

v (s; �)2ds

�dy ! �1; as x! �1

(2.12)

knowing that G <1 because fgh has �nite second moment.From (2.12) we have

v (x; �)2d ln fgh (x; �)

dx+ 2v0 (x; �) v (x; �) = 2� (2.13)

where the superscript 0 denotes di¤erentiation with respect to x: Settingh (x; �) = v2 (x; �) ; then from the equivalent equation

d

dxffgh (x; �)h (x; �)g = 2�fgh (x; �)

we obtain

h (x; �) = f�1gh (x; �)

�2�

Z x

0fgh (s; �) ds+G

�1�

(2.14)

and

v (x; �) =ph (x; �) (2.15)

= f�1=2gh (x; �)

�2�

Z x

0fgh (s; �) ds+G

�1�1=2

Next, we still have to showZ x

0exp

��2Z y

0

1

v (s; �)2ds

�dy = �1; as x! �1

By exploiting fgh (x; �) = G�1v (x; �)�2 exp

�2R x0

1

v (s; �)2ds

�;we have

I1 : =

Z x

0

dy

exp�2R y0 v

�2 (s; �) ds (2.16)

= G�1Z x

0

dy

fgh (y; �) v2 (y; �)

= G�1Z x

0

dy�2�R y0 fgh (s; �) ds+G

�1�

such thatlim

x!�1I1 = �1 for � � 0

Thus, the proposed ergodic di¤usion model in (2.5) is

St = exp (�t+Xt)

SSE: NON-GENERALIZED HYPERBOLIC DIFFUSION 11

with

Xt = X0 +

Z t

0�ds+

Z t

0v (Xs; �) dWs (2.17)

= X0 +

Z t

0�ds+

Z t

0f�1=2gh (Xs; �)

�2�

Z Xs

0fgh (r; �) dr +G

�1�1=2

dWs

with invariant density fgh (�; �) for � � 0:Finally, applying Ito�s formula to the model, we obtain St as

dSt = St

��b0 (t) +

1

2v2 (lnSt � b (t))

�dt+ v (lnSt � b (t)) dWt

�(2.18)

where b (t) = �t:

Remark 4. It is well known that under certain non-degeneracy and local in-tegrability conditions, a stochastic integral-di¤erential equation can be trans-formed into a driftless equation (see Shreve98 [38]) or a constant-di¤usionequation (see Rogers79 [36] ). But in most cases with the presence of driftfunctions, we will have to use the methods in Ait-Sahalia02 [3] or in Lo88[28].

3. MCMC ESTIMATION OF PARAMETERS

To reduce the computational complexity that comes from computing thenumerical integral

R s0 fgh (r; �) dr involved in Model II, we will only estimate

the model

lnSt � �t = Xt = X0 +Z t

0v�Xs; ~�

�dWs; t � 0 (3.1)

de�ned in (2.1) � (2.4) with

v��; ~��=

s�2

f (�; �) (3.2)

Here S = fSt; t � 0g denotes the underlying stochastic process for the ob-served SSE index and X = fXt; t � 0g denotes that for the detrended index~D = flnPt � �t; t = 1; :::; 4129g : (See Figure 1 for the plot of ~D.)

3.1. Milstein Discretization Scheme. To approximate the likelihood func-tion and the unknown posterior p

�~�jX

�for the parameter vector ~� =

(�; �; �; �; �; �) based on the discrete observation X = fXt; t = 1; 2; :::; ngof the detrended index,we start from the Milstein approximation scheme(of strong convergence of order one) by discretizing model (3.1) for a smallchange �t in time t into

Xt+�t = Xt + v�Xt; ~�

��Wt +

1

2v�Xt; ~�

� @v �Xt; ~��@Xt

h(�Wt)

2 ��ti

12 XIONGZHI CHEN

0 500 1000 1500 2000 2500 3000 3500 4000 4500­3

­2

­1

0

1

2

3

4

5

6

7

Number of Trading Days from 01/01/1991

Lin

ea

rly D

etr

en

ed

 In

de

x

Linearly Detrended Logarithmic SSE Index

Figure 1. Detrended Index

This can be rewritten into

Xt+�t �Xt +1

2v�Xt; ~�

� @v �Xt; ~��@Xt

�t+v�Xt; ~�

�2@@Xtv

�Xt; ~�

� (3.3)

=�v�Xt; ~�

�p�t��+

0@12v�Xt; ~�

� @v �Xt; ~��@Xt

�t

1A �2 + v�Xt; ~�

�2@@Xtv

�Xt; ~�

�where

@v�Xt; ~�

�@Xt

:=@v�x; ~�

�@x

������x=Xt

:= @@Xtv�Xt; ~�

�and � is standard

normal.To obtain the conditional density after discretization, we use the results

from Elerian98 [18] and set8>>>>>>>>>>>>>><>>>>>>>>>>>>>>:

At =1

2v�Xt; ~�

� @v �Xt; ~��@Xt

�t

Bt = �v�Xt; ~�

�2@@Xtv

�Xt; ~�

� +Xt �At�t =

1

�t�@@Xtv

�Xt; ~�

��2yt+1;�t =

Xt+1 �BtAt

(3.4)

Therefore yt+1;�t =��+

p�t�2 and

h (Xt+�tjXt) =1

jAtj�2�Xt+1 �Bt

At; 1;�t

�

SSE: NON-GENERALIZED HYPERBOLIC DIFFUSION 13

where �2 (�; d; l) is a non-central chi-squared density function with d degreesof freedom and location parameter l. More speci�cally

�2 (z; d; l) =1

2exp

�� l + z

2

��zl

�d=4�1=2I�1=2

�plz�

where I�1=2 (w) =

r2

�wcoshw.

In our case with d = 1, we have

�2 (z; 1; l) =1

2exp

�� l + z

2

��zl

��1=4I�1=2

�plz�

=1

2exp

�� l + z

2

�z�1=4l1=4

p2

p�pp

lzcosh

�plz�

=1p2�exp

�� l + z

2

�z�1=2 cosh

�plz�

Thus

h�Xt+�tjXt; ~�

�(3.5)

=

exp

��0:5

��t +

Xt+�t �BtAt

��cosh

�r�t (Xt+�t �Bt)

At

�p2� jAtj

s�Xt+�t �Bt

At

�

This saves a bit of computation time when computing the uniform residualsfor the model test done later.With the Milstein discretization, the likelihood function ~LM

�~�jX

�of ~�

given observation X is approximated by

LM

�~�jX

�:=

nQt=1h�Xt+�tjXt; ~�

�(3.6)

which further approximates the unknown p�~�jX

�(by the Bayesian rule).

However, based on the observation X (of the detrended index), the ap-

proximate likelihood function LM�~�jX

�generates huge quantities which

are even beyond the computational capabilities of Mathematica and Mat-lab. Thus we use another scheme by approximating the unknown posterior

p�~�jY

�based on the transformed observation Y as follows.

14 XIONGZHI CHEN

We set8>>>>>>>>>><>>>>>>>>>>:

g�Xt; ~�

�=1

2v�Xt; ~�

� @v �Xt; ~��@Xt

ct = v�Xt; ~�

�p�t

dt = g�Xt; ~�

��t =

1

2v�Xt; ~�

� @v �Xt; ~��@Xt

�t

Yt;�t = Xt+�t �Xt + g�Xt; ~�

��t = Xt+�t �Xt + dt

to obtain

Yt;�t = ct�+ dt�2 = dt

"��+

ct2dt

�2� c2t4d2t

#(3.7)

where � is standard normal. Here Z =��+

ct2dt

�2has density

h�z; ~��=1

2exp

��rt + z

2

��z

rt

��1=4I�1=2 (

prtz) (3.8)

where rt = c2t =�4d2t�and I�1=2 (w) =

r2

�wcoshw (see Elerian98 [18]).

Thus the density of Yt;�t is

h��y; t; ~�

�=

1

jdtjh

�y

dt+c2t4d2t

�(3.9)

and the approximate likelihood given the observationsY = fYt;�t; t = 1; :::; ngis

LM

�~�jY

�=

nQt=1h��y; t; ~�

�(3.10)

Next, we compute the functions needed in the Milstein discretization 7.Firstly we have

@Xtv�Xt; ~�

�= @Xt

8><>:vuut �2

f�Xt; ~�

�9>=>; (3.11)

= �12

�2f�2�Xt; ~�

� d

dXtf�Xt; ~�

�v�Xt; ~�

�= �0:5v

�Xt; ~�

� d

dXtln f

�Xt; ~�

�7It is obvious that for the observed detrended index, we have �t = 1.

SSE: NON-GENERALIZED HYPERBOLIC DIFFUSION 15

Secondly we have

d

dxln fgh (x; �)

=d

dx

�lnK��1=2

��

q�2 + (x� �)2

�+�� 1=22

ln��2 + (x� �)2

�+ � (x� �)

�= � +

(�� 1=2) (x� �)�2 + (x� �)2

+d

dxlnK��1=2

��

q�2 + (x� �)2

�= � +

(�� 1=2) (x� �)l (x; �; �)

��K�+1=2

��pl (x; �; �)

�+K��3=2

��pl (x; �; �)

��2�K��1=2

��pl (x; �; �)

� � � (x� �)pl (x; �; �)

from the fact that

d

dxlnK��1=2

��

q�2 + (x� �)2

�(3.12)

=

�12

�K�+1=2

��q�2 + (x� �)2

�+K��3=2

��q�2 + (x� �)2

��K��1=2

��q�2 + (x� �)2

�� � (x� �)q

�2 + (x� �)2

=�12

�K�+1=2

��pl (x; �; �)

�+K��3=2

��pl (x; �; �)

��K��1=2

��pl (x; �; �)

� � � (x� �)pl (x; �; �)

wherel (x; �; �) = �2 + (x� �)2 (3.13)

(see Karsten99 [27] for properties of the modi�ed Bessel function of thesecond kind).Consequently

@

@Xtv�Xt; ~�

�= �0:5v

�Xt; ~�

� d

dxln f (Xt; �) (3.14)

= �0:5v�Xt; ~�

���� +

(�� 1=2) (Xt � �)l (Xt; �; �)

��K�+1=2

��pl (Xt; �; �)

�+K��3=2

��pl (Xt; �; �)

��2K��1=2

��pl (Xt; �; �)

� � (Xt � �)pl (Xt; �; �)

9=;

16 XIONGZHI CHEN

3.2. MCMC Methods and Metropolis-Hastings Algorithm. To es-timate the parameter vector ~� in Model I by Markov chain Monte Carlo(MCMC) methods we �rst remove the constraints on the ranges of the com-ponent parameters (except j�j < �) by this batch of transformations:

8>>>><>>>>:� = exp (��) ; �� 2 R� = exp (��) ; �� 2 R� = exp (��) ; �� 2 R

� =exp (��)� 1exp (��) + 1

exp (��) ; �� 2 R

(3.15)

Let the new parameter vector be denoted by ~� = (��; ��; ��; �; �; ��) =�~��1;~��2;~��3;~��4;~��5;~��6

�2 R6 and still let � =

�~��1;~��2;~��3;~��4;~��5

�for notational

simplicity. We then have

v�x; ~�

�=

sexp (��)

f (x; �)

By the Bayesian formula, the posterior density ~p�~�jY

�for the parameter

vector ~� given the observation Y is approximated by

p�~�jY

�=

p�Yj~�

���~��

Rp�Yj~�

���~��d�/ �

�~��LM

�~�jY

�(3.16)

where ��~��is the prior density for ~�, p

�Yj~�

�is the joint density for the

observation Y given ~� and LM�~�jY

�is as in (3.10).

To implement the MCMC methods to estimate ~�, we set the prior ��~��to

be a radially symmetric uniform distribution centered at zero with compactsupport as large as required. We propose the next value of the parameter

vector ~�0by generating it from the multivariate normal proposal N

�~�n;�

�with mean vector ~�n and a pre-assigned constant non-singular covariancematrix �. Here ~�n is the current value of the parameter vector ~� at then-th iteration in the Metropolis-Hastings (MH) algorithm that drives theproposed Markov chain. Under such settings, the acceptance probability for

SSE: NON-GENERALIZED HYPERBOLIC DIFFUSION 17

~�0given ~�n is simply

��~�0j~�n

�= min

8<:1; p�~�0jY�q�~�nj~�

0�p�~�njY

�q�~�0j~�n

�9=; (3.17)

= min

8<:1; p�Yj~�0

���~�0�q�~�nj~�

0�p�Yj~�n

���~�n

�q�~�0j~�n

�9=;

� min

8<:1; LM�~�0jY���~�0�q�~�nj~�

0�LM

�~�njY

���~��q�~�0j~�n

�9=;

= min

8<:1; LM�~�0jY�

LM

�~�njY

�9=;

where p�Yj~�

�is as in (3.16), q

�~�0j~�n

�= N

�~�n+1; ~�n;�

�. Note that

N��; ~�n;�

�here also denotes a multivariate normal density with mean vec-

tor ~�n and covariance matrix �.The Metropolis-Hastings (MH) algorithm (see, for example, Chib95 [12])

is summarized below:

Step 1: Given the current state ~�n, generate a candidate ~�0from the

proposal density q��j~�n

�Step 2: Calculate the acceptance probability �

�~�0j~�n

�Step 3: 8Generate a random number u from the uniform distribution

on (0; 1) : If u � ��~�0j~�n

�, set ~�n+1 = ~�

0. Otherwise, set ~�n+1 = ~�n.

Step 4: Repeat the previous steps to obtain a chainn~�0; ~�2; :::

o; where

~�0 denotes the initial state of ~�. Discard the burn-in valuesn~�0; ~�0; :::~�m

ofor some pre-assigned m 2 N (probably depending on the acceptancerates) obtained while the chain converges in distribution. Then the

remaining values,n~�m+1; ~�m+1; :::

o; form a correlated Markov chain

sampled from its stationary density which is p�~�jY

�.

As long as the chain converges, the post burn-in values, sayn~�m+1; ~�m+1; :::; ~�n1

ofor su¢ ciently large n1 2 N are used to estimate the unknown parameters

8Note that the acceptance criterion adopted here is the most widely used one.

18 XIONGZHI CHEN

by Monte Carlo methods. In this paper, the ergodic mean

~��=

1

n1 �m

n1Xi=m+1

~�i ! Ep(~�jX)

�~�jY

�; a:s: (3.18)

which is exactly the Monte Carlo estimate 9 of the unknown ~�, is usedas an estimate of the unknown ~�: The symbol "! a:s:" denotes " strongconvergence " and the expectation is taken with respect to the posterior

density p�~�jY

�: (For more details on Monte Carlo methods in �nance, see

Glasserman04 [32]).

3.3. Convergence Checking. In the actual implementation of the MCMCmethods, the sampled path, denoted by

n~�i : i = 1; 2; � � � ; N

o; forms a con-

vergent Markov chain whose stationary density is the unknown posterior

p�~�jY

�: The target quantity Ep(~�jY)

h'�~��iis estimated by the ergodic

average, i.e., the Monte Carlo (MC) estimate

~'N =1

N

NXi=1

'�~�(i)�

(3.19)

where ' (�) is a (pre-assigned) real-valued function. Ep(~�jY) [�] again denotes

the expectation with respect to p�~�jY

�. For the purpose of parameter

estimation in this paper, ' is just the identity function.The most important part in MCMC simulation is convergence checking

(since the best we can do is not to reject the null hypothesis that the chainhas converged), which can be done by a combination of checking the MonteCarlo standard errors (MCSEs) and the Gelman-Rubin (GR) convergencediagnostic.

3.3.1. MCSE Convergence Criterion. Roberts96 [35] states the followingconvergence theorem:

pN�~'N � Ep(~�jY)

h'�~��i�

d! N�0; s2'

�(3.20)

for some s2' � 0; whered! denotes convergence in distribution and N (a; a1)

denotes a normal density with mean a and variance a1. Thus the accuracy of

the ergodic average as an estimate of Ep(~�jY)

h'�~��iis essentially measured

by s2': a quantity that can only be empirically estimated from the sample.In practice, s2' is estimated by the so called " batch mean ". For it, the

9Monte Carlo estimates are essentially just the average of the samples from the targetquantity, regardless of ergodicity. Theoretical convergence of the Markov chains in thesimulation and ergodicity of the means obtained from the post burn-in samples here areensured by the detailed balance condition built into the Metropolis-Hastings algorithm.

SSE: NON-GENERALIZED HYPERBOLIC DIFFUSION 19

MCMC algorithm is run for N = m� n iterations for su¢ ciently large n sothat

yk =1

n

knXi=(k�1)n+1

'�~�(i)�; k = 1; 2; :::;m (3.21)

are approximately independently distributed as

N�Ep(~�jX)

h'�~��i; s2'=n

�Then s2' is estimated by

s2' =n

m� 1

mXk=1

(yk � ~'N ) (3.22)

Subsequently the standard error of ~'N can be estimated byqs2'=N; which

is called the Monte Carlo standard error (MCSE). This is used to measurethe mixing performance of the MH algorithm and as a convergence diagnosticfor the MCMC algorithm.

3.3.2. Gelman-Rubin Statistic. A single Markov chain may converge locallyto some mode of the unknown posterior and the MCSE is not able to di-agnose whether the proposed unknown posterior has been su¢ ciently sam-pled. Therefore we have to run multiple chains in the MCMC simulationto ensure that a large proportion of the unknown posterior is sampled andexplore whether all chains converge to the same stationary distribution.Consequently, we adopt the famous Gelman-Rubin (GR) statistic 10 (seeGelman92 [20], Brooks00 [11], and Brooks98 [10]) for convergence diagnosisof multiple-sequence simulations.Now we describe how to construct the GR statistic in both univariate

and multivariate cases. We independently simulate I � 2 sequences oflength 2N; each beginning with di¤erent initial values that are su¢ ciently 11

widely dispersed with respect to the unknown yet stationary distribution (towhich the I independent Markov chains are supposed to converge). Thenwe discard the �rst N iterations of each simulated sequence and retain onlythe last N . For the purpose of our simulation here, we compute variance

between the I sequence means, which are denoted by �(i)

� . We de�ne

B =1

I � 1

IXi=1

��(i)

� � ����

(3.23)

10In the literature, the GR statistic is often called the "Potential Scale ReductionFactor (PSRF)".

11It is possible to measure how "su¢ ciently" the initial values are dispersed. SeeDempster77 [14], where a method to choose su¢ ciently widely dispersed initials is given.However due to the time constraints for the project, the author was not able to use themethod suggested therein.

20 XIONGZHI CHEN

where ~�(i)t denotes the t-th observation of ~� from the i-th chain and

�(i)

� =1

N

2NXt=N+1

~�(i)t ; �

�� =

1

I

IXi=1

�(i)

�

Further, we de�ne W; the mean of the I within-sequence variances, s2(i), by

W =1

I

IXi=1

s2(i) (3.24)

where

s2(i) =1

N � 1

2NXt=N+1

�~�(i)t � �(i)�

�2Finally if ~� is univariate, then we de�ne

V =N � 1N

W +

�1 +

1

I

�B (3.25)

and

R =V

W(3.26)

Otherwise, we de�ne

Rp =N � 1N

+

�1 +

1

I

�� (3.27)

where � is the largest eigenvalue of the positive de�nite matrix W�1B. SeeBrooks98 [10] for more details on (3.26), (3.27).The key property of R and Rp is that they both converge to 1 when all

individual Markov chains in the MCMC simulation converge to the samestationary density. Consequently they can be used as convergence diag-nostics. Another interesting relationship between them is that, if R (k)denotes the GR statistic for the k-th component of a d-dimensional mul-tivariate parameter vector ~� in a multiple-chain MCMC simulation, then

max1�k�dnR (k)

o� Rp. Since each R (k) � 1; k = 1; :::; d if the initial val-

ues for the Markov chains in the simulation are su¢ ciently widely dispersed(see Brooks00 [11]), we do not have to check the convergence for each R (k) ;1 � k � 6 when the convergence of Rp is con�rmed.

3.3.3. Con�dence Interval under the GR Scheme. Under the Gelman-Rubindiagnostic scheme, a 100 (1� p)% central posterior con�dence interval forthe parameter ~� is obtained based upon the �nal N of 2N iterations asdescribed below (see Brooks98 [10]):

S1: From each individual chain, take the empirical 100 (1� p)% inter-val, that is, the 100

p

2% and the 100 (1� p)% points of the N simu-

lation draws to form I within-sequence interval length estimates.

SSE: NON-GENERALIZED HYPERBOLIC DIFFUSION 21

S2: From the entire set of IN observations gained from all chains, com-pute the empirical 100 (1� p)% interval to obtain a total-sequenceinterval length estimate.

S3: Evaluate R de�ned as

Rinterval =length of total-sequence interval

mean length of within-sequence intervals

3.4. Implementation and Estimates. Since the approximate posteriorp�~�jY

�is proportional to LM

�~�jY

�and the prior �

�~��is chosen to be

a radially symmetric uniform distribution centered at zero, we see that the

modes of p�~�jY

�depends on those of LM

�~�jY

�. To choose initial values

that are su¢ ciently widely dispersed with respect to p�~�jY

�, we estimated

all 3 modes of the empirical density of the detrended index (see Figure 2for this density and its nonlinear least squares �t by a GH density). For

­4 ­2 0 2 4 6 80

0.02

0.04

0.06

0.08

0.1

0.12

0.14

Detrended Index

Pro

ba

bility

Empirical Density of Detrended Indexagainst Its Nonlinear Least Squares Fit

Empirical Density of the Detrended IndexNonlinear Least Squares Fit

Figure 2. EPDF of Detrended Index

each mode a group of two independent chains are simulated. Unfortunatelythis simulation scheme gives negative z�s in (3.8), which forces the authorto choose the following initialization strategy.Firstly by �tting the empirical squared di¤usion of the detrended index

by v��; ~��in (2.6) in the least square sense (within the working precision of

Matlab), we obtain the initial estimate

~�0 = (�0; �0; �0; �0; �0; �0)

= (�0:3090;�2:7787;�19:2004; 3:2323; 3:6693;�3:7583)Figure 3 plots the squared di¤usion function for X = fXt; 1 � t � 4129gbased on the estimated parameter vector from chain 2 in the simulation.Then we simply start 6 independent Markov chains whose initial states

are the 6 di¤erent vectors generated by a symmetric uniform distribution

22 XIONGZHI CHEN

­3 ­2 ­1 0 1 2 3 4 5 6 70

1

2

3

4

5

6

7x 10­3 Compared Squared Diffusions for Chain 2

Detrended Index

Sq

ua

red

 Vo

latility

Squared Empirical Diffusion FunctionModel Predicted Squared Diffusion Function

Figure 3. Squared Di¤usion Function in Model 2

0 10 20 30 40 50 60 70 80 90 1000

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

Number of Groups in Iterations (Group size 500)

Ma

xim

um

 Co

mp

on

en

t M

CS

E

Maximum of MCSE for 6 Component Paramters for Individual Chains

Figure 4. Max MCSEs for 6 Chains

centered at ~�0. The tuning parameter is set to be � = 0:0015; partially basedon the dimension of the parameter space (see Haario01 [23], Roberts08 [34])and by the need to keep the acceptance rates between 0:2 and 0:5 (seeJohannes03 [25], Chib95 [12], and Tse04 [42]).At around 50; 000 iterations MCSEs suggest that all six independent

chains have converged to their own stationary distributions and the MC-SEs remain at around 10�3 afterwards. See Figure 4 for the plot of themaximum component MCSEs for 6 individual chains for the �rst 50; 000iterations.But the GR statistic does not convergence to 1 even after 350; 000 it-

erations and the average acceptance rates are too low for six chains afteraround 50; 000 iterations. So the simulation has to be terminated since the 6

SSE: NON-GENERALIZED HYPERBOLIC DIFFUSION 23

0 0.5 1 1.5 2 2.5 3 3.5

x 105

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Number of Iterations

Acce

pta

nce

 Ra

tes

Acceptance Rates Against Number of Iterations for 6 Chains

Figure 5. Acceptance Rates for 6 Chains

� � � � � �Chain 1 �0:633729 �3:003546 �19:157304 3:041060 4:578130 �3:509746Chain 2 �0:358308 �2:672644 �20:055300 2:611961 4:382647 �5:049936Chain 3 �0:412318 �1:712247 �18:745429 2:597826 3:120015 �5:662547Chain 4 �0:423069 �1:950044 �18:693538 2:999596 3:721765 �4:576278Chain 5 �0:277084 �1:368601 �19:774321 3:635318 3:143871 �4:097122Chain 6 �0:389012 �2:301538 �19:248898 2:943287 4:338962 �4:459357Table 1. Monte Carlo Estimates for Parameter Vector for 6 Chains

chains might have converged to di¤erent stationary distributions and furtheriterations will not alter the chains enough. See Figure 5 for the acceptancerates and the �gures for the badly mixed sample paths in the appendix.We take the last 10% of the �rst 50; 000 simulated samples (which we

call the "e¤ective samples" ) from each chain in the simulation to reduceor remove the e¤ects of initialization in the Markov chains, and obtain 6MC estimates for the parameter vector ~�; which suggests 6 possible models.Table 1 summarizes the MC estimates for each component parameter foreach individual chain. The corresponding MCSEs for the 100-th batch areprovided in Table 2 . Note that all numbers in Table 2 should be multipliedby 10�3.With Table 2, the 95% Bayesian con�dence interval (CI) for each estimate

in Table 1 can be very accurately estimated by (3.20), but is unnecessary tobe computed here because the uniform residual test in the next subsectionrejects all 6 possible models .

3.5. Test of Model. To test the proposed model

Xt = X0 +

Z t

0v�Xs; ~�

�dWs; t � 0

24 XIONGZHI CHEN

� � � � � �Chain 1 0.0482 0.0635 0.0278 0.0544 0.2010 0.2914Chain 2 0.0002 0.0006 0.0001 0.0001 0.0004 0.0012Chain 3 0.0003 0.0012 0 0.0001 0.0001 0.0018Chain 4 0.0004 0.0016 0.0001 0.0001 0.0007 0.0024Chain 5 0.0843 0.4031 0.0190 0.0774 0.1154 0.1500Chain 6 0.1628 0.9247 0.1062 0.0403 0.2049 0.7743

Table 2. MCSEs at the 100 th Group

Reject Null p-Value Chi2 Stat DFChain 1 Yes 1:8573E � 161 777:0412 9Chain 2 Yes 2:8942E � 149 720:3632 9Chain 4 Yes 9:476E � 152 731:9177 9Chain 6 Yes 8:8394E � 152 732:0581 9Table 3. Uniform Residual Test Statistics for 4 Models

based on the observed detrended indexnln ~Pt � �t; t = 1; :::; 4129

o, we sim-

ply follow the methods in Bibby97 [5] and compute the uniform residuals

Ut

�~�(j)est

�= P~�(j)est

(Xt+1 � yjXt = x) ; x; y 2 R; j = 1; ::; 6; t = 1; ::; 4128

for each Monte Carlo estimate ~�(j)est of ~� obtained from the 6 chains, where

P~�(j)est(�j�) is exactly the conditional (transition) distribution based on (3.5).

Based on Bibby97 [5], if the models are appropriate, then Ut�~�(j)est

�; t =

1; ::; 4128 for each �xed j should be independent and identically uniformlydistributed on the unit interval. This can be easily seen from the Milsteindiscretization, the property of the standard Brownian motion, and the in-variance property of independence (between random vectors) under Boreltransformations.Due to the computational inability of both Mathematica and Matlab to

handle the badly mixed chain 3 and chain 5 (as revealed by the samplepaths given in the appendix), we test only the other four models obtainedfrom chain 1, chain 2, chain 4 and chain 6 by the uniform residual test (see

Bibby95 [4], Bibby97 [5]). We directly test the null hypothesis: Ut�~�(j)est

�;

t = 1; ::; 4128 for each �xed j are independent and identically uniformly

distributed in (0; 1) 12 (rather than testing �2P4128t=1 ln

�Ut

�~�(j)est

��for such

�xed j as done in Bibby97 [5] ).Test results for these 4 chains are summarized in Table 3 , where "p-Value"

denotes the signi�cance level at which the null hypothesis is rejected, "Chi2

12This is easily done by Matlab command "chi2gof" applied to testing standard uniformdistributions.

SSE: NON-GENERALIZED HYPERBOLIC DIFFUSION 25

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1­0.4

­0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

Quantiles of Standard Uniform

Qu

an

tile

s o

f U

nifo

rm R

esid

ua

ls

QQ Plot of the Uniform Residuals against Standard Uniform

Figure 6. QQ Plot of Residuals for Model 2

Stat" the chi-squared statistic, and "DF" the degrees of freedom. Supple-

mentary to these statistics, the quantile plot of Ut�~�(j)est

�; t = 1; ::; 4130 for

j = 2 is provided in Figure 6.These test results clearly show that all 4 possible models are rejected sta-

tistically because of the high chi-squared statistics and degrees of freedom 13

compared to the sample size 4128. Thus under the scheme of approximatingthe unknown posterior for the parameter vector by the likelihood functionobtained from the Milstein discretization, we reject the null hypothesis that" the underlying stochastic process for the detrended index in the tradingperiod from 01/01/1991 to 11/14/2007 is an ergodic, GH di¤usion withsome GH invariant density", i.e., the detrended index is not an ergodic GHdi¤usion.

4. CONCLUSION AND DISCUSSION

It is no doubt that the ergodic GH di¤usion can not �t the detrended indexsince for the ergodic GH di¤usion its invariant density, the GH density, isunimodal while the empirical density of the detrended index is tri-modal.Moreover, the detrended index still have a very strong linear trend whichrequires a non-zero drift function in the proposed model, but the proposedGH di¤usion is driftless, thus the GH di¤usion is inappropriate. Uniformresidual test results also reject the assumption that the underlying stochasticprocess for the detrended index is an ergodic GH di¤usion since obviouslythe residuals from the model are not independent and identically uniformlydistributed on the unit interval.The high acceptance rates in the �rst 50; 000 iterations might be the result

of correctly estimated initial states for the Markov chains in the simulation

13Note the degrees of freedom in the chi-square goodness-of-�t test is sensitive to thenumber of bins.

26 XIONGZHI CHEN

or due to the small tuning parameter which results in a narrow scope ofsampling. Since we approximate the unknown posterior by the likelihoodfunction obtained from the Milstein discretization scheme and the initialstates for the Markov chains are obtained based on the empirical squareddi¤usion, it might be that these initial states capture the modes of theunknown posterior su¢ ciently well before the chains wandered away fromthe high acceptance region. Also, a small tuning parameter might be thecause of small MCSEs in the simulation, since the proposal states for theMarkov chains in the simulation might be concentrated locally around someregion of the unknown posterior.Finally, due to the high volatility of the SSE index, it unknown to the

author yet whether the posterior of the unknown parameter vector basedon the transformed observation approximates the exact unknown posteriorwell or not.

References

[1] Yacine Aït-Sahalia 1996: Nonparametric pricing of interest rate derivative securities.Econometrica, Vol 64, No. 3 (May, 1996), 527-560

[2] Yacine Aït-Sahalia 1999: Transition densities for interest rate and other nonlineardi¤usions. The journal of �nance, Vol IV, No 4, August 1999.

[3] Yacine Aït-Sahalia: Maximum Likelihood Estimation of Discretely Sampled Di¤u-sions: A Closed-From Approximation Approach, Econometrica, Vol. 70, No. 1 (Jan-uary, 2002), 223�262

[4] B.M. Bibby, M. Sorensen 1995: Martingale estimation functions for discretely ob-served di¤usion processes. Bernoulli 1(2), 1995, 017-039

[5] B.M. Bibby, M. Sorensen 1997: A hyperbolic model for stock prices. Finance andStochastics 1, 25-41, 1997

[6] B.M. Bibby 2001: Simpli�ed estimating functions for di¤usion models with a high-dimensional parameter. Scandinavian Journal of Statistics, Vol 28:99-112, 2001

[7] B.M Bibby, et al 2004: Estimating Functions for Discretely Sampled Di¤usion-TypeModels. Working paper

[8] B.M. Bibby, IB M. Skovgaard, M. Sorensen 2005: Di¤usion-type models with givenmarginal distribution and autocorrelation function. Bernoulli 11(2), 2005, 191-220

[9] Jaya P. N. Bishwal 2008: Parameter Estimation in Stochastic Di¤erential Equations.Springer Verlag 2008

[10] Stephen P. Brooks and Andrew Gelman 1998: General methods for monitoring con-vergence for iterative simulations. Journal of Computational and Graphical Statistics,Vol. 7, Number 4, 1998

[11] S.P. Brooks and P. Giudici 2000: Markov chain Monte Carlo convergence assessmentvia two-way analysis of variance. Journal of Computational and Graphical Statistics,Vol. 9, No 2, 2000

[12] S. Chib, E. Greenberg 1995: Understanding the Metropolis-Hastings Algorithm. TheAmerican Statistician, November 1995, Vol. 49, NO. 4

[13] Rama Cont 2001: Empirical properties of asset returns: stylized facts and statisticalissues. Institute of Physics Publishing 2001 PII:S1469-7688(01)21683-2

[14] A. P. Dempster; N. M. Laird; D. B. Rubin 1977: Maximum Likelihood from Incom-plete Data via the EM Algorithm. Journal of the Royal Statistical Society. Series B(Methodological), Vol. 39, No. 1. (1977), pp. 1-38

[15] Bjorn Eraker 2001: MCMC analysis of di¤usion models with application to �nance.Journal of Business and Economic Statistics. April 2001, Vol 19, No. 2

SSE: NON-GENERALIZED HYPERBOLIC DIFFUSION 27

[16] Eberlein E. and U. Kller 1995: Hyperbolic distributions in �nance. Bernoulli 1, 282-299

[17] Ola Elerian, et al (2001). Likelihood Inference for Discretely Observed Nonlinear Dif-fusions. Econometrica, Vol 69

[18] Ola Elerian, 1998: A note on the existence of a closed form conditional transition den-sity for the Milstein scheme. Economics Discussion Paper 1998-W18 Nu¢ eld College,Oxford

[19] H.J. Engelbert and W. Schmidt (1984). On solutions of one-dimensional stochasticdi¤erential equations without drift. Probability Theory and Related Fields

[20] Andrew Gelman, Donald B. Rubin, 1992: Inference from iterative simulation usingmultiple sequences. Statistical Science, 1992, Vol 7, No. 4, 457-511

[21] Guang-lu Gong 1979: Introduction to stochastic di¤erential equations (Chinese). Bei-jing University Press

[22] P. Gopikrishnan, H. E. Stanley, et al 1999: Scaling of the distribution of �uctuationsof �nancial market indices. Physical Review E, Vol 60, Number 5, Nov. 1999

[23] H. Haario, et al, 2000: An adaptive metropolis algorithm. Bernoulli 7(2), 2001, 223-242[24] B. Jessen, R Poulsen, 1999: A comparison of approximation techniques for transition

densities of di¤usion process. Working paper, Center for analytical �nance, Universityof Aarhus

[25] M. Johannes, N. Polson 2003: MCMC methods for continuous time �nancial econo-metrics. Working Paper.

[26] Yury A. Kutoyants 2004: Statistical inference for ergodic di¤usion processes. SpringerVerlag 2004

[27] Karsten Prause 1999: The generalized Hyperbolic Model: Estimation, FinancialDerivatives and Risk Measures. PhD Thesis

[28] Andrew W. Lo 1988. Maximum likelihood estimation of generalized Ito processes withdiscretely sample data. Econometric theory, 4, 1988, 231-247

[29] Terence C Mills (1999): The econometric modelling of �nancial time series. Cam-bridge University Press

[30] D.B. Madan, M. Yor 2002: Making Markov martingales meet marginals: with explicitconstructions. Bernoulli 8(4), 2002, 509-536

[31] Bernt Oksensal 2005: Stochastic Di¤erential Equations. Springer 2005[32] Paul Glasserman 2004: Monte Carlo Methods in Financial Engineering. Spriner-

Verlag New York 2004[33] Min-ping Qian, Guang-lu Gong 1997: Introduction to stochastic processes. Beijing

University Press[34] G. O. Roberts, J.S. Rosenthal, 2008: Examples of Adaptive MCMC. Working Paper.[35] G. O. Robers 1996: Markov chain concepts related to sampling algorithms. Markov

Chain Monte Carlo in Practice.[36] L.C.G. Rogers, David Williams, 1979: Di¤usions, Markov processes and martingales,

Vol 2. Cambridge University Press 1979[37] T. H. Rydberg 1999: Generalized hyperbolic di¤usion processes with applications in

�nance. Mathematical �nance, Vol 9, No. 2 (April 1999), 183-201[38] I. Karatzas, S. Shreve 1998: Brownian Motion and Stochastic Calculus. Springer 1998[39] A.V. Skorokhod 1989: Asymptotic methods in the theory of stochastic di¤erential

equations. Vol 78. Translation of the mathematical monographs[40] H. Sorensen 2002: Estimation of di¤usion parameters for discretely observed di¤usion

processes. Bernoulli 8(4), 2002, 491-508[41] D.W. Stroock, S.R.S. Varadhan 1979: Multidimensional di¤usion processes. Springer-

Verlag 1979[42] Y.K. Tse, et al 2004: Estimation of hyperbolic di¤usion using the Markov chain Monte

Carlo method. Quantitative Finance, Vol 4 (2004) 158-169

28 XIONGZHI CHEN

5. APPENDIX

5.1. Numerical Results. We present some numerical results in the MCMCsimulation. The e¤ective sample paths are the last 10% of the �rst 50; 000iterations of the MH algorithm.

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000­0.8

­0.6

­0.4Sample path for alpha for chain 1

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000­0.6

­0.4

­0.2Sample path for alpha for chain 2

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000­0.5

­0.4

­0.3Sample path for alpha for chain 3

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000­0.5

­0.4

­0.3Sample path for alpha for chain 4

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000­0.3

­0.28

­0.26Sample path for alpha for chain 5

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000­0.6

­0.4

­0.2Sample path for alpha for chain 6

Figure 7. E¤ective Sample Paths in 6 Chains for �

SSE: NON-GENERALIZED HYPERBOLIC DIFFUSION 29

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000­3.2

­3

­2.8Sample path for beta for chain 1

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000­2.8

­2.6

­2.4Sample path for beta for chain 2

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000­1.8

­1.7

­1.6Sample path for beta for chain 3

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000­2.2

­2

­1.8Sample path for beta for chain 4

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000­1.4

­1.35

­1.3Sample path for beta for chain 5

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000­2.5

­2Sample path for beta for chain 6

Figure 8. E¤ective Sample Paths in 6 Chains for �

30 XIONGZHI CHEN

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000­19.4

­19.2

­19Sample path for delta for chain 1

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000­20.2

­20

­19.8Sample path for delta for chain 2

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000­18.8

­18.7

­18.6Sample path for delta for chain 3

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000­18.8

­18.7

­18.6Sample path for delta for chain 4

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000­19.8

­19.78

­19.76Sample path for delta for chain 5

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000­19.4

­19.2

­19Sample path for delta for chain 6

Figure 9. E¤ective Sample Paths in 6 Chains for �

SSE: NON-GENERALIZED HYPERBOLIC DIFFUSION 31

0 500 1000 1500 2000 2500 3000 3500 4000 4500 50003

3.1

3.2Sample path for lamda for chain 1

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000

2.6

2.8Sample path for lamda for chain 2

0 500 1000 1500 2000 2500 3000 3500 4000 4500 50002.55

2.6

2.65Sample path for lamda for chain 3

0 500 1000 1500 2000 2500 3000 3500 4000 4500 50002.8

3

3.2Sample path for lamda for chain 4

0 500 1000 1500 2000 2500 3000 3500 4000 4500 50003.6

3.65

3.7Sample path for lamda for chain 5

0 500 1000 1500 2000 2500 3000 3500 4000 4500 50002.8

3

3.2Sample path for lamda for chain 6

Figure 10. E¤ective Sample Paths in 6 Chains for �

0 500 1000 1500 2000 2500 3000 3500 4000 4500 50004

4.5

5Sample path for miu for chain 1

0 500 1000 1500 2000 2500 3000 3500 4000 4500 50004

4.5

5Sample path for miu for chain 2

0 500 1000 1500 2000 2500 3000 3500 4000 4500 50003

3.2

3.4Sample path for miu for chain 3

0 500 1000 1500 2000 2500 3000 3500 4000 4500 50003.6

3.8

4Sample path for miu for chain 4

0 500 1000 1500 2000 2500 3000 3500 4000 4500 50003.1

3.2

3.3Sample path for miu for chain 5

0 500 1000 1500 2000 2500 3000 3500 4000 4500 50004

4.5

5Sample path for miu for chain 6

Figure 11. E¤ective Sample Paths in 6 Chains for �

32 XIONGZHI CHEN

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000­3.6

­3.4

­3.2Sample path for sigma for chain 1

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000­5.5

­5

­4.5Sample path for sigma for chain 2

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000­5.7

­5.65

­5.6Sample path for sigma for chain 3

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000­5

­4.5

­4Sample path for sigma for chain 4

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000­4.15

­4.1

­4.05Sample path for sigma for chain 5

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000­5

­4.5

­4Sample path for sigma for chain 6

Figure 12. E¤ective Sample Paths in 6 Chains for �

SSE: NON-GENERALIZED HYPERBOLIC DIFFUSION 33

­3 ­2 ­1 0 1 2 3 4 5 6 70

0.005

0.01Compared Squared Diffusions for Model 1

­3 ­2 ­1 0 1 2 3 4 5 6 70

0.005

0.01Compared Squared Diffusions for Model 2

­3 ­2 ­1 0 1 2 3 4 5 6 70

0.005

0.01Compared Squared Diffusions for Model 3

­3 ­2 ­1 0 1 2 3 4 5 6 70

0.005

0.01Compared Squared Diffusions for Model 4

­3 ­2 ­1 0 1 2 3 4 5 6 70

0.005

0.01Compared Squared Diffusions for Model 5

­3 ­2 ­1 0 1 2 3 4 5 6 70

0.005

0.01Compared Squared Diffusions for Model 6

Empirical Squared VolatilityModel Based Squared Volatility

Empirical Squared VolatilityModel Based Squared Volatility

Empirical Squared VolatilityModel Based Squared Volatility

Empirical Squared VolatilityModel Based Squared Volatility

Empirical Squared VolatilityModel Based Squared Volatility

Empirical Squared VolatilityModel Based Squared Volatility

Figure 13. Simulated Squared Volatilities for 6 Models

34 XIONGZHI CHEN

0 500 1000 1500 2000 2500 3000 3500 4000 4500­5

0

5

10Simulated Detrended Index from Model  1

0 500 1000 1500 2000 2500 3000 3500 4000 4500­5

0

5

10Simulated Detrended Index from Model  2

0 500 1000 1500 2000 2500 3000 3500 4000 4500­5

0

5

10Simulated Detrended Index from Model  3

Detrended IndexSimulation from Model 3

Detrended IndexSimulation from Model 2

Detrended IndexSimulation from Model 1

Figure 14. Simualted Detrended Index for Models 1~3

SSE: NON-GENERALIZED HYPERBOLIC DIFFUSION 35

0 500 1000 1500 2000 2500 3000 3500 4000 4500­5

0

5

10Simulated Detrended Idex from Model  4

0 500 1000 1500 2000 2500 3000 3500 4000 4500­5

0

5

10Simulated Detrended Idex from Model  5

0 500 1000 1500 2000 2500 3000 3500 4000 4500­5

0

5

10Simulated Detrended Idex from Model 6

Detrended IndexSimulation from Model 6

Detrended IndexSimulation from Model 5

Detrended IndexSimulation from Model 4

Figure 15. Simualted Detrended Index for Models 4~6

36 XIONGZHI CHEN

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.5

1

Standard Unifmorm QuantilesU

R Q

uantile

s

QQ Plot of Uniform Residuals against Standard Uniform for Model 1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.5

1

Standard Unifmorm Quantiles

UR

 Quantile

sQQ Plot of Uniform Residuals against Standard Uniform for Model 2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.5

1

Standard Uniform Quantiles

UR

 Quantile

s

QQ Plot of Uniform Residuals against Standard Uniform for Model 4

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.5

1

Standard Uniform Quantiles

UR

 Quantile

s

QQ Plot of Uniform Residuals against Standard Uniform for Model 6

Figure 16. QQ Plot of Residuals of 4 Tested Models