Modelling and Detecting Long Memory in Stock Returns MSc student Ciprian Necula Doctoral School of...

Modelling and Detecting Long Memory in Stock Returns

MSc student Ciprian Necula

Doctoral School of Finance and Banking

Academy of Economic Studies Bucharest

Presentation contents

• modelling and detecting long memory in time series

• data, methodology and empirical results

• a continuous trading model with long memory

• pricing derivative securities in presence of long memory

Modelling long memory

• ARFIMA(p,d,q)

tt LxdLL )(1)( 2,0~ iidt

the differencing parameter can take any real value

0 11

k

kd Lkd

dkL

where is Euler’s gamma function

Hosking (1981) shows that

• if and the roots of are outside the unit circle the process is stationary and invertible

• the autocorrelations of such a process decline at a hyperbolic rate to zero, a much slower rate of decay than the exponential decay of the ARMA process

• for positive d the sum of the absolute values of the autocovariance function is infinite; this is the case of long memory or long range dependence

5.0,5.0d LL ,

Consider now a time series of stock returns tr

Starting form the classical model:

ttr

we will focus on the model:

ttd rL 1

5.0,5.0d

Consequences of the long memory property:

• the presence of long memory in asset returns contradicts the weak form of the market efficiency hypothesis, which statesthat, conditioned on historical returns, future asset returns are unpredictable

• Mandelbrot (1971) suggests that pricing derivative securities with martingale methods may not be appropriate if the underlying follows a continuous stochastic process that exhibits long memory

Testing for Long Memory

• ADF – null hypotesis I(1)

Diebold and Rudebusch (1991) and Hassler and Wolter (1994) find that ADF tests tend to have low power against the alternative hypothesis of fractional integration

• KPSS – null hypotesis I(0)

Lee and Schmidt (1996) show that KPSS test can be used to distinguish short memory and long memory stationary processes

KPSS(1992)

Kpss statistics: T

ttS

lST 122

1

where and is the residual from regressing the series against a constant or a constant and a trend.

T

iitS

1

i

is a consistent estimator of the “long run variance” of , lS 2i T

TSE

T

1lim2

1

10

2 ,2)(T

jjjlklS

where and is a kernel function depending on the bandwidth parameter l.

T

jtjttj T 1

1

kernels: the Bartlett kernel and the Quadratic Spectral kernel.

a data dependent procedure to estimate the optimal bandwidth parameter l , approach explored by Andrews (1991) and Newey and West (1994)

FDFDonaldo, Gonzalo and Mayoral (2002) proposed a Fractional Dickey-Fuller test (FDF) for testing the null hypothesis of against the alternativehypothesis .

0dI

011 0, dddI

FDF is based on the t-statistic of the OLS estimator of in the regression:

ttd

td xxo 1

1

If then 5.00 d 1,0ˆ Nt

R/S testLet and let be the OLS estimator of the coefficient of the trend from regressing the series against a constant and a trend

0

1xx

T T ̂tx

the classical R/S statistic: txtxT

rsc tTt

tTt

minmax1

0

Lo(1991) generalized R/S statistic:

txtxTlS

rslo tTt

tTt

minmax12

Cavaliere (2001) generalized R/S statistic:

txtxTlS

rscav tTt

tTt

ˆminˆmax12

the null hypotesis against , 1I dI 1 0d

where is the usual estimate of the variance of the series and is a consistent estimator of the “long run variance” of

xxt xxt lS 2

0

Estimating the degree of fractional differencing Log Periodogram Estimator

Consider peridogram of a series :tx 2

12

1

T

tt

it xxeT

I

Geweke, J. and S. Porter-Hudak(1983) proposed as an estimate of

the OLS estimator of from the regression:d

b

jj

j baI

2sin4lnln 2 mj ,...,1

the bandwidth is chosen such that for , butm T m 0T

m

perform tests both using the t ratios based on the standard deviations

of the regression and using the aymtotic distribution obtained by Robinson (1995a)

Approximate MLE estimators

Sowell (1992) derives an exact Maximum Likelihood Estimator of the ARFIMA(p, d, q) process. However the Sowell estimator is computationally burdensome

In this paper we used two approximate MLE estimators:

• approximate Whittle estimator proposed by Fox and Taqqu (1986)

• an approximate wavelet MLE proposed by Jensen (2000)

Data, methodology and empirical results

• 7 international indices: United States – S&P500 Index and NASDAQ Index; France - CAC40 Index; United Kingdom – FTSE100 Index, Japan -Nikkei 225 Index, Singapore- Straits Times Index, Taiwan - Weighted Index

• 3 Romanian indices: BET, BETC, RASDAQ

• periods up to June 2002

The tests and the estimation are implemented in Mathcad 2000

The estimation procedures require either the peridogram or the wavelet coefficients. These are in fact the Fourier Transform and the Wavelet transform of the data series. Since the algorithms implemented in Mathcad to compute this two transforms (Fast Fourier Transform respectively Fast Wavelet Transform) requires that the number of inputs be a power of 2 , we have to reduce our samples to the largest power of two.

The tests procedures proved to be a much bigger burden for the computer than the estimation procedures. So, due to lack of computing power, when conducting a test we reduced thesample size up to 2000 observations

Tests

ADF 0 KPSS-44.5492 B QS B QS 0.1 0.05 0.01

0.01 0.67326 0.59253 0.079532 0.066981 KPSS level 0.347 0.463 0.739-6.17234 KPSS trend 0.119 0.146 0.2161.996661

FDF 0.4 0.3 0.2 0.1-16.5623 -12.3267 -8.1272 -3.942976.53E-62 3.25E-35 2.2E-16 4.02E-05-6.11141 -6.11857 -6.13003 -6.143921.969368 1.974024 1.98309 1.994297

R/S classicalB QS B QS 0.1 0.05 0.01

1.411028 1.564693 1.473786 1.706952 1.60778 R/S Lo 1.62 1.747 2.001R/S Cavaliere 1.55 1.682 1.956

Robinson LM -0.2 -0.1 0 0.1 0.212.80773 4.235423 -1.54002 -5.42192 -8.225931.48E-37 2.28E-05 0.123555 5.9E-08 1.9E-16

Estimations

GPH 0.45 0.5 0.55 0.8 Robinson classical modified Wavelet MLE Whittle0.161045 0.066315 0.053136 -0.02907 -0.04832 0.041141 -0.01497962 0.0336681.852118 0.914977 0.905625 -1.62411 -1.77656 1.512735 3.8691490.064045 0.36023 0.365161 0.104391 0.075641 0.130347 0.0001090.956287 0.49324 0.495428 -0.833290.338927 0.621843 0.620298 0.404684

USA SP500 daily

Critical values for the KPSS test

Critical values for the R/S test

level trend

Lo Cavaliere

Tests

ADF 1 KPSS-33.414 B QS B QS 0.1 0.05 0.01


FDF 0.4 0.3 0.2 0.1-14.4142 -10.1771 -5.96343 -1.751922.11E-47 1.25E-24 1.23E-09 0.039894-6.22063 -6.22954 -6.24365 -6.261151.943318 1.950565 1.963409 1.979693



Robinson LM -0.2 -0.1 0 0.1 0.212.68383 4.705447 -0.70349 -4.52956 -7.281547.27E-37 2.53E-06 0.481753 5.91E-06 3.3E-13

Estimations

GPH 0.45 0.5 0.55 0.8 Robinson classical modified Wavelet MLE Whittle-0.08499 -0.09868 -0.09806 0.002709 0.020564 0.057924 0.036740098 0.026444-0.68131 -0.96988 -1.34438 0.112093 0.573225 1.614605 2.1829670.495717 0.332165 0.178901 0.910756 0.566492 0.106396 0.029038-0.43454 -0.61553 -0.75689 0.0588870.663896 0.538205 0.449115 0.953042



UK FTSE100 daily

level trend

Lo Cavaliere

Tests

ADF 0 KPSS-33.1345 B QS B QS 0.1 0.05 0.01


FDF 0.4 0.3 0.2 0.1-12.0961 -8.7279 -5.35457 -1.964025.54E-34 1.3E-18 4.29E-08 0.024764-5.03898 -5.04337 -5.05019 -5.056771.970652 1.975549 1.985225 1.996269



Robinson LM -0.2 -0.1 0 0.1 0.216.34629 7.779204 1.80793 -2.28051 -5.168024.62E-60 7.3E-15 0.070617 0.022577 2.37E-07

Estimations

GPH 0.45 0.5 0.55 0.8 Robinson classical modified Wavelet MLE Whittle0.031958 0.097909 0.067245 0.077239 0.120972 0.075926 0.022791441 0.0440010.192528 0.733993 0.662454 1.889172 1.935546 1.214816 1.9040880.847367 0.463121 0.507829 0.059152 0.052923 0.224436 0.0568990.119501 0.431842 0.355604 0.963570.904878 0.665856 0.722137 0.335261



Taiwan Weighted index daily

level trend

Lo Cavaliere

Tests

ADF 0 KPSS-25.7844 B QS B QS 0.1 0.05 0.01


FDF 1 0.4 0.3-18.5222 -3.86458 -0.569836.84E-77 5.56E-05 0.284397-4.86661 -5.01102 -5.027642.045159 1.955509 1.975536



Robinson LM -0.2 -0.1 0 0.1 0.226.94749 16.79774 9.375312 3.826367 -0.036316.1E-160 2.54E-63 6.9E-21 0.00013 0.971031

Estimations

GPH 0.45 0.5 0.55 0.8 Robinson classical modified Wavelet MLE Whittle0.050982 0.01659 0.007108 0.195572 0.249901 0.345542 0.220783804 0.1930010.444139 0.192866 0.105293 4.669356 3.998419 5.528668 10.032540.657036 0.847102 0.916164 3.42E-06 6.38E-05 3.23E-08 1.1E-230.190636 0.073171 0.037589 2.4397850.848811 0.94167 0.970015 0.014696



Romania BET daily

level trend

Lo Cavaliere

Tests

ADF 0 KPSS-24.1307 B QS B QS 0.1 0.05 0.01


FDF 1 0.4 0.3-17.3954 -3.82789 -0.70274.47E-68 6.46E-05 0.24112-5.38662 -5.53575 -5.551362.060288 1.956066 1.973778



Robinson LM -0.2 -0.1 0 0.1 0.228.02967 17.64668 10.24652 4.095692 0.1390457.1E-173 1.08E-69 1.23E-24 4.21E-05 0.889415

Estimations

GPH 0.45 0.5 0.55 0.8 Robinson classical modified Wavelet MLE Whittle0.088351 0.085298 0.12068 0.148007 0.292333 0.277366 0.187315626 0.2095320.545883 0.717317 1.11177 3.673721 4.67733 4.437849 11.705810.585265 0.473342 0.266499 0.000251 2.91E-06 9.09E-06 1.19E-31

0.33037 0.376219 0.638177 1.8464040.74112 0.706754 0.523358 0.064834



Romania BETC daily

level trend

Lo Cavaliere

Tests

ADF 0 KPSS-40.0337 B QS B QS 0.1 0.05 0.01


FDF 1 0.4 0.3 0.2 0.1-30.4091 -20.3003 -17.3931 -14.581 -11.84022.1E-203 6.38E-92 4.66E-68 1.86E-48 1.21E-32-5.54924 -5.85611 -5.86392 -5.87459 -5.88622.402077 2.123701 2.116916 2.107335 2.09235



Robinson LM -0.2 -0.1 0 0.1 0.21.685306 -3.51757 -5.11351 -8.77614 -10.11890.091929 0.000436 3.16E-07 1.69E-18 4.55E-24

Estimations

GPH 0.45 0.5 0.55 0.8 Robinson classical modified Wavelet MLE Whittle0.143385 0.175931 -0.01728 -0.10716 -0.16671 -0.31997 -0.09287454 -0.118430.672634 1.010667 -0.12449 -1.82576 -2.02817 -3.89261 -2.916840.501485 0.312655 0.900973 0.068471 0.042543 9.92E-05 0.0035360.460951 0.657857 -0.07502 -1.016420.644834 0.51063 0.940199 0.309432



Romania RASDAQ daily

level trend

Lo Cavaliere

A continuous trading model with long memory

Fractional Brownian Motion

If the fractional Brownian motion (fBm) with Hurst parameter H isthe continuous Gaussian process , with mean and whose covariance is given by:

10 H

RttBH , 0tBH 0tBE H

HHH

HHH ststsBtBEstC222

2

1,

The Hurst parameter determines the sign of the covariance of the future and past increments. This covariance is positive when , zero when andnegative when .

5.0H 5.0H5.0H

for the fractional Brownian motion has long range dependence 5.0H

Since for the fractional Brownian motion is neither a Markov process, nor a semimartingale, we can not use the usual stochastic calculus to analyze it.

5.0H

Lin (1995) developed a pathwise integration theory for fractional Brownian motion and Rogers (1997) proved that the market mathematical models based on thisintegration theory can have arbitrage

Duncan, Hu and Pasik-Duncan (2000) and Hu and Oksendal (2000) introduced a new kind of integral, a generalization of the Ito integral, called fractional Ito

integral. They derive a fractional Ito lemma, and introduced two new concepts:quasi-conditional expectation and quasi-martingale

A fractional Black-Scholes market Let .Consider a fractional Black-Scholes market that has two investment possibilities:

• a money market account

5.0H

TtMdttrMtdM 0,10,

where r represent the constant riskless interest rate

• stock whose price satisfies the equation

TtSStdBtSdttStdS H 0 ,00 ,

Where are constants0,

Hu and Oksendal (2000) have shown that this market does not have arbitrage and is complete.

They compute the risk-neutral measure and under this measurewe have that:

TtSStBdtSdttrStdS H 0 ,00 ,

A formula for the price of a European option at is alsoderived

0t

Pricing derivative securities

We will denote by the quasi-conditional expectation with respect to the risk-neutral measure.

tE~

Necula(2002) have shown that in a fractional Black-Scholes market we have the following results:

Theorem 1 (fractional risk-neutral evaluation)The price at every of a bounded claim is given by

Tt ,0 F

FEetF ttTr ~

Theorem 2 (fractional Black-Scholes equation)

The price of a derivative on the stock price with a bounded payoff is given by , where is the solution of the PDE: TSf tStD , StD ,

02

22122

rD

S

DrS

S

DStH

t

D H

SfSTD ),(

Theorem 3 (fractional Black-Scholes formula)

The price at every of an European call option with strikeprice is given by

).0[ Tt

K

)()()())(,( 2)(

1 dNKedNtStStC tTr

HH

HH

tT

tTtTrK

tS

d22

222

12

)(ln

HH

HH

tT

tTtTrK

tS

d22

222

22

)(ln

where is the cumulative probability of the standardnormal distribution

N

Theorem 4 (The Greeks)

1dNS

C

2dNeK

C tTr

HH tTdSfC 22

1

2dNKetTr

C tTr

1222

2 1df

tTSS

CHH

122

122 df

tT

SHtdNrKe

t

CHH

HtTr

2

2

2

1 z

ezf

where

-0.2

-0.18

-0.16

-0.14

-0.12

-0.1

-0.08

-0.06

-0.04

-0.02

0

price

-0.01

-0.008

-0.006

-0.004

-0.002

0

0.002

0.004

0.006

0.008

delta

1.0r 2.0 55.0H 5.0t 1T

Conclusions:

• Using a wide range of test and estimation procedures we have investigated whether stock returns exhibit long memory. Some evidence of long range dependence was found in daily returns of S&P500, NASDAQ, FTSE100, Singapore ST and Taiwan WI indices and in weekly returns of Nikkei 225 and Singapore ST indices. Strong evidence of long memory was found in daily returns of Romanian BET and BETC indices.

• If the underlying follows a continuous stochastic process that exhibits long memory, pricing derivative securities with martingale methods it is not appropriate.

Modelling and Detecting Long Memory in Stock Returns MSc student Ciprian Necula Doctoral School of...

Documents

Transcript of Modelling and Detecting Long Memory in Stock Returns MSc student Ciprian Necula Doctoral School of...