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Physica A 344 (2004) 134–137

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doi:10.1016/j

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A comparison between several correlatedstochastic volatility models

Josep Perelloa,�, Jaume Masolivera, Napoleon Anentoa,b

aDepartament de Fısica Fonamental, Universitat de Barcelona, Diagonal, 647, E-08028 Barcelona, SpainbGaesco Bolsa, SVB, S.A., Diagonal, 429, 08036-Barcelona, Spain

Received 4 December 2003

Available online 23 July 2004

Abstract

We compare the most common stochastic volatility models such as the Ornstein–Uhlenbeck

(OU), the Heston and the exponential OU models. We try to decide which is the most

appropriate one by studying their volatility autocorrelation and leverage effect, and thus

outline the limitations of each model. We add empirical research on market indices confirming

the universality of the leverage and volatility correlations.

r 2004 Elsevier B.V. All rights reserved.

PACS: 02.50.Ey; 02.50.Ga; 89.65.Gh; 02.50.Cw

Keywords: Volatility autocorrelation; Leverage; Stochastic volatility models

The multiplicative diffusion process is considered to be the most popular model infinance. We thus have that the log-price change without average describes thediffusive process _X ðtÞ ¼ sx1ðtÞ (in the Ito sense), where s is the volatility (assumed tobe constant) and x1 is Brownian noise with hx1ðtÞx1ðt

0Þi ¼ dðt � t0Þ. However, thismodel is unable to capture most of the statistical properties of real markets, thosecalled stylized facts.

see front matter r 2004 Elsevier B.V. All rights reserved.

.physa.2004.06.103

nding author.

dress: perello@ffn.ub.es, josep.perello@ub.edu (J. Perello).

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J. Perello et al. / Physica A 344 (2004) 134–137 135

The stochastic volatility (SV) models are a natural way out to avoid theinconsistencies of the log–Brownian model by still assuming that X follows adiffusion process but now with a random s. However, the dynamics of s has notbeen definitively attached to any specific SV model. In this sense, we want to discernbetween the most common SV models. Among others we have:

(a) The Ornstein–Uhlenbeck (OU) SV model [1,2]

_sðtÞ ¼ �aðs� mÞ þ kx2ðtÞ; sðtÞ ¼ m þ k

Z t

�1

e�aðt�t0Þx2ðt0Þdt0 : ð1Þ

(b) The Heston SV model [3,4] (with V s2)

_V ðtÞ ¼ �aðV � m2Þ þ kffiffi½

p�Vx2ðtÞ ; ð2Þ

V ðtÞ ¼ m2 þ k

Z t

�1

e�aðt�t0Þffiffi½

p�V ðt0Þx2ðt

0Þdt0 : ð3Þ

(c) And, finally, the exponential Ornstein–Uhlenbeck (expOU) SV model [5] (withY lnðs=mÞ)

_Y ðtÞ ¼ �aY þk

mx2ðtÞ; Y ðtÞ ¼ m þ

k

m

Z t

�1

e�aðt�t0Þx2ðt0Þdt0 : ð4Þ

Note that the Heston model does not provide a closed equation for the volatility.This can be a serious drawback in case we want to deal with analytical expressions.In addition, we suppose that noises are correlated (i.e., hx1ðtÞx2ðt

0Þi ¼ rdðt � t0Þ with�1prp1). This allows us to explain some stylized facts.

The leverage effect [6] is measured by: LðtÞ hdX ðt þ tÞ2 dX ðtÞi=hdX ðtÞ2i. For themodels above we have

LOUðtÞ ¼ 2rn

ffiffi½

p�2að1 þ n2e�atÞ

ð1 þ n2Þ2m

� �e�at; LHðtÞ � 4r

ak

e�at ; ð5Þ

LexpOUðtÞ ¼2rk

m2en

2=2 expð2n2e�at � atÞ ; ð6Þ

for t40 and LðtÞ ¼ 0 for negative t. It seems impossible to derive an analyticalexpression for the Heston leverage because V ðtÞ does not have a closed expression(cf. Eq. (3)). The leverage given in Eq. (5) is exact only when m2 ¼ k2=2a but Ref. [4]shows that this is not true for the Dow Jones. We will have thus to simulate theprocess to compute the leverage effect in each specific Heston case. Eqs. (5) and (6)are plotted and compared with data in Fig. 1. The data is too noisy to assert which isthe most appropriate model. In any case, the three models have the desiredexponential decay with a characteristic time of the order of few days. We also notethat, without the correlation coefficient r, there is no leverage and data confirms thatro0 [7].

The volatility autocorrelation is CðtÞ ½hdX ðtÞ2 dX ðt þ tÞ2i � hdX ðtÞ2ihdX ðt þ

tÞ2i�= Var½dX ðtÞ2�. In this case, we can exactly derive this correlation for each model.

ARTICLE IN PRESS

−0.2

−0.1

0

vola

tility

-ret

urn

corr

elat

ion

Dow JonesDAXS&P500Nikkei

−30

−20

−10

0

10

leve

rage

Nikkei dataOU fitHeston fitexpOU fit

NIKKEI

−40 0 40 80 120trading days

−30

−20

−10

0

10

leve

rage

DAX dataOU fitHeston fitexpOU fit

DAX

−40 −20 0 20 40 60 80trading days

−40 −20 0 20 40 60 80 100trading days

−60

−40

−20

0

20

leve

rage

S&P 500 dataOU fitHeston fitexpOU fit

S&P 500

Fig. 1. The leverage effect for several daily price indices. We also add the leverage function LðtÞ fit for the

different SV models. The S&P 500 is much more liquid.

J. Perello et al. / Physica A 344 (2004) 134–137136

We have

COUðtÞ ¼n2e�atðn2e�at þ 1Þ

4n2ð2 þ n2Þ þ 1; CHðtÞ ¼ n2e�at ; ð7Þ

CexpOUðtÞ ¼expð4n2e�atÞ � 1

3 expð4n2Þ � 1: ð8Þ

The volatility autocorrelation observed in empirical data shows at least two differenttime scales (see Fig. 2 and Ref. [8]). The shortest one coincides with the leveragecorrelation time while the second scale is of the order of years, around 10 timesbigger than the leverage time. The OU and Heston models are not capable ofreproducing the slowest time decay but the expOU model has a nontrivial decay withthe two time scales:

CexpOUðtÞ � e�at ðat41Þ; CexpOUðtÞ � e�ðk=mÞ2t ðato1Þ ; ð9Þ

where the faster decay is given by the diffusive coefficient k=m and the slow decay ofthe volatility is provided by the reversion coefficient a (cf. Eq. (4)).

Therefore, we conclude that the expOU model appears to be as more realistic.Further investigations on this model are required to confirm this assertion. However,an alternative choice is to sophisticate the OU and Heston models. A second time

ARTICLE IN PRESS

0 20 40 60 800

0.05

0.1

0.15

0.2

0 200 400 600 800 1000

trading days

0.001

0.01

0.1

NikkeiDAXS&P-500Dow Jones

20 40 60 80 1000

0.1

0.2

0.25

DAX

0 200 400 600 800 1000

trading days

0

0.025

0.05

0.075

Vol

atili

ty a

utoc

orre

latio

n S&P-500Heston and OU fitexpOU fit

0

0.05

0.1

0.15

0.2

Vol

atili

ty a

utoc

orre

latio

n

NikkeiHeston and OU fitexpOU fit

10 20 30 40 500

0.05

0.1

0.15

200 400 600 800 1000

DAXexpOU fit

Heston and OU fit 0.05

0.15

Fig. 2. The volatility correlation for several daily indices. We add two exponential fits one with a short

range (similar to the leverage time scale) and a second one for the larger time lags.

J. Perello et al. / Physica A 344 (2004) 134–137 137

scale can be generated by including a third SDE for m. In a previous work [8], wehave studied this possibility for the OU model with some success by taking:_mðtÞ ¼ �amðm � yÞ þ kmx3ðtÞ. A similar analysis can be performed for the Hestonmodel.

This work has been supported in part by Direccion General de Proyectos deInvestigacion under contract No. BFM2003-04574, and by Generalitat de Catalunyaunder contract No. 2001SGR-00061.

References

[1] E.M. Stein, J.C. Stein, Rev. Financial Stud. 4 (1991) 727–752.

[2] J. Masoliver, J. Perello, Int. J. Theor. Appl. Finance 5 (2002) 541–562.

[3] S.L. Heston, Rev. Financial Stud. 6 (1993) 327–343.

[4] A. Dragulescu, V. Yakovenko, Quant. Finance 2 (2002) 443.

[5] J.P. Fouque, G. Papanicolaou, K.R. Sircar, Int. J. Theor. Appl. Finance 3 (2000) 101–142.

[6] J.-P. Bouchaud, A. Matacz, M. Potters, Phys. Rev. Lett. 87 (2001) 228701.

[7] J. Perello, J. Masoliver, Phys. Rev. E 67 (2003) 037102.

[8] J. Perello, J. Masoliver, J.-P. Bouchaud, Applied Mathematical Finance 11 (2004) 27–50.