Fat tails and colored noise in financial derivatives

Physica A 314 (2002) 736–742www.elsevier.com/locate/physa

Fat tails and colored noise in !nancial derivativesJosep Perell#o, Jaume Masoliver ∗

Departamento de F� sica Fonamental, Universitat de Barcelona, Diagonal 647, 08028 Barcelona, Spain

Abstract

We study the e(ect of heavy tails and correlations on the price of the one of the simplest !-nancial derivative: the European call option. We see that both e(ects have opposite and nontrivialconsequences on the price of the derivatives.c© 2002 Elsevier Science B.V. All rights reserved.

PACS: 89.65.Gh; 02.50.Ey; 05.10.Gg

Keywords: Martingale option pricing method; Heavy tails; Correlated stocks

Pricing derivatives lies, from the very beginning, at the heart of mathematical !nanceas scienti!c discipline. Focussing on options and after the landmark work of Blacket al. [1], the stock share S(t) of any option is usually assumed to be driven bythe geometric Brownian motion which implies that return, R(t)=ln S(t)=S(t0), obeys thearithmetic Brownian motion:

R= � + ��(t) ; (1)

whose solution is R(t)=�(t− t0)+�∫ tt0�(t′) dt′ and where � is the mean return rate,

� is the volatility and �(t) is zero-mean Gaussian white noise. Note that the initialreturn is zero and the integral of �(t) is the Wiener process W (t).However, the market model given by Eq. (1) does not fully account for all properties

observed in real markets. Some of them are quite crucial such as the existence of heavytails, self-scaling and mild correlations in the increments of return (see for instance,Ref. [2]). These properties undoubtedly have to a(ect option pricing and the theoreticalprice for the option obtained through the assumption that markets are governed byEq. (1) would not be adjusted to reality. Our main objective here is to elucidate howthe above properties of real markets may a(ect the option price.

∗ Corresponding author. Fax: +34-93-402-1149.E-mail address: jaume@(n.ub.es (J. Masoliver).

0378-4371/02/$ - see front matter c© 2002 Elsevier Science B.V. All rights reserved.PII: S 0378 -4371(02)01151 -2

J. Perell�o, J. Masoliver / Physica A 314 (2002) 736–742 737

We restrict the analysis to European call options. An European call option is aderivative giving to its owner the right but not the obligation to buy a share at a !xeddate T (the maturity) for a certain price K (the strike or striking price). The call optioncontract is basically speci!ed by its gain at maturity [S(T )−K]+ ≡ max[S(T )−K; 0].

If X (t) follows the equation:

X = F(t) ;

where F(t) is a zero-mean random process, the Brownian model can be generalized toinclude other driving noises and thus trying to account for some more realistic marketsfeatures. We therefore, propose a generic market model (cf. Eq. (1))

R(t) = �(t − t0) +∫ t

t0F(t′) dt′ ; (2)

where the integral of F(t) should be interpreted in the sense of Ito [1]. We also de!nethe zero-mean return X (t) by

X (t) =∫ t

t0F(t′) dt′ ; (3)

which is required to avoid arbitrage opportunities, i.e., money pro!ts for free with-out taking any risk are not allowed. To this end, it suKces that X (t) be a noise ofunbounded variation [3].We now return to options. The Black–Scholes (B–S) option price can be obtained,

in a shorter way, by martingale methods [4]. The equivalent martingale measure theoryimposes the condition that, in a “risk-neutral world”, the stock price S(t) discards anyunderlying process that allows for arbitrage opportunities. Thus, if at time t we buya European call on stock having price S, then in terms of the risk-neutral densityp∗(R; t|t0), is [4]

C(S; t) = e−r(T−t)∫ ∞

ln(K=S)(SeR − K)p∗(R; T |t) dR ; (4)

where T and K are the maturity and strike of the option. This density is the probabilitydensity function (pdf) of a process R∗(t) which we call risk-neutral return and suchthat 〈expR∗(t) | t0〉= exp r(t − t0), where r is the risk-free interest rate ratio.Martingale pricing allows us to obtain the fair option price when the market obeys a

random dynamics di(erent than that of the geometric Brownian motion. One only hasto know the return distribution of the underlying. Nevertheless, knowing an analyticalexpression of the pdf pX (x; t|t0) may be, in practice, very diKcult. There are however,many situations in which we know the characteristic function of the model althoughone is not able to invert the Fourier transform and obtain the density pX (x; t | t0). 1 Forthese cases, we will develop an option pricing based on a combination of martingalemethods and harmonic analysis.

1 This is the case of non-Gaussian models such as L#evy processes [5] and some generalizations [6] or thecase of stochastic volatility models (see for instance, Ref. [7]) among others.

738 J. Perell�o, J. Masoliver / Physica A 314 (2002) 736–742

The characteristic function (cf) of a random variable is the Fourier transform ofits probability density function, that is: ’X (!; t | t0) = 〈exp i!X (t) | t0〉. After severalmanipulations, we !nally get (see [8] for more details)

C(S; t) = S − K2e−r(T−t)

−K�e−r(T−t)

∫ ∞

0’X (!; T |t)

[cos!�(T − t) + sin!�(T − t)

!

]d!

1 + !2 ;

(5)

where �(t) ≡ ln S=K + rt − ln’X (−i; t).We now apply the formalism developed hitherto to pricing options whose underlying

assets have probability distributions showing fat tails. First market models incorporatingheavy tails were L#evy distributions [5]. However, these models have no !nite momentsbeyond the !rst one. This is a severe limitation specially for the practitioner. In a recentwork, we have presented a market model that explains the appearance of fat tails andself-scaling but still keeping all moments !nite [6]. In that model we assume that allrandom changes in the stock price are given by a continuous superposition of di(erentshot-noise sources, each source corresponding to the arrival of new information. OurX (t) does not allow arbitrage since, as we have proved in Ref. [8], X (t) has unboundedvariations. Assuming the gamma distribution of jumps, the characteristic function ofX (t) explicitly reads [6]

’X (!; t) = exp{a�

[1− Re

{F(�;−�; 1− �;− i!(bt)1=�

�

)}]}; (6)

where F(a; b; c; z) is the hypergeometric function and Re(·) is the real part. The sub-stitution of this equation into Eq. (5) results in the price of the call. In Fig. 1, weobserve that heavy tails overprice by far the call. This con!rms the intuition under-standing that fatter tails implies higher probability for large changes, more risk and,!nally, a higher price for the option. The relative di(erence �=(CTail−CBS)=CTail forOTM (S=K ¡ 1), ATM (S=K = 1), and some ITM options (S=K ¿ 1) � increases asT − t increases, while for options well in the money � becomes smaller with T − t. 2There is thus a “crossover” which, in our example, is located around S=K = 1:03. Wealso note that as S=K → ∞, �→ 0 very slowly which implies the great persistence ofthe heavy-tail e(ect.We next summarize the e(ect of colored noise on options. The classical option

formula of Black, Scholes and Merton is based on the “eKcient market hypothesis”which means that the market incorporates instantaneously any information concerningfuture market evolution. Indeed, market eKciency is closely related to the assumptionof totally uncorrelated price variations (white noise). However, as empirical evidenceshows, real markets are not eKcient, at least at short times [2]. White processes areconvenient mathematical objects valid only when the observation time is much larger

2 The initials ITM, OTM and ATM, respectively, correspond to “in the money”, “out the money” and “atthe money”.


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Black-Scholes call -maturity 30 d.heavy tail call -maturity 30 d.

deterministic call -maturity 30 d.

C/K

Fig. 1. The heavy tail and their relative price di(erences in terms of S=K . We take r = 5% yr−1 andparameters estimated from S&P-500 (1988–1996) with � = 3:54× 10−3 day−1=2.

than the auto correlation time of the process, “ineKciencies” (i.e., correlations, delays,etc.) occur.In Ref. [9], we have derived a nontrivial option price by relaxing the eKcient market

hypothesis and allowing for a !nite, nonzero, correlation time of the underlying noiseprocess. The O–U process is the simplest generalization of Gaussian white noise (1).We thus assume that stock price is not driven by Gaussian white noise �(t) but byO–U noise V (t). In other words,

R(t) = � + V (t) V (t) =1#[− V (t) + ��(t)] ; (7)

where V (t) is O–U noise in the stationary regime, which is a Gaussian colored noisewith zero mean and correlation: 〈V (t1)V (t2)〉 = (�2=2#) exp − |t1 − t2|=#. When # = 0,this correlation goes to �2$(t1− t2) and thus recover the GBM model given by Eq. (1).


In the opposite case when # =∞, R(t) = �t and stock evolves as a riskless security.However, since the velocity V (t) is a non tradable asset, its evolution is ignored andthe resulting “projected” process is driven by a noise of unbounded variation whosezero-mean return (3) reads [9]

X (t) =√%(T − t)�(t) ;

where dot denotes times derivative and %(t)=�2[t− #(1− exp t=#)]. Note that we needto specify the !nal condition of the process because the volatility

√% is a function of

the time to maturity T − t and this implies that the projected model depends on eachparticular contract. Its corresponding characteristic function is

’X (!; T | t) = exp[− %(T − t)!2=2]

and substituting the characteristic function into Eq. (5), we derive

COU (S; t) = S N (dOU1 )− Ke−r(T−t) N (dOU2 ) ; (8)

where N (z) is the probability integral and

dOU1 =ln S=K + r(T − t) + %(T − t)=2√

%(T − t) ; dOU2 = dOU1 −√%(T − t) :

Note that the O–U price in Eq. (8) has the same functional form as B–S price when�2t is replaced by %(t) (cf. [1]). When # = 0, the variance becomes %(t) = �2t andthe price reduces to the B–S price. When # = ∞, there is no random noise and thisreduces Eq. (8) to a deterministic price

Cd(S; t) = max[S − Ke−r(T−t); 0] : (9)

From Fig. 2, we observe that colored driving noise underprices the call. This is con-sistent with the idea that correlations imply more predictability and therefore, less riskand, !nally, a lower price for the option. We can also see that the relative di(erenceD=(CBS −COU )=CBS increases as T − t decreases and, contrary to the fat tail case, Dquickly fades away with moneyness and it is negligible for ITM options showing nocrossover.From Figs. 1 and 2, we conclude that, taking the B–S price CBS as a benchmark,

heavy tails overprice by far options while colored driving noise underprices it. Thescheme is

S¿CTail¿CBS¿COU ¿Cd ;

where S is the stock price when the option is bought and Cd is the deterministic price(9). We have also seen that the relative di(erence with B–S price is in both cases moreimportant for OTM and ATM options (S=K6 1) than for ITM options (S=K ¿ 1) but,in any case, there is a greater persistence of heavy tail e(ects.


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C/K

Fig. 2. The correlated call prices and the relative price di(erences in terms of S=K . We take r = 5% yr−1

# = 2 days and with � = 3:54× 10−3 day−1=2.

Acknowledgements

The authors acknowledge helpful discussions with Miquel Montero on the resultsherein derived for the market model showing fat tails. This work has been supportedin part by Direcci#on General de Investigaci#on under contract No. BFM2000-0795, byGeneralitat de Catalunya under contract No. 2000 SGR-00023 and by Societat Catalanade F#Ssica (Institut d’Estudis Catalans).

References

[1] J. Perell#o, J.M. PorrUa, M. Montero, J. Masoliver, Physica A 278 (2000) 260–274.[2] R. Cont, Quantitative Finance 1 (2001) 223–236.


[3] J.M. Harrison, R. Pitbladdo, S.M. Schaefer, J. Business 57 (1984) 353–365.[4] J.M. Harrison, D. Kreps, J. Econ. Theory 20 (1979) 381–408;

J.M. Harrison, S. Pliska, Stochastic Process. Appl. 11 (1981) 215–260.[5] R.N. Mantegna, E.H. Stanley, Nature 376 (1995) 46–49.[6] J. Masoliver, M. Montero, J.M. PorrUa, Physica A 283 (2000) 559–567;

J. Masoliver, M. Montero, A. McKane, Phys. Rev. E 64 (2001) 01110.[7] J. Masoliver, J. Perell#o, Int. J. Theor. Appl. Finance, in press.[8] J. Perell#o, J. Masoliver, Physica A 308 (2002) 420–442.[9] J. Masoliver, J. Perell#o, to appear in Quantitative Finance.

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Fat tails and colored noise in financial derivatives