The European Journal of Finance 3, 311–324 (1997) Could nonlinear dynamics contribute to i

chaotic p ocesses a e, fo any p actical pu poses, indistinguishable f om stochastic processes.

The ?rst investigations in the ?eld of nonlinear dynamics have concentrated

on the cha

r acte

r

ization of the asymptotic inva

r

iant sets, called st

r

ange

attr actor s (Eckmann and Ruelle, 1985), on which the tr ajector ies of chaotic systems settle after a transient time. It has been shown that the distribution of points on such sets looks almost the same at all level of magni?cation, i.e. that these sets ar e self-similar (Mandelbr ot, 1982). This featur e has been descr ibed in a statistical mechanical fr amewor k by means of static invar iants like the gener alized fr actal dimensions. The main tests aiming at discover ing the existence of deter ministic chaos in physical systems ar e based on the computation of generalized fractal dimensions by two different approaches: the ?xed-size (Gr assber ger and Pr ocaccia, 1982) and the ?xed-mass (Badii and Politi, 1984) methods. Numerical experiments have demonstrated that the latter method is better suited to deal with high-dimensional systems (Kostelich and Swinney, 1987).

Besides testing for a signature of chaos, the scienti?c community has devoted its attention towar ds an impr oved under standing of the dynamic featur es of chaotic motion in physical systems. In this framework, two different modelling approaches have been developed and intensively analysed. The ?rst one, which requires less data than the algorithm for computing fractal dimensions, consists in ?tting local maps to the obser ved points in or der to r econstr uct the tr ajector ies in the phase space (Far mer and Sidor owich, 1987). It has been demonst ated that this method allows one to obtain eliable sho t-te m predictions of the evolution of chaotic systems, even if their dimensionality is high.

The second modelling appr oach, called symbolic dynamics (Alekseev and Yakobson, 1981) performs a physically meaningful partition of phase-space and orders the partition elements in hierarchical order of importance on a logic tree (Badii et al., 1992). This coar se-gr ained representation of the dynamics allows one to identify the most impor tant topological invar iants of the system, i.e. for bidden tr ansitions between par tition elements in the phase-space, in a str aightfor war d way and to constr uct a hier ar chy of Mar kov models that appr oximates the topology and the metr ic featur es of the attr actor up to the accuracy that is available in the experimental signal (Badii, 1990). This method has been successfully applied to an experimentally generated signal (Badii et al. 1994) but requires a low-dimensional dynamics and a large amount of noise-free data.

Evidence of low-dimensional chaotic behaviour has been observed in several physical systems such as classical many-body systems, nonlinear optics, lasers, convection, chemical reactions, electronic circuits and magnetism (Vohra et al., 1992). It has been sur mised that nonlinear dynamics could allow shor t-ter m pr edictions in economics (Far mer and Sidor owich, 1987; Far mer, 1994), or at least explain some qualitative featu es of the ma kets (de G auwe and Dewachter, 1993). It has been repor ted that many ?nancial time series are not entir ely r andom walks (e.g. Taylor, 1986; Hsieh, 1991; Kaplan, 1993). However, the ability of nonlinea dynamics to captu e an unde lying dete ministic 312Darbellay and Finardi

str uctur e in ?nancial time ser ies is far fr om being obtained (Br ock e t al .,1992).

The great majority of studies of deterministic chaos in ?nance has revolved ar ound tests based on the calculation of the cor r elation dimension with the Grassberger and Procaccia (1983) algorithm. Investigations with extensive data sets have concluded negatively as to the presence of chaos in stocks or foreign exchange mar kets (Hsieh, 1991; Vassilicos e t al ., 1993; Guillaume, 1994). It is impor tant to r ealize that these tests always r equir e lar ge amounts of data (Eckmann and Ruelle, 1992) and are prone to be deceived by noise. It has been found that if an even small r andom per tur bation is added to a chaotic signal,this could render the correlation dimension procedure incapable of distinguish-ing the lar ge chaotic component (Gr

anger and Ter ¨a svir ta, 1993). Besides the attr actor dimensions, such as the cor elation dimension, ther e ar e other quantities of inter est, namely the Lyapunov exponents (Gr assber ger e t al .,1991). Unfor tunately, the number of points needed to estimate Lyapunov exponents is about the squar e of that needed to estimate the dimension (Eckmann and Ruelle, 1992).

The novelty of this paper is twofold. First, we use the Badii and Politi (1984)algorithm, whose behaviour is believed to be superior to the Grassberger and Procaccia (1983) algorithm when the dimension of the system becomes larger.Second, as this test could also be deceived by noise, we build a pr edictive model and test dir ectly whether any useful for ecasts could be made, namely whether the timing of hedge tr ades in the cour se of a business day could be impr oved.

In this study we have worked with intra-day data, also known as tick-by-tick data or high fr equency data, fr om the for eign exchange and the inter est r ate mar kets. We have per for med intensive analyses of the dimensionality of such time se ies in o de to investigate the feasibility of the two modelling appr oaches mentioned above, i.e. the appr oximation with local maps or with symbolic dynamics. Since no evidence of the existence of low-dimensional chaos could be found, we concentr ated our attention on the shor t-ter m prediction approach with local maps, because it has less stringent requirements in ter ms of dimensionality of the system and data quality. In ?nance, one is primarily interested in knowing whether forecasts can lead to improved returns.Here we consider the problem of timing hedge trades. Depending on the ?ow of client or der s, a given por tfolio of tr ading positions, also known as a tr ading book, has to be (re)hedged. Once the decision to hedge has been taken, the risk manager has some leeway on choosing the actual time of the hedge tr ade(s).The question is: within a predetermined time interval, say a few hours, how long should the risk manager wait to obtain a better price for his hedge trade? The strategy of not waiting will be our benchmark, as it generates, of course, neither gains nor losses. For this timing pr oblem, unlike in pr opr ietar y tr ading, no transaction costs need to be accounted for, as the hedge trade has to be done anyway. The pr actical aim of this paper is thus to ?nd out whether for ecasts based on nonlinear dynamics could help in saving costs thr ough impr oved timing.

313Could nonlinear dynamics contribute to intra-day risk management?

314Darbellay and Finardi Tests have been done with the intra-day series of two foreign exchange rates, namely USD/DEM and USD/JPY,1and three shor t-term (3 months) interest rate futures, for the USD, the DEM and the JPY. The data are tick-by-tick quotes as pr ovided by Reuter s between ear ly 1994 and mid-1995. For cur r encies this means several millions of quotes, for interest rate several hundred thousands. In Section 2 we describe the method of time-delay embedding reconstruction. In Section 3 we outline our analyses of the dimensionality of foreign exchange and interest rates time series. In Section 4 we describe the hedging strategy that we have designed in order to take advantage of the possible existence of chaotic dynamics in the foreign exchange and interest rates markets. Finally, our results are summarized in Section 5.

2.PHASE-SPACE RECONSTRUCTION

Consider a ?nancial time series x(t), where t denotes the discrete time. We will assume that this scalar time series is a realization of a multidimensional (usually nonlinear) dynamical system. In order to understand and predict the evolution of x(t) we will attempt to reconstruct the trajectories of a vector y(t) of which x(t) is a scalar measur ement. Below we give the r ationale for doing so and explain the resulting algorithm.

Differ entiable dynamics, in its discrete for mulation, is the study of deter mi-nistic time evolutions of the form

y(t?1)?f(y(t))(1)

The vector of functions f is assumed to be differentiable. The vector y(t) runs over a multidimensional phase space, in our case R n. The bounded region of the phase space, also known as the state space, where the trajectory of y(t) settles after a transient period is called an attractor. The attractor is known as chaotic or str ange if the tr ajector y has sensitive dependence on the initial conditions (Eckmann and Ruelle, 1985). Chaotic systems look random by strongly amplify-ing the uncertainty about the initial conditions.

Takens (1981) has demonstr ated that in the absence of noise, i.e. exter nal random in?uences, it is possible to reconstruct a multidimensional state space, in our case R E, from a scalar time series. More precisely, the time evolution

t ? x(t)?(x(t), x(t??), x(t?2?),..., x(t?(E?1)?))(2)

in R E is a faithful image of the original time evolution t?y(t) in R n. This is known as time-delay embedding.2A minimal requirement for unfolding the attractor is that the embedding dimension E?D, where D is the dimension of the attractor 1USD is the ISO code for the US Dollar, DEM for the German Mark and JPY for the Japanese Yen 2Time-delay embedding appear s similar to tr aditional state-space models with lagged vector s. Unlike the latter, which are also referred to as spectral methods, time-delay embedding is based on the assumptions that the state of the system can be described by a small number of variables, the inter nal degrees of freedom, and that the evolution is gover ned by some difference equation, or differential equation for continuous systems (Grassberger et al., 1991). Spectral methods are best suited if there are no nonlinear ities and if the number of degrees of freedom is large. Predictive models based on spectral methods and their generalizations can easily suffer from over-?tting.

in R E . Takens has shown that E ?2D ?1 guar antees a pr oper embedding. In general, this relation is less useful than it might seem, because D is not known a priori . Furthermore, Takens’ theorem gives little guidance as to how to choose the embedding coor dinates in R E because, in dynamical systems theor y, the quality of the r econstr uction is invar iant under a change of coor dinates. In practice, however, the presence of noise in the data blurs the trajectories and breaks this invariance.

Filtered embedding has been developed to reduce the effect of noise. An idea is to use a linear ly tr ansfor med ver sion of the lag vector. The singular value decomposition is such a method (Br oomhead and King, 1986; Sauer, 1992). It consists in ?nding out the principal direction of the reconstructed attractor. In order to take into account the nonlinear nature of the data, nonlinear schemes have been intr oduced as well (Fr aser and Swinney, 1986; Far mer and Sidor -owich, 1988; Gr assberger et al ., 1991; Schreiber, 1993). On the other hand, the data should be ?lter ed no mor e than necessar y, other wise some r elevant information might be thrown away. In Section 3 below, we use both ?ltered and un?lter ed data. The ?lter ed data ar e obtained by aver aging over 1 minute inter vals. In section 4, we ar e inter ested in for ecasting pr ice changes, i.e.returns, in which case it makes no sense to use tick-by-tick price data. We thus tr ansfor med the tick-by-tick pr ice data by aver aging over non-over lapping inter vals of length T , and calculating the logarithmic returns

u (t )?log [ˉx (t )/ˉx (t ??)](3)where the bar denotes the averaging over the original time series x (t ) and where ?is a multiple of T . The embedding lag ?was chosen as the ?rst minimum of the mutual information I (u (t ), u (t ??)) (Fraser and Swinney, 1986). Unlike the auto-cor r elation, the mutual infor mation measur es the gener al statistical depend-ence, both linear and nonlinear, between two var iables. The idea of using the ?r st minimum of I as the time lag is to ?nd the closest yet most independent coordinates.

All that remains is to choose the embedding dimension E . A large number of degr ees of fr eedom is not compatible with the appr oach chosen her e and is computationally expensive when looking for nearest neighbours, as will become appar ent below. For the computation of the dimension of the signal we have chosen 4?E ?12, while, for the local pr edictions we concentr ated on the following r ange: 5?E ?15. We wor ked with data ser ies of 10000 to 50000points.

3.LOOKING FOR A STRANGE ATTRACTOR IN FINANCIAL TIME SERIES The numer ical evaluation of fr actal dimensions in exper imental time ser ies is usually per for med after the signal has been r econstr ucted in an embedding space with dimension E (see Section 2). Given a partition of the E -dimensional space in a regular grid of boxes of size e centred at the points b j , the dimension D can be oper ationally de?ned to descr ibe the decr ease of the mass p (e )contained in a box of size e , as e?0, i.e. p (e )?e D .

315Could nonlinear dynamics contribute to intra-day risk management?

Fig. 2.Same as in Fig. 1, for a time series of USD/DEM tick-by-tick data. The parameters in the calculation are the same as in Fig. 1, except for the embedding dimension which ranged from 4 to 12. Notice that the estimated fractal dimension appears to be a monotonic function of the embedding dimension, indicating that the signal is not governed by low-dimensional chaotic dynamics.

the embedding r econstr uction. Each line in the pictur e r epr esents a differ ent choice of the near est neighbour or der (fr om 50 to 200 in multiples of 10). As intensive numerical analyses have indicated (Kostelich and Swinney, 1987), the most precise estimate of the dimension is usually obtained over a ‘plateau’ of embedding dimensions and near est-neighbour s or der s. Hence, the pictur shows that ou best estimate of the dimension of the H′e non map (i.e.

0.06) is obtained with embedding dimensions E ranging from 4 to 9, while

ger values of E the conver gence of the algor ithm is mor e dif?cult, especially for low near est-neighbour s or der s. This effect has been alr eady ibed in the liter ature (Kostelich and Swinney, 1987). Our estimate of the mation dimension of the H′e non map at standar d par ameter values is consistent with the dimension 1.26 commonly epo ted in the lite atu (Grassberger and Procaccia, 1983).

Fig. 3.Same as in Fig. 2, for a time series of USD/DEM foreign exchange rates averaged over one minute intervals. The same parameters have been used in the calculation as in Fig. 2. Also in this case, no evidence of the existence of a strange attractor can be seen.

We have analysed sever al ?nancial time ser ies, namely, the logar ithmic returns of the USD/DEM and the USD/JPY exchange rate and those of one short-term Eurodollar futures contract. We have worked with both tick-by-tick data as well as pr ices aver aged over one minute. All computations have been car r ied out with the following parameters: n?50000 points, m?2000 reference points, embedding dimensions ranging from 4 to 12 and nearest-neighbours orders from 50 to 200. In all cases, we have found qualitatively the same results, namely, no

gence of the calculated infor mation dimensions, i.e. no evidence for chaotic dynamics in the ?nancial markets. For example, in Figs 2 and 3 we show the estimates of the attractor dimension obtained from two different time series: one with USD/DEM tick-by-tick data and one with the mid-prices averaged over one minute. As the embedding dimension is incr eased, the estimates of the

mation dimension of the r econst r ucted signal g r ow fo r all nea r

est-neighbour s or der s and do not conver ge to any ?nite constant value. The sampled points ?ll unifor mly the whole embedding space, indicating that the dynamics of the system does not possess any chaotic featur e. Similar r esults have been found for the USD/JPY foreign exchange data and for the time series of the USD interest rate futures.

In order to make sure that the lack of convergence of the estimates is not due to an insuf?cient number of data points used or to a mistake in our program, we un a simulation with no mally dist ibuted noise, using the same ameter s as in the pr evious calculations. The r esults ar e shown in Fig. 4.Notice that the picture is similar to the previous two. Thus, it appears that from

Fig. 4.Same as in Figs 2 and 3, for a time series drawn from a normal distribution.The same parameters have been used in the calculation as in Fig. 2. Notice, that,as in Figs 2 and 3, the estimates diverge as the embedding dimension is increased, indicating that the signal has been generated by a noisy dynamics.

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