Why study financial time series?

Why study financial time series? M . D . London, A . K . Evans and M.J. Turner Institute of Simulation Sciences, S E R C , Hawthorn Building, De Montfort University, Leicester L E I 9BH

Abstract This article gives an introduction to quantitative finance: why financial time series are interesting and what we stand to gain by studying them. The first part motivates and describes the role that complex systems, selforganised criticality and universality might play in unravelling the mystery surrounding the stylised and non-trivial empirical regularities observed in markets. The second part of the paper then describes each of these empir­ ical regularities in turn with some mathematical background and popular statistical models.

1

Stock prices a n d s t o c h a s t i c t i m e series

T h e erratic n a t u r e of financial processes such as stock market and exchange r a t e prices is an inescapable feature of modern life. Prices are now quoted on the hourly news a n d affect us all in some way. Figure 1 shows a graph of an American stock market index, the New York Average (NYA) with daily observations from 1980-89. One event clearly stands out from t h e rest: t h e crash of 1987 t h a t left many economists a n d investors puzzled and dejected [8, 75]. A mathematician might describe t h e time series in figure 1 as stochastic or random; another word for erratic. In recent decades stochastic time series (or signals) and therefore stochastic modelling have become a n increasingly impor­ t a n t p a r t of t h e mathematical sciences due t o t h e number of naturally occurring systems in which stochastic signals are inherent. Examples of such systems in­ clude traffic, neural activity, heart r a t e variability, diffusion, earthquakes, forest fires, speciation a n d extinction, light emission from quasars a n d countless more (see for example [5, 79, 69, 12, 14]). In economies t h e behaviour of stock prices, including apparent anomalies like market crashes, emerges as t h e result of a lot of complicated interactions. Inter­ actions between companies, financial agents, governments a n d j u s t chance [80]. Emergence a n d complexity are words, often used synonymously, t h a t describe a new concept of systems whose behaviour cannot be understood in a reductionist manner [5]. 'Complex systems' are systems with many (often identical) interact­ ing parts, in which simple rules (which specify how t h e system evolves) lead t o complicated and unpredictable behaviour. T h u s complexity, like chaos ([69] a n d 68

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NYA 1980-89

3000

time in days

Figure 1. The New York Average from 1980-89, showing the crash of late 1987 at around day 2000. defined formally below), has a m a t h e m a t i c a l definition t h a t is not t h e same as its English language definition a n d which is often twofold. It essentially describes the variation within a structure formed by a complex system or t h e degree t o which such variation is present. We might intuitively associate this with t h e entropy of the system b u t the two concepts are not t h e same. Entropy, which describes the a m o u n t of disorder (or randomness) in t h e system, is not max­ imised in systems with high complexity. Complexity is a more interesting kind of variation a n d one t h a t is difficult t o quantify. For example t h e landscape of t h e E a r t h has a high degree of complexity, b u t its entropy is far from m a x i m u m . In fact, the e a r t h would look very boring if its entropy were m a x i m u m . Understanding these microscopic interactions in complex systems t u r n s out t o be quite difficult. T h e most i m p o r t a n t reason for this is t h a t t h e main way in which we test our models is t o compare synthesised d a t a with macroscopic empirical d a t a : d a t a in which all the i m p o r t a n t microscopic workings have al­ ready taken place and been realised in such a way t h a t we cannot easily work backwards to see what they were. In finance we are often left guessing a t what series of events could have led t o t h a t bubble or crash. Seismologists are often left wondering what series of small tectonic movements could have initiated t h a t large earthquake, and motorists are often frequently bemused by t h a t frustrating phenomenon of the traffic suddenly coming t o a violent halt only t o clear again straight away revealing a clear road.

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After the event, it is feasible t h a t one could work out the causes for such rare and extreme events. T h e point is, t h a t no one could have predicted t h e m before­ hand. In this way, financial processes are similar t o other naturally occurring phenomenon such as earthquakes a n d some e a r t h q u a k e modelling [5] has proven t o be useful in understanding m a n y complex processes. There are two main ways t o approach t h e modelling process: the first is t h e imitation approach. T h a t is, t o study t h e statistics of t h e real d a t a a n d t r y t o imitate t h e d a t a within some m a t h e m a t i c a l framework; t o u n d e r s t a n d the process by being able t o re-create it statistically. Analogies may be drawn between t h e constituent p a r t s of t h e financial process a n d t h e m a t h e m a t i c a l mechanisms t h a t synthetically mimic t h e process a n d this may lead t o some new insight. Sometimes a model may b e justified by means of some other argument based on assumptions a b o u t real markets as is the case with t h e Efficient Market Hypothesis (EMH, which will b e described later). T h e alternative is t o try t o formulate a financial market microcosm in which t h e desired behaviour 'emerges' as it does in real life. To achieve this, appro­ priately simplified rules must b e chosen t o govern the evolution of t h e system. This approach offers a n a t u r a l framework for interpretation, b u t there are two obstacles t o overcome: firstly, rules must be selected t h a t give you all of the right behaviour (reproduce t h e statistics) a n d secondly, t h e rules must be simple enough t o make mathematical analysis of t h e model possible. It is often surpris­ ingly difficult t o analytically show why t h e simplest of models behave the way they do. Some might argue t h a t a further difficulty in finance is t h a t agents of t h e economies differ greatly in ability, importance and influence. Or t h a t psychol­ ogy adds a layer of complexity t o modelling financial processes t h a t is impossible to overcome because people in general do not behave in a consistent manner. Or perhaps t h a t rare events like t h e d e a t h of Princess Diana or t h e recent terrorism in America cannot be foreseen a n d yet have a n i m p o r t a n t impact on m a n y finan­ cial processes. We argue here t h a t t h e magnitudes of such events are p a r t of a continuum, ranging from t h e everyday least significant events t o t h e rarest a n d most devastating. W i t h their great variety a n d number, these events m a y be governed by some universal laws. We argue t h a t if statistical regularity exists, it may be so because t h e large events are related t o t h e small events by some n a t u r a l mechanism. This is a big claim and one t h a t will need t o be defended carefully. We begin by building a diverse coalition of examples of systems in which similar claims have been justified. Modern financial research is linked t o work in a diverse range of disciplines including seismology, psychology, biological physics and electronics t o list b u t a few. Many useful metaphors are becoming available a n d hence a b e t t e r un­ derstanding of the general principles t h a t underly such processes; for example chaos. This brings us t o t h e concept of universality.

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Universality

Separating the general from the particular is a big p a r t of what science is a b o u t [29]. A universality describes a situation in which surprisingly many particulars can be ignored and a general result still prevails. More precisely, a universality class is a class of systems whose macroscopic behaviours are qualitatively t h e same although the details of their construction may differ considerably. This has applications in chaos theory [69], statistical physics [12] a n d other fields. For example, a system in which large samples of independent r a n d o m variates are taken from any distribution and added together is governed by t h e central limit theorem (CLT) (discussed again later) a n d t h e resulting behaviour of t h e system can be described by the class of L-stable or Levy distributions of which the Gaussian distribution is a particular case. This is one of t h e most celebrated universality classes. Many n a t u r a l phenomenon follow this distribution: t h e heights of h u m a n s , t h e error after t h e best model has been fit t o a d a t a set, diffusion in disordered media [12]: any system in which there is a n element of chance potentially obeys this law. Another example is t h e Fibonacci sequence and t h e golden ratio [43]. T h e sequence begins by taking two numbers a\ and a a n d creating t h e rest of t h e se­ quence using t h e rule, a \ = a + a„-\. For example: a={l,1,2,3,5,8,13,21...}. It can easily b e shown t h a t the ratio, α „ + ι / α „ , tends t o , (1 + \ / 5 ) / 2 « 1.618 regardless of t h e choice of a\ a n d 0 2 . Much of n a t u r e ' s geometry is governed by these so called golden ratios [20]. For example, shell spirals, t h e branching of plants, t h e number of petals on different types of flowers, t h e p a t t e r n s of seed heads a n d pine cones, the arrangement of leaves a n d m a n y ratios of measure­ ments of the h u m a n body [43]. There is evidence t o suggest t h a t building things with these ratios in mind is a very efficient way of doing things [23]. Further­ more, this fact is now being reflected by h u m a n t a s t e , from a r t t o music t o t h e h u m a n face, it appears t h a t we find geometries t h a t obey this ratio pleasing t o the eye. Some work (including [71] and a n 'art-house' film called ' π ' ) has argued t h a t t h e golden ratio a p p e a r s in financial time series t h o u g h this is very difficult t o verify a n d not generally accepted. 2

n +

n

Chaos theory, another example, is a great achievement in t h e u n d e r s t a n d i n g of the mathematical principles a t work in our universe. It shows us how un­ der certain conditions, simple systems can behave unpredictably. A system, / , defined by, xt+i = f{xt)), for some nonlinear function / , is chaotic when ini­ tially close orbits diverge exponentially, \ft(xo) — ftfeo + f)| ~ exp(Ai), or more precisely t h a t the average value of this measure λ is positive [7, 69], (2.1) assuming t h a t / ' exists. Although t h e r e are m a n y types of system t h a t exhibit chaos, if we know t h a t they are chaotic then we can expect certain regularities, particularly concerning the route t o chaos for example, period doubling and win-

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dows of order governed by universal ratios [69]. We have a good understanding of how chaos works and can t h u s now explain why it is difficult t o predict so many n a t u r a l systems like weather [55, 76], population dynamics [63], fluid mechanics [49] a n d healthy heart r a t e variability [47]. Fractal geometry [58] is another example of a ubiquitous phenomenon which can be seen and embraced given a change of thinking. Often the origin of fractals is linked with chaos, since chaotic orbits may create fractal objects in some phase space [7]. However, many systems and mechanisms t h a t produce fractals, such as Diffusion Limited Aggregation (DLA) [21] a n d non-chaotic complex systems [5, 14], are known not t o b e chaotic indicating t h a t we do not have t h e complete picture as yet. One promising new mechanism known as criticality [5] might be one of the missing pieces of the puzzle. Criticality is currently less well understood. It reveals itself t h r o u g h power law probability distributions with infinite second moments, fractal noise a n d Zipf's law [5]. A simple example recently quoted is known as ' p o t a t o logic' [5,14]. If one were t o throw frozen p o t a t o e s a t a wall repeatedly and t h e n carefully weigh all of t h e fragments on the surrounding floor, then it would soon become clear t h a t t h e size and frequency of the fragments are related by t h e simple power law, / oc l / s i z e ° , a = 2 (actual experiments carried out by [67]). T h e r e is no obvious or intuitive reason t h a t this should be so. W h a t ' s more, in [67] the experiment was repeated with chunks of gypsum, soap a n d other substances, t h e results were also power laws. Similarly earthquakes, traffic j a m s , forest fires and many more n a t u r a l systems obey this same power law with exponents 1 < a < 3 [5, 14]. This is one of t h e new hot topics in m a t h e m a t i c s . T h e above examples illustrate t h a t understanding universality is a b o u t under­ standing certain mathematical mechanisms. For instance stability under some mathematical operation (like addition, or convolution), or the n a t u r e of some condition like equation (2.1). Understanding such mechanisms will explain why certain details t h a t don't affect the universality class t h a t the system falls into. For example, in the case of t h e (Gaussian) CLT, one can see the u n i m p o r t a n c e of the moments of order greater t h a n two by looking a t t h e limiting form of t h e characteristic function (Fourier transform of t h e P D F ) , (see appendix 13.) T h e i m p o r t a n t thing a b o u t universality is t h a t all we have t o do t o u n d e r s t a n d a whole (universality) class of processes is t o invent t h e simplest 'toy' model which contains the relevant basic features necessary t o belong t o the class. We can t h e n use this simplest model t o u n d e r s t a n d and make inference a b o u t t h e whole class of processes. Universality is relevant in financial modelling for two reasons. Firstly different markets trading different financial instruments in different countries have certain dynamical features in common, a n d they are not trivial (examples of universality classes already understood). These features will be discussed in t h e latter half of this paper. Secondly, other n a t u r a l a n d social systems exhibit similar dynamical behaviour as well. This t e m p t s us t o consider t h e possibility t h a t some very general principles are a t work. T h e exposition of such a universality class t o describe financial processes is t h e holy grail of financial research. Before we d o

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so, we go back one hundred years t o t h e first a t t e m p t s t o justify models of price processes.

3

A s t a r t i n g point: t h e r a n d o m walk

In 1900 a French mathematician by t h e n a m e of Louis Bachelier submitted a P h D thesis titled, ' T h e Theory of Speculation' in which he proposed a r a n d o m walk model for asset prices [2]. He concluded t h a t t h e price of a commodity today is t h e best estimate of its price in t h e future. T h e r a n d o m behaviour of commodity prices was again noted by Working [81] in a n analysis of time series d a t a . Kendall [41] a t t e m p t e d to find periodic cycles in indices of security a n d commodity prices. He did not find any. Prices appeared to be yesterday's price plus some r a n d o m change. He suggested t h a t price changes were independent and t h a t prices followed random walks. T h e majority of early financial research is reported [80] t o have been in agreement: asset price changes a r e r a n d o m and independent, so prices follow random walks. T h e r a n d o m walk model is often referred t o as Wiener Brownian motion ( W B M ) n a m e d after t h e Mathematician Norbet Wiener and t h e Scottish botanist R o b e r t Brown. Brown [13] noticed t h e erratic motion of a small particle suspended in a fluid a n d Wiener [62] later provided a rigorous mathematical framework with which t o work with what we now know as stochastic processes. Bachelier also noticed t h a t t h e size of price movements was proportional t o t h e price, thus his model proposed t h a t t h e log of price changes should b e Gaussian distributed. This behaviour is often described mathematically by a model of t h e form [80], JO

— = σάΧ + μάί

(3.1)

where S is t h e price a t time t, μ is a drift t e r m which reflects t h e average r a t e of growth of t h e asset, σ is t h e volatility a n d dX is a sample from a normal distribution. There are problems with this notation and a b e t t e r notation [33] is, ^

= aN(t){dtf'

2

+ μάί

(3.2)

In other words t h e relative price change of an asset is equal t o some r a n d o m element plus some underlying trend component. More precisely this model is a log-normal r a n d o m walk. T h e Brownian motion model has t h e following impor­ t a n t properties: 1. Statistical stationarity of price increments. Samples of Brownian motion taken over equal time increments have identical m o m e n t s . 2. Scaling of price. Samples of Brownian motion corresponding t o different time increments c a n be re-scaled using a simple power laws such t h a t they too have t h e same descriptive statistics (moments). For example, denoting

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B(t) t o be a Brownian process, (\B(t

+ NT) - B(t)\« > ~ A(

\ (t B

+ r) - B(t)\" )

(3.3)

3. Independence of price changes taken between non-over lapping intervals. 3.1

Efficient m a r k e t s ?

It is common knowledge amongst traders and anyone with an interest in the market t h a t the distribution of price changes has fatter tails t h a n t h e Gaussian and t h a t volatility exhibits clustering (see [26] for a good review). One only has to look a t t h e raw d a t a t o see this (Figure 1). So why did people think t h a t prices should follow random walks? It is often stated t h a t asset prices should follow random walks because of the Efficient Market Hypothesis ( E M H ) . T h e E M H states t h a t the current price of an asset fully reflects all available information relevant to it and t h a t new information is immediately incorporated into t h e price. T h u s in a n efficient market, t h e modelling of asset prices is really a b o u t modelling t h e arrival of new information. New information must be independent and random, otherwise it would have been anticipated and would not b e new. In this way, asset prices are a Markov process [33]. T h e E M H implies independent price increments but why should they be Gaussian distributed? Perhaps t h e Gaussian P D F is chosen because macroscopic price movements are presumed t o be an aggregation of smaller (microscopic) ones, and this aggregation is governed by t h e Central Limit Theorem [12]. T h e E M H assumes t h a t there is a rational and unique way t o use the available information and t h a t all agents possess this knowledge. Moreover, t h e E M H (very ambitiously) assumes t h a t this chain reaction happens instantaneously. In an efficient market, only t h e revelation of some d r a m a t i c information can cause a n extreme event, yet post-mortem analyses of crashes typically fail t o (convincingly) tell us what this information must have been [48]. 3.2

Price response to information.

T h e E M H assumes a simple relationship between t h e reaction of t h e m a r k e t t o information a n d t h e importance of t h a t information. Trivial information should cause trivial price adjustments a n d cannot be responsible for crashes, which can only be instigated by very i m p o r t a n t information. In reality however, avalanches (bubbles a n d crashes) are known t o occur for n o good reason. Panic selling (or buying) motivated by h u m a n fear, greed a n d imitation are also familiar. T h e r e is an obvious feedback mechanism a t work in financial systems: agents' trading decisions regarding a given financial instrument are based on (amongst other things) t h e price of t h a t financial instrument. And it is t h e agents' collective actions t h a t form its new price. This means t h a t t h e response of t h e price t o

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information is more complicated: agents respond t o information a n d t h e n t o each others responses and so on. To gain insight into t h e way information is used in markets, it is useful t o know something a b o u t t h e trading mechanisms a n d tools available t o agents of the economy and how they are used t o m a n i p u l a t e risk. T h e following is a brief description of how risk, information, agents and trading contracts form economies.

4

S t r a t e g y : microscopic m a r k e t a c t i v i t y

Years ago Portfolio theory focused on rates of r e t u r n with t h e occasional caveat, 'subject t o risk'. Modern Portfolio theory assumes t h a t there must b e a t r a d e off between risk and return. A simple argument outline in [80] illustrates why this should be so. We must first assume the existence of a (virtually) risk free investment, for example, depositing money in a reputable bank. Suppose now t h a t we could b e a t t h e bank by investing in equities without taking any e x t r a risk. Clearly no one would invest any money in t h e bank. W h a t would t h e bank d o a b o u t t h a t ? It would have t o raise its interest rates, t o a t t r a c t more investment in t h e bank, t o make more money from its own lending t o pay t h e interest on t h e accounts in credit and t o make it more difficult for a person t o make money by borrowing from t h e bank and investing it in equities. (There are transaction costs and differences between buying and selling prices b u t these do not really affect this argument.) T h e market is full of arbitrageurs whose j o b it is t o seek out irregularities and mispricings and profit from t h e m . T h e concept of arbitrage is i m p o r t a n t in finance [80]. Often stated as, 'there's no such thing as a free lunch', it means t h a t opportunities t o make a risk free profit cannot exist for very long before prices move t o eliminate t h e m . T h u s investing in non-safe investments is a b o u t speculation. As markets have grown and evolved, new trading contracts have emerged called derivatives [80], which use various clever tricks t o m a n i p u l a t e risk. T h e value of a derivative is derived from, b u t not t h e same as, some underlying asset or price index. These special deals really j u s t increase t h e n u m b e r of moves financial agents have at their disposal t o ensure t h a t t h e b e t t e r players win. Anyone t h a t has played a g a m e like noughts and crosses or d r a u g h t s a large number of times will soon reach t h e stage where each game is t h e same a n d a stalemate. If you think you are intellectually superior a n d you want t o take advantage of it then you should suggest a game of chess. T h e r e are m a n y different kinds of derivative t r a d e d on t h e world's markets today [80]. To illustrate t h e basic principles we introduce t h e simplest a n d most common one: the option. An option is t h e right (but not t h e obligation) t o buy (a call) or sell (a p u t ) a financial instrument a t a prescribed time in t h e future (the expiry date) a t a given price, known as t h e strike price, or exercise price. Options may be used as a more pure means of speculating. If the expiry d a t e

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arrives a n d the price has not gone t h e way you speculated t h e n you simply do not exercise t h e option. All you have lost is t h e initial premium t h a t you paid for the option; you are only paying for t h e right t o speculate. If volatility is high then the potential gains are unbounded b u t t h e loss is always limited t o t h e initial premium paid. T h u s , t h e value of a n option is a function of volatility, t h e strike price, a n d t h e time left till expiry. Options may also be used as a means of hedging, t h a t is as an insurance policy. Suppose an investor owns shares in a particular company. If he thinks there is a chance t h a t the share price may take a dive, but he doesn't want t o risk liquidating his stock, t h e n he may wish t o buy some put options in t h e same company. If t h e price goes down, he has t h e right t o sell his stock at a price (hopefully) higher t h a n the current price. He can then buy it straight back generating a profit and his long t e r m investment resumes. If t h e price does not take a dive then t h e investor simply carries on with his long t e r m investment losing t h e premium paid on t h e options b u t profiting from having left his long term investment going. A sensible question t o ask a t this point is: how much should one pay for an option? Before the introduction of t h e Black-Scholes (BS) formula [9] in 1973, options were valued quite subjectively. T h e Black-Scholes formula, which is an algebraic equation (derived from a partial differential equation), assumes t h a t the price of t h e share follows a log-normal random walk a n d works by taking advantage of t h e fact t h a t t h e value of t h e option and t h e value of the under­ lying asset are correlated. For further reading on t h e m a t h e m a t i c s of financial derivatives and a derivation of t h e Black-Scholes equation see [53, 80]. It would seem t h a t t h e way t o use t h e BS formula is t o estimate t h e p a r a m ­ eters, the interest rate, t h e exercise price of option, time till expiry, t h e price of the underlying a n d the volatility, substitute t h e m into the formula and then estimate the value of t h e derivative product. It t u r n s out t h a t this is no longer the most common use of option models. This is partly due t o t h e fact t h a t , in practice, it is difficult t o measure t h e volatility of the underlying asset. How­ ever, despite these difficulties, option prices are still quoted in t h e market. This suggests t h a t , even if we do not know t h e volatility, t h e market does. So, hav­ ing substituted into t h e formula, t h e interest rate, t h e exercise price, t h e time till expiry and t h e price of t h e underlying (all of which are easy t o measure), all t h a t remains is the derivative price a n d t h e volatility. Since (according t o t h e model) there is a one-to-one correspondence between t h e volatility and t h e option price, we could equally well t a k e t h e option price quoted on t h e market and substitute t h a t in instead t o estimate t h e market's opinion of t h e volatility over the remaining life of t h e option. This estimation of t h e volatility from t h e BS formula a n d t h e price of t h e option is called t h e implied volatility [80, 18]. A common empirical feature of t h e implied volatility is t h a t it isn't constant across the exercise prices [80, 18]. T h a t is, if t h e other p a r a m e t e r s of t h e model are fixed, t h e n t h e prices of options across the exercise prices should reflect a uniform value for t h e volatility. T h e volatility of options with exercise prices far from the current underlying price (so called in-the-money options) is often

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greater t h a n those with exercise price close t o t h e current price (at-the-money). This is known as the volatility 'smile', there are other less common possibilities as well such as t h e volatility 'frown' [80, 18].

5

Adaptive markets

T h e BS model has many such shortcomings, including t h e major failure of as­ sumption involving the distribution for t h e price increments of t h e asset. These shortcomings will b e discussed in detail in t h e second half of this paper. Jessica J a m e s [37] (head of research in t h e strategic risk m a n a g e m e n t advisory group a t t h e first bank of Chicago, London) explains t h a t t h e Black-Scholes formula has not lost popularity since its shortcomings (assumption of log-normal distribu­ tion) became a p p a r e n t , because of t h e way in which it is used. T h e Black-Scholes volatility a n d the price of an option are now so closely linked in t h e market t h a t a n option is usually quoted in option volatilities or 'vols' (which are displayed on traders screens all across t h e world). Traders know t h a t market distributions are not log-normal, so they adjust t h e option price t o take account of this. Traders make this adjustment by allowing t h e estimated volatility, which determines t h e price of t h e option, t o vary with t h e options strike price as described above. This variation can have a n u m b e r of characteristic shapes. This crucial piece of flexibility removes a t a stroke, most of the limitations of t h e model, a n d allows options t o b e valued based on a variety of estimated future price distributions. Common variations are t h e 'vol smile', 'frown' a n d 'smirk'. A paper by P o t t e r s , Cont and Bouchaud [72] verifies, by studying t h e prices of options on liquid markets t h a t t h e market has empirically corrected t h e BlackScholes formula t o account for the 'fat tails' a n d correlations in t h e scale of fluctuations. P o t t e r s et al suggest a replacement of Gaussian price increments with increments t h a t are t h e product of two r a n d o m variables, one of which contains time correlations. This highlights one reason why t h e economy is so fascinating. It is greater t h a n t h e sum of its p a r t s . It is of course driven by h u m a n beings buying and selling assets a n d other financial instruments, b u t these people do not decide and cannot comprehend the affect they are collectively going t o have on t h e econ­ omy (they wish they could). T h e observed universal characteristics of m a r k e t s t h a t emerge through h u m a n activity do not seem understandable t h r o u g h t h e reductionist approach of observing the actions of individuals.

6

Empirical regularities

T h e remainder of this paper is devoted purely t o exposing facts a n d does not focus on any theory of how those facts might have come a b o u t . Its purpose is t o establish empirical regularities, a n d suggest known m a t h e m a t i c a l distributions and mechanisms t h a t are consistent with observation. We will use several real life d a t a sets [1] t o make inferences a b o u t financial time series in general. These

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Price Series Dow Jones Industrial Average Financial Times Industrial Index New York Average New Zealand forty share Japanese topix French F r a n c / U S $ Belgian F r a n c / U S $ Japanese Yen/US$ Netherlands Guilder/US$

financial

t i m e series?

Acronym DJIA14 FTIID NYA NZ40D TOPXD FRF BEG JPY NLG

Period 1987-00 1933-00 1960-89 1970-00 1950-00 1971-00

Frequency 14 o b s / d a y Daily Daily Daily Daily Daily

Size 50000 17936 8339 7895 13948 7408

» Η

Table 1. Details of the data sets used throughout this thesis.

d a t a sets are described in Table 1 a n d will often be referred t o thereafter by their acronyms.

7

De-trending the data

T h e raw price d a t a t h a t we analyse contains a trend component. A graph of any asset price time series over a long time looks a bit like y = e . It always s t a r t s in t h e b o t t o m left corner of t h e graph and ends u p in the t o p right and grows exponentially, like the t o p left graph in figure 2. We are not interested in studying this feature of the d a t a . Exchange r a t e time series do not display such underlying trends b u t t h e following de-trending process proposed for t h e asset price series can safely b e applied t o exchange r a t e d a t a sets without undesirable side affects. T h e way in which the trend is removed is as follows. x

Let p(t) denote the price a t time t a n d let xr(t) denote the de-trended log price increment over a sampling interval Τ measured in days, x {t) T

= logp(r) - (a + blogp(t

- T)) « logp(t)

- log(t - Γ ) ,

(7.1)

where a and b are constants t h a t describe the line of auto-regression. Formally, xr(t) are t h e Auto-Regressive 1 (AR1) residuals [70] of t h e log of t h e price, though in practice for daily d a t a there is not much difference between this a n d t h e log increment of t h e price. Taking t h e A R 1 residuals is m o r e i m p o r t a n t when analysing higher frequency d a t a in which there is more serial correlation. Figure 3 shows the de-trended log price increments of t h e NYA from 1960-89 and Gaussian r a n d o m numbers. B o t h sets have been normalised for comparison (and shifted vertically so we can see t h e m separately). Before any analysis is carried out, it is clear t h a t r a r e events h a p p e n m o r e often t h a n t h e y would if t h e d a t a were Gaussian and t h a t t h e large a n d small events seem t o cluster together. This is t h e first clue t h a t a simple r a n d o m walk will not b e sufficient t o explain

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79 Auto-regression

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F i g u r e 2. Illustration of the detrending process. Data used in this example is the NYA. p(t) is the price at time t time steps in days, x(t) is the de-trended increments and X(t) = t'-

Σ

x

all features of financial time series. In t h e following sections we scrutinize t h e d a t a and catalogue t h e empirical facts.

8

Probability Density Function ( P D F )

For processes t h a t follow a log-normal r a n d o m walk, increments xr(t) will b e Gaussian distributed on all scales. Figure 4 shows the probability density func­ tion ( P D F ) on t h r e e time scales: daily, weekly a n d monthly, for t h e Dow Jones In­ dustrial Average (DJIA) index a n d the Financial Times Industrial Index ( F T I I D ) . T h e figures also show the Gaussian best fit as t h e dashed curve. T h e distribu­ tions have a sharply peaked mean and 'fat tails' as compared t o t h e Gaussian. A distribution t h a t has these characteristics is mathematically referred t o as be­ ing leptokurtic a n d t h e extent t o which this condition occurs is quantified by a measure called the kurtosis defined as,

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Figure 3. De-trended New York Average daily price increments (top) versus the Random walk hypothesis (bottom). DJIA TOPXD BEF

22 182 5

NYA NZ40D NLG

50 47 5

FTIID FRF JPY

10 7 13

Table 2. Kurtosis for the daily increments of the datasets.

_ ((*-μ) ) 4

J r

/4

2

,

(8.1) '

where μ and μ' are the expected value of χ and x respectively and (.) denotes the averaging operator. This measure κ is defined such t h a t it is zero for t h e Gaussian. For a n independent and identically distributed (IID) process t h e kurtosis should scale according to, κ(Τ) ~ 1 / Γ [77], where Τ again, is t h e time scale measured in days. A study [17] has suggested t h a t t h e kurtosis scales much more slowly a n d according t o a power law, κ ~ T~& with β « 0.5. Figure 5 shows the kurtosis of xr(t) against the sampling increment size, T. T h e dashed curves represent t h e power law best fits and β (inset) is t h e scaling exponent for the power law. T h e curves fit t h e d a t a b u t not always incredibly well. T h e 2

2

F i g u r e 4. PDF for the de-trended Dow Jones Industrial Average (top) and the Finan­ cial Times Industrial Index (bottom) with Gaussian best fit for comparison (dashed). The horizontal axis measures the log price increments and the vertical axis is the esti­ mated probability.

exponents axe β w 0.3 for the intra-daily DJIA14, in the range 0.5 < β < 0.6 for the daily stock market indices a n d in t h e range 0.9 < β < 1 for t h e exchange rates. T h e kurtosis of the financial d a t a does decay more slowly t h a n a n IID process, and the extent to which this h a p p e n s is much greater for t h e stock markets t h a n the exchange rates, particularly for the higher frequency DJIA14. T h e significance of a positive kurtosis is t h a t the likelihood of rare events is much greater t h a n t h e log-normal model predicts. This observation a p p e a r s t o b e a t odds with the Efficient Market Hypothesis (EMH) because it implies t h a t t h e individual contributions t o a price increment (the affect of t h e actions instigated by the arrival of new information mentioned above in t h e E M H definition) are not random. T h e r e is however another possibility, t h a t they are r a n d o m b u t do not have finite variance.

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DJIA14

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Figure 5. Kurtosis (vertical axis) of xr(t) against sampling interval size Τ (horizontal axis) for eight financial time series (see individual captions for which).

M . D . L o n d o n et al. 8.1

83

S t a b l e (fractal or L e v y ) d i s t r i b u t i o n s

There is another class of stable distributions (distributions for which t h e sum of two Independently a n d Identically Distributed (IID) r a n d o m variables is identi­ cally distributed) which do not necessarily have finite variances a n d or means. These are commonly known as Levy distributions [51, 78], named after P a u l Levy. It t u r n s out t h a t these distributions d o have fatter tails a n d sharper peaks at the mean t h a n the Gaussian. T h e distribution is normally specified t h r o u g h its characteristic function [77] (Fourier transform of the P D F ) ,
0 < μ < 2.

(8.2)

This is because except for a few values of μ there is no closed form for t h e P D F (the inverse Fourier transform of $(&))· W h e n μ = 0 t h e P D F is a d e l t a function, μ = 1 gives the Cauchy distribution and μ — 2 is t h e Gaussian [77]. T h e r e are also closed form expressions for μ = 1/2 and μ = 1/3 [12]. It can b e shown [52] t h a t this distribution has t h e property t h a t its tails follow t h e asymptotic power law, 1 which is relevant from t h e point of view of complexity a n d criticality theory. Stable distributions enjoy t h e self-affinity property, P

^ )

= ^ P { ^

I

) ,

(8-3)

verifying t h a t such processes are fractal, a property which will b e discussed again and again on progressively more detail t h r o u g h o u t this chapter. Are t h e increments, x(t) Levy distributed r a n d o m variables? Figure 6 shows a Levy distribution fit t o t h e same P D F shown in Figure 4, of t h e DJIA14 d a t a set. T h e p a r a m e t e r μ was ' h a n d picked' t o make t h e fit near t h e mean good. And from the graph, the shape of t h e Levy distribution is similar t o the empirical distribution around t h e mean. On closer inspection of t h e tails however a discrepancy reveals itself. They are too fat: they decay more slowly t h a n those of the empirical distribution. Regression a n d iterative m e t h o d s normally used t o estimate t h e p a r a m e t e r s of the stable distribution such as t h e Mc-Culloch [64] m e t h o d and t h e Koutrouvelis [45, 46] are numerically unstable giving non-sensible answers (like infinity or not-a-number) which confirms t h a t t h e overall shape of t h e distribution is not consistent with a stable distribution even though t h e centre a n d t h e tails may (separately and inconsistently) display stable distribution a t t r i b u t e s . Some studies [44, 34, 61] have proposed the use of a t r u n c a t e d Levy flight as a model for t h e P D F of market returns. T h e p a r a m e t e r μ can t h e n b e chosen t o fit t h e centre of t h e distribution and the multiplicative truncation factor means t h a t they decay more like the empirical P D F , according t o a n equation of t h e form (8.4).

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Figure β. PDF: L^vy distributions and FTIID stock market index. 8.2

Leading order behaviour of tails

This leads us t o the issue of the scaling (decaying) behaviour of t h e tails of the empirical P D F . Some research [11] report a power law, p(x) ~ x~&, a n d others t h a t the tails decay like those of t h e log-normal distribution ([2] a n d references in [19]). This issue is linked t o t h e m a t t e r of whether t h e second moment of the empirical distribution is finite. In general, if t h e distribution is continuous a n d its tails decay faster t h a n 1 / x then t h e process's second moment exists [12, 77]. Figure 7 shows the log magnitude (of price increment) against log frequency. T h e curves are nearly straight and therefore t h e tails are close t o following a power law though the t r u e r a t e of decay is very slightly faster. T h e best fit lines have slopes, β, ranging from j u s t under three t o j u s t over four a n d there is n o noticeable difference between t h e stock m a r k e t s a n d t h e exchange rates. T h e r a t e at which t h e tails of the P D F decay determines whether or not t h e m o m e n t s of t h e distribution exist. A study [17] and others have proposed t h a t t h e empirical distribution is characterised by a t r u n c a t e d LeVy distribution whose tails would decay as, 3

p(x) ~ x~° exp

(8.4)

Another way of illustrating t h e likelihood of t h e iVth moment converging is t o plot this moment for progressively larger sized samples of daily increments a n d

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Figure 7. The tails of the PDF: log(size) against log(freq) for eight financial time series, (see individual titles.)

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track its route t o convergence graphically. Figure 8 shows this performed for t h e first and second (top and b o t t o m ) moments of the NYA daily index. It is clear t h a t the first moment converges whereas t h e second moment has apparent dis­ continuities. T h e frequency and magnitude of these 'jumps' determines whether the second moment converges. In t h e case of the NYA d a t a set it is difficult t o say whether the second moment may exist. If it does it certainly converges very slowly. For a distribution with power law tails and an exponent less t h a n three, these discontinuities would carry on appearing for ever a n d d o m i n a t e t h e behaviour of the index.

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Figure 8. The first and second moments (top and bottom) of the NYA plotted sequentially.

8.2.1

Finite or infinite moments: Fractal time series

Fractals are objects with non-integer dimension a n d self-affinity. (See [69] for a more thorough introduction t o fractals.) Time series are fractal if their statistics scale according t o a power law; they are statistically similar over different time scales. From figure 4 we see t h a t the d a t a has a similar distribution over different time scales. And from figures 7 a n d 8 t h a t it is difficult t o determine whether the second moment of t h e empirical distribution converges. It is generally agreed in t h e quantitative finance community t h a t t h e second moment does converge albeit very slowly [17, 11]. A time series characterised by a P D F with an infinite

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(divergent) second moment has n o characteristic scale a n d this is how it (perhaps trivially) inherits its fractal s t a t u s . [70, 15, 16]. I t ' s statistical regularity can only b e expressed by means of a fractal dimension. T h e fractal dimension describes t w o things, 1. How densely t h e object fills its space, 2. T h e structure of the object as the scale changes. For physical (or geometric) fractals, this scaling law takes place in space. A fractal time series scales statistically, in time. T h e statistic usually used t o describe fractal time series is t h e box counting fractal dimension DB [69], which is a measure of how densely t h e fractal fills its image space. It works by covering t h e signal with t h e (as m a n y as) necessary Ν boxes of size S , as S decreases t h e number of boxes required t o cover t h e signal will increase. If t h e time series is fractal then t h e increase will behave as a power law,

Ν

S~

DB

oc

(8.5)

T h e importance of establishing fractal n a t u r e of any time series is t h a t fractals require a degree of organisation. T h e y m a y have local randomness, b u t always global statistical determinism revealed by power law scaling properties. A p a r t from t h e trivial case where t h e process is r a n d o m a n d governed by t h e stable laws mentioned above, this places some restrictions o n how t h e underlying process operates, a n d thus helps us t o develop models. Since we have verified t h a t financial time series a r e n o t examples of I I D stable processes (organised by t h e CLT), their fractal n a t u r e points t o some other mechanism for their organisation. We talk a b o u t t h e scaling properties of fractal a n d financial time series in more detail in sections 10 a n d 11 b u t before this we must investigate another basic related property t h a t h a s thus far n o t been considered.

9

Correlation

T h e second downfall of the r a n d o m walk model is t h a t price increments a r e n o t independent. T h e Auto-Correlation Function (ACF), A(t), which c a n b e defined as t h e inverse Fourier transform of t h e power spectrum [33], is used t o measure the structure of t h e dependence in t h e price increments: m

=

(x(t + t')x(t'))-(x(t'))(x(t

+ t'))

(

9

1

)

^x(t')
«

C J dk\f{k)\

2

exp(-ikt)

(9.2)

where C is a normalisation constant a n d / is t h e Fourier transform. T h e statistics of t h e power spectrum of a fractal time series c a n b e related t o its fractal dimension a n d also t h e scaling behaviour of its m o m e n t s (discussed

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FTIID

DJIA14 1 0.8 0.6 0.4 0.2 0

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Figure 9. Autocorrelation function of the increments of eight financial time series.

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t Figure 10. Autocorrelation function of the absolute increments of eight financial time series.

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again in more detail in sections 10 a n d 11). If t h e power spectrum scales accord­ ing t o a power law, |/(fc)| oc

(9.3)

2

and the variance of the increments a t a time scale Τ scales according to,

<4> ~

T, G

(9.4)

then t h e Fourier fractal dimension Dp and G can be directly calculated from q using [25] D

F

= 2-G

= ^ ± ,

(9.5)

and t h e box counting dimension DB is b o u n d by Dp by [25] D
F

+1/2.

(9.6)

T h u s t h e fractal n a t u r e of a time series is intimately related t o t h e structure of its time correlations. T h e definition of independence is t h a t t h e autocorrelation of any continuous function of the increments takes the value one a t t h e origin a n d zero elsewhere. Figure 9 shows the A C F of eight financial time series (see individual captions). T h e graphs show t h a t t h e daily stock market indices contain short-lived autocor­ relations in t h e price increments, significant over just a few days. T h e intra-daily stock market index, the DJIA14, has a significant negative autocorrelation a t lag one, b u t nothing after t h a t . And t h e daily exchange r a t e indices show no evi­ dence of autocorrelation at all. T h e ACFs of t h e absolute value of the increments are more consistent amongst the different d a t a sets and decays slowly. T h e r e are correlations in t h e magni­ tudes, the squares, and hence the volatility of t h e price increments. This is often referred t o by people in finance as volatility clustering. Large increments tend to be followed by more large increments a n d small increments by small b u t not necessarily of the same sign. In higher frequency exchange r a t e d a t a autocorre­ lations in the raw increments are reported t o exist for very short time scales u p t o approximately 15 minutes [17]. It t u r n s out there is some very short-term correlation in t h e signs of the in­ crements. For example, for stock market d a t a the probability of a daily price increment having the same sign as the previous one is 0.6 instead of 0.5. These kinds of statistical p a t t e r n s are investigated thoroughly in [54] using t h e condi­ tioning entropy measure [74]. It is worth noting t h a t a Markov process defined by the rule t h a t t h e probability of two particular consecutive values having the same sign is 0.6, has exactly t h e same A C F s t r u c t u r e as t h e signed increments of the financial d a t a . (See section 15 for details of this experiment.) Cont [17] shows t h a t the autocorrelations in t h e square of t h e price increments can account for the anomalous scaling of t h e kurtosis (figure 5). If g(T = Ν τ) is the autocorrelation of x\ defined as,

M . D . L o n d o n et al.

(xUt)x r(t + NT))2

9(T)

=

91

(x* (t))(x (t

+ Nr))

2

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T

Var[x2]

(9.7)

where r is some lower time scale t o s t a r t a t (in our case one day) a n d κ(ΛΓτ) is excess kurtosis defined by (8.1) then,

*=i

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F i g u r e 11. Actual (solid), best fit (light dash) and estimated (dark dash) kurtosis against T. Figure 11 shows four examples (two stock m a r k e t a n d two exchange r a t e indices) of the empirical scaling of the kurtosis defined by equation (8.1) with best fit and the estimated kurtosis defined by equation (9.8). T h e fit is not tremendous b u t close enough t o d e m o n s t r a t e t h a t there is a definite link between the correlations in volatility and t h e slowly decaying kurtosis.

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Global scaling p r o p e r t i e s o f a 'financial walk'

We now r e t u r n t o t h e scaling properties of financial time series. In t h e previous section(s) we examined t h e leptokurtic n a t u r e of t h e P D F (of price increments) and the correlations structure of t h e increments and their magnitudes. T h e analysis carried out thus far is fairly traditional. Next we show t h a t it is possible t o explain properties of the time series missed by t h e previous analysis by looking a t the global scaling properties of a financial walk, a type of r a n d o m walk with signed price increments in place of r a n d o m numbers. In section 11 we then exploits the ideas further t o investigate t h e phenomenon of multiscaling. 10.1

Re-scaled range analysis ( R S R A ) : Measuring m e m o r y

Η. E. Hurst (1900-1978) was a hydrologist who had worked on t h e river Nile d a m n project in the early twentieth century. He studied records of t h e river overflows t h a t had been kept by the Egyptians and noticed a statistical phenomenon t h a t s t a n d a r d statistical analysis did not seem t o cater for. Large overflows seemed to be followed by more large overflows until abruptly, t h e system would change t o low overflows, which tended t o b e followed by more low overflows. T h e r e were cycles b u t with no predictable period. Furthermore, there were no significant autocorrelations. Hurst decided t o develop his own methodology. T h e story is related in [70]. Hurst was aware of Einstein's [24] work on Brownian motion. Einstein found t h a t t h e distance the particle covers increased with t h e square root of time, i.e. R(t) α

\it,

(10.1)

where R(t) is t h e range covered by t h e walk a t time t. For a time series, t h a t is in one dimension, R(t) is defined t o be the m a x i m u m value of the process minus t h e minimum u p t o t h a t time. T h a t is t h e a m o u n t of t h e real line t h e I D random walk has covered. T h e scaling property (10.1) is j u s t a manifestation of t h e result,

for uncorrelated (not necessarily independent) i j . From t h e A C F of the price increments (figure 9) we should expect t h a t the range of a financial walk scales similarly t o the normal r a n d o m walk. Hurst's idea was t o use this property (equation (10.1)) t o test t h e Nile River's overflows for randomness. Hurst's contribution was t o generalise this equation to (10.2) (R/S) = Cn , H

n

where S is t h e s t a n d a r d deviation for the same η observations a n d C is a constant. At this stage we define a Hurst process t o b e a process t h a t obeys equation (10.2) for some Η called the Hurst exponent. T h e R/S value of equation (10.2)

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is referred t o as t h e rescaled range because it has zero m e a n a n d is expressed in terms of local s t a n d a r d deviations. If t h e system were independently distributed, t h e n Η = 0.5. Hurst found t h a t t h e exponent for t h e Nile River was Η — 0.91. T h e rescaled range was increasing a t a faster r a t e t h a n t h e square root of time. This meant t h a t t h e system was covering more distance t h a n a r a n d o m process would, a n d therefore t h e annual discharges of t h e Nile h a d t o be correlated. This scaling law (10.2) behaviour is t h e first connection between Hurst pro­ cesses a n d fractal geometry. As previously s t a t e d this is i m p o r t a n t because, a p a r t from t h e trivial case where Η = 0.5, this behaviour requires a certain a m o u n t of organisation on t h e p a r t of t h e underlying process. T h e source of this organisation is t h e obvious question a n d a n avenue for research. In t h e case of t h e River Nile there is clearly no intelligent mechanism organising t h e statis­ tics of t h e overflows. Addressing these t y p e of questions is now a big research area and m a n y researchers are considering interesting new explanations like 'self organised criticality' (SOC) [5]. We have already s t a t e d t h a t Η = 0.5 is consistent with a n independently distributed system, b u t w h a t do other values of Η m e a n ? 0.5 < Η < 1 implies a persistent time series, a n d a persistent time series is characterised by positive correlations. 0 < Η < 0.5 indicates anti-persistence which means t h a t t h e time series covers less ground t h a n a r a n d o m process. For a system t o cover less distance t h a n a r a n d o m process, it must reverse itself more often t h a n a r a n d o m process, or be deterministic with slow growth. One can think of a smooth deterministic function as t h e limit of a stochastic process t h a t reverses itself very quickly b u t deviates only slightly from its line of best fit. Antipersistence may b e loosely thought of as a m e a n reverting tendency although this is not formally correct because t h e system may have no stable m e a n as such. T h e Hurst exponent, H, for processes governed by stable (often called fractal or Levy) distributions (equation 8.2) is related t o t h e dimension, by Η = Ι / μ and t o t h e fractal dimension by Η = 2 - D. 10.1.1

Applying RSRA t o financial time series

In this section we apply this analysis t o our financial d a t a sets t o see if they are, (a) Hurst processes, and (b) r a n d o m (IID). Peters [70] r e p o r t s strong evidence t h a t t h e (daily) Dow Jones Industrial Average financial d a t a set ( J a n 2 1888 through Dec 31 1991) is a persistent Hurst process for periods u p t o four years. A different study [35] concludes t h a t Hurst exponents for financial time series are, in general, not significantly different t o 0.5. T h e test for significance was originally based by Hurst [36] and Feller [28] on t h e Binomial null hypothesis and t h e result, (R/S)

= (ΛΓπ/2) / . 1

N

2

(10.3)

This result has been refined slightly in [70] and a hypothesis test (based on t h e fact t h e R / S values are normally distributed r a n d o m variables) was developed.

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Figure 12. Re-Scaled Range Analysis results for eight financial time series.

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T h e R / S analysis results for t h e four daily stock market indices a n d four exchange r a t e price time series are shown in figure 12. P l o t t e d in t h e graphs is the log of the rescaled range against t h e log of t h e time scale T , thus if equation 10.2 is obeyed then t h e graph should b e a straight line with Η as t h e slope. For the precise methodology used t o perform R S R A see 17. T h e Hurst exponents are in t h e range 0.51 < Η < 0.64. T h e lines fit pretty well a n d there is no evidence of any kinks in the line. A kink in t h e line would represent a feature of the d a t a t h a t is nearly periodic: one with a distribution of wavelengths with a spread t h a t is much less t h a n t h e size of t h e wavelength. We therefore conclude t h a t our six randomly selected financial processes are Hurst processes. Figure 13 shows RSRA for the DJIA14 intra-daily d a t a set. One line repre­ sents t h e rescaled range for the increments a n d t h e other (the steeper of t h e two) t h e rescaled range for t h e absolute value of t h e increments. T h e Hurst exponent for the former is approximately equal t o 0.518. R/S

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Iog10(t)

Figure 13. Re-scaled Range Analysis performed on the increments (bottom) and the absolute increments (top) of the DJIA14 intra-daily data set. Peters [70] also describes R S R A of high frequency 'tick' d a t a a n d concludes t h a t high frequency d a t a is far more dominated by short-term memory effects and it becomes especially i m p o r t a n t t o take AR(1) residuals prior t o performing RSRA, since auto-regressive behaviour can cause a bias in t h e R / S . This implies t h a t day traders merely react t o most recent events. After this is done, markets do not follow r a n d o m walks even a t the 3-minute level, though a t this time scale

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the difference is very small. At high frequencies we only see pure stochastic processes t h a t resemble white noise, and as we step back and look at lower frequencies, a global s t r u c t u r e becomes apparent. T h e structure is characterised by volatility which behaves similarly t o t h e River Nile overflows: large increments followed by large for a r a n d o m length of time and then small by small for another r a n d o m length of time. 10.1.2

Volatility: a Hurst process too?

In the same book [70], Peters describes t h e same R S R A of (log differences of) a volatility times series a n d finds t h a t volatility is an anti-persistent process; reversing itself more often t h a n a r a n d o m process would. T h i s would m e a n a large increase in volatility has a high probability of being followed by a decrease of unknown magnitude. This is easily verified for the datasets studied above. (See figure 14.) Note t h a t the volatility time series is calculated on non-overlapping time intervals of t h e original time series. T h o u g h the magnitudes of t h e original increments are persistent (positively autocorrelated), the volatility time series is anti-persistent with negative autocorrelations. 10.2

T h e J o k e r Effect: n o n - p e r i o d i c c y c l e e x p l a i n s H u r s t p r o c e s s e s

After his discovery, Hurst analysed all the d a t a he could get his hands on includ­ ing rainfall, sun spots, mud sediments, tree rings a n d others [70, 36]. In all cases, Hurst found Η to be greater t h a n 0.5. He was intrigued t h a t Η often took a value of a b o u t 0.7. Hurst suspected t h a t some universal phenomenon was taking place so to investigate, he carried out some experiments using numbered cards. T h e values of the cards were chosen t o simulate a P D F with finite moments, i.e. 0, ± 1 , ± 3 , ± 5 , ± 7 and ± 9. He first verified t h a t time series generated by summing the shuffled cards gave Η = 0.5. To simulate a biased r a n d o m walk, he carried out the following steps. 1. Shuffle t h e deck and cut it once, noting the number, say n. 2. Replace the card and re-shuffle the deck. 3. Deal out 2 hands of 26 cards, A a n d B. 4. Replace t h e lowest η cards of deck Β with the highest η cards of deck A, thus biasing deck Β t o t h e level n . 5. Place a joker in deck Β a n d Shuffle. 6. Use deck Β as time series generator until t h e joker is cut, t h e n create a new biased h a n d . Hurst did 1000 trials of 100 hands a n d calculated Η = 0.72 j u s t as he h a d done in n a t u r e . This is an incredible result. T h i n k of t h e process involved: first t h e

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Figure 14. ReScaled Range Analysis results for the volatility of eight financial time series.

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bias of each hand, which is determined by a random cut of the pack; then, the generation of the time series itself, which is another series of r a n d o m cuts; a n d finally, t h e appearance of t h e joker, which again occurs a t r a n d o m . Despite all of these random events Η = 0.72 would always appear. This is called t h e 'joker effect'. T h e Joker Effect, as illustrated above, describes a tendency for d a t a of a certain magnitude t o be followed by more d a t a of approximately the same mag­ nitude, b u t only for a fixed and r a n d o m length of time. A n a t u r a l example of this phenomenon is in weather systems. Good weather a n d b a d weather t e n d s t o come in waves or cycles (as in a heat wave). This does not mean t h a t weather is periodic, which it is clearly not! We use t h e t e r m non-periodic cycle t o describe cycles of this kind. If markets are Hurst processes then they exhibit trends t h a t per­ sist until the economic equivalent of t h e joker comes along t o change t h a t bias in m a g n i t u d e a n d / o r direction. -Peters [70] In other words RSRA can, along with t h e P D F and P S D F , help t o describe a stochastic time series t h a t contains within it, many different short-lived trends or biases (both in size and direction).

11

Local scaling p r o p e r t i e s a n d multifractals

Having shown t h a t certain intricate correlation structures can lead t o Hurst pro­ cesses and thus fractal time series, we now explore the possible structures for the implied organisation. T h e motivation for this is quite clear: t o u n d e r s t a n d how strict the constraints placed on t h e underlying process need t o b e t o achieve this kind of behaviour. T h a t is, t h e extent t o which it is special and therefore impor­ t a n t t o observe such scaling behaviour. Self-affinity, or statistical self-similarity, a key characteristic of fractals can b e described by t h e following condition. Denoting again, χτ = X(t + T) — X(t), t o be the de-trended price increment a t a t i m e scale T , t h e probability distribution, rescaled by a lag-dependent factor ξ{Τ) can be written as [10, 30, 59], (11.1) where G{u) is a time independent scaling function. For example, if X(t) is a 'normal' random walk, then ξ(Τ) = ayf{T) a n d G(u) = exp(-u /2)/y/2n. Equation (11.1) implies t h a t all moments of χτ t h a t are finite scale similarly (for detail see section 16), 2

m (T) q

= (\χ \η τ

=

(11.2)

Α ξ(Τ) , 9

ν

where A is a ς-dependent number. Often ξ(Τ) behaves as a simple power law: ξ(Τ) oc Τ . In this case of a monofractal process we have, m , oc Τ « , with q

ζ

ς

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99

= ςζ. Here, t h e p a r a m e t e r ζ is called t h e Holder exponent a n d t h e function ζ(ς) the scaling function. Holder exponents tell us a b o u t t h e regularity of the signal. For 0 < ζ < 1, the signal is continuous a n d for values close t o zero t h e signal is similar t o white noise. For ζ = 0.5 the signal is consistent with Brownian motion. For 0.5 < ζ < 1, t h e signal is smoother t h a n Brownian motion. W h e n ζ > 1 the signal breaks u p and becomes discontinuous. A normal random walk (one with a step distribution with a finite second m o m e n t ) , or Wiener Brownian motion ( W B M ) , has ζ = 0.5 a t all points a n d has a scaling function, C(
(11-3)

T h e r e is another class of signals t h a t possess this monofractal property, known as fractional Brownian motions (FBM) a n d their discrete analogue, fractional Gaussian noises (FGN) inspired by Mandelbrot a n d Ness [60]. F B M s are Gaus­ sian fractionally integrated processes, t h a t behave as, P&,T)

= ± G

,

(11.4)

defined by t h e convolution of Gaussian noise η(ί) a n d some power law function oft: rt

B {t)= H

Jo

{t-s)"-^(s)ds,

0
(11.5)

where Η denotes again the Hurst exponent. This is equivalent t o t h e following definition via t h e Fourier transform, denoted by B. Β„(ω)

= (ίω)-°- - η(ω),

0 < Η < 1,

5 Η

(11.6)

which is also a widely accepted definition of fractional differ-integration [66, 73, 42]. It is useful t o know t h a t this is also t h e same as filtering t h e power spectrum of η: \\B„(t)\\

= BB'

= ^

.

m

(11.7)

From this equation one can see t h a t t h e smoothing works by filtering o u t t h e power of the signal a t high frequencies. T h e correlation between two equal and non-overlapping increments of F B M is [60, 22], C(t) = ((B (0) H

- B {-t)) H

(B (t)

- B„(0))

H

)/B {tf

= 2 "2

H

1

- 1,

(11.8)

which is (notably) independent of t, illustrating the self-affine n a t u r e of t h e process. T h e autocorrelation a t a lag t is given by [60], C(t) = \(\t

+ 1 | " - 2 | i | " - |t - 1| ") , 2

2

2

(11.9)

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which is clearly zero for Η = 1/2 and for Η φ 1/2 can b e approximated by t h e power law, C(t) ~ t ~ . 2H

(11.10)

2

Therefore F B M has a scaling function, C(i) = C<7 = Hq. 11.1

(11.11)

M u l t iscaling

W h e n the function ξ(α) behaves as (11.2) with ζ φ ζς then the process χτ is said t o be multiscaling [30, 56, 59, 10, 68, 38, 3]. One of t h e implications of this is t h a t there is, instead of one Holder exponent (ζ) t o describe t h e whole process, now an intricate mixture of local (coarse) Holder exponents. We can expose mixtures of Holder exponents by examining multiscaling prop­ erties because different values of q p u t a n emphasise on different sized χ values. T h a t is, when we examine the scaling of | x r | ' for large q, only the large χ values are contributing. T h e extent t o which this h a p p e n s is related t o t h e regularity of t h e signal. Smooth signals have small increments t h a t are of similar m a g n i t u d e . Jagged signals t h a t go u p a n d down more violently have a greater variety of magnitudes and therefore a greater capacity for the larger ones t o dominate. W h a t is t h e scaling function of financial d a t a like? Figure 15 shows t h e partition function, defined as: 9

S (T) V

oc Tm (T) q

= Σ

\XT\" OC T « » > ,

(11.12)

+ 1

for values of q ranging from 1.5 t o 4 in steps of 0.5. T h e dashed lines represent the theoretical scaling slopes for W B M as described by equation (11.3). Since the lines are straight, t h e moments scale as power laws, indicating t h a t t h e d a t a are self-affine. T h e scaling function is not t h e same as t h a t of t h e r a n d o m walk or W B M and more importantly, it is not linear in q. In t h e case of the DJIA14, t h e higher frequency index, there is a clear crossover period caused by t h e s t r o n g negative autocorrelations in the higher frequency (half hourly) d a t a . We therefore conclude t h a t our financial d a t a are indeed multifractal. Some recent papers [30, 56, 17, 19, 4] suggest t h a t this is a fruitful avenue of research in the sense t h a t a lot of organisation is required for this property t o hold: t h e distribution of Holder exponents for instance. T h e scaling function ζ ( ς ) , determined by t h e moments, can b e used t o esti­ m a t e the multifractal spectrum which tells us t h e distribution of Holder exponents [30]. T h e multifractal spectrum, / ( a ) , and the scaling function, ζ , are related through t h e Legendre transform [30, 56, 59], ν

f(a)=mm[qa-\(q)).

(11.13)

Ot

Examples of systems known t o exhibit multiscaling are turbulence [32], In­ ternet traffic [27], gestural expressionist art [65] and many others [57]. Whereas

M . D . L o n d o n et al.

Figure 15. The partition function for the DJIA data set.

101

102

Why study

financial

t i m e series?

there is strong theoretical evidence for multiscaling in turbulence [32], there is as yet n o theory connecting t h e phenomenon with financial time series. This is a direction for future research. Bacry et al [3, 4] show how it is possible t o estimate t h e necessary parameters t o fit a multifractal random walk t o a n empirical d a t a set. A paper by Bouchaud et al [10] however warns t h a t multifractal scaling may be a trivial property resulting from correlations in t h e volatility. T h e y describe a model t h a t is asymptotically monofractal by construction b u t t h a t exhibits mul­ tiscaling similar t o financial d a t a due t o cross over effects caused by correlations in volatility. T h e relationship between volatility correlations and multiscaling remains unclear and a n avenue for future research. Bouchaud et al suggest t h a t t h e cumulants better describe their model a n d perhaps also financial time series since the cumulants of their process scale ex­ actly as power laws whereas t h e moments scale as sums of power laws with similar exponents; hence t h e transient scaling behaviour.

12

S u m m a r y a n d c o n c l u d i n g remarks

T h e first half of this paper motivates research into financial d a t a ; stock indices and exchange r a t e price series. It explains how certain apparently universal em­ pirical observations of such d a t a sets might suggest similarities between stock markets and dynamical driven out-of-equilibrium systems. This view is influ­ enced by the concept of criticality developed (mainly) in statistical physics over the last 30 years in order t o describe a class of cooperative phenomenon such as magnetism a n d melting, b u t also in other areas to describe sand piles, forest fires, traffic j a m s a n d many others. Before the modelling can begin, one needs a thorough understanding of t h e empirical facts a successful model would need t o b e able t o capture. T h e second half of this paper has examined several financial d a t a sets a n d reviewed m a n y research papers t o t r y t o establish as many of t h e empirical regularities associated with financial time series as possible. Much of the research is in agreement a n d t h e main conclusions are summarised below. • T h e empirical P D F has Levy style features b u t exponents are not consistent for a good fit b o t h near t h e mean and in t h e tails; it is one or t h e other. Tails of distribution behave as, p(x) ~ x~ , with 3 < α < 4 (figures 4 and 6). a

• T h e kurtosis of increments scales roughly according t o a power law, κ ~ T~ , in t h e time scale of t h e increment. Exponents are β w 0.3 for t h e intra-daily DJIA14, in t h e range 0.5 < β < 0.6 for t h e daily stock market indices a n d in t h e range 0.9 < β < 1 for t h e exchange rates (figure 5). 0

• T h e probability of moving t h e same direction as yesterday is approximately 0.6 instead of 0.5.

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• T h e autocorrelation function of t h e signed increments is not significantly different t o zero for time-scales above half and hour, (figure 9). • T h e autocorrelation function of t h e absolute increments decays slowly, reaching zero typically a t a r o u n d 500 days (2 years) (figure 10). There is additionally evidence of: • Slight persistence in increments, though not statistically significant (figure 12). • Strong anti-persistence in t h e volatility increments (figure 14). • Multiscaling (figure 15). Several areas seem obvious main targets for future research. T h e first is t h e statistical issue surround t h e phenomenon of multiscaling a n d how it is affected by t h e volatility clustering. T h e second a n d p e r h a p s most i m p o r t a n t area is t h e need for more simple 'toy' models like those described in [6, 18, 40, 50, 3 1 , 39] t o u n d e r s t a n d how t h e concept of criticality a n d universality of m a r k e t s is m a d e possible. This will b e t h e subject of a forthcoming paper.

Appendix 13

T h e Central Limit T h e o r e m (CLT)

Consider a number Ν of independent and identically distributed (IID) r a n d o m variables x with η = 1,..., Ν, with probability density function, p(x). Let, n

Ν

ΧΝ

Σ »r»=l

( )

Χ

=

131

Given t h a t t h e first two moments, (x) a n d ( x ) are finite, t h e mean a n d variance of t h e X depend linearly on N, 2

X = (x)N,

X*-X

2

= ((χ

2

) - (χ) )ΛΓ. 2

(13.2)

Up t o a translation of reference we may suppose t h a t (x) = 0. XN behaves as 1 2 ΛΓ / and it is t h e distribution of t h e variable, XN/N / t h a t a d m i t s a limiting l 2 form. T h e distribution of X/N l ,f(X), given by the joint distribution for t h e s u m m a n d s is, 1

2

f(X) = J...Jftp(x )dx 6(^-NV j^x y 2

n

n

n

(13.3)

Using a n integral representation of t h e i-function, 1

f °° +

δ(χ) = — / 2 7 r

exp(-ikx)dk, Jk=-oo

(13.4)

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equation (13.3) may be written as,

f( ) x

= -^j*

dkexp(ikX/\/N)^J

dxpWexpi-ikx/VN)^

.

(13.5)

This expression involves t h e Fourier transform (characteristic function) p(k) of p(x). For large Ν only t h e behaviour of p(k) close t o A; = 0 is important, since

\p{k)\ < Ip{k/VN)

N

= [1 - \{x )k /N 2

2

+ 0(N ' )] 3 2

N

-> e x p ( - ( x ) f c / 2 ) 2

2

(13.6)

T h e Gaussian P D F is then recovered by integrating 13.5 over k.

14

D e - t r e n d i n g process for financial t i m e series

Here we justify using the auto-regressive one log difference de-trending process (section 7) for all the financial time series even though the exchange rates do not show the same exponential trends as t h e stock markets. T h e are several reasons t h a t this is a good idea. T h e first is for consistency: consistent t r e a t m e n t of all d a t a sets and consistency with most other published quantitative research. An­ other reason is a general argument often used by physicists for signal processing. T h e details follow below b u t t h e general argument goes something like this. Most systems are translationally time invariant a n d as such can be written so t h a t its solutions are t h e eigenvectors of a linear operator t h a t commutes with a time translation operator. Autonomous systems have solutions in t h e form of exponentials (including complex). Also, any two linear operators t h a t c o m m u t e share eigenvectors, so in this way any operator t h a t commutes with the time translation operator is autonomous and has exponential eigenvectors; therefore the corresponding system has exponential solutions. If A is a linear operator such t h a t , Ax = Ax,

(14.1)

then χ is a n eigenvector of A a n d A is a n eigenvalue. Suppose a system is governed by a equation like, Af(t)

= cf(t),

(14.2)

where c is a constant. For example, the differential equation, fA+u>f(t)=3f(t),

(14.3)

or a stochastic differential equation, then solutions t o t h e system are eigenvalues of A. A system is translationally invariant (autonomous), when if f(t) is a solution, f(t - a) is also a solution. For example, | = 3 ,

(14.4)

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is a u t o n o m o u s b u t , (14.5) isn't.

14.1

Statement

Af = cf is autonomous if a n d only if (iff) A commutes with T , i.e., Q

[A,T ]

= AT -T A

a

where T

a

a

(14.6)

is the time translator operator, T f(t)

= f{t -

a

14.2

= 0,

a

a).

(14.7)

Proof

We must first prove t h a t if f(t) is the solution of t h e system Af = cf where A commutes with t h e time translation operator T , then f(t — a) is also a solution. Say /(<) solves Af = cf, a n d [A, T ] = 0, then, T Af = cT f, a n d A(T f) = c(T f), b u t , T f = f(t-a), so Af(t-a) = cf(t-a), i.e. f(t-a) is a solution. a

a

a

a

a

a

a

We must also prove t h a t whenever f(t) and f(t—a) are solutions, [T , A] = 0. If Af(t) = cf(t) and Af(t — a) = cf(t — a) then t h e latter can be written in terms if t h e time translator operator as, A(T f(t)) = c(T f(t)) = T (cf(t)) = T Af{i). T h u s , AT = T A = 0. a

a

a

14.3

a

a

a

a

Theorem

If [^4, B] = 0 then every eigenvector of A is an eigenvector of B.

14.3.1

Proof

If χ is an eigenvector of A then, Ax

(14.8)

BAx

cx cBx

(14.9)

ABx

cBx.

(14.10)

If eigenvectors are non-degenerate (each eigenvector has unique eigenvalue) t h e n we say, Bx is an eigenvector of with eigenvalue c, so Bx is a multiple of x, so χ is an eigenvector of B.

106

15

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A u t o c o r r e l a t i o n f u n c t i o n of a Markov p r o c e s s

A Markov process defined by t h e rule: t h e probability t h a t t h e price movement a t time t + 1 has the same sign as t h e price movement a t time t is equal to ρ has the following properties. Denote r t o b e t h e r a n d o m variable t h a t is t h e length of a sequence of con­ secutive movements in t h e same direction, then t h e mean of the process is, (r) = p / ( l - p) and t h e s t a n d a r d deviation is, σ = p / ( l — p ) . Empirically t h e mean values of r are in the range, 1.35 > (r) > 1.65 indicating t h a t t h e ρ value is in t h e range, .0.574 > ρ > 0.623. T h e s t a n d a r d deviations are a r o u n d 3.75 which is larger t h a n (r) and for this reason we should not expect any near period cycles. 2

τ

Figure 16 shows t h a t the correlation structure of the de-trended increments of the F T I I D index is similar t o t h a t of a Markov process defined with t h e same ρ value (p = 0.5836). T h e autocorrelation s t r u c t u r e depends also of course on the shape of the conditional distributions, which in this experiment were taken to be uniform. T h e point of this section in the appendix was j u s t t o show t h a t this feature (the Markov feature) can capture t h e correlation s t r u c t u r e of the daily financial time series. Autocorrelation plots.

Figure 16. (solid).

Autocorrelation of FTIID (dashed) versus synthetic Markov process

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107

P r o o f of e q u a t i o n (11.2)

From equation (11.1) t h e moments m

q

m

q

= (\x \i)

are defined as,

=

T

(16.1) (16.2) (16.3) (16.4)

17

R / S analysis applied

T h e following is a step by step methodology for applying R/S analysis t o stock market d a t a . Note t h a t t h e AR(1) [70] notation used below stands for a u t o regressive process with 1-daily linear dependence. T h u s taking AR(1) residuals of a signal is equivalent t o plotting the signals 1 day out of phase and taking t h e day t o day linear dependence out of t h e d a t a . 1. P r e p a r e the D a t a . Take AR(1) residuals of log ratios. T h e log ratios account for the fact t h a t price changes are relative, i.e. depend on price. T h e AR(1) residuals remove any linear dependence, serial correlation, or short-term memory which can bias the analysis. V

=

Xt

=

t

logiPt/Pt-!) Vt-ic

+

(17-1) (17.2)

mVt-y)

T h e AR(1) residuals are taken t o eliminate any linear dependency. 2. Divide this time series (of length N) u p into A sub-periods, such t h a t t h e first and last value of time series are included i.e. Α χ η = Ν. Label each sub-period I with a — 1,2,3,..., A. Label each element in J with N where k = 1 , 2 , 3 , n . For each I of length n, calculate t h e mean a

0

k
η

(17.3)

3. Calculate the time series of accumulated departures from t h e mean for each sub interval. k

(17.4)

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4. Define t h e range as Ri

a

= max(yife, ) - π ή η ( 1 * , ) 0

(17.5)

α

where 1 < A; < n . 5, Define t h e sample s t a n d a r d deviation for each sub-period as

(17.6)

6. Each range, Rj is now normalised by dividing by its corresponding Sj . Therefore the re-scaled range for each I is equal t o Ri /Sj From step 2 above, we have A contiguous sub-periods of length n. Therefore the average R/S value for each length η is defined as a

a

a

a

a

(17.7) Q=l

7. T h e length η is then increased until there are only two sub-periods, i.e. η = N/2. We then perform a least squares regression on log(n) as the independent variable and log(R/S) as t h e dependent variable. T h e slope of t h e equation is t h e estimate of t h e H u r s t exponent, H.

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M . D . L o n d o n et al. 7. G. L. Baker a n d J. P. Gollub. Chaotic dynamics bridge University Press, 1990.

109 - an introduction.

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