During the twelfth and thirteenth centuries at least 50 cathedrals were built within the geographical limits of contemporary France. For the society of that time this represented an overpowering investment the magnitude of which was probably (in relative terms) comparable to the one required for the construction of the railroad network in the second half of the nineteenth century. Was it a rational investment? Obviously the answer very much depends on how economic rationality is defined. In the eyes of the people of the twelfth century it was undoubtedly the right decision, for no other activity was deemed more worthwhile than to celebrate the glory of God. For twentieth-century Americans or Europeans it is certainly difficult to imagine to what extent religion permeated the society of that time. Education, common law, intellectual life, government, every aspect of social life was deeply influenced by religion. For instance, about one half of all the books published in the seventeenth century were concerned with religious topics (Bendix 1978); although we do not have a similar statistic for the twelfth century it can be assumed that the proportion was even higher.

One might think that nowadays at last, the definition of “rational economic behavior” is fairly well established, but that is probably only because we lack the benefit of hindsight. It is no longer the celebration of the Glory of God which is our ultimate aspiration, but it is hardly more rational to consider that no expense should be spared for the preservation of health; after all, once we are dead, the people of the twelfth century would reason, do we not share the Glory of God?

Johann Gregor Mendel was born in Hyncice, in what is now the Czech Republic, on July 22, 1822. After studying science at Vienna (1851–1853), he became Abbot at Brno (1868). The question of how traits were passed fromone generation to the next was at that time extensively investigated by several scientists, but with fairly inconclusive results. Unlike others Mendel studied only one trait at a time and he studied several generations instead of just two or three. He also managed to set apart accidental factors, such as the influence of foreign pollen. It is estimated that in the course of his investigation he observed about 28,000 peas, a figure which attests to the thoroughness of his investigation. He also devoted much time to meteorological observations; in addition to his two celebrated papers on hybridization he wrote nine articles on meteorological questions. This part of his activity is less well known because it did not lead to path-breaking discoveries, but it is interesting to observe that it continued a well-established tradition. Before himse veral other great scientists, such as Kepler (1571–1630), Descartes (1596–1650), or Lavoisier (1743–1794), had devoted a substantial part of their scientific activity to meteorological studies, without, however, being able to make significant inroads. This short account of Mendel's accomplishments encapsulates several of the themes that we develop in this chapter, such as the emphasis on thorough and systematic experimental work or the classification of scientific problems according to their degree of complexity.

As one knows the term econophysics designates the investigation of economic problems by physicists.

In the previous chapters we have seen that the dynamics of stock prices is a complex subject, and that a definitive model has yet to be constructed. The complexity of the entire financial system is even greater. Not only is the trading of financial securities complex, but additional sources of complexity come from the issuing of financial contracts on those fluctuating financial securities.

An important class of financial contracts is derivatives, a financial product whose price depends upon the price of another (often more basic) financial product [22, 45, 73, 74, 122, 127]. Examples of derivatives include forward contracts, futures, options, and swaps. Derivatives are traded either in over-the-counter markets or, in a more formalized way, in specialized exchanges. In this chapter, we examine the most basic financial contracts and procedures for their rational pricing. We consider idealized markets and we discuss the underlying hypothesis used in obtaining a rational price for such a contract.

Forward contracts

The simplest derivative is a forward contract. When a forward contract is stipulated, one of the parties agrees to buy a given amount of an asset at a specified price (called the forward price or the delivery price K) on a specified future date (the delivery date T). The other party agrees to sell the specified amount of the asset at the delivery price on the delivery date. The party agreeing to buy is said to have a long position, and the party agreeing to sell is said to have a short position.

In Chapter 14, we considered the option-pricing problem in ideal frictionless markets. Real markets are often efficient, but they are never ideal. In this chapter, we discuss how the complexity of modeling financial markets increases when we take into account aspects of real markets that are not formalized in the ideal model. These aspects are addressed in the literature as market microstructure [26] or market imperfections [127].

The terminology used in the economics literature suggests a clear parallel with similar scenarios observed in physical sciences. For example, it is much easier to construct a generalized description of the motion of a mechanical system in an idealized world without friction than in the real world. A similar situation is encountered when we compare equilibrium and non-equilibrium thermodynamics. In this chapter, we show that knowledge of the statistical properties of asset price dynamics is crucial for modeling real financial markets. We also address some of the theoretical and practical problems that arise when we take market imperfections into account.

Discontinuous stock returns

The existence of a portfolio containing both riskless and risky assets – replicating exactly the value of an option – is essential in determining the rational price of the option under the assumption that no arbitrage opportunities are present. Whether a portfolio is replicating or not depends on the statistical properties of the dynamics of the underlying asset. In the previous chapter, we saw that a replicating portfolio exists when the price of the underlying asset follows a geometric Brownian motion, but we also saw that this case cannot be generalized.

Physicists are currently contributing to the modeling of ‘complex systems’ by using tools and methodologies developed in statistical mechanics and theoretical physics. Financial markets are remarkably well-defined complex systems, which are continuously monitored – down to time scales of seconds. Further, virtually every economic transaction is recorded, and an increasing fraction of the total number of recorded economic data is becoming accessible to interested researchers. Facts such as these make financial markets extremely attractive for researchers interested in developing a deeper understanding of modeling of complex systems.

Economists – and mathematicians – are the researchers with the longer tradition in the investigation of financial systems. Physicists, on the other hand, have generally investigated economic systems and problems only occasionally. Recently, however, a growing number of physicists is becoming involved in the analysis of economic systems. Correspondingly, a significant number of papers of relevance to economics is now being published in physics journals. Moreover, new interdisciplinary journals – and dedicated sections of existing journals – have been launched, and international conferences are being organized.

In addition to fundamental issues, practical concerns may explain part of the recent interest of physicists in finance. For example, risk management, a key activity in financial institutions, is a complex task that benefits from a multidisciplinary approach. Often the approaches taken by physicists are complementary to those of more established disciplines, so including physicists in a multidisciplinary risk management team may give a cutting edge to the team, and enable it to succeed in the most efficient way in a competitive environment.

Motivation

Since the 1970s, a series of significant changes has taken place in the world of finance. One key year was 1973, when currencies began to be traded in financial markets and their values determined by the foreign exchange market, a financial market active 24 hours a day all over the world. During that same year, Black and Scholes [18] published the first paper that presented a rational option-pricing formula.

Since that time, the volume of foreign exchange trading has been growing at an impressive rate. The transaction volume in 1995 was 80 times what it was in 1973. An even more impressive growth has taken place in the field of derivative products. The total value of financial derivative market contracts issued in 1996 was 35 trillion US dollars. Contracts totaling approximately 25 trillion USD were negotiated in the over-the-counter market (i.e., directly between firms or financial institutions), and the rest (approximately 10 trillion USD) in specialized exchanges that deal only in derivative contracts. Today, financial markets facilitate the trading of huge amounts of money, assets, and goods in a competitive global environment.

A second revolution began in the 1980s when electronic trading, already a part of the environment of the major stock exchanges, was adapted to the foreign exchange market. The electronic storing of data relating to financial contracts – or to prices at which traders are willing to buy (bid quotes) or sell (ask quotes) a financial asset – was put in place at about the same time that electronic trading became widespread. One result is that today a huge amount of electronically stored financial data is readily available.

Concepts, paradigms, and variables

Financial markets are systems in which a large number of traders interact with one another and react to external information in order to determine the best price for a given item. The goods might be as different as animals, ore, equities, currencies, or bonds – or derivative products issued on those underlying financial goods. Some markets are localized in specific cities (e.g., New York, Tokyo, and London) while others (such as the foreign exchange market) are delocalized and accessible all over the world.

When one inspects a time series of the time evolution of the price, volume, and number of transactions of a financial product, one recognizes that the time evolution is unpredictable. At first sight, one might sense a curious paradox. An important time series, such as the price of a financial good, is essentially indistinguishable from a stochastic process. There are deep reasons for this kind of behavior, and in this chapter we will examine some of these.

Arbitrage

A key concept for the understanding of markets is the concept of arbitrage – the purchase and sale of the same or equivalent security in order to profit from price discrepancies. Two simple examples illustrate this concept. At a given time, 1 kg of oranges costs 0.60 euro in Naples and 0.50 USD in Miami.

The statistical properties of the time evolution of the price play a key role in the modeling of financial markets. For example, the knowledge of the stochastic nature of the price of a financial asset is crucial for a rational pricing of a derivative product issued on it. The full characterization of a stochastic process requires the knowledge of the conditional probability densities of all orders. This is an incredible task that cannot be achieved in practice. The usual empirical approach used by physicists is performed in two steps. The first concerns the investigation of time correlation and power spectrum, while the second concerns the study of the asymptotic pdf.

The most common stochastic model of stock price dynamics assumes that ln Y(t) is a diffusive process, and the ln Y(t) increments are assumed to be Gaussian distributed. This model, known as geometric Brownian motion, provides a first approximation of the behavior observed in empirical data. However, systematic deviations from the model predictions are observed, the empirical distributions being more leptokurtic than Gaussian distributions (Fig. 8.1). A highly leptokurtic distribution is characterized by a narrower and larger maximum, and by fatter tails than in the Gaussian case. The degree of leptokurtosis is much larger for high-frequency data (Fig. 8.2).

Based on theoretical assumptions and empirical analyses, several alternative models to geometric Brownian motion have been proposed.