Introduction

Physicists like using power-laws to fit data. The reason for this is that complex, collective phenomenon often give rise to power-laws which are furthermore universal, that is to a large degree independent of the microscopic details of the phenomenon. These power-laws emerge from collective action and transcend individual specificities. As such, they are unforgeable signatures of a collective mechanism. Examples in the physics literature are numerous. A well-known example is that of phase transitions, where a system evolves from a disordered state to an ordered state: many observables behave as universal power laws in the vicinity of the transition point (see e.g. Goldenfeld (1992)). This is related to an important property of power-laws, namely ‘scale invariance’: the characteristic length scale of a physical system at its critical point is infinite, leading to self-similar, scale-free fluctuations. Similarly, power-law distributions are scale invariant in the sense that the relative probability to observe an event of a given size s0 and an event ten times larger is independent of the reference scale s0. Another interesting example is fluid turbulence, where the statistics of the velocity field has scale invariant properties, to a large extent independent of the geometry, the nature of the fluid, of the power injected, etc. (see Frisch (1997)).

Introduction

Aim of the chapter

The aim of this chapter is to introduce the general theory of derivative pricing in a simple and intuitive, but rather unconventional, way. The usual presentation, which can be found in all the available books on the subject, relies on particular models where it is possible to construct riskless hedging strategies, which replicate exactly the corresponding derivative product. Since the risk is strictly zero, there is no ambiguity in the price of the derivative: it is equal to the cost of the hedging strategy. In the general case, however, these ‘perfect’ strategies do not exist. Not surprisingly for the layman, zero risk is the exception rather than the rule. Correspondingly, a suitable theory must include risk as an essential feature, which one would like to minimize. The following chapters thus aim at developing simple methods to obtain optimal strategies, residual risks, and prices of derivative products, which take into account in an adequate way the peculiar statistical nature of financial markets, that have been described in Chapters 6, 7, 8.

Strategies in uncertain conditions

A derivative product is an asset the value of which depends on the price history of another asset, the ‘underlying’. The best known examples, on which the following chapters will focus in detail, are futures and options.

Aim of the chapter

The easy access to enormous financial databases, containing thousands of asset time series, sampled at a frequency of minutes or sometimes seconds, allows one to investigate in detail the statistical features of the time evolution of financial assets. The description of any kind of data, be it of physical, biological or financial origin requires, however, an interpretation framework, needed to order and give a meaning to the observations. To describe necessarily means to simplify, and even sometimes betray: the aim of any empirical science is to approach reality progressively, through successive improved approximations.

The goal of the present and subsequent chapters is to present in this spirit the statistical properties of financial time series. We shall propose some plausible mathematical modelling, as faithful as possible (though imperfect) to the observed properties of these time series. The models we discuss are however not the only possible models; the available data is often not sufficiently accurate to distinguish, say, between a truncated Lévy distribution and a Student distribution. The choice between the two is then guided by mathematical convenience. In this respect, it is interesting to note that the word ‘modelling’ has two rather different meanings within the scientific community. The first one, often used in applied mathematics, engineering sciences and financial mathematics, means that one represents reality using appropriate mathematical formulae.

Introduction

This chapter takes on finance. The first part offers an overview of the major products in modern finance. It describes some of the possible complex characteristics of each of these products, and suggests tricks on how to take these particularities into account in data analysis. The second part of the chapter turns to the practical functioning of financial markets: who are the players who buy and sell, and how are the markets organized. This chapter is essentially a source of definitions and references, covering the basic knowledge necessary to understand our book as a whole, knowledge familiar to practitioners in finance, but perhaps new to people outside the field. More details on the all the subjects alluded to in this chapter can be found in standard textbooks (see further reading at the end of the chapter).

Financial products

Cash (Interbank market)

Putting cash in the bank is the simplest form of investment. Banks pay interest rates on deposits as well as charge interest rates on loans. Interest is paid at the end of a given period, quoted in annual rate, although actual deposits and loans can have any lifetime.

Since the 1980s an increasing number of physicists have been using ideas from statistical mechanics to examine financial data. This development was partially a consequence of the end of the cold war and the ensuing scarcity of funding for research in physics, but was mainly sustained by the exponential increase in the quantity of financial data being generated everyday in the world's financial markets.

Jean-Philippe Bouchaud and Marc Potters have been important contributors to this literature, and Theory of Financial Risk and Derivative Pricing, in this much revised second English-language edition, is an admirable summary of what has been achieved. The authors attain a remarkable balance between rigour and intuition that makes this book a pleasure to read.

To an economist, the most interesting contribution of this literature is a new way to look at the increasingly available high-frequency data. Although I do not share the authors' pessimism concerning long time scales, I agree that the methods used here are particularly appropriate for studying fluctuations that typically occur in frequencies of minutes to months, and that understanding these fluctuations is important for both scientific and pragmatic reasons. As most economists, Bouchaud and Potters believe that models in finance are never ‘correct’ – the specific models used in practice are often chosen for reasons of tractability. It is thus important to employ a variety of diagnostic tools to evaluate hypotheses and goodness of fit.

Plain Monte-Carlo

Motivation and basic principle

The calculation of the price of options can only be done analytically in some very special cases: the statistics of the underlying contract must be ‘simple enough’ (for example, the Black-Scholes log-normal process), and the option contract itself must not be too ‘exotic’. For example, ‘American’ options, that can be exercised at any time, cannot be priced exactly, even in the simple framework of the Black-Scholes model.

On the other hand, as emphasized throughout this book, the fluctuations of the underlying assets are in general non-Gaussian, with complex volatility-volatility and return-volatility correlations. Furthermore, real world options often involve clauses that are difficult to deal with analytically. It can therefore be compulsory to rely on numerical methods to price and hedge these options. Aversatile and commonly used method is the Monte-Carlo method, that we first explain in the context of ‘plain vanilla’ European options. Suppose that the interest rate r is zero, and that one chooses to Δ-hedge the option. In this case, the price of the option today is equal to the average pay-off of the option, over the detrended distribution of price returns (this price is called the ‘risk-neutral’ price, see the discussion in Section 15.2.2).

The most naive method to compute the option price would be to generate an ensemble of paths according to a certain (zero-drift) stochastic model, and to compute the average pay-off over this ensemble.

Divisibility and the continuous time limit

Divisibility

We have discussed in the previous chapter how the sum of two iid random variables can be computed if the distribution of these two variables is known. One can ask the opposite question: knowing the probability distribution of a variable X, is it possible to find two iid random variables such that X = δX1 + δX2, or more precisely such that the distribution of X can be written as the convolution of two identical distributions. If this is possible, the variable X is said to be divisible.

We already know cases where this is possible. For example, if X has a Gaussian distribution of variance σ2, one can choose X1 and X2 to be have Gaussian distribution of variance σ2/2. More generally, if X is a Lévy stable random variable of parameter aµ (see Eq. (1.20)), then X1 and X2 are also Lévy stable random variables with parameter aµ/2. However, all random variables are not divisible: as we show below, if X has a uniform distribution in the interval [a, b], it cannot be written as X = δX1 + δX2 with δX1, δX2 iid.

Risk measurement and diversification

Measuring and controlling risks is now a major concern in many modern human activities. The financial markets, which act as highly sensitive (probably over sensitive) economical and political thermometers, are no exception. One of their rôles is actually to allow the different actors in the economic world to trade their risks, to which a price must therefore be given.

The very essence of the financial markets is to fix thousands of prices all day long, thereby generating enormous quantities of data that can be analysed statistically. An objective measure of risk therefore appears to be easier to achieve in finance than in most other human activities, where the definition of risk is vaguer, and the available data often very poor. Even if a purely statistical approach to financial risks is itself a dangerous scientists' dream (see e.g. Fig. 1.1), it is fair to say that this approach has not been fully exploited until the very recent years, and that many improvements can be still expected in the future, in particular concerning the control of extreme risks. The aim of this chapter is to introduce some classical ideas on financial risks, to illustrate their weaknesses, and to propose several theoretical ideas devised to handle more adequately the ‘rare events’ where the true financial risk resides.

Scope of the book

Finance is a rapidly expanding field of science, with a rather unique link to applications. Correspondingly, recent years have witnessed the growing role of financial engineering in market rooms. The possibility of easily accessing and processing huge quantities of data on financial markets opens the path to new methodologies, where systematic comparison between theories and real data not only becomes possible, but mandatory. This perspective has spurred the interest of the statistical physics community, with the hope that methods and ideas developed in the past decades to deal with complex systems could also be relevant in finance. Many holders of PhDs in physics are now taking jobs in banks or other financial institutions.

The existing literature roughly falls into two categories: either rather abstract books from the mathematical finance community, which are very difficult for people trained in natural sciences to read, or more professional books, where the scientific level is often quite poor. Moreover, even in excellent books on the subject, such as the one by J. C. Hull, the point of view on derivatives is the traditional one of Black and Scholes, where the whole pricing methodology is based on the construction of riskless strategies.

Non-linear correlations and dependence

Non identical variables

The previous chapter presented a statistical analysis of financial data that implicitly assumes homogenous time series. For example, when we constructed the empirical distributions of price returns ηk, we pooled together all the data, thereby assuming that P1(η1), P2(η2), …, PN(ηN) are all identical.

It turns out that the distributions of the elementary random variables P1(η1), P2(η2), …, PN(ηN) are often not all identical. This is the case, for example, when the variance of the random process itself depends upon time – in financial markets, it is a wellknown fact that the daily volatility is time dependent, taking rather high levels in periods of uncertainty, and reverting back to lower values in calmer periods. This phenomena is called heteroskedasticity. For example, the volatility of the bond market has been very high during 1994, and decreased in later years. Similarly, the stock markets since the beginning of the century have witnessed periods of high volatility separated by rather quiet periods (Fig. 7.1). Another interesting analogy is turbulent flows which are well known to be intermittent: periods of relative tranquility (laminar flow), where the velocity field is smooth, are interrupted by intense ‘bursts’ of activity.

In this chapter and chapter 13 we shall consider a number of open economy models that exhibit dynamic behaviour. We shall start with the very simplest – the income–expenditure model considered at the beginning of all courses on macroeconomics. This model assumes a fixed exchange rate. Simple as it is, it will allow us to set the scene and illustrate, in the simplest possible terms, how instability may occur, but is less likely to occur in an open economy in comparison to a closed one. We then do the same in the context of the IS-LM model we discussed in chapter 10, extending it to the open economy, but considering the situation under both a fixed and a flexible exchange rate. This forms the basis of the Mundell–Fleming model. This model was originally concerned with the relative impact of monetary and fiscal policy under fixed and floating exchange rate regimes, but with perfect capital mobility. It has become the standard model of open economy macroeconomics, and so we shall look into its dynamic properties in some detail – for models with some (but not perfect) capital mobility and for situations of perfect capital mobility. We shall find that the assumption about the degree of capital mobility is quite important to the dynamic results. As in earlier chapters, we shall be particularly interested in what happens out of equilibrium, and hence in the dynamic forces in operation in an open economy.