salviati: Greetings, Sagredo, Simplicio, my good friends. I can hardly believe that it was only yesterday that we resolved to meet and talk about this book. How the time drags when I am not in pleasant company such as yours.

simplicio: Greetings to you, most courteous Salviati, and well met, well met I say. My mind is already racing in anticipation. I have not forgotten, and could not forget, our wonderful discussions with Professor Galileo in Tuscany, and I am convinced that on this occasion too you have found something worth the labour of a Dialogue (Galilei, 1632).

sagredo: For my part I am also delighted to see you both again. In the company of two such philosophers as yourselves I never fail to find inspiration and illumination. Now, would you care to tell me the nature of the subject, Salviati?

salviati: Certainly, certainly, shall I come to the point: I feel that a change is happening, just as it was when we met with Professor Galileo.

simplicio: Change! Ha!

sagredo: Now, now, Simplicio … Let's hear this out. The book is about a change, is it? But I don't understand even the title.

Perhaps everyone has asked themselves whether there is anything in the universe that is perfectly isolated, separate and complete within itself. Some will answer that the universe itself is the only thing, because the universe alone is unique. However, the physics of elementary particles and cosmology suggests the possibility that many universes similar to our universe exist and interact with each other through quantum gravity.

The appearance of a discussion of the universe or universes in a book on economic theory may seem abrupt and even odd, but if we recast the point in terms that are more familiar it will not seem so peculiar. When strangers meet by chance, at a party or on a train for example, it is not uncommon for them to find that they have shared acquaintances. ‘It's a small world!’, they will exclaim, in genuine surprise. In fact, this happens all the time, so it is not so much the frequency that stimulates our wonder, but the reasons which underlie it, and these are the subject of long-standing and unsolved questions.

In an attempt to shed light on this matter, sociologists have conducted many kinds of experiments, and needless to say this attempt requires an understanding of human networks. However, it is just as important to extend our outlook and to consider the more abstract and far-reaching question: ‘What is a network?’

In the previous chapters, we have described various outcomes derived from the basic research, such as a growth theory of companies, business networks and agent simulations. In this chapter, by changing our perspective, we will review what consequences are expected from the research outcomes, in other words the practical applications of econophysics. Real economic data is abundant in Japan compared with other countries, and for this reason alone the Japanese economy is of particular importance in our work. It is also convenient in other ways. The practical applications of our work are currently under development, and the concrete content of these applications is being improved through ongoing conversations with companies in Japan. Indeed, the contents of this chapter should be regarded as an interim statement, which might be greatly altered in the future. However, we think that it is of real interest even at this stage to describe some of our suggested applications. We will concentrate on three topics, namely, the methodology for developing a business strategy, the management of the propagation of credit risk and the encouragement of innovation in business models, and offer them as of potential value to central government civil servants, acquisition specialists, investment bankers, staff working in the management planning division of companies, financial departments, materials departments, credit divisions, and rating companies, as well as business administrators.

Development of business strategies

In this section we will begin by explaining the elements of corporate finance theory, and then move on to discuss issues relating to the practical applications of that theory.

This book is one outcome of the new field of econophysics, and explains a wide range of recent findings relating to the dynamics of companies. While economics and physics each have long histories of their own, and their methods and purposes are obvious, econophysics, which has only a twenty-year track record, is still unfamiliar to many. Indeed, an emerging interdisciplinary approach in which the economy is studied with the tools of physics may provoke doubts as to whether the methods of a hard science can tell us anything about phenomena in which human beings are essential players. However, economics has in fact mimicked physics since the nineteenth century. This is particularly true of those who developed modern economics, the ‘neoclassical’ economists. The old masters such as Alfred Marshall and Léon Walras all drew inspiration from Newtonian mechanics. The fundamental concept of ‘equilibrium’, known to all students of the subject, is, of course, borrowed from physical science.

Thus, a moment's reflection shows us that the relation between physics and economics is long-standing and far closer than is commonly realised. Nevertheless, the recent development of econophysics represents a significant development. While traditional economics learned from classical mechanics, which analyses behaviours such as that of a ball thrown in the air or the motion of a weight at the end of a spring, econophysics looks to the statistical methods of the modern physicist.

In the preceding chapter we examined the statistical laws and patterns observed in distributions of company sizes, and it was noted that the tails of the distributions of flow and stock variables quantifying company size both obey Pareto's law.

Indeed, as was mentioned in subsection 2.1.2, Pareto's power law is commonly found in many phenomena observed within the natural sciences, as well as the social and economic sciences. Most researchers in these fields would accept that there is no single mechanism giving a universal explanation for all these phenomena, but rather that different mechanisms would be found for each of a variety of classes of phenomena.

Company size obviously changes over time: last year's sales are not equal to this year's; the previous quarter's total assets are different from those in this quarter; and debt increases or decreases according to levels of borrowing from financial institutions or from investors. Thus we can see that observations on the distribution of company size are a snapshot of the state of a collection of companies, each of which is individually subject to fluctuations.

Our analysis is now being led towards the examination of temporal change in company size in terms of flow and stock variables, with a view to providing an account of the appearance of size distributions seen as a consequence of fluctuations. Therefore, our next step is to understand fluctuations in company size, in other words the growth or contraction of companies.

salviati: Greetings, Simplicio, Sagredo. You are both looking very thoughtful; which I hope is a good sign and shows that you have made good progress with econophysics. In any case, I am very eager to hear what kind of impression the book has left upon you?

simplicio: Well, I thought the most interesting coffee break bit is …

sagredo: Oh heavens above; I sometimes wonder if you are a serious person, Simplicio. I haven't forgotten that it was you who made our good friend Galileo go down on his knees before the Pope.

simplicio: That is a long time ago.

sagredo: Indeed, and it is still fresh in my memory.

salviati: Now, now, friends, let us keep our eye on the matter in hand. Sagredo, you first; tell me what you think.

sagredo: Well, in spite of what you said about it being written as non-technically as possible and for a wide range of readers, I found much of the argument quite complicated. In any case, I learned of many concepts that were new to me, for example the Pareto distribution, the fat tail, Gibrat's law, complex networks and open innovations, to name a few.

salviati: Good. At least you are now free from the restrictive view of the normal Gaussian distribution. Incidentally, our good friend Eugene Stanley has a very inspiring thing to say about the current global economic crisis in relation to the Pareto distribution (Stanley, 2009).

Financial derivatives – and options in particular – form an important component of financial instruments. Considerable negative criticism has been directed at options and derivatives in light of the 2008 economic crisis, some of it being justified and some being off the mark. As long as there are assets and liabilities, there will be a need to protect the future value of assets, as well as of finding ways for maximizing returns on assets. Derivatives play a central role in achieving these twin objectives.

Given the uncertainties of the financial markets, there is a strong demand from banks, financial organizations, and investors for predicting the future behavior of securities. Derivative instruments, and options in particular, are a response to this need of the market and are widely traded in the financial markets.

Options and other derivatives of underlying financial securities have contributed significantly to the explosion of the capital markets and their general principles are discussed. There are three broad categories of derivatives, namely forwards, futures, and options. Option theory is developed for equities using a path integral formulation of white noise.

A series expansion of an option's price is defined for a generic case, in powers of the underlying security's volatility. The volatility expansion is of great generality and will turn out to be crucial in developing approximation schemes for a variety of interest rates and coupon bond options.

Options on interest rates and coupon bonds share many general properties with equity options, but are far more complex and have a much richer internal structure. There is a great variety of interest rate derivatives, which comprise over 50% of the total derivatives' markets. Options on interest rates are primarily based on interest rate caps and interest rate swaptions. Coupon bonds and options are defined and it is shown that swaptions are a special case of coupon bond options. Various put–call parity relations are derived for interest rates and coupon bond options.

The HJM (Heath–Jarrow–Morton) model of interest rates is based on stochastic calculus and is briefly discussed using a path integral formulation. The HJM model serves as an example for demonstrating the point of departure of the quantum finance formulation of interest rates from the one based on stochastic calculus.

Introduction

Coupon bond options and interest derivatives comprise a major subfield of finance. To convey some of the key features of this subfield, the US credit derivatives' market – being globally the largest – is briefly reviewed. Figure 4.1(a) gives the notional value of outstanding credit derivatives and Figure 4.1(b) gives a breakdown of the diverse variety of swap derivatives most frequently used in the US capital markets. Since 2001, the global credit derivatives' market had grown at a phenomenal annual rate of over 100%; from relatively insignificant beginnings at the turn of the new millennium, by 2006 the notional value of credit derivatives had reached US$26 trillion.

The industry standard for pricing an interest rate caplet is Black's formula, which was derived in Section 8.2 from the Libor Market Model. The underlying Libor forward interest rates fL(t, x) are known to be nonlinear, as discussed in Section 6.11.1. A different price of the caplet, namely the linear pricing formula, was derived in Section 9.7 using the bond forward interest rates.

An empirical study is carried out of the linear caplet pricing formula. The main purpose is to ascertain how important are the differences in the Libor Market Model and bond forward interest rates – using the pricing of caps and caplets as an example. In particular, the linear caplet price is compared with the market price of caps and caplets to obtain an estimate of the importance of the nonlinear effects that are the hallmark of the Libor Market Model.

Historical volatility and correlation of forward interest rates are used for predicting the linear caplet price; another approach is to predict the linear price from a parametric formula of the effective volatility using market caplet prices. The study shows that bond forward interest rates generate prices of a caplet and cap with fairly large errors, greater than 17%.

Introduction

The price of a mid-curve caplet has been obtained in Eq. (9.36) based on the bond forward interest rates' model. The result is called the linear caplet price to distinguish it from Black's caplet price, which is an exact result of the Libor Market Model.

The US financial system has been going through major transformations during the last 30 years and is thought to be a precursor of the changes that all mature financial systems will go through. The US financial system has been an innovator of new and novel financial instruments and the impact of these instruments on the economy can be studied by examining their effect on the US economy.

The primary focus of this book is the debt market. The US debt markets are analyzed as an exemplar of the characteristics of a global leader of the debt markets.

Another reason for studying the US debt market is that extensive data on financial instruments and on the derivatives market are available in the public domain. All the data for the graphs and diagrams in this appendix are taken from publicly listed sources.

Growth of US debt market

One of major changes in the structure of the US economy during the last 30 years has been the increasing importance of the financial sector in generating corporate earnings. As shown in Table B.1, the fraction of corporate earnings from the financial sector has grown almost 400% over the last 60 years and 50% over the last 20 years.

The total debt of the US, both internal and external, has undergone a dramatic increase. As shown in Figure B.1 (a) the total US public and private debt over the last 30 years has grown from US$3 trillion in 1975 to US$42 trillion in 2005, close to the entire world's 2005 Gross Domestic Product (GDP) of about US$44 trillion.

At present, mainstream theoretical finance is almost completely dominated by stochastic calculus. Quantum finance, in contrast, presents a formulation of finance that is completely independent of stochastic calculus. Quantum finance addressed only a few problems of finance. The onus thus fell on me to demonstrate that a wide class of problems of finance, which at present are understood in terms of stochastic calculus, have a natural quantum finance formulation and generalization. It needed to be established that the quantum formulation of finance is equivalent to the one based on stochastic calculus and – if it is to be useful for the practitioners – is conceptually transparent and computationally tractable. Furthermore, it needed to be shown that quantum finance is not merely equivalent to stochastic calculus but goes beyond it; namely, that models based on quantum finance are more general than those based on stochastic calculus.

Interest rates and coupon bonds and their derivatives form a major component of the debt market. They are, by far, the most complex and intricate types of financial instruments that also have a rich mathematical structure. Debt instruments provided an ideal testing ground for quantum finance. I gravitated to the study of interest rates and coupon bonds, since these financial instruments provide an excellent arena for exhibiting the main features of quantum finance – as well as for illustrating its point of departure from stochastic calculus.

No attempt has been made to survey the subject of debt instruments, for which there exists a vast corpus of literature. Instead, this book has focused on only new results that have been obtained by applying quantum finance to debt instruments.