Libor L(t, T) is one of the primary interest rate instruments in the capital markets, the other being Euribor. The term Libor will be used generically for all interest rates on fixed deposits. The Libor Market Model (LMM) is defined in the framework of quantum finance and leads to a key generalization: the Libors, for different future times, are imperfectly correlated. A major difference between a forward interest rates' model and the LMM lies in the fact that the LMM is calibrated directly from the observed market values for L(t, T). The short distance Wilson expansion of the Gaussian quantum field A(t, x) driving the Libors yields a derivation of the Libor drift term that incorporates imperfect correlations of the different Libors. The logarithm of Libor ϕ(t, x) is defined and leads to a quantum field theory of Libor. The Lagrangian and Feynman path integral are obtained for the log Libor quantum field ϕ(t, x).

Introduction

Interest rates can be modeled using either the zero coupon bonds B(t, T) or the simple interest Libor L(t, T). Both these approaches are, in principle, equivalent but are quite different from an empirical, computational, and analytical point of view.

One can take the view that there exists a single set of underlying forward interest rates f(t, x) that can be used for modeling both L(t, T) and B(t, T). The HJM approach, in fact, takes this view and the HJM's quantum finance generalization goes a long way in accurately modeling interest rate instruments.

The correlation of two different coupon bond options is studied in the framework of bond forward interest rates discussed in Chapter 5. Coupon bond options are discounted using the money market numeraire. The correlation is studied for illustrating the mathematics required for pricing more complex instruments, including a more general version of the volatility expansion. The correlation of coupon bonds can lead to the definition of new derivative instruments.

Introduction

Exotic equity options often combine a basket of equities that are correlated; the price of the options reflect the effects of equity correlations, which are also required for hedging a portfolio of equities.

The correlation of coupon bond options has many new features not present in the pricing of a single coupon bond option. The calculation for the coupon bond option correlation generalizes the pricing formula obtained for the coupon bond option. The correlation results extend in a straightforward manner to the correlation of swaptions.

A major new feature of the coupon bond option correlation is that – not being traded in the financial markets – it does not have a martingale evolution; in particular, the drift is not fixed by the martingale condition. Instead, the drift for the individual coupon bond option has to be evaluated from market data.

The forward bond numeraire can no longer be used to simplify the option price calculations since the two coupon bond options, in principle, have different maturities. It turns out that the most efficient approach for evaluating the correlation function is to use the money market numeraire for discounting the value of the individual coupon bond options.

The prices of Libor options are obtained for the quantum finance Libor Market Model. The option prices show new features of the Libor Market Model arising from the fact that, in the quantum finance formulation, all the different Libor payments are coupled and (imperfectly) correlated.

Black's caplet formula for quantum finance is given an exact derivation. The coupon and zero coupon bond options as well as the Libor European and Asian swaptions are derived for the quantum finance Libor Market Model. The approximate Libor option prices are derived using the volatility expansion developed in Section 3.14.

The BGM–Jamshidian expression for the Libor interest rate caplet and swaption prices is obtained as the limiting case when all the Libors are exactly correlated.

Introduction

The Libor option prices are obtained from the Libor zero coupon bonds BL(t, T) – obtained from the Libor ZCYC curve ZL(t, T) discussed in Section 7.9 – and the benchmark three-month Libor L(t, T). For notational convenience, Libor zero coupon bonds BL(t, T) will be denoted by B(t, T).

All the options are defined to mature at future calendar time T0, with present time given by t0 = T−k; the notation of present being denoted by t0 is used to simplify the notation. It is natural for these options to choose B(t, T0) as the forward bond numeraire. In other words, the forward bond numeraire is B(t, TI+1) with I = −1 and Libor drift is calculated for this numeraire. Libor calendar and future time are shown in Figure 6.1 and Libor times t0 = T−k, T0, and Tn are shown in Figure 8.1.

European options on coupon bonds are studied in the framework of bond forward interest rates f(t, x) studied in Chapter 5 as a linear (Gaussian) quantum field. One of the advantages of the Gaussian formulation is that the coupon bond option has a representation that is tractable and allows for various analytical approximation schemes. More precisely, including the payoff function for the coupon bond option into the path integral for the option price makes it nonlocal and nonlinear. A perturbation expansion using Feynman diagrams gives a closed-form approximation for the price of European and Asian coupon bond options. The approximate bond option is studied for two limiting cases, namely (a) the industry standard one-factor exponential volatility HJM formula and (b) the BGM–Jamshidian model's swaption price.

Introduction

Coupon bonds and interest rate swaps are the most important derivatives of the debt markets and options on these instruments are widely traded. The pricing of European options on coupon bonds is studied in some detail using the quantum finance approach. The volatility of the forward interest rates is a small quantity, of the order of 10−2/year, and hence provides a small parameter for obtaining a volatility expansion for the option price. A perturbation expansion for the option price is developed, using Feynman diagrams, in a power series to fourth power in the forward interest rates' volatility.

A perturbative study of the coupon bond option price has been discussed in Section 8.5 for the Libor forward interest rates similar to the one that is the main focus of this chapter.

A quantum field theory of forward interest rates is developed as a natural generalization of the HJM model: the forward interest rates are allowed to have independent fluctuations for each future time. The forward interest rates are modeled as a two-dimensional Gaussian quantum field, leading to forward interest rates that have a finite probability of being negative. The model is consistent not for the interest rate sector but only for the bond sector and is consequently called the bond forward interest rates. The concept of a quantum field is briefly discussed in Appendix A. 7. The ‘stiff’ quasi-Gaussian model, together with the concept of market time, describes the forward interest rates. A differential formulation of forward interest rates' dynamics is obtained. Using a singular property of the forward interest rates' quantum field, a generalization of Ito calculus follows from the Wilson expansion. A derivation of a risk-neutral measure for zero coupon bonds is obtained based on the differential martingale condition.

Introduction

The complexity of the forward interest rates is far greater than that encountered in the study of stocks and their derivatives. A stock, at a given instant in time, is described by only one random variable (degree of freedom) S(t) and which is usually modeled using stochastic differential equations. In the case of interest rates, it is the entire interest rates yield curve f(t, x) that undergoes a random evolution. Clearly, at each instant, the most general random evolution is that the forward interest rates f(t, x), for each of future time x, should be an independent random variable.

The pricing formulas for coupon bond options derived in Chapter 11 are employed for an empirical study. This chapter studies the realization of swaptions as a special case of coupon bond options. Similar to the analysis of interest rate caplets in Chapter 10, by considering swaptions as a special case of coupon bonds, one in effect is using bond forward interest rates to model the swaptions. An empirical study of the swaption market is carried out in some detail and an efficient computational procedure is developed for analyzing swaption data. Empirical results of the swaption price, swaption volatility, and swaption correlation are compared with the predictions of the quantum finance model that generates, up to a scaling factor, the market swaption prices to an accuracy of over 90%.

Introduction

Interest rate swaptions have a deep and liquid market and arguably are today the most liquid option on interest rates; one of their major components is the highly liquid European swaptions. Swaption pricing is a nonlinear problem that has been widely studied using numerical techniques.

The book consists of three major themes. Any one of the three components can be read without many gaps in the analysis.

1. The introductory chapters are primarily intended for readers who are unfamiliar with the fundamental concepts of finance. The principles and mathematical expressions for debt instruments, which are analyzed in later chapters, are reviewed in Chapter 2, 3, and 4. Options are briefly discussed and the Black–Scholes option theory is given a path integral formulation.

2. A major subject matter of the book is the theory of coupon bonds. A quantum field theory of the bond forward interest rates f(t, x) is developed in Chapter 5 and forms a core chapter. It provides a model for the study of coupon and zero coupon bonds. Many of the derivations in later chapters are based on the quantum finance model of bond forward interest rates.

3. The quantum finance formulation of Libor interest rates is another major topic. The Libor Market Model is formulated in Chapter 6; the nonlinear Libor forward interest rates fL(t, x) that it is based upon are transformed into logarithmic Libor interest rates ϕ(t, x). In Chapter 7 some empirical properties of the Libor Market Model are studied and in Chapter 8 the prices of Libor options are obtained by using techniques of quantum field theory. A derivation of the Libor Market Model's nonlinear drift term is given in Chapter 15, based on the Libor Hamiltonian and state space of ϕ(t, x).

We will begin with several standard ideas from finance: no-arbitrage, the time value of money, and the Modigliani–Miller theorem, and then will introduce the newer concepts of liquidity and reversible trading, market instability, value in uncertain markets, and Black's idea of noise traders. More fundamentally, we'll formulate the efficient market hypothesis (EMH) as a Martingale condition, reflecting a hard-to-beat market. We'll also formulate hypothetical stationary markets, and show that stationary markets and efficient markets are mutually exclusive. A dynamic generalization of the neo-classical notion of value to real, nonstationary markets is presented. We will rely heavily on our knowledge of stochastic processes presented in Chapter 3. Orientation in finance market history is provided in the books by Bernstein (1992), Lewis (1989), Dunbar (2000), and Eichengreen (1996).

What does no-arbitrage mean?

The basic idea of horse trading is to buy a nag cheap and unload it on someone else for a profit. An analog of horse trading occurs in financial markets, where it's called “arbitrage.” The French word sounds more respectable than the Germanic phrase, especially to academics and bankers.

The idea of arbitrage is simple. If gold sells for $1001 in Dubai and for $989 in New York, then traders should tend to sell gold short in Dubai and simultaneously buy it in New York, assuming that transaction costs and taxes are less than the total gain (taxes and transaction costs are ignored to zeroth order in theoretical finance arguments). This brings us to two points.

Reductionism and holism

It was a fad in certain circles in the 1990s to announce that “reductionism is dead,” and that holism is necessary in order to understand biology and social systems. To discuss this we must first define the terms (The American Heritage Dictionary).

Holism is a theory or belief emphasizing the importance of the whole and the interdependence of its parts. In classical mechanics, for example, we understand the solar system in terms of its parts (the sun and the planets), and the interactions (“relationships”) are nonlinear. In quantum mechanics we understand complicated molecules like DNA in terms of the binding of smaller molecules. The genetic code is understood in terms of its building blocks: the four letter (computer) alphabet constructed from the bases A, C, G, and T. One understands the bases in terms of simpler molecules, the molecules in terms of atoms, and the atoms in terms of electrons and nuclei. This is reductionism: everything can be understood to be constructed systematically from electrons and positively charged nuclei. Yet, there is something that we cannot calculate from first principles using quantum mechanics: the behavior or the genes as a classical computer with three-letter words constructed from a four-letter alphabet. Quantum theory is linear, and the phase coherence cannot be destroyed from within, meaning that classical behavior cannot be derived from quantum theory without introducing a nonquantum assumption of destruction of phase coherence. This leads into the famous measurement problem clarified by von Neumann.