The economic crisis of 2008 has shown that the capital markets need new theoretical and mathematical concepts to describe and price financial instruments. Focusing on interest rates and coupon bonds, this book does not employ stochastic calculus – the bedrock of the present day mathematical finance – for any of the derivations. Instead, it analyzes interest rates and coupon bonds using quantum finance. The Heath-Jarrow-Morton and the Libor Market Model are generalized by realizing the forward and Libor interest rates as an imperfectly correlated quantum field. Theoretical models have been calibrated and tested using bond and interest rates market data. Building on the principles formulated in the author's previous book (Quantum Finance, Cambridge University Press, 2004) this ground-breaking book brings together a diverse collection of theoretical and mathematical interest rate models. It will interest physicists and mathematicians researching in finance, and professionals working in the finance industry.
The economic crisis of 2008 has shown that the capital markets need new theoretical and mathematical concepts to describe and price financial instruments. Focusing on interest rates and coupon bonds, this book does not employ stochastic calculus – the bedrock of the present day mathematical finance – for any of the derivations. Instead, it analyzes interest rates and coupon bonds using quantum finance. The Heath-Jarrow-Morton and the Libor Market Model are generalized by realizing the forward and Libor interest rates as an imperfectly correlated quantum field. Theoretical models have been calibrated and tested using bond and interest rates market data. Building on the principles formulated in the author's previous book (Quantum Finance, Cambridge University Press, 2004) this ground-breaking book brings together a diverse collection of theoretical and mathematical interest rate models. It will interest physicists and mathematicians researching in finance, and professionals working in the finance industry.
The economic crisis of 2008 has shown that the capital markets need new theoretical and mathematical concepts to describe and price financial instruments. Focusing almost exclusively on interest rates and coupon bonds, this book does not employ stochastic calculus - the bedrock of the present day mathematical finance - for any of the derivations. Instead, it analyzes interest rates and coupon bonds using quantum finance. The Heath-Jarrow-Morton and the Libor Market Model are generalized by realizing the forward and Libor interest rates as an imperfectly correlated quantum field. Theoretical models have been calibrated and tested using bond and interest rates market data. Building on the principles formulated in the author's previous book (Quantum Finance, Cambridge University Press, 2004) this ground-breaking book brings together a diverse collection of theoretical and mathematical interest rate models. It will interest physicists and mathematicians researching in finance, and professionals working in the finance industry.
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.
This book applies the mathematics and concepts of quantum mechanics and quantum field theory to the modelling of interest rates and the theory of options. Particular emphasis is placed on path integrals and Hamiltonians. Financial mathematics is dominated by stochastic calculus. The present book offers a formulation that is completely independent of that approach. As such many results emerge from the ideas developed by the author. This work will be of interest to physicists and mathematicians working in the field of finance, to quantitative analysts in banks and finance firms and to practitioners in the field of fixed income securities and foreign exchange. The book can also be used as a graduate text for courses in financial physics and financial mathematics.
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 B L (t, T) – obtained from the Libor ZCYC curve Z L (t, T) discussed in Section 7.9 – and the benchmark three-month Libor L ( t, T ). For notational convenience, Libor zero coupon bonds B L (t, T) will be denoted by B ( t, T ). All the options are defined to mature at future calendar time T 0 , with present time given by t 0 = T − k ; the notation of present being denoted by t 0 is used to simplify the notation. It is natural for these options to choose B ( t, T 0 ) as the forward bond numeraire. In other words, the forward bond numeraire is B ( t, T I +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 t 0 = T − k , T 0 , and T n are shown in Figure 8.1.
Interest rate instruments form a major component of the capital markets. The Libor market model (LMM) is the finance industry standard interest rate model for both Libor and Euribor, which are the most important interest rates. The quantum finance formulation of the Libor market model is given in this paper and leads to a key generalization: all the Libors, for different future times, are imperfectly correlated. A key difference between a forward interest rate model and the LMM lies in the fact that the LMM is calibrated directly from the observed market interest rates. The short distance Wilson expansion [Phys. Rev. 179, 1499 (1969)] of a Gaussian quantum field is shown to provide the generalization of Ito calculus; in particular, the 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 phi(t,x) is defined and provides an efficient and compact representation of the quantum field theory of the Libor market model. The Lagrangian and Feynman path integrals of the Libor market model of interest rates are obtained, as well as a derivation given by its Hamiltonian. The Hamiltonian formulation of the martingale condition provides an exact solution for the nonlinear drift of the Libor market model. The quantum finance formulation of the LMM is shown to reduce to the industry standard Bruce-Gatarek-Musiela-Jamshidian model when the forward interest rates are taken to be exactly correlated.
Two underlying themes run through this book: first, defining and analyzing the subject of quantitative finance in the conceptual and mathematical framework of quantum theory, with special emphasis on its path-integral formulation, and, second, the introduction of the techniques and methodology of quantum field theory in the study of interest rates. No attempt is made to apply quantum theory in re-working the fundamental principles of finance. Instead, the term ‘quantum’ refers to the abstract mathematical constructs of quantum theory that include probability theory, state space, operators, Hamiltonians, commutation equations, Lagrangians, path integrals, quantized fields, bosons, fermions and so on. All these theoretical structures find natural and useful applications in finance. The path integral and Hamiltonian formulations of (random) quantum processes have been given special emphasis since they are equivalent to, as well as independent of, the formalism of stochastic calculus – which currently is one of the cornerstones of mathematical finance. The starting point for the application of path integrals and Hamiltonians in finance is in stock option pricing. Path integrals are subsequently applied to the modelling of linear and nonlinear theories of interest rates as a two-dimensional quantum field, something that is beyond the scope of stochastic calculus. Path integrals have the additional advantage of providing a framework for efficiently implementing the mathematical procedure of renormalization which is necessary in the study of nonlinear quantum field theories. The term ‘Quantum Finance’ represents the synthesis of the concepts, methods and mathematics of quantum theory, with the field of theoretical and applied finance.