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Physical Review E

Feynman perturbation expansion for the price of coupon bond options and swaptions in quantum finance. II. Empirical

Belal Ehsan Baaquie, Cui Liang

Published Jan 11, 2007• 18 citations
Publisher

Abstract

The quantum finance pricing formulas for coupon bond options and swaptions derived by Baaquie [Phys. Rev. E 75, 016703 (2006)] are reviewed. We empirically study the swaption market and propose an efficient computational procedure for analyzing the data. Empirical results of the swaption price, volatility, and swaption correlation are compared with the predictions of quantum finance. The quantum finance model generates the market swaption price to over 90% accuracy.

DOI
10.1103/physreve.75.016704

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The quantum finance pricing formulas for coupon bond options and swaptions derived by Baaquie [Phys. Rev. E 75, 016703 (2006)] are reviewed. We empirically study the swaption market and propose an efficient computational procedure for analyzing the data. Empirical results of the swaption price, volatility, and swaption correlation are compared with the predictions of quantum finance. The quantum finance model generates the market swaption price to over 90% accuracy.
European options on coupon bonds are studied in a quantum field theory model of forward interest rates. Swaptions are briefly reviewed. An approximation scheme for the coupon bond option price is developed based on the fact that the volatility of the forward interest rates is a small quantity. The field theory for the forward interest rates is Gaussian, but when the payoff function for the coupon bond option is included it makes the field theory nonlocal and nonlinear. A perturbation expansion using Feynman diagrams gives a closed form approximation for the price of coupon bond option. A special case of the approximate bond option is shown to yield the industry standard one-factor HJM formula with exponential volatility.
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.
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.
An option price was derived exactly in Baaquie (2019) using the quantum oscillator. The derivation of the oscillator option price that is based on evaluating the pricing kernel using the Feynman path integral was briefly reviewed. The oscillator option price is a generalization of the Black-Scholes option price and contains new parameters. In the current paper, the hedging parameters are exactly derived for the generalized option price and are seen to be far more complicated than the Black-Scholes hedging parameters. The parameters in the oscillator option price do not require the daily value of implied volatility for pricing the option since these parameters encode the information that is missing in the Black-Scholes option price.