A statistical generalization of microeconomics has been made in Baaquie (2013), where the market price of every traded commodity, at each instant of time, is considered to be an independent random variable. The dynamics of commodity market prices is modeled by an action functional and the focus of this paper is to empirically determine the action functionals for different commodities. The correlation functions of the model are defined using a Feynman path integral. The model is calibrated using the unequal time correlation of the market commodity prices as well as their cubic and quartic moments using a perturbation expansion. The consistency of the perturbation expansion is verified by a numerical evaluation of the path integral. Nine commodities drawn from the energy, metal and grain sectors are studied and their market behavior is described by the model to an accuracy of over 90% using only six parameters. The paper empirically establishes the existence of the action functional for commodity prices that was postulated to exist in Baaquie (2013)

The linearized Hamiltonian model is proposed to extend the London Interbank Offered Rate (LIBOR) Market Model (LMM). Firstly, we studied the Hamiltonian of LMM in the framework of quantum finance, and the nontrivial upper triangle form of LIBOR drift is derived. The linearized Hamiltonian is derived to improve the explanatory capability of the model for market data. Our approach uses one more parameter to explain the initial condition and the model can be used to calibrate LIBORs with extremely high accuracy. Furthermore, the market time index is required for applying the model to multi-LIBOR, and the results imply that the LIBOR future time lattice becomes shorter as one goes from near future to distant future.

A statistical generalization of microeconomics has been made in Baaquie (2013). In Baaquie et al. (2015), the market behavior of single commodities was analyzed and it was shown that market data provides strong support for the statistical microeconomic description of commodity prices. The case of multiple commodities is studied and a parsimonious generalization of the single commodity model is made for the multiple commodities case. Market data shows that the generalization can accurately model the simultaneous correlation functions of up to four commodities. To accurately model five or more commodities, further terms have to be included in the model. This study shows that the statistical microeconomics approach is a comprehensive and complete formulation of microeconomics, and which is independent to the mainstream formulation of microeconomics.

The industry standard Black-Scholes option pricing formula is based on the current value of the underlying security and other fixed parameters of the model. The Black-Scholes formula, with a fixed volatility, cannot match the market's option price; instead, it has come to be used as a formula for generating the option price, once the so called implied volatility of the option is provided as additional input. The implied volatility not only is an entire surface, depending on the strike price and maturity of the option, but also depends on calendar time, changing from day to day. The point of view adopted in this paper is that the instantaneous rate of return of the security carries part of the information that is provided by implied volatility, and with a few (time-independent) parameters required for a complete pricing formula. An option pricing formula is developed that is based on knowing the value of both the current price and rate of return of the underlying security which in physics is called velocity. Using an acceleration Lagrangian model based on the formalism of quantum mathematics, we derive the pricing formula for European call options. The implied volatility of the market can be generated by our pricing formula. Our option price is applied to foreign exchange rates and equities and the accuracy is compared with Black-Scholes pricing formula and with the market price.

We study the range accrual swap in the quantum finance formulation of the LIBOR market model (LMM). It is shown that the formulation can exactly price the path dependent instrument. An approximate price is obtained as an expansion in the volatility of Libor. The Monte Carlo simulation method is used to study the nonlinear domain of the model and determine the range of validity of the approximate formula. The price of accrual swap is analyzed by generating daily sample values by simulating a two dimension Gaussian quantum field.