This paper applies Zipf rank approach to measure how long it will take for the individual economy to reach the final state of equilibrium by using local-level data of China's urban areas. The indicators, the gross domestic product (GDP) per capita and the market capitalization (MCAP per capita of 150 major cities in China are used for analyzing their convergence. Besides, the power law relationship is examined for GDP and MCAP. Our findings show that, compared to the classical approaches:β-convergence and σ-convergence, the Zipf ranking predicts that, in approximately 16 years, all the major cities in China will reach comparable values of GDP per capita. However,the MCAP per capita tends to follow the periodic fluctuation of the economic cycle, while the mean-log derivation (MLD) confirms the results of our study. Moreover, GDP per capita and MCAP per capita follow a power law with an average value of α = 0.41 which is higher than α= 0.38 obtained based on a large number of countries around the world.

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.

This paper develops a model to describe the unequal time correlation between rate of returns of different stocks. A non-trivial fourth order derivative Lagrangian is defined to provide an unequal time propagator, which can be fitted to the market data. A calibration algorithm is designed to find the empirical parameters for this model and different denoising methods are used to capture the signals concealed in the rate of return. The detailed results of this Gaussian model show that the different stocks can have strong correlation and the empirical unequal time correlator can be described by the model's propagator. This preliminary study provides a novel model for the correlator of different instruments at different times.

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.

The simulation of the Libor Market Model (LMM) is extensively studied in the frame work of quantum finance.The imperfectly correlated Libor rates are simulated based on a Gaussian quantum field and a recursion equation of nontrivial stochastic drift. The Libor options are studied using both the simulation method and the analytical formula. The caplet price of simulation is compared with Black's caplet formula which can be exactly derived from the LMM. The invariance of caplet price for different forward bond numeraire is verified by using the simulation. The simulation results for coupon bond options and swaptions are compared with the approximate price, which are limited for the reason that the approximate price is derived using the small volatility expansion. The simulation method is shown to have great potential in the application of pricing interest rate instruments.