DeepPricing:pricing convertible bonds based on financial time-series generative adversarial networks

Convertible bonds are an important segment of the corporate bond market,however,as hybrid instruments,convertible bonds are difficult to value because they depend on variables related to the underlying stock,the fixed-income part,and the interaction between these components.Besides,embedded options,such as conversion,call,and put provisions are often restricted to certain periods,may vary over time,and are subject to additional path-dependent features of the state variables.Moreover,the most challenging problem in convertible bond valuation is the underlying stock return process modeling as it retains various complex statistical properties.In this paper,we propose DeepPricing,a novel data-driven convertible bonds pricing model,which is inspired by the recent success of generative adversarial networks(GAN),to address the above challenges.The method introduces a new financial time-series generative adversarial networks(FinGAN),which is able to reproduce risk-neutral stock return process that retains the unique statistical properties such as the fat-tailed distributions,the long-range dependence,and the asymmetry structure etc.,and then transit to its risk-neutral distribution.Thus it is more flexible and accurate to capture the dynamics of the underlying stock return process and keep the rich set of real-world convertible bond specifications compared with previous model-driven models.The experiments on the Chinese convertible bond market demonstrate the effectiveness of DeepPricing model.Compared with the convertible bond market prices,our model has a better convertible bonds pricing performance than both model-driven models,i.e.Black-Scholes,the constant elasticity of variance,GARCH,and the state-of-the-art GAN-based models,i.e.FinGAN-MLP,FinGAN-LSTM.Moreover,our model has a better fitting capacity for higher-volatility convertible bonds and the overall convertible bond market implied volatility smirk,especially for equity-liked convertible bonds,convertible bonds trading in the bull market,and out-of-the-money convertible bonds.Furthermore,the Long-Short and Long-Only investment strategies based on our model earn a significant annualized return with 41.16%and 31.06%,respectively,for the equally-weighted portfolio during the sample period.

Eigen microstates and their evolutions in complex systems

Emergence refers to the existence or formation of collective behaviors in complex systems.Here,we develop a theoretical framework based on the eigen microstate theory to analyze the emerging phenomena and dynamic evolution of complex system.In this framework,the statistical ensemble composed of M microstates of a complex system with N agents is defined by the normalized N×M matrix A,whose columns represent microstates and order of row is consist with the time.The ensemble matrix A can be decomposed as■,where r=min(N,M),eigenvalueσIbehaves as the probability amplitude of the eigen microstate U_I so that■and U_I evolves following V_I.In a disorder complex system,there is no dominant eigenvalue and eigen microstate.When a probability amplitudeσIbecomes finite in the thermodynamic limit,there is a condensation of the eigen microstate UIin analogy to the Bose–Einstein condensation of Bose gases.This indicates the emergence of U_I and a phase transition in complex system.Our framework has been applied successfully to equilibrium threedimensional Ising model,climate system and stock markets.We anticipate that our eigen microstate method can be used to study non-equilibrium complex systems with unknown orderparameters,such as phase transitions of collective motion and tipping points in climate systems and ecosystems.