Research On Pricing Of Convertible Bonds Under Mixed Fractional Brownian Motion

Since 2017,China’s convertible bond market has been booming.This is because China Securities Regulatory Commission revised the implementation rules of nonpublic offering of shares in the same year and put forward a number of new regulatory rules.However,this new regulation did not restrict convertible bond financing,which led many companies to turn their eyes to convertible bond,a financial product with high flexibility.In 2022,the actual issuance of convertible bonds has reached 271.7billion yuan.Obviously,issuing convertible bonds has become a good choice for enterprises to raise funds.But even so,the market scale of Chinese convertible bond companies is still smaller than that of developed countries.Establishing effective pricing model suitable for Chinese convertible bond market is significant to promote the benign development of Chinese convertible bond market.Based on the review of domestic and foreign literature and the current mainstream pricing methods of convertible bonds,combined with the value change characteristics of financial assets such as autocorrelation and non-stationary increment,this paper decides to use mixed fractional Brownian motion stochastic differential equation to define the value of convertible bonds,and the overall pricing idea is based on the structural analysis method.The value of convertible bond is divided into pure bond value and option value.The discount cash flow method is adopted to calculate the partial value of pure bonds.The interest rate of several bonds with similar credit rating and issuing time and issuing period as the discount rate is selected.The specific calculation process of the partial value of convertible bond option is to derive the partial differential equation(PDE)of convertible bond option driven by mixed fractional Browan motion by ITO formula and no arbitrage principle,and then convert PDE into Cauchy problem of heat conduction equation by variable substitution method,and then solve it by classical solution formula of heat conduction equation.In addition,this paper also takes into account the option value of additional terms,and finally obtains the value of convertible bonds under the mixed fractional environment.In addition,since the outbreak of COVID-19 in late 2019,the medical industry has faced a severe test,and medical enterprises also need to resort to convertible bonds for financing.Therefore,in order to further verify the effectiveness of the pricing model constructed in this paper,12 large-scale medical convertible bonds were selected for comparative analysis.Then,in order to explore the adaptability of this model to the long-term value of convertible bonds,this paper took Jianfan convertible bonds as a case and selected the time node of one year after its listing for case analysis.The research results of this paper show that the mixed fractional pricing model has better data fitting effect than the traditional B-S model,and the method based on PDE in this paper is effective for the pricing of convertible bonds under this model.The main purpose of this paper is to provide a new model for the pricing of convertible bonds and a new way of thinking and financing for enterprises.Finally,this paper discusses the limitations of the model and future research areas.It is suggested that subsequent studies can consider the impact of stock price jumps,transaction costs and stock dividends on the model on the basis of this paper,and consider using machine learning methods to predict the value of convertible bonds,rather than simply relying on the pricing model.