| Foreign exchange(FX)options are an important tool for financial institutions and import and export enterprises to hedge exchange rate risk.At the same time,it is also an important means for China to deal with foreign exchange risks and improve the financial market.Reasonable pricing of FX options and improved pricing accuracy have significant practical significance for enhancing exchange risk resistance and improving the development of financial markets.Therefore,the FX option pricing model has become the research object of scholars' attention.Garman and Kohlhagen proposed the classic BS-GK model based on the BlackScholes model.Although the Black-Scholes model has been modified,it still follows the assumption that the asset price movement follows a normal distribution.As we all know,this assumption is not in line with reality,resulting in the pricing effect of BS-GK model is not satisfactory.Therefore,finding a FX option pricing model more in line with market conditions has become a major research hotspot in current financial engineering and financial mathematics.The bifractional Brownian motion(biFBM)model is a modification of the Brownian motion model.It is a more general Gaussian process.It is a semi-martingale under certain conditions,and it takes into account long memory and self-similarity,which is more in line with the actual market situation.In this paper,the statistical analysis of the logarithmic rate of return of the USD / CNH shows that the FX rate not only has non-normal distribution characteristics such as sharp peaks and thick tails,but also has long memory.In order to make the pricing model better reflect these characteristics and make up for the shortcomings of the BS-GK model,this paper chooses biFBM model for research.First,using the Wick-Itó formula and the classic heat conduction equation,the PDE(Partial Differential Equation)and pricing formulas for the FX option prices satisfied by the model are obtained.Secondly,the rationality of the pricing formula is explained through numerical simulation.Finally,the market data is used to calculate the MSE(Mean Square Error)and the PMSE(Percentage Mean Square Error)to compare the pricing accuracy of the model and the BS-GK model.Through empirical research,it is shown that the pricing effect of the biFBM model for in-sample data and out-sample data(in the money)is slightly lower than the BS-GK model,but significantly better than the BS-GK model for at the money and out of money of out-sample data.The biFBM model can characterize long memory in asset prices,but the trend of price changes is uncertain,often reversing,and asset prices may even undergo drastic changes.The biFBM model clearly fails to capture these changes.The CGMY model is a more general exponential Lévy model,which has an infinite activity rate.It can capture dramatic changes in asset prices and small jumps of different amplitudes.Also,it has a steady-state distribution and has a better simulation effect on asset prices.Theoretically it is a good foreign exchange option pricing model.Therefore,this paper chooses the CGMY model for further research.First,the FPDE(Fractional Partial Differential Equation)governing the FX option price under this model is derived,and the pricing formula is obtained through methods such as Fourier transform and inverse transform.Secondly,the validity of the formula is demonstrated through numerical experiments.In this process,the generalized Laguerre-Gauss quadrature formula and a new scaling parameter are used to solve the computational difficulties.Finally,in order to deepen the understanding of the model,a numerical example is conducted to analyze the influence of parameters on the option price,hoping to provide more useful references for the study of FX option pricing under the CGMY model. |
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