Bayesian inference approach to inverse problems in a financial mathematical model

Abstract

This paper investigates an inverse problem of option pricing in the extended Black–Scholes model. We identify the model coefficients from the measured data and attempt to find arbitrage opportunities in financial markets using a Bayesian inference approach. The posterior probability density function of the parameters is computed from the measured data. The statistics of the unknown parameters are estimated by a Markov Chain Monte Carlo (MCMC), which explores the posterior state space. The efficient sampling strategy of an MCMC enables us to solve inverse problems by the Bayesian inference technique. Our numerical results indicate that the Bayesian inference approach can simultaneously estimate the unknown drift and volatility coefficients from the measured data.

Keywords:

• Inverse problem
• option pricing
• Bayesian inference approach

2010 Mathematics Subject Classifications:

• 65L09
• 65M32
• 62F15
• 91G60

Acknowledgments

The author would like to thank the referee for carefully reading our manuscript and for giving such constructive comments which substantially helped to improve the quality of our paper.

Disclosure statement

The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.

ORCID

Yasushi Ota http://orcid.org/0000-0003-0534-0031

Yu Jianghttp://orcid.org/0000-0003-4551-9335

Masaaki Uesakahttp://orcid.org/0000-0001-6377-7472

Additional information

Funding

The first author would like to acknowledge the supports from Grant-in-Aid for Scientific Research (C) 18K03439 of the Japan Society for the Promotion of Science (JSPS). The second author was supported by the National Natural Science Foundation of China [grant no. 11971121]. The third author would like to acknowledge the supports from Grant-in-Aid for Scientific Research [15K21766 and 15H057403] of the Japan Society for the Promotion of Science (JSPS).