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 Markov Chain Monte Carlo (MCMC), which explores the posterior state space. The efficient sampling strategy of 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.

Parameters identification for an inverse problem arising from a binary option using a Bayesian inference approach

“…The above studies provided point estimates of unknown parameters by exact determination or least squares optimization, without rigorously examining and considering the measurement errors in the inverse solutions. In [25], we considered the Option Problem, which has an initial condition Φ(x, T ) in ( 1) such as…”

“…And we attempt to derive the option pricing equation of so-called Dupire type which we were unable to derive in [25], and to simultaneously estimate the drift and volatility coefficients from the measured data. We can apply results of this study to the problem that we estimate the market risk from the pricing of derivatives such as interest rate.…”

Parameters Identification for Inverse Option Problems Using Markov Chain Monte Carlo Methods

“…The above studies provided point estimates of unknown parameters by exact determination or least squares optimization, without rigorously examining and considering the measurement errors in the inverse solutions. In [25] we reconstruct the parameters not by linearizing the inverse problems but by applying Bayesian inference to IOP.…”