Harnack Inequality And No-arbitrage Analysis

No-arbitrage problem has always been a hot and difficult issue in the field of financial economics which has attracted wide attention from researchers both at home and abroad,and achieved remarkable progress in recent decades.Carciola et al.considered a multi-dimensional B-S market model,and obtained its corresponding option pricing equation,a kind of Kolmogorov operator equation,by using a-hedging.Then proceeded from the fundamental solution,the Harnack inequality of the Kolmogorov operator equation is proved by the optimal control theory,and the no-arbitrage analysis of the self-financing financial market is obtained.Since a known fundamental solution of the operator is a prerequisite for using the optimal control method to prove the Harmack inequality,and once the fundamental solution is unknown,it is obvious that how to study the problem is meaningful.In this paper,by means of the gradient estimation method,we give a proof of the Harnack inequality,which is independent of the fundamental solution,and the arbitrage of the financial market is also studied.Firstly,for the sake of comparison,we consider the same option pricing equation.By constructing Harnack quantity and using the gradient estimation,we prove the Hamack inequality of this kind of Kolmogorov operator equation,and then the non-arbitrage of the self-financing financial market is further analyzed.Secondly,we begin with the simplified option pricing equation,then the proof of the Harnack inequality of the simplified option pricing equation is also obtained.Finally,by the proved Harnack inequality in this paper,a simple example analysis is given based on the practical transaction data of the SSE 50ETF options.The example illustrates that it is correct to describe a non-arbitrage behavior in a self-financing market based on the Harnack inequality.