Risk-neutral Density Estimation Using Bernstein Polynomial With Application To Finance

In option pricing,the risk-neutral method is a very important pricing method.This paper proposes a non-parametric method to estimate the risk-neutral density function of the underlying asset based on the Bernstein polynomial with positive coefficients.This article uses(m+1)Beta distribution mixture model approximates the risk-neutral density function,and derives the pricing formula of European options.When estimating model parameters,the problem of estimating coefficients is transformed into a quadratic programming problem,and the KKT condition is used to give the display expression of the coefficient estimated value,then use the change point estimation method to determine the optimal order of the model.In the numerical simulation,the lognormal distribution and the nearly lognormal distribution are selected as the true risk neutral density function,and the method proposed in this paper is compared with the mixed model of the double lognormal distribution and the cubic spline method for simulation.The results show that the Bernstein polynomial method is effective and robust.As an application,this paper applies the proposed Bernstein polynomial model to variance swap pricing in financial risk management.