A BS-Bachelier Integrated Model And Its Application To Natural Gas Options

In April 2020,WTI crude oil futures and spot prices fell to negative,causing a great sensation.The price of the underlying asset falling to a negative value has caused a certain impact on the derivatives pricing system,so it is necessary to take into account the special situation that the underlying asset price may be negative in derivative pricing.In the event of negative price of WTI crude oil,the Chicago Mercantile Exchange(CME)modified the pricing rules and changed the option pricing model from Black-Scholes(BS,the underlying asset price is subject to the Geometric Brownian motion)to Bachelier(the underlying asset price is subject to the Brownian motion)to support possible negative prices.Based on the pricing rules proposed by CME,this paper establishes a BS-Bachelier Integrated Model.The conversion between the models is realized by setting two boundary values,which allow the underlying asset price to be negative,and then study the impact of negative price on option pricing.In terms of the model,first of all,according to the initial state,the price process of the underlying asset is divided into two situations:(1)The initial pricing model is the BS model;(2)The initial pricing model is the Bachelier model.Then,according to these two cases,the risk-neutral pricing formula and partial differential equations satisfied by option prices are written respectively.Finally,finite difference method is used to solve the corresponding partial differential equations,and Monte Carlo method is used to verify the accuracy of the numerical results.In terms of model application,this paper applies the BS-Bachelier integrated model to natural gas options where the underlying asset price may be negative.Firstly,through numerical calculations,this paper believes that the BS-Bachelier integrated model can better deal with the extreme situation that the underlying asset price is negative,and solves the problem that the BS model cannot price derivatives whose underlying asset price is negative.Furthermore,by analyzing the impact of important parameters on the model,this paper also obtains the following results.(1)If the underlying asset price is within the switching boundary range of the upper and lower mechanism,the option price is more sensitive to the change of the switching boundary value and the selection of the initial pricing model;(2)Regardless of the value of the underlying asset,the price of the call option is always positively correlated with the change of volatility or maturity and negatively correlated with the strike price;(3)The rho of the negative-priced asset call option is negative.Finally,this paper uses delta dynamic hedging method to manage the risk of negative price asset option.