A Study On The Pricing Of Discrete Monitoring Arithmetic Mean Asian Options Based On Gauss-Hermite Orthogonal Non-uniform Distribution Algorithm

Option,is a right that gives the purchaser a contract to buy or sell a certain un-derlying asset at a certain price at a certain time in the future.With the improvement of financial market demand complexity,it is difficult to meet the special needs of cus-tomers only by using standard options.In order to meet the special needs of the market and customers,but also to avoid the own risks faced,besides trading familiar Euro-pean and American options,many financial companies have also pioneered the design of a large number of new varieties that are usually traded in the over-the-counter mar-ket from standard option changes,combinations,and derivatives——non-standardized derivative securities,called "new options" with path dependence characteristics.That is,the price of an option not only depends on the price of the underlying asset on its expiry date,but also depends on the path of the price of the underlying asset during the validity period of the option.Asian options are one of the representative products,and it is also the most active new type of option in the financial derivatives market today.There are several reasons why Asian options could develop rapidly:First,according to the characteristics of path dependence,it can prevent the price of the underlying asset from being manipulated and controlled market risk;Second,compared to the strategy of using standard option combinations to replicate the structure of Asian options,the cost of directly constructing Asian options is cheaper.The path dependence characteristics make the pricing model of Asian options show a relatively large difference compared with that of standard options,and the pricing problem is far more complicated than that of European options.Around the Asian option pricing research,most revolves the continuous Asian options as the research object,but only discrete monitoring arithmetic average Asian options are actually traded in the financial market.However,there is no precise closed solution for dis-crete monitoring arithmetic average Asian options.Thus,many scholars and financial practitioners have proposed effective numerical simulation pricing algorithms for dis-crete monitoring arithmetic average Asian options based on various models.Therefore,this article selects discrete Asian options as the research object,which is more in line with the development needs of my country's financial market and more in line with the needs of the market.In this study,we will construct a pricing algorithm for Eu-ropean discrete monitoring arithmetic average Asian options.This paper mainly uses the Gauss-Hermitian quadrature algorithm to construct the relevant grid.Then,under some assumptions of Forsyth et al.,a non-uniform allocation scheme and Lagrangian multiplier method are used to select the optimal number of arithmetic averages that need to be allocated for each grid point.Because fixed strike price Asian options and floating strike price Asian options can be connected by an equation,and European dis-crete monitoring arithmetic average Asian options European discrete Asian put options can be priced by the average put option price.Therefore,here mainly uses European fixed-price Asian call options as the research object to propose approximate pricing algorithms.For other types of discrete arithmetic average Asian options,we can also approximate pricing through the algorithm proposed in this article.The final numer-ical results show that the algorithm has quadratic time convergence in the number of nodes,which is comparable to that proposed by Hsu et al.More interestingly,our algorithm has linear time convergence in terms of monitoring times,which means that when we have a lot of monitoring time,our algorithm runs faster than the algorithm of Hsu et al.