Valution For Optimal Two Stoppings Problems Of American Options Based On Swing Option

Multiple exercisable American options (MEAO) may be considered asgeneralizations of American-style options as they provide the holder more than oneexercise right. In the past decade, many examples in which MEAOs are directly orindirectly used start to emerge in finance sectors ranging from insurance to energyindustries; swing options are a particular example among many others. As the pricingof American options, the valuation of MEAOs is a problem in stochastic optimalcontrol. For American options the solution provides both a value and optimizingexercise price, or the optimal stopping time. For MEAOs, the solution resulted from apricing problem also gives both a value and optimizing policy. For pricing the generalAmerican option the method is to find the optimal stopping time within the validityperiod of the contract. So the point to solve the pricing problem is how to find thestopping times stipulated in the contract and make the overall profit maximum.Through the previous conclusions, in fact, American options with n exercise rightscan be simplified as n American options with one exercise right. Given thisconclusion makes the complex method of the optimal exercise strategy greatlysimplified.Since it is difficult to solve the analytical solution for general American option,we, instead, come to solve the numerical solution for general American option. Threemain and most accepted methods are Binary Tree method, Finite Difference methodand Monte Carlo simulation. Furthermore, the Least-squares Monte Carlo algorithmproviding a convenient method to estimate continuing value is widely adopted infollow-up studies.On the basis of the existing research on the multiple optimal stopping times, thispaper is aimed to provide a more optimized numerical simulation method on this kindof options.I use the Least-Square Monte Carlo algorithm to simulate pricing formulaof American put option with two exercise rights. The Least-Squares Monte Carloalgorithm uses least square method to regress the sample paths of the cross sectiondata on each moment, and then calculate the estimation of the conditional expectationvalue of continuing value. The continuing value at one time point can be expressed by a conditional expectation, and I calculate the Least-square estimation of the continuing value, let: Lj(X) is given basis function. The coefficient of basis function Lj(X) can be calculated by least square method under certain samples. In this way, the continuing value of options can be expressed by linear combination of basis functions.I mark the time point as stopping time when its exercising return is higher than the continuing value. And then on each sample path, I choose the two earliest time points as optimal stopping times. The value of American options with two exercise rights is the average of the discount value of stopping times of every sample path.The main conclusion of this paper is that the pricing formula of American option with two exercise rights by using the Least-Squares Monte Carlo algorithm is: Here is the discount of exercising option on τＪ→in the path of cωj, τＪ→=(τＪ1, τＪ2), τＪ1=inf｛t∈S; Yij>0｝, τＪ2=inf｛t∈S,t＞τＪ1; Yij>0｝.In the theorem, I also proved the convergence of this pricing formula:The use of this idea on the algorithm is not sophisticated and it will has a vital significance if this algorithm is applied to solving the motion of the relatively complex option pricing problems.