| This paper deals with a finite difference method of option pricing equation,considering two problems: one is about the numerical method of Black-Scholes optionpricing partial differential equations, the other is about the numerical method of nonlinearpartialdifferentialequationswhentransactioncostsaretakenintoaccount.Consideringthe B-S option pricingmodel, we discretize the partial equation bytakinga mixed difference format : firstly ,utilizing a proper equivalent transform , the B-Sequation is transformed into a standard parabolic equation, then through taking divideddifference with respect to the time variable ,taking five-point differential formula withrespect to the space variable, and taking into a parameter ?, the partial equation can bediscretized into a resistant mixed difference format. By analyzing, the truncated error ofthis numerical methodis o ( h 4? k)whichis higherthanothermethods. What's more,weprove the convergence and stability of this method by Fourier analysis. In the end, thismethod proves to be good by doing some illustrative examples. Compared with othermethods, the relative error of this one is obviously smaller and it follow the rule that theexpiration time is longer, the relative error is smaller which shows that this numericalmethod is fit for the option pricing with longer expiration time. The numerical examplesalso show that the option price is relative to?, with ? smaller, the relative error issmaller, and when ? is zero, the relative error is smallest, then the mixed differenceformat develops into an explicit difference format which is more simple and easier tosolve.Inaword,thisnumericalmethodiseffectiveanduseful.Considering the option pricing model with transaction costs, we also take into afive-point difference format. There is an obstacle in proving the convergence and stabilitybecauseofthemodifiedvolatility.Butthisdoesn'taffecttheutilityofthemethod,becausethenumericalexamplesshowthattheoptionpricecoincideswiththefinancemarket. |
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