Study On Paris Option Pricing Using Crank-Nicolson Difference Method

In 1997,two American scholars,Robert Merton,a professor at Harvard Busi-ness School,and Myron Scholes,a professor at Stanford University,were awarded the 29th Nobel Prize in Economics.The Black-Scholes option pricing model they created and developed laid the foundation for the rational pricing of derivative fi-nancial instruments,including stocks and bonds.With the continuous development of financial industry,the pricing of financial derivatives such as options has become one of the main topics of financial quantitative research.Owing to the difference of option trading mode,direction and subject matter,there are different ways to classify options.For example,according to different exercise dates,options can be divided into three types:European option,American option and Bermuda option.With the continuous development of options,in order to adapt to various mar-ket demands,exotic options emerge as the times require.They are more complex derivative securities than ordinary options.Barrier option is a kind of exotic option.Its return depends on whether the price of the underlying asset reaches a certain level in a certain period of time.The Paris option introduced in this paper is also a special barrier option.It can activate the knock-in or knock-out characteristics only when the underlying price continues to break through the barrier in a certain period of time.At present,it has been widely used in speculation and hedging strategies in various markets,especially in the foreign exchange market.The main content of this paper:Firstly,based on the Black-Scholes framework,according to the characteristics of Paris options,the partial differential equation and the corresponding boundary conditions of Paris options are derived.Secondly,the dimensionality of the partial differential equation of call Paris option is reduced by knocking out the call Paris option continuously upward,and the three-dimensional partial differential equation is transformed into two-dimensional partial differential equationjwhich is convenient for further calculation.Finally,the partial differential equation is solved by Crank-Nicolson difference,and the pricing of Paris options is obtained.It is proved that Crank-Nicolson difference scheme is unconditionally stable and has convergence advantage.