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The Application of Stochastic Processes in Options Pricing Model

Since English botanist Brown first observed the inconsistent mobility of pollen floating in liquid in 1827, stochastic motion has become one of the primary research topics today, and the use of stochastic processes in financial markets has never stopped [1]. The change in the process in any time interval do not affect the change in any other time intervals that do not overlap with of Brownian Motion, which possesses stationary independent increments. Therefore, this paper obtains the Black-Scholes Merton formula from the stochastic process through layers of derivation and analyses its limitations in terms of its existence. Therefore, this paper will derive the BSM formula and the logical relationships between them, step by step, starting from the stochastic process and discuss the problems and limitations of the BSM formula. The experiential results demonstrate that the proposed method can achieve better performance.