Abstract: The parametric estimation of the diffusion process has naturally born deficiency, with the first impression firmly entrenched. Besides, it's a pretty tough job to set the diffusion function. However, by the nonparametric, prior information of the data generating process is not a must anymore, instead, it estimates the diffusion function based on discrete data directly, through which irons out the shortcomings mentioned above. Upon the continuous time process, this paper develops a nonparametric stock option model. It lifts all the restrictions for the diffusion function of the underlying process, constructs its nonparametric estimator helped by discrete data to match the density function, and further calculates its equilibrium price. This essay theoretically demonstrates the consistency and asymptotic normality of diffusion estimation. The empirical studies deliver a very clear message: compared with the actual market price, this presented method has a high accuracy of simulation, especially in the market turbulence, under which it estimates much better than the classic B-S model.

Key words: continuous time model, nonparametric kernel estimation, bandwidth, option pricing, stock index options