Pricing Parisian and Parasian options analytically

In this paper, two analytic solutions for the valuation of European-style Parisian and Parasian options under the Black-Scholes framework are, respectively, presented. A key feature of our solution procedure is the reduction of a three-dimensional problem to a two-dimensional problem through a coordinate transform designed to combine the two time derivatives into one. Compared with some previous analytical solutions, which still require a numerical inversion of Laplace transform, our solutions, written in terms of double integral for the case of Parisian options but multiple integrals for the case of Parasian options, are both of explicit form; numerical evaluation of these integrals is straightforward. Numerical examples are also provided to demonstrate the correctness of our newly derived analytical solutions from the numerical point of view, through comparing the results obtained from our solutions and those obtained from adopting other standard finite difference approaches.