Pricing Parisians and barriers by hitting time simulation

Abstract

Parisian options are not exchange traded, but there are various applications of Parisian optionality in the fields of real option theory, convertible bond valuation and credit risk. Especially the valuation of consecutive Parisian options is complicated and there exist no explicit formulas for these contracts. So far valuation can be done by numerically inverting Laplace transforms or by PDE methods. This paper develops a Monte Carlo method by exploiting the Markovian nature of the underlying value process. As a result, the Parisian option value can be written as an expression that can be solved by Monte Carlo integration, where the Parisian times are the random variables that need to be simulated. The Parisian times cannot be simulated directly as there exists no explicit distribution function. Therefore, these times are approximated by the simulation of hitting times in a special way. The quality of this approximation can be controlled and is a trade-off between accuracy and computation time.