American Option Pricing under Uncertainty

摘要

A mathematical model for American put option with uncertainty is presented and the randomness and fuzzi-ness are are evaluated by both probabilistic expectation and fuzzy expectation defined by a possibility measure from the viewpoint of fuzzy expectation, taking account of decision-maker's subjective judgment. An optimal-ity equation for the optimal stopping problem in a fuzzy stochastic process is derived and an optimal exercise time is given for the American put option. Two models in American options with uncertainty, the case with an expiration date and the perpetual option case, are demonstrated and it is shown that the optimal fuzzy price is a solution of the optimality equation under a reasonable assumption. The writer's/seller's permissible range of optimal expected price in the American put option are presented and the meaning and properties of the optimal expected prices are discussed in two numerical examples. Finally, three models with jumps are also discussed.