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Randomized Stopping Times and American Option Pricing with Transaction Costs
Author(s) -
Chalasani Prasad,
Jha Somesh
Publication year - 2001
Publication title -
mathematical finance
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.98
H-Index - 81
eISSN - 1467-9965
pISSN - 0960-1627
DOI - 10.1111/1467-9965.00107
Subject(s) - optimal stopping , stochastic game , arbitrage , upper and lower bounds , economics , transaction cost , stopping time , portfolio , mathematical economics , duality (order theory) , value (mathematics) , mathematics , microeconomics , financial economics , mathematical analysis , statistics , discrete mathematics
In a general discrete‐time market model with proportional transaction costs, we derive new expectation representations of the range of arbitrage‐free prices of an arbitrary American option. The upper bound of this range is called the upper hedging price, and is the smallest initial wealth needed to construct a self‐financing portfolio whose value dominates the option payoff at all times. A surprising feature of our upper hedging price representation is that it requires the use of randomized stopping times (Baxter and Chacon 1977), just as ordinary stopping times are needed in the absence of transaction costs. We also represent the upper hedging price as the optimum value of a variety of optimization problems. Additionally, we show a two‐player game where at Nash equilibrium the value to both players is the upper hedging price, and one of the players must in general choose a mixture of stopping times. We derive similar representations for the lower hedging price as well. Our results make use of strong duality in linear programming.