penalty method

American-style options are important derivative contracts in today's worldwide financial markets.  They trade large volumes on various underlying assets, including stocks, indices, foreign exchange rates, and futures.  In this work, a penalty approach is derived and examined for use in pricing the American style of Asian option under the Merton model.  The Black–Scholes equation incorporates a small non-linear penalty factor.  In this approach, the free and moving boundary imposed by the contract's early exercise feature is removed in order to create a stable solution d