Penalty method for pricing American-style Asian option with jumps diffusion process

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 domain.  By including Jump-diffusion in the models, they are able to capture the skewness and kurtosis features of return distributions often observed in several assets in the market.  The performance of the schemes is investigated through a series of numerical experiments.