Under the Black-Scholes model, it can be shown that the American style put option price can be decomposed into the sum of the European put option and an early exercise premium. Similarly in a jump-diffusion market, the price of an American-style option (call with dividends, put with/without dividends, exchange options with dividends) can also be decomposed into the sum of the European put option and an early exercise premium. However the early exercise premium in this case can be further decomposed into two terms. One term arises the diffusion part of the dynamics of the stock prices. The second term arises from the jump part of the dynamics of the stock prices. This second term can be interpreted as rebalancing costs should the jumps bring the stock prices back from the early exercise region to the continuing region. However, as with the case of the pure-diffusion model, no closed-form solution for the early exercise premium exists, but we can express it in the form of an integral representation which can be solved numerically (or using transform methods) through a series of linked equations.
Contact: Gerald Cheang
GHL Cheang, C Chiarella & A Ziogas, 'The Representation of American Option Prices under Stochastic Volatility and Jump-Diffusion Dynamics', Quantitative Finance, 2011.
GHL Cheang & C Chiarella, 'Exchange Options under Jump-Diffusion Dynamics', Applied Mathematical Finance, 18:245-276, 2010.