Abstract: Installment options are designed for an investor who is willing to pay a little extra for the opportunity to terminate a contract and reduce losses caused by a void investment position. Because of this extra privilege, installment options are weakly path-dependent. American continuous-installment put options are discussed in this paper. In addition to the right to terminate the contract by stopping the payments at any time, the holder also has the right to exercise the option at any time until maturity; this leads to three potential choices during the holding period:cancel, exercise, or hold on. The mathematical pricing model of this option can be formulated as a parabolic variational inequality, which is a free boundary problem. The existence and uniqueness of the solution can be solved using the penalty method and regular PDE arguments. Different from the standard American put option, this option has two free boundaries irrespective of dividends. Attention is focused on properties of the free boundaries, such as monotonicity, smoothness, and location.

Key words: installment option, American put option, option pricing, free boundary