Penalty American options
We present a new American-style option whereby on the event of exercise before expiry, the holder pays the writer a fee (which will be referred to as a 'penalty'). The valuation of the option is not straightforward as it involves determining when it is optimal for the holder to exercise the option, leading to a free boundary problem. As most options in the traded markets have short maturities, accurate and fast valuations of such options are important. We derive analytic approximations for the value of the option with short times to expiry (up to 3 months) and its optimal exercise boundary. Some properties of the option, such as the put-call relationship, are explored as well. Numerical experiments suggest that our solutions both for the optimal exercise boundary and option value provide very accurate results.