No abstract

We study the numerical Adomian decomposition method for the pricing of European options under the well-known Black–Scholes model. However, because of the nondifferentiability of the pay-off function for such options, applying the Adomian decomposition method to the Black–Scholes model is not straightforward. Previous works on this assume that the pay-off function is differentiable or is approximated by a continuous estimation. Upon showing that these approximations lead to incorrect results, we provide a proper approach, in which the singular point is relocated to infinity through a coordinate transformation. Further, we show that our technique can be extended to pricing digital options and European options under the Vasicek interest rate model, in both of which the pay-off functions are singular. Numerical results show that our approach overcomes the difficulty of directly dealing with the singularity within the Adomian decomposition method and gives very accurate results.

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 [Formula: see text] 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.