2 MATHEMATICAL MOEL OF SHOUT OPTIONS
In this section, we derive a new partial integro-differential inequality
(PIDI) for complicated shout options pricing on the assumption that the
price of the underlying asset follows the jump-diffusion model and
construct the mathematical model by combining specific features and
terminal conditions.
The simple form of shout options is defined in John C. Hull [10].
Shout options is such an option that the holder can shout to the option
seller during the life of the option. At the end of the life of the
option, the option holder receives either the usual payoff from a
European option or the intrinsic value at the time of the shout,
whichever is greater. That is, the payoff function of simple shout
options is
\begin{equation}
g\left(S,K\right)=\left\{\begin{matrix}\&\max\left\{K-S_{T},0\right\},\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \text{if}\ \text{no}\ shout,\\
\&\max\left\{S_{\tau}-S_{T},0\right\}+K-S_{\tau},\text{if}\ \text{shouting}\ \text{occur}\ \text{at}\ \text{the}\ \text{time}\ \tau,\tau\in\left(0,T\right),\\
\end{matrix}\right.\ \nonumber \\
\end{equation}where \(S\) denotes the price of underlying asset,\(\ T\) is the
maturity time and\(\ \tau\) is any time of the period.
The more complicated shout options is defined in Windcliff [4].The
holder could have multiple rights according to the contract rules, in
some cases with a limit placed on the number of rights which may be
exercised within a given time period. That is, the holder has rights to
convert the holding option contract into another one with the same form
but less flexibility by exercising the options.