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.