2.2 Some results from Theorem2.1
According to the Theorem2.1 we can obtain some results including the simplified form of the PIDI (2.2), discrete jump distribution and completeness and so on.
When the jump sizes follow specific continuous distributions, the PIDI (2.2) can be transformed into the partial differential inequality.
Corollary 2.1(Simplified form of the PIDI (2.2)) Assuming that the jump sizes \(Y_{i}\ \)of underlying asset price follow exponential distribution, that is, \(f\left(y\right)=\theta e^{-\theta y}\), the PIDI (2.2) can be simplified as follows
\begin{equation} -\text{rV}+\frac{\partial V}{\partial t}+\text{aS}\frac{\partial V}{\partial S}+\frac{b}{2}S^{2}\frac{\partial^{2}V}{\partial S^{2}}\leq 0\ ,\ \ \ \nonumber \\ \end{equation}
where\(\ a=r+\lambda\left(\frac{1}{\theta}-1-\beta\right),b={1+\sigma}^{2}-\frac{2}{\theta}+\frac{2}{\theta^{2}}.\)
Proof: For\(\left[V\left(t,\left(y+1\right)S\right)-V\left(t,S\right)\right]\)using Taylor expansion and substituting into the PIDI (2.2), we obtain
\begin{equation} -\text{rV}+\frac{\partial V}{\partial t}+\left[r-\text{βλ}+\lambda\int_{0}^{\infty}{\left(y-1\right)f\left(y\right)}\text{dy}\right]S\frac{\partial V}{\partial S}+\left[\sigma^{2}+\int_{0}^{\infty}{\left(y-1\right)^{2}f\left(y\right)}\text{dy}\right]\frac{S^{2}}{2}\frac{\partial^{2}V}{\partial S^{2}}\leq 0.\ \ \ \ \ \ \ \ \ \ \ \ \ \ (2.10)\nonumber \\ \end{equation}
If \(Y\) follows the exponential distribution with parameter\(\ \theta\), then
\begin{equation} \int_{0}^{\infty}{\text{yf}\left(y\right)}dy=EY=\frac{1}{\theta}\ ,\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (2.11)\nonumber \\ \end{equation}\begin{equation} \int_{0}^{\infty}{y^{2}f\left(y\right)}dy=EY^{2}=\text{DY}+E^{2}Y\ =\frac{1}{\theta^{2}}+\frac{1}{\theta^{2}}=\frac{2}{\theta^{2}}\ .\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (2.12)\nonumber \\ \end{equation}
Let
\begin{equation} a=r+\lambda\left(\frac{1}{\theta}-1-\beta\right),\nonumber \\ \end{equation}\begin{equation} b={1+\sigma}^{2}-\frac{2}{\theta}+\frac{2}{\theta^{2}}\text{\ .}\nonumber \\ \end{equation}
Substituting (2.11) and (2.12) into (2.10), we obtain
\begin{equation} -\text{rV}+\frac{\partial V}{\partial t}+\text{aS}\frac{\partial V}{\partial S}+\frac{b}{2}S^{2}\frac{\partial^{2}V}{\partial S^{2}}\leq 0\ .\ \ \ \nonumber \\ \end{equation}
When the jump sizes are discrete random variables, the PIDI followed by the price of complicated shout options could be obtained easily.
Remark 2.1(Discrete jump distribution). There are modifications of Theorem 2.1 for the case when the jump sizes \(Y_{i}\ \)have a probability mass function\(p\left(y_{1}\right),\cdots,p\left(y_{m}\right)\ \)rather than a density \(f\left(y\right)\) under risk-neutral measure. In (2.2), the term\(\int_{0}^{+\infty}{V\left(t,\text{yS}\right)f\left(y\right)}\text{dy}\)would be replaced by\(\sum_{m=1}^{M}{p(y_{m})V\left(t,y_{m}S\right)}\).
The mathematical model of complicated shout options (Theorem 2.1) always holds for an arbitrary shout options with different payoff functions.
Remark 2.2(Completeness). In this subsection, we have contracted the value for complicated shout put options on the underlying asset driven by jump-diffusion processes. It is clear from the analysis that the same argument would work for an arbitrary shout options with payoff\(h\left(S\left(T\right)\right)\) at time \(T\) written on a stock modelled this way. One could simply replace the put payoff by the function of \(h\) in equation (2.9).The partial integro-differential inequality (2.2) and the inequality (2.3) would still apply, although now with the terminal condition\(\text{\ V}\left(S,K,U,T\right)=h(S)\).
The following Remark2.3 shows that the previous research on shout option is only a specific case of this paper and Remark 2.4 gives the significance of inequality (2.3).
Remark 2.3 (Consistency). If there is not jump during the life of the contract, that is, the Eq. (2.1) does not contain the jump term, then the PIDI (2.2) reduce to PDE in [4].
Remark 2.4 (Sufficiency). Shout options holders have the right to exercise at any time or shout any times during the life of the contract. In view of this situation, the second inequality in (2.2) ensures that the seller of shout options has sufficient amount to meet the buyer’s exercise.