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.