and the terminal condition
\(V\left(S,K,U,T\right)=\left\{\par \begin{matrix}\&0,\ \ \ \ \ \ \ \ \ S\rightarrow+\infty,\\ \&K-S,S\rightarrow-\infty,\\ \end{matrix}\right.\ \)
where the value \(\ \tilde{V}\) of the contract the holder receives upon shouting for complicated shout options is defined as
\(\tilde{V}\left(S,K,U,t\right)=\left\{\par \begin{matrix}\&V\left(S,K,U+1,t\right)+D\left(S,K,U,t\right),\text{if}\ U+1\leq U_{\max},\\ \&-\infty,\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \text{otherwise.}\\ \end{matrix}\right.\ \) (2.4)
Proof: Under the risk-neutral measure, the underlying asset price \(S\) follows the following stochastic process
\begin{equation} \text{dS}\left(t\right)=\left(r-\text{βλ}\right)\text{Sdt}+\text{σS}\left(t\right)\text{dW}\left(t\right)+\left(y-1\right)S\left(t\right)\text{dQ}\left(t\right).\nonumber \\ \end{equation}
The Itô formula implies
\begin{equation} e^{-\text{rt}}\left[V\left(t,S\left(t\right)\right)-V\left(0,S\left(0\right)\right)\right]\nonumber \\ \end{equation}\begin{equation} =\int_{0}^{t}{e^{-\text{rk}}\left[-\text{rV}\left(k,S\left(k\right)\right)+\frac{\partial V}{\partial t}\left(k,S\left(k\right)\right)+\left(r-\text{βλ}\right)S\left(k\right)\frac{\partial V}{\partial S}\left(k,S\left(k\right)\right)+\frac{1}{2}\sigma^{2}S^{2}\frac{\partial^{2}V\left(k,S\left(k\right)\right)}{\partial S^{2}}\right]}\text{dk}\nonumber \\ \end{equation}\begin{equation} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ +\int_{0}^{t}e^{-\text{rk}}\text{σS}\left(k\right)\frac{\partial V}{\partial S}\left(k,S\left(k\right)\right)\text{dW}\left(k\right)\nonumber \\ \end{equation}\begin{equation} +\int_{0}^{t}e^{-\text{rk}}\left[V\left(k,S\left(k\right)\right)-V\left(k,S\left(k-\right)\right)\right]\text{dk.\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }(2.5)\nonumber \\ \end{equation}
If \(S\) has a jump at time k, then the underlying asset price satisfies
\begin{equation} S\left(k\right)=\text{yS}\left(k-\right).\nonumber \\ \end{equation}
We examine the last term in (2.5)
\begin{equation} \int_{0}^{t}e^{-\text{rk}}\left[V\left(k,S\left(k\right)\right)-V\left(k,S\left(k-\right)\right)\right]\text{dk}\nonumber \\ \end{equation}\begin{equation} =\int_{0}^{\infty}{\int_{0}^{t}{e^{-\text{rk}}\left[V\left(k,\text{yS}\left(k-\right)\right)-V\left(k,S\left(k-\right)\right)\right]\text{dk}}}\text{dy}\nonumber \\ \end{equation}\begin{equation} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ =\int_{0}^{\infty}{\int_{0}^{t}{e^{-\text{rk}}\left[V\left(k,\text{yS}\left(k-\right)\right)-V\left(k,S\left(k-\right)\right)\right]d\left(N\left(k\right)-\text{λk}\right)}}\text{dy}\nonumber \\ \end{equation}\begin{equation} \ \ \ \ \ +\int_{0}^{t}{e^{-\text{rk}}\left[V\left(k,\text{yS}\left(k-\right)\right)-V\left(k,S\left(k-\right)\right)\right]\text{λdk.}}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (2.6)\nonumber \\ \end{equation}
Substituting (2.6) into (2.5), we obtain
\begin{equation} \text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }d\left(e^{-\text{rt}}V\left(t,S\left(t\right)\right)\right)=e^{-\text{rt}}\left\{-\text{rV}\left(t,S\left(t\right)\right)+\frac{\partial V}{\partial t}\left(t,S\left(t\right)\right)+\left(r-\text{βλ}\right)S\left(t\right)\frac{\partial V}{\partial S}\left(t,S\left(t\right)\right)\ +\frac{1}{2}\sigma^{2}S^{2}\frac{\partial^{2}V\left(t,S\left(t\right)\right)}{\partial S^{2}}+\lambda\int_{0}^{\infty}{\left[V\left(t,y_{m}S\left(t\right)\right)f\left(y\right)-V\left(t,S\left(t\right)\right)\right]\text{dy}}\right\}\text{dt}+e^{-\text{rt}}\ \text{σS}\left(t\right)\frac{\partial V}{\partial S}\left(t,S\left(t\right)\right)\text{dW}\left(t\right)\nonumber \\ \end{equation}\begin{equation} +\int_{0}^{\infty}{e^{-\text{rt}}\left[V\left(t,\text{yS}\left(t-\right)\right)-V\left(t,S\left(t-\right)\right)\right]d\left(N\left(t\right)-\text{λt}\right)\text{.\ }}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (2.7)\nonumber \\ \end{equation}
Using Itô formula for\({\text{\ \ }e}^{-\text{rt}}\pi\left(t\right)\), the following equation can be obtained
\(d\left(e^{-\text{rt}}\pi\left(t\right)\right)=e^{-\text{rt}}\left[-\text{rπ}\left(t\right)\text{dt}+\text{dπ}\left(t\right)\right]\)
\(=e^{-\text{rt}}\left[\pi\left(t\right)\text{dS}\left(t\right)-\text{rS}\left(t\right)\pi\left(t\right)\text{dt}\right]\ \)
\(=e^{-\text{rt}}\left[-\text{σS}\left(t\right)\pi\left(t\right)\text{dW}\left(t\right)+\pi\left(t\right)S\left(t-\right)d\left(Q\left(t\right)-\text{βλt}\right)\right]\)
\begin{equation} =e^{-\text{rt}}\left[-\text{σS}\left(t\right)\pi\left(t\right)\text{dW}\left(t\right)+\pi\left(t\right)S\left(t-\right)\int_{0}^{\infty}{d\left(N\left(t\right)-\text{λdt}\right)}\right]\ .\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (2.8)\nonumber \\ \end{equation}
Let
\begin{equation} \pi\left(t\right)=\frac{\partial V}{\partial S}\text{\ .}\nonumber \\ \end{equation}
Considering (2.7) and (2.8), we obtain
\(d\left[e^{-\text{rt}}V\left(t,S\left(t\right)\right)-e^{-\text{rt}}\pi\left(t\right)\right]\)
\(=d\left[e^{-\text{rt}}V\left(t,S\left(t\right)\right)\right]-d\left[e^{-\text{rt}}\pi\left(t\right)\right]\)
\begin{equation} =e^{-\text{rt}}\left\{-\text{rV}\left(t,S\left(t\right)\right)+\frac{\partial V}{\partial S}\left(t,S\left(t\right)\right)+\left(r-\text{βλ}\right)S\left(t\right)\frac{\partial V}{\partial S}\left(t,S\left(t\right)\right)+\frac{1}{2}\sigma^{2}S^{2}\frac{\partial^{2}V\left(t,S\left(t\right)\right)}{\partial S^{2}}\right.\ +\left.\ \lambda\left[\int_{0}^{\infty}{V\left(t,\text{yS}\left(t\right)\right)f\left(y\right)\text{dy}-V\left(t,S\left(t\right)\right)}\right]\right\}\text{dt}\nonumber \\ \end{equation}
Therefore, the following inequality can be established
\begin{equation} -\text{rV}+\frac{\partial V}{\partial t}+\left(r-\text{βλ}\right)S\frac{\partial V}{\partial S}\ +\frac{1}{2}\sigma^{2}S^{2}\frac{\partial^{2}V}{\partial S^{2}}+\lambda\left[\int_{-1}^{+\infty}{V\left(t,\text{yS}\right)f\left(y\right)\text{dy}-V\left(t,S\right)}\right]\leq 0.\nonumber \\ \end{equation}
Because the remaining times of contract held by the shout options holder after exercising the shout right is shorter than that of the original contract, and the number of shouts is less, there is
\(\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\tilde{V}\leq V.\)
For random variable\(\ U\), when \(U+1\leq U_{\max}\), the new contract value \(\tilde{V}\left(S,K,U,t\right)\) with residual shout numbers \(U\) corresponds to the original contract value\(V\left(S,K,U+1,t\right)\) with shout numbers \(U+1\) and the dividend function \(D\left(S,K,U,t\right)\) due to shout, that is
\begin{equation} \tilde{V}\left(S,K,U,t\right)=\left\{\begin{matrix}\&V\left(S,K,U+1,t\right)+D\left(S,K,U,t\right),\text{if}U+1\leq U_{\max},\\ \&-\infty,\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \text{otherwise.}\\ \end{matrix}\right.\ \nonumber \\ \end{equation}
At the maturity time\(\ t=T\) we can get the following equation
\(V\left(S,K,U,T\right)=\ g\left(S,K\right).\)
We will restrict attention in this work to shout put options, then the payoff function is given by
\(g\left(S,K\right)=\max\left(K-S,0\right),\) (2.9)
then for shout put option
\(V\left(S,K,U,T\right)=\max\left(K-S,0\right),\)
So the terminal condition of shout put options is
\(V\left(S,K,U,T\right)=\left\{\par \begin{matrix}\&0,\ \ \ \ \ \ \ \ \ S\rightarrow+\infty,\\ \&K-S,S\rightarrow-\infty.\\ \end{matrix}\right.\ \)