2.1. Mathematical model of shout options
So far research on shout options remains only on the assumption that the
price of the underlying asset follows either Brownian motion or
geometric Brownian motion. Now, Compared with geometric Brownian motion,
jump-diffusion model can describe the underlying asset more accurately.
Therefore, this paper assumes that the underlying asset follows
jump-diffusion model, that is
\(\text{dS}\left(t\right)=\left(r-\text{βλ}\right)\text{Sdt}+\text{σS}\left(t\right)\text{dW}\left(t\right)+(y-1)S\left(t\right)\text{dQ}\left(t\right),\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (2.1)\)
where \(r\) is risk free rate, \(\sigma\ \)is volatility of underlying
asset,\(W\left(t\right)\ \)denotes Brownian
motion.\(\ S\left(t-\right)\ \)is the value of \(S\) immediately
before the jump. \(Q\left(t\right)\ \)is compound Poisson process with
intensity\(\ \lambda\ \)and mean size\(\ \beta\).
Suppose the jump sizes Y have a
density\(\text{\ f}\left(y\right)\). In this case, the average jump
size\(\text{\ β}=\text{EY}=\int_{0}^{+\infty}yf\left(y\right)\text{dy}\).
For example, the jump size \(Y\ \)of underlying asset price follows
log-normal distribution, that is
\begin{equation}
f\left(y\right)=\frac{1}{\sqrt{2\pi}\text{yγ}}e^{-\frac{\left(logy-\alpha\right)^{2}}{2\gamma^{2}}},\nonumber \\
\end{equation}then the average jump size\(\ \beta=e^{\alpha+\gamma^{2}/2}\).
Under the assumption that the price of the underlying asset follows
jump-diffusion model, we derive a new partial integro-differential
inequality (PIDI) for complicated shout options pricing, that is,
theorem 2.1 is a new result in this paper.
Theorem 2.1(The mathematical model for complicated shout put
options) . When the underlying asset price \(S\) follows
the jump-diffusion model (2.1) and the jump size has a density
function \(\text{\ f}\left(y\right)\), the shout put options
value V satisfies the PIDI (2.2) and the inequality (2.3)
\begin{equation}
\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_{0}^{+\infty}{V\left(t,\text{yS}\right)f\left(y\right)dy-V\left(t,S\right)}\right]-\text{rV}\leq 0,\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (2.2)\nonumber \\
\end{equation}\begin{equation}
\tilde{V}\leq V\ ,\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (2.3)\nonumber \\
\end{equation}