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}