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# How can I drive FPDE of American Option Price from FMLS Model?


# How can I drive FPDE of American Option Price from FMLS Model?












Under the risk neutral measure $\mathbb{Q}$. The FMLS model assumes that the log value of the underlying i.e., $\bar{x}_t=\ln S_t$. with dividend yield $D$ follows a stochastic differential equation of the maximally skewed LS process: $$d\bar{x}_t=(r-D-v)dt+\sigma dL^{\alpha -1}_t$$ where $r$ and $D$ are the risk free intrest and the dividend yield, respectivly. $t$ is the current time. and $v=\sigma ^{\alpha}sec \frac{\alpha \pi}{2}$ is a convexity adjustment. $dL^{\alpha -1}_t$ denotes the maximally skewed log-stable process, which is a special case of the Levy-$\alpha$-stable process $dL^{\alpha ,\beta}_t$ , where $\alpha \in \left( 0,2 \right]$ is the tail index describing the deviation of the LS process from the Brownian motion, and $\beta \in \left[ -1,1 \right]$ is the skew parameter. Remark that in the maximally skewed LS process, the skew parameter $\beta$ βis set to −1, in order to achieve finite moments for index levels and negative skewness in the return density. To ensure that the underlying return has the support on the whole real line, the tail index $\alpha$ needs to be restricted to $(1, 2]$.

let $\bar{V}\left( \bar{x},t;\alpha \right)$ be the price of American puts, with $\bar{x}$ xbeing the log underlying price defined as $\bar{x}=\ln S$ and $\alpha$ being the tail index.

Now how can i show that $\bar{V}\left( \bar{x},t;\alpha \right)$ should be governed by

$$\frac{\partial \bar{V}}{\partial t}+\left( r-D+\frac{1}{2}{{\sigma }^{\alpha }}\sec \frac{a\pi }{2} \right)\frac{\partial \bar{V}}{\partial \bar{x}}-\frac{1}{2}{{\sigma }^{\alpha }}\sec \frac{a\pi }{2}{}_{-\infty }D_{{\bar{x}}}^{\alpha }V\bar{V}-r\bar{V}=0,\,\,\,for\,\,{\bar{x}\in \left( {{x}_{f}},+\infty \right)}$$

where ${x}_{f}$ is the logarithm of the optimal exercise price, i.e., ${{x}_{f}}\left( t;\alpha \right)=\ln {{S}_{f}}\left( t;\alpha \right)$ and ${}_{-\infty }D_{{\bar{x}}}^{\alpha }$ is the one-dimensional Weyl factional operator defined as

$${}_{-\infty }D_{{\bar{x}}}^{\alpha }f\left( x \right)=\frac{1}{\Gamma \left( n-\alpha \right)}\frac{{{\partial }^{n}}}{\partial {{x}^{n}}}\int\limits_{-\infty }^{x}{\frac{f\left( y \right)}{{{\left( x-y \right)}^{n-\alpha -1}}}dy},\,\,\,\,\,\,n-1\le \Re \left( \alpha \right)<n $$

I appreciate any help.

Thanks.