Estimating a Three-Point Shock Model from Even Moments

A stylized inventory-shock model assumes the one-step PnL jump $X$ takes values $-a$, $0$, and $+a$ with probabilities $p/2$, $1-p$, and $p/2$, respectively, where $a>0$ is unknown.

From data, the empirical second moment is $2$ and the empirical fourth moment is $10$.

Use the method of moments to estimate $p$ and $a$.