User:Lehalle/Notebook

I've got a french notebook too.

Wiki tricks

• Wikipedia:Manual of Style (mathematics)

My mathematical notes

Diffusion with a jump volatility

differentiating a portfolio $\Pi ^{+}=V^{+}-\Delta _{1}S$ (when the volatility is $\sigma ^{+}$), we obtain:

$d\Pi ^{+}=dV^{+}-\Delta _{1}dS=\partial _{S}V^{+}\cdot dS+\partial _{t}V^{+}dt+{1 \over 2}\partial _{S}^{2}V^{+}d\langle S\rangle +\underbrace {(V^{-}-V^{+})dq} _{d\alpha }-\Delta dS$

The $d\alpha$ term capture the possible jump of volatility (which has no direct instantaneous impact on $S$, but has on $V^{+}$, because it could turn it into $V^{-}$). this term can only be captured in expectation, and because $\mathbf {E} (dq)=\lambda ^{-}dt$, we obtain the desired Black Scholes equations ?