User:Lehalle/Notebook

I've got a french notebook too.

Wiki tricks

 * Manual of Style (mathematics)

Diffusion with a jump volatility
differentiating a portfolio $$\Pi^+=V^+ - \Delta_1 S$$ (when the volatility is $$\sigma^+$$), we obtain:

$$d\Pi^+ = dV^+ - \Delta_1 dS = \partial_S V^+ \cdot dS + \partial_t V^+ dt + {1\over 2}\partial^2_{S} V^+ d\langle S\rangle + \underbrace{(V^- - V^+)dq}_{d \alpha} -\Delta dS$$

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 ?