User:Lehalle/Notebook

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I've got a french notebook too.

[edit] Wiki tricks

[edit] My mathematical notes

[edit] Diffusion with a jump volatility

differentiating a portfolio Π + = V + − Δ1S (when the volatility is σ + ), 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 dα 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 ?