# Problems with a Black-Scholes modified equation

I haven't really studied much financial mathematics until about 2 months ago so I'm quite new to this stuff, so I'm sorry if this is a trivial question. At the moment I'm trying to work out what the terms of a modified Black-Scholes equation are equal to, so if someone could help me out I'd appreciate it. The equation is as follows:

$u(0,S_{0}) = \mathbb{E}^{\mathbb{Q}_{BS}}(\Phi) + \frac{\lambda}{2}(\mathbb{E}^{\mathbb{Q}_{BS}}((\Phi*)^2) - (\mathbb{E}^{\mathbb{Q}_{BS}}(\Phi*))^2)$

where $\Phi* = s\partial_s\Phi - \Phi$, $\Phi$ is the payoff, and $\mathbb{Q}_{BS}$ is the risk-neutral probability for the BS equation.

My question is, how do I find out what $\Phi*$ is equal to? In particular, what is $\mathbb{E}^{\mathbb{Q}_{BS}}(\Phi*)$ equal to? There is a lot of material on calculating $\mathbb{E}^{\mathbb{Q}_{BS}}(\Phi)$ so I know what that is equal to, but I have no idea how to find what the other term is equal to ($\lambda$ is just an arbitrary value so I don't need to worry about that).

• To calculate the expected value first solve $\partial_s \Phi$ and then calcualte the dynamics of $S^2$ using Ito's lemma. $E((\Phi*)^2)$ can than be caluclated using basic stochastic calculus and by solving an simple ODE. – Phun Apr 30 '16 at 9:31
• what is $\Phi$? – Gordon Apr 30 '16 at 14:26
• Sorry, $\Phi$ is the payoff, I'll edit the comment now. Also, if $\Phi$ is the payoff, how do I solve for $\partial_{s}\Phi$? I honestly have no idea :/ – ThePlowKing Apr 30 '16 at 22:25