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).

Thanks in advance.

  • $\begingroup$ 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. $\endgroup$
    – Phun
    Apr 30, 2016 at 9:31
  • $\begingroup$ what is $\Phi$? $\endgroup$
    – Gordon
    Apr 30, 2016 at 14:26
  • $\begingroup$ 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 :/ $\endgroup$ Apr 30, 2016 at 22:25
  • $\begingroup$ Where does this problem come from? If we know more background, we may have a clue. $\endgroup$
    – Gordon
    Apr 30, 2016 at 23:46
  • $\begingroup$ Honestly, it just came from my supervisor. He's given me a few questions to work out since he knows I'm new to financial math but since he's gone away for a month I can't really check with him so I thought I'd try post it here. The only thing I know about this problem is that it's supposed to be a modified BS equation, and the information I know I listed above. I guess it's possible he's left out some information to solve it, so I may have to wait for him to come back so I can ask him... $\endgroup$ May 1, 2016 at 0:01


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