Suppose a stock follows the stochastic differential equation


such that $W_t$ is a wiener process, $\omega\in\mathbb{R}^+$, and $P_t,S_t\in\mathbb{R}$. If the value of an option is $V(t,S_t)$, what is the value of $dV(t,S_t)$ given by Itô's Lemma (i.e. similar to the stochastic derivation of the Black-Scholes formula)? Any help would be much appreciated.

So far I have,

$$dV=\left(\frac{\partial V}{\partial t}+\frac{P-S}{\omega}\frac{\partial V}{\partial S}+\frac{S^2}{2}\frac{\partial^2 V}{\partial S^2}\right)dt+S\frac{\partial V}{\partial S} dW_t,$$

but I'm not sure whether this is correct or how best to simplify.

  • 1
    $\begingroup$ Assuming $P$ is a constant, your expression for $dV$ looks correct to me. $\endgroup$ Sep 1, 2021 at 14:17
  • $\begingroup$ Thanks for your response. $P$ and $S$ are both functions of $t$, so does this change the expression for $dV$? $\endgroup$
    – UNOwen
    Sep 1, 2021 at 14:19
  • 1
    $\begingroup$ In your expression, $S$ is not a function of $t$ but it is indexed by $t$, yes? In any case, this does not change the expression for $dV$; unless $P$ is governed by a stochastic differential equation as well. $\endgroup$ Sep 1, 2021 at 14:26
  • $\begingroup$ Yes, so $P$ is deterministic and $S$ is indexed by $t$. I assume setting $dV$ to $r\left(V-\frac{\partial f}{\partial S} S\right)dt+\frac{\partial V}{\partial S} dS$ would recover an equation in the form of Black Scholes for risk free rate $r$? $\endgroup$
    – UNOwen
    Sep 1, 2021 at 14:31
  • $\begingroup$ So you want to calculate the hedge portfolio? $\endgroup$ Sep 1, 2021 at 14:35


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