# Ito's Lemma in option pricing for a stock satisfying $dS=\frac{P-S}{\omega}dt+SdW_t$

Suppose a stock follows the stochastic differential equation

$$dS=\frac{P-S}{\omega}dt+SdW_t,$$

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.

• Assuming $P$ is a constant, your expression for $dV$ looks correct to me. Sep 1, 2021 at 14:17
• Thanks for your response. $P$ and $S$ are both functions of $t$, so does this change the expression for $dV$? Sep 1, 2021 at 14:19
• 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. Sep 1, 2021 at 14:26
• 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 $r$? Sep 1, 2021 at 14:31
• So you want to calculate the hedge portfolio? Sep 1, 2021 at 14:35