Question about greeks and put options

I am trying to establish the relationship between put options and greeks through a general form problem. Suppose that there is a company that at market opening trades its shares at a price $$p_1$$.Now assume that we price a put option with a given delta $$\delta$$ and a given gamma $$\gamma$$ at a price of $$p_2$$. Now suppose that the stock price decreased by an amount of $$\epsilon$$ soon after the market opens, enough so that the value of the option increases by $$p_3$$. What general form will solving for $$\epsilon$$ take using $$\gamma$$ and $$\delta$$?

• To a first approximation $p_3 \approx \delta\cdot\epsilon + 0.5\cdot\gamma\cdot \epsilon^2$. (If we neglect higher order terms, the passage of time $\Delta t$ and the change in IV if any.). That's just a result of the basic definition of $\delta$ and $\gamma$. – noob2 Jun 11 at 13:07