# Will the Delta of an Option always be the same irrespective of the underlying stock price?

Suppose, under the Black Scholes model we keep all the parameters the same except that we vary the asset price. Will the Delta of the option always remain the same?

• Assuming I understand your question correctly, the delta will differ based on the stock price. You can just check this in BS world by plugging in different stock prices into the delta equation. Basically the ratio of the stock price and strike price of the given option leads to different delta values. For example, for a high strike call option, if the stock price is very low, the delta will be low since the option value barely changes for a unit change in stock price. And for an at the money option, the delta will be higher since a change in stock price will affect the option value more. Jul 14 '19 at 5:22

You can just check this yourself with the BSM formulas. You fill out the regular $$d_{1}$$ formula in Excel and hold all variables fixed except for the $$S_{t}$$. Excel example is provided below the formulas.
$$\Delta_{Call}=N(d_{1})$$ $$\Delta_{Put}=-N(-d_{1})$$
$$d_{1}={ln(S_{t}/K)+(r+\sigma^2/2)t\over\sigma\sqrt{t}}$$
As you can see in the image when $$S_{t}$$ increases (holding other factors fixed), the delta of the call option increases and the delta of the put option decreases.