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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?

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    $\begingroup$ 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. $\endgroup$ – Slade Jul 14 at 5:22
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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.

enter image description here

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You can think of the gamma (second derivative, convexity) as the change in delta cause by the change in the price of the underlying. In other words, the gamma tells you the adjustment in the delta hedge necessitated by the change in the pricer of the underlying. So if the option is either far in the money or far out of the money then the gamma is zero and a small change in the price of the underlying does not change the delta and does not necessitate an adjustment of your selta hedge. But if the option is close to being at the money, then you have a gamma, and have to adjust the delta hedge in order to flatten the exposure to the underlying.

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