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I found this post online which is plotting different results for option value and greeks depending on spot price.

Why would someone want to do calculate the value of the option with different spot prices? Is not delta what you would use to see the how the option changes with respect to spot price.

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closed as off-topic by LocalVolatility, Sanjay, skoestlmeier, byouness, Helin Mar 1 at 23:15

This question appears to be off-topic. The users who voted to close gave this specific reason:

  • "Basic financial questions are off-topic as they are assumed to be common knowledge for those studying or working in the field of quantitative finance." – LocalVolatility, Sanjay, skoestlmeier, byouness, Helin
If this question can be reworded to fit the rules in the help center, please edit the question.

  • $\begingroup$ Why has the question downvoted? Could you please give a reason? $\endgroup$ – joseprupi Feb 28 at 13:26
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    $\begingroup$ Didn't down-vote but this question is clearly off-topic for being too basic. You are essentially asking why we are interested in the actual solution if we could just as well use a first order approximation for it. $\endgroup$ – LocalVolatility Feb 28 at 13:30
  • $\begingroup$ I guess my question is about assuming a different spot price would have the same volatility, and therefore the sense of the results $\endgroup$ – joseprupi Feb 28 at 13:46
  • $\begingroup$ Your standard Black-Scholes delta implicitly assumes that volatility on the strike doesn't change as the spot moves as well. If your actual question is sth. like "what are the moving dynamics of the volatility smile as the spot changes and how do they affect delta", then I suggest you don't edit this question which already has two answers but ask again, trying to be clearer about what it is you are looking for. $\endgroup$ – LocalVolatility Feb 28 at 13:51
  • $\begingroup$ Thanks for your comments, I have deleted the question asking if it is correct to use same volatilities with different spot prices and accepted one of the answers as I think makes overall clearer. With the answers I understand this is a correct approach. $\endgroup$ – joseprupi Feb 28 at 14:08
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"Why would someone want to do calculate the value of the option with different spot prices?"

Because Delta is the derivative of the option price function wrt spot. Since the option price function is not linear in spot according to the BS model you cannot get the option prices for different spot just by Delta. What you are asking about is simply using a first order approximation.

Here is a (primitive) graph demonstrating the problem:enter image description here

As @Bikenfly mentions a keyword here is Gamma. Using Delta and Gamma will then be a second order approximation

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Understand gamma - delta is an approximation of the price change, but as you extrapolate out, the approximation gets worse. Gamma shows how much delta changes based upon a change in the underlying spot price. So delta and gamma together will give a better approximation. However, gamma is also an approximation of the delta change and also changes as the spot price changes.

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