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Can the Delta be used to calculate the option premium given a certain target?
One thing I would like to clear up here is that Black Scholes is just a model that makes some assumptions about the dynamics of the underlying + a few other things and with some rather complicated math, out pops the Black Scholes formula. Black Scholes gives you the "real" price under the assumptions of the model. Your definition of what a "real" price entails will depend on what assumptions you make. With that being said, Black Scholes is popular for pricing European options because of the simplicity and speed of using an analytic formula as opposed to having a more complex model that can only be evaluated using a numerical method, as DumbCoder mentioned (should note that, for many other types of derivative contracts, e.g. American or Bermudan style exercise, the Black Scholes analytic formula is not appropriate). The other important thing to note here is that the market does not necessarily need to agree with the assumptions made in the Black Scholes model (and they most certainly do not) to use it. If you look at implied vols for a set of options which have the same expiration but differing strike prices, you may find that the implied vols for each contract differ and this information is telling you to what degree the traders in the market for those contracts disagree with the lognormal distribution assumption made by Black Scholes. Implied vol is generally the thing to look at when determining cheapness/expensiveness of an option contract. With all that being said, what I'm assuming you are interested in is either called a "delta-gamma approximation" or more generally "Greek/sensitivities based profit and loss attribution" (in case you wanted to Google some more about it). Here is an example that is relevant to your question. Let's say we had the following European call contract: Popping this in to BS formula gives you a premium of $4.01, delta of 0.3891 and gamma of 0.0217. Let's say you bought it, and the price of the stock immediately moves to 55 and nothing else changes, re-evaluating with the BS formula gives ~6.23. Whereas using a delta-gamma approximation gives: The actual math doesn't work out exactly and that is due to the fact that there are higher order Greeks than gamma but as you can see here clearly they do not have much of an impact considering a 10% move in the underlying is almost entirely explained by delta and gamma.