How does an option's premium depend on its strike price?

Question

(NB: In this question's title, and elsewhere throughout this post, I have used the word "options" as a stand-in for both "call options" and "put options", for the sake of simplicity. In other words, I am interested in the answer to this question for both call options and put options.)

In its simplest form, my question is this: If we keep all the other variables that could affect an option's premium constant, what is the relationship between the option's premium and its strike price?

In other words, what would a graph with premium on the vertical axis and strike price on the horizontal axis look like?

I realize that it could be that, even when we limit ourselves to, say, call options, there is no single answer to my question that will hold under all possible combinations of the other variables. For example, I imagine that this relationship would change qualitatively depending on whether the call option is in the money or out of the money. I hope, however, that the number of possible combinations of variable value regimes that one must consider is not astronomical.

I also realize that, it could be very well be the case that, to do even minimal justice to this question, a much lengthier treatment than can be reasonably expected from a Money StackExchange answer would be necessary. If this is the case, I would appreciate pointers to such a treatment.

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^{1 For a fixed agent, the meaning of the word "premium" depends on whether the agent is buying or selling the option. Therefore, one could approach this question either from the option's buyer's or the option's seller's point of view. I imagine, however, that there is a pretty straightforward relationship between these two points of view, so the answer to this post's question would basically apply to both of them. More specifically, I imagine that it is never the case that the "seller's premium" is higher (in absolute value) than the "buyer's premium", because otherwise one could make money simply by buying and then immediately re-selling the same option.}

Answer 1

Suppose you have the choice between two call options: One that pays out if the stock ends up above $10 at expiration, and one that pays off if the stock goes above $15 at expiration. Which one would be more valuable? Obviously the stock has a higher chance of payout with a $10 strike vs a $15 strike, so the $10 call would be more valuable (higher expected present value of payoff). In general, as strike goes up, the value of a call goes down. (and vice-versa for puts)

So that handles the directional relationship - how about the numerical relationship. Without going too deep into the math, we look at the two extremes: Deep ITM and Deep OTM.

Deep ITM calls (extremely low strike) have a very small chance of expiring below the strike, so the payout will be the difference between the strike and the stock price (you buy the stock for K and sell it for S, so your net payout is S-K). Since the value of a call changes by $1 for every $1 that the strike goes down (a $1 rise in K reduces the payout by $1), the "slope" of the graph in this region is nearly -1.

What about Deep OTM calls (extremely high strike)? They have a very small chance of expiring above the strike, so they have virtually no value. Since they still have no value if the underlying moves in this region, the "slope" of this line is nearly zero.

So you can picture a graph that starts at S for a zero strike call, has a slope of -1, slopes down, and gradually goes along the x-axis (no value) for extremely high strikes:

Source

I imagine that this relationship would change qualitatively depending on whether the call option is in the money or out of the money.

Not really. The relationship between strike and premium is continuous at-the-money, as you can see.