The Golden Relationship

Consider for instance an OTM call option. Price of call for strike K and time T is C(X, T, K), where X is the underlying's current price. We are here interested to the case when the difference between underlying price X and strike k is equal to the call premium. Let us denote K* such strike, and refer to it as the Golden strike. Associated with strike K* is the Golden Premium C(X,T,K*). For a time T, K* verifies (K* depends on T):

C(X,T,K*)=K*-X.


I have been looking into this since this morning, and found some interesting things (one of the results makes the golden number appear in a nice relationship).

My questions are:

1. Do you know anything related to the golden premium and strike, etc
2. Do you know if this was studied somewhere?
3. Do you have any comments (for or against) studying the golden things.

My aim is to see if we can make money out of the results.

Thank you

 

Here's a graph of what you looking at. I've never seen it discussed or studied. Honestly I don't the intuition behind it, would you care to elaborate?

 

Thanks for the response. Could you please explain the charts you included? My guess is that you were trying to find K* for a given stock and associated option parameters. Please provide the parameters and/or explain your chart and points if you can.

In the course of analyzing some strategies I designed, I had to make a departure from my thinking about strikes and premium in the usual way. In general people think what is the premium for a given strike.

In solving the problems I have been studying, I had to think the opposite way. For a given premium what strikes could I buy? I am thinking of premium as a budget, and strikes as the commodities to buy.

Now this does not answer directly your question, but the above relationship appears in many places and leads to interesting insight (and numbers in particular the golden number which is related to fibonacci numbers) in the solution process.

The results I obtained so far are quite striking, and I think will help me decide on the design and selection of strategies to trade options.

 

X-axis is the strike.
Blue line is the price of a call option (I picked some parameters like stock at $100, 1 month to expiration and 30% vol )
Red line is 100 - strike .

The point where two curves intersect is your "golden strike" , C(X,T,K*)=K*-X

I don't really see how you can have an analytical solution for K*, even in the case of Black-Scholes. I think you would have to just use a numerical method, but you obviously came up with something different. Guess your math is better than mine. Could you explain how you figured this out and what it has to do with the golden number?

By golden number I assume you refer to golden ratio, 1.618 ... ?

 

You have a good intuition, and that is what's most important. It is probable that I have more math tools than you, but human intelligence is greater than all tools.

By the way the golden number does not appear in the solution formula, it appears somewhere else. That somewhere else is the thing I want to keep in the inside for the moment. If I were to say where the number is at this point, I will also reveal some of the edges that I am building. I may share some of all of the information in the future with a select group. If so, I will be happy to put you in that group.

Let me make some remarks:

1. The golden strike will always be out of the money (because premium is positive, therefore K*-X is positive because of the equality, which means K* greater than X).
2. There is only one Golden strike per expiration.
2. You are right that one does not need a formula. There are approximation however. If computed numerically, one can do so for instance in log (I) steps, where I is an interval that brackets K*. For instance even if the largest strike is 1000, you can do it in less than 10 steps for sure, at each point you zero in on the K*. Your graphical representation is even better as it is visual. The calculation of K* is however not what is most important.
3. You asked of examples that might be useful. I will give this quick example.

Suppose that you held stock, and you wanted to write a covered call. Suppose further that you are bullish, but you believe that the stock will behave in a random manner and therefore not sure where it will end at the end of expiration. You want to take some profits by writing a call, and keep space for the stock to move. If K is the short strike, then C(K) is the call sale proceeds, and K-X is what you would hope to make from the movement of the stock. If you decide to split your potential profits equally, then the short strike is the golden strike.

Note that if you weight them equally then the golden strike minimizes the sum of the areas under the two curves. The area of the call has to be measured right to left, and the other from left to right.

The covered call is not the reason I was interested in it, but knowing the properties associated with that strike help understand many things. So, obtaining the value of K* is not what most important, but rather studying its behaviour, its relation to other strikes, and across other variables, etc, etc, that is more important.

I agree that to see some "hidden things", mathematical analysis is an advantage, but I also think the combined wisdom and knowledge of a forum is more powerful than what a single individual can achieve alone.

I noticed that you have just a few posts under your belt? May I ask what got you interested in this question. You are the only one who responded so far, so I am guessing that the majority may not have seen an interest in this topic for themselves.

Look forward to your response.

 

1) Do you believe that the "upside" golden strike price maximizes the rate of return from capital gains and option premium in a covered call writing strategy?
2) Why not short-sell put options at the "downside" golden strike price instead?
3) If you're focused on premium income, why not just sell the at-the-money strike price?
4) Maybe your "method" has some merit for trading option-strangles struck at the golden strike prices.

 

1) I have not looked at it. I know it maximizes the sum of the areas under the curves plotted by the other poster. It may do what you have in mind, but I cannot tell if yes or not. Could you check it?
2) I do not trade CCs/NPs (I know NP is "equivalent" to CC).
3) I did not understand what you meant. Do you have in mind CC or other strategy. Why the ATM?
4) Why do you think so?

 

It does? How did you figure the are under the option curve?

 

For the example you posted, you get it by a look of the eye.
It is a general result in a field known in a discipline known as equilibrium analysis. The minimization of that area is equivalent to intersecting the two curves.


I am going to suggest plotting another curve. It is coming in a next post.

 

dd4nyc,

Here is the follow up. I thought to think in terms of premium rather than strikes. Define S as the difference between K-X and the premium C.

S+C=K-X, which means that S=K-X-C. This quantity is called S because it is can be thought of as a form of slack.

When S=0, this corresponds to the golden strike There is no slack). To the right and to the left of the golden strike, S is either positive or negative.

Now what I want to plot is the value of the premium C versus the value of the slack S. This should be a curve where the X-axis is the slack and the Y-axis is the premium.

One needs a table of S and C values, and then put it into excel to plot it.