Help me solve this option formula

[Bostrom]

I'm working on a homemade option strategy where I want to sell when the leverage effect i maximized.

Example:
Stock XYZ costs 100$, a 110$ call option costs 5$. The maximum point where you should sell is at 35% stock return where you get 400% return on your option which is the stock return compunded 5,36 times( (1+0,35)^5,36 = (1+4) ).
If the stock return is less or more than 35% the exponent will be less than 5,36. I'm currently using a Excel spreadsheet that finds the optimal point by looking at each percent unit but I haven't come up with a formula yet.
I've come this far:
(1+rs)^X = (1+ro) -> log(1+ro)/log(1+rs) = X
where rs = stock return, ro = option return, X = exponent that should be maximized by changing rs.

I guess I should continue by setting the derivative to 0 but the calculations are too complicated for me. Can somebody help?

I also wonder if this type of strategy has been used before? Can't find anything on Google.

[Fish]

First, you need to define “ro” in terms of “rs” so the function has 1 variable and not 2. Assuming the option is in the money, we have:

ro = 20*(rs - 0.15)

Where “20” is the leverage from the option (stock price / option price) and “0.15” is the value of “rs” for the option to break even, that is: (Strike price - stock price + option premium)/(stock price).

Now the expression becomes:
X = log(1+20*(rs-0.15)) / log(1+rs)

The first derivative of the RHS is zero when:
20*log(1+rs)/(1+20*(rs-0.15)) - log(1+20*(rs-0.15))/(1+rs) = 0

You can plug in rs=0.35 and see that the first derivative is zero:
20*log(1.35)/(1+4) - log(1+4)/1.35 = 1.2004-1.1922 ≈ 0

Using the second derivative to verify that this is a local maximum is an exercise left to the reader.

Hope this helps. Use the expressions for “20” and “0.15” above to come up with a more generalized formula. Check and understand the math before making any investing decisions based on this!!

[plantingtheseed]

great explanation. Before solving for the second derivative , please note verifying the solution (rs=.35) into the original eq. yields 4 -> log(5)/log(1.35) =5.36 > 0 (check) which is a local maximum. always caveat emptor w/ regards to options. good luck!

[Bostrom]

Thanks, I've checked the math. Is it also possible to solve for rs so I know at what stock price it's time to sell?

[xmj]

How did you derive the max 35% stock return -> 400% options return?