How can I represent this observation regarding options in a formula?

By observing how an option's expiration P/L changes as its underlying asset price changes, we can discover the following system of equations:

$$\begin{cases}S_{Long} = C_{Long} + P_{Short} \\ S_{Short} = P_{Long} + C_{Short} \\ S_{Long} = -S_{Short} \\ C_{Long} = -C_{Short} \\ P_{Long} = -P_{Short}\end{cases}$$, where the $$S$$'s are stocks, the $$C$$'s are call options, and the $$P$$'s are put options. Linear algebra reveals that any one of these equations could be removed without loss of information.

By observing how an option's expiration P/L changes as its strike price changes, we can discover the following similar system of equations:

$$\begin{cases}S_{Long} = P_{Long} + C_{Short} \\ S_{Short} = C_{Long} + P_{Short} \\ S_{Long} = -S_{Short} \\ C_{Long} = -C_{Short} \\ P_{Long} = -P_{Short}\end{cases}$$, again with one linear dependence shared between all five equations. It can be seen in the linked data that the call options and put options have exactly swapped behaviors, leading to the slight variation between the two systems.

How can we mathematically represent this swap and harmonize the systems? My intuition is that partial derivatives are at play, and that this may lead to a partial differential algebraic system of equations. If so, I'll probably only understand a thoroughly curated answer, as I'm just preparing to begin studying PDEs.

• According to your I./1. and II./2. equations $S_{Long} = C_{Long} + P_{Short} = S_{Short}$. Is this what you meant to write? Aug 3 '20 at 20:30
• Also, what do you mean by "How can we mathematically represent this swap and harmonize the systems"? Seems like you already wrote down the mathemathical representations; as long as the equations are consistent, I guess they are harmonius? Aug 3 '20 at 20:31
• @Giskard What is there is what I meant to write, but I don't know if it's the correct thing to write. As is, the two systems cannot simply be merged, as your $S_{Long} = S_{Short}$ comment demonstrates. Rather, in light of the linked data, I'd say something like "as the underlying stock price varies and the strike price is held constant, $S_{Long} = C_{Long} + P_{Short}$, while as the strike price varies and the underlying stock price is help constant, $S_{Short} = C_{Long} + P_{Short}$." This is the relationship I'm trying to make explicit to harmonize the two systems. Aug 3 '20 at 21:21
• If I’m reading your notation correct, you are writing that a short position in a stock is the negative of the long position. That’s a tautology: a short position is economically equivalent to holding a negative amount of an asset. You can’t do very much by adding tautological equations to a system of equations. Aug 3 '20 at 23:38
• @BrianRomanchuk I would explain it as "the way to algebraically represent that a short position is economically equivalent to holding a negative amount of an asset is to write $S_{Long} = -S_{Short}$, $C_{Long} = -C_{Short}$, and $P_{Long} = -P_{Short}$." The math won't know this unless you make it explicit, right? The one tautology I do note in the question is the linear dependence, so one of these equations is redundant relative to the rest. Aug 3 '20 at 23:48