Combining discontinuous opposing returns into a single continuous function

Question

See the function here.

It's my intent to measure the expected log return of the optimal trade with regards to long/short positions in a trading range. The trading range has an equilibrium where the maximal gains of the long position equate to the maximal gains of a short position which results in neither side gaining any ground. If the close price is greater than the equilibrium, use the long position as a measure of expected future returns. If it's less than the equilibrium, use the short position. If it's equal, 0.

Currently the way I've structured things, this is a discontinuous function with strong jumps around 0 which doesn't make any real world sense to me. I would rather like to see some sort of smooth transition between 0 and the regime change. I just don't know how to go about doing that. I'm also unsure if a linear mapping is reasonable or if some sort of curve/distribution would make more sense and why.

Answer 1

For an simple approximation (which is not particularly useful if you want values to be asymptotically close around $N$),

$$\ln\left(\frac{\max\left(L,H\right)}{0.01}\right)\tanh\left(k\left(x-N\right)\right)$$

or

$$\ln\left(\frac{\max\left(L,H\right)}{0.01}\right)\operatorname{erf}\left(k\left(x-N\right)\right)$$

are suitable. If you want an asymptotic approximation (which is very accurate),

$$R_{N}=\lim_{k\rightarrow\infty}R_{O}\left(1-e^{-k\left(x-N\right)^{2}}\right)$$

such that $R_N=R_O$, $\forall x\in\mathbb{R}\backslash\{N\}$ and $R_N=0$ when $x=N$. Taking large values of $k$, yields a continuous solution. See https://www.desmos.com/calculator/45djjacr1i. I hope this helps.