When µ < r, immediate sale wins unless support supplies a local-time push. Jun Maeda shows how that upward push can make waiting worthwhile. His claim about selling at resistance follows from what happens when the price also meets a downward push there.
This is a theory paper. It reports no empirical result and makes no claim about returns. The figures that follow come from the paper's numerical examples, apart from our own backtest in the last section.
The price process
Maeda uses a geometric multi-skew Brownian motion: geometric Brownian motion with an extra push at each of two fixed levels. Support L pushes the price upward through local time, with βL in (0,1). Resistance H > L pushes it downward, with βH in (−1,0). Local time accumulates only when the price occupies a level, so these pushes act on a set of Lebesgue measure zero. The process has no memory or regime flag; the levels keep their assigned roles. A holder of one unit solves the perpetual problem sup E[e^{−rτ}S_τ] under the physical measure, with r > µ.
The choice of measure matters. As Maeda explains, local time persists under any equivalent change of measure. Once either skew is non-zero, no equivalent martingale measure exists. His results give timing rules at a subjective discount rate, with no valuation claim.
The reason to wait begins at support. Over a short horizon t, expected local time grows like √t, against drift and discounting costs of order t. The support push wins over that horizon, leaving support strictly inside the continuation band. With base parameters r = 0.05, µ = 0.01, σ² = 0.04 and L = 1, βL = 0.05 gives a band of (0.975163, 1.025150), width 0.049987. Raising βL to 0.90 expands it to (0.625261, 1.465402). At βL = 0, the band disappears and immediate sale is optimal.
When does resistance become the exit?
First Maeda calculates the "L-band", the continuation interval obtained with the skew at H switched off. If that band ends at or below H, resistance cannot affect the decision: regime A. At βL = 0.30 and H = 3.0, the band remains (0.85690699, 1.15417976) for every βH in {0, −0.3, −0.6, −0.9}, identical to eight decimals.
Beyond that case, resistance separates two outcomes. In regime B, the continuation band straddles H. Smooth fit holds at both ends, while the slope breaks at H. In regime C, the upper boundary lands exactly at b = H; an inequality replaces smooth fit there.
The dividing line is permeability, π(β) = (1+β)/(1−β), the determinant of the matrix carrying the value function across a level. When the support-only band extends beyond H, sale exactly at H is optimal if and only if π(βH) ≤ V′(H−), equivalently βH ≤ β_H. For βL = 0.9 and H = 1.15, β_H = −0.392. Moving resistance toward support raises the strength required: at βL = 0.90, β*_H is −0.1498 for H = 1.30, −0.51463 for H = 1.10 and −0.67222 for H = 1.05. The transition is continuous. Its limiting slope dβH/db takes the values 0.752, 0.668 and 0.522 at those three H values, so the threshold does not jump.
Theorem 5.2 settles dominance for opposite-sign skews with nothing further to check. Maeda also shows why a solver's answer can look convincing and be wrong. Give both levels support strength, βL = βH = 0.60, with H = 2.2: the boundary system finds a root satisfying smooth fit to machine precision, although V − g = −0.1360 at x = 1.4346. The true solution has disjoint bands (0.7313, 1.3113) and (1.6089, 2.8848). The comparison argument in Section 7 identifies when the merged band really wins, at or below Hc = 1.79301900 in this example, and resolves the spurious root.
The conditions behind the trade
How often would a holder sell at resistance under the paper's own parameter draws? Of 400 random draws, with βL uniform on (0.05, 0.95), βH uniform on (−0.95, −0.05) and log(H/L) uniform on (0.02, 0.9), 301 fell in regime A, 18 in B and 81 in C. Selling at resistance applies in about one draw in five. In three out of four draws, resistance has no effect on the decision. These proportions belong to that sampling box; the paper does not estimate where real levels sit within it.
A trader would first need an estimate of support strength. The paper's forward examples share one base set, with µ/r = 0.2 and σ = 0.2. We did not find an estimate of βL for any real asset. Without one, the reported bands have no calibrated use. Width 0.050 is about 5% of support; width 0.840 allows a stock to move from 0.625 to 1.465 of its support level while the holder stays in.
Permanence is another demanding assumption. L and H must remain fixed and known across an infinite horizon. Maeda says a broken support level becoming resistance would require an auxiliary state that records the price's history, which his model deliberately omits. Real levels break and get redrawn. An estimate from a trailing window loses relevance when its level resets; after a reset, the optimality proof no longer describes the process being traded.
The payoff also assumes µ < r, linearity and frictionless execution. There are no transaction costs, impact or partial sales, and the paper names finite maturity as an extension. It models no costs. Since the support push supplies the value of waiting, costs reduce that value directly.
What exits reveal
Maeda poses an inverse question: can a holder's exits tell us where the levels are? For support, a band ending below resistance can give an exact answer. His closed-form inversion maps (0.856907, 1.154180) to L = 1.000000 and βL = 0.300000. Resistance strength is recoverable from the rule only in regime B. Regime C reveals H and the one-sided bound βH ≤ β*_H. As the paper puts it, "a mildly sticky resistance level and an impenetrable one generate identical behaviour." In regime A, "a level the holder arranges not to visit leaves no trace in the stopping rule."
This limit concerns exercise data. The paper notes that threshold estimators using high-frequency occupation times can still recover skew parameters; the obstacle here is inferring resistance strength from exits.
Our minute-bar construction lost 3.5%
Maeda reports no trading performance to compare with a backtest. Our construction used 1-minute bars from 1 January 2020 to 1 July 2024 and lost 3.50% in total. Sharpe was -0.21, Sortino -0.24 and maximum drawdown -10.30%. Annualised volatility was 3.76%, giving a Calmar of -0.08. Each year, the universe comprised the 60 highest-volume US stocks and ETFs, excluding ADRs.
We entered long at the 10:00 close. Pivots from 20 trailing sessions supplied L and H, with ±0.15% zones and at least 6 independent touches per level. Trailing minute moments supplied drift and volatility; r was set at 5% annually. Smoothed zone-exit frequencies served as proxies for βL and βH. We admitted a calibration only when it produced positive support skew, negative resistance skew and µ < r. The rule then solved for the band and sold at the next minute close following a close at or beyond either boundary. Any remaining position was closed at 15:55. We capped position size at 10% and leverage at 4.0. Costs were $0.0040 a share, with a $1.00 minimum. Fills used bar prices; impact and financing were not modeled.
Those returns describe our construction, one automated pass rather than a verdict on Maeda's work. Its connection to the theorem depends throughout on our proxies: finite-width zones for point levels, exit frequencies for local-time coefficients and a session close for a perpetual horizon. To isolate the exit rule's value, it would need comparison with paired 60-minute and frozen-resistance exits. We did not run the paired 60-minute and frozen-resistance exits. This run therefore cannot establish whether the band beat a timed sale. The gap is in our implementation and has no bearing on Maeda's theorems.
A zone-exit skew estimate for US equities would change our view if it stayed stable across level resets and was large enough to put β*_H within reach. For now, the model has no measured βL for any traded asset.
Our backtest stops at 2024-07-01, and everything after that date is deliberately left untouched so the same strategy can be checked out of sample later.