At β = 0.05, Maeda's single-support premium is 6.2 basis points before trading costs. His model proves that "buy at support" is optimal in every exit regime with one support and one resistance. The size of the upward push at support determines whether that rule is worth trading, yet the paper never estimates it.
This is pure theory, with no market data or backtest. Its numerical results come from a brute-force check. Our figures therefore have no reported trading performance from the paper to compare against.
Why wait to buy?
The price follows geometric Brownian motion, with an extra push at a finite set of levels. At each level z, β_z in (−1, 1) scales the local time accumulated there. Positive β pushes price upward at a touch and represents support; negative β pushes it downward and represents resistance. In a companion paper, Maeda solves the selling problem under µ < r: v(x) is the value of holding a share and choosing when to sell. Without those pushes, discounted price is a supermartingale, so the holder sells immediately.
Buying on its own gives a degenerate problem. Under µ < r, waiting always reduces the expected discounted purchase price. Maeda instead makes the buying reward the exit premium W(x) = v(x) − x, the value of gaining the right to sell later. Within the exit continuation region, v is a discounted martingale and x a strict discounted supermartingale. Thus (L − r)W = (r − µ)x > 0, and the premium grows in expectation.
Waiting to buy pays except, potentially, where β > 0. At such a level, W has a concave corner with jump Δ(W) = −2β. Waiting incurs a local-time loss of order √t, which outweighs the drift gain of order t. As Maeda writes, "One buys exactly where the value of waiting to sell comes from."
To find the entry value, the paper takes the least concave majorant of finitely many points, one for each support and including the origin, at F = ψ_r/φ_r. It then multiplies by φ_r, the decreasing fundamental solution. With one support L and one resistance H, there is just one support point and the entry region is exactly {L}. The rule needs no free boundary or smooth fit. Below L, value is (G(a) − L)(x/L)^α₂, where a is the exit stop: the buyer waits for price to rise back to L.
The trade at L
The numerical example sets r = 0.05, µ = 0.01, σ² = 0.04, L = 1 and β_L = 0.9. In regime A (H = 1.60, β_H = −0.30), the investor buys when price hits 1. The exit is the first touch of either the 0.6253 stop or the 1.4654 upper sale level; W(L) is 0.182971. In regime C (H = 1.15, β_H = −0.50), the sale level sits at H = 1.15, the stop is 0.7003 and the premium drops to 0.109221. Across regimes A, B and C, the buy level stays put while the value changes.
A brute-force concave majorant using 4×10^5 grid points agrees with the closed form to 3×10^-10 on [0.3, 3]. That verifies the algebra. It provides no evidence about markets.
With two supports, entry can occur at one level or both. The choice turns on W(L)/W(H) relative to a window formed from hitting-time discounts. For H = 1.3, β_L = 0.9 and β_H = 0.3, the ratio is 2.5287, outside the [0.8934, 1.718] window, and the investor buys only at L. Negative β levels draw no entry. When both skews are negative, W and u vanish to within 2×10^-12.
How large is the push?
Maeda notes that setting every β to zero restores plain geometric Brownian motion and gives W ≡ 0. The local-time push supplies the whole trade. Its size matters sharply: the single-level premium per unit of price climbs from 6.2×10^-4 at β = 0.05 to 0.202 at β = 0.95. The headline example takes β_L = 0.9, near the top of that grid. Under the paper's r = 0.05, µ = 0.01 and σ² = 0.04, a single support with β = 0.05 offers 6.2×10^-4 of price, or 6.2 basis points.
Costs then get a large say. The transaction-cost remark establishes only that a costly entry region remains inside the support set. The paper says it "may be empty, in which case never buying is optimal." It leaves the hull formula with costs open. Against a 6.2 basis point premium, a 2 bps half-spread on each side consumes about two thirds. Our minute-bar figures below charge commissions only; they do not deduct a 2 bps half-spread.
Estimating β is the other obstacle. The levels and coefficients are known and fixed in Maeda's model. The paper also observes that entry decisions give no information about β_H beyond what the exit band already reveals. Its support strength never decays, and the model is memoryless. Maeda contrasts this with Henderson et al.: in one of three regimes, the buying set lies in (L, A]; when the + regime's decay rate vanishes, it is an interval [a, b].
The proof itself holds up.
Entry duality needs only four properties of W from Lemma 3.1, two of which carry the mechanism. The paper explicitly says that smoothness of W at the band edges is irrelevant.
Our minute-bar version
We built a strategy around the idea using 1-minute bars for the top 50 US non-ADR stocks by dollar volume. Our run covers 2020-01-01 to 2024-07-01. Over 2020-01-01 to 2024-07-01, our construction returned 40.71% in total, with a 0.21 Sharpe on 42.42% annualised volatility and a 0.10 Calmar. Beta to SPY was 1.36; 35.46% of trades won. A 0.21 Sharpe alongside 1.36 beta looks like levered market exposure. This weak result comes from one automated pass and says nothing about whether real supports have an upward push.
Our figures include commissions of $0.004 a share, with a minimum $1 per order. Fills use the bar price without slippage. We charge neither market impact nor financing on leverage of up to 4.0.
We set L at the 20-session low and H at the 20-session high, freezing both from completed daily bars before each session. A completed minute bar that touches L ± 0.1 ATR (14-session) prompts a buy at the next minute's open. The stop is L − 0.5 ATR, the sale signal is H, and the position lasts at most three sessions. The control buys at L + 0.75(H − L). We match each control to a support entry in the same stock, within 10 sessions, in the same 30-minute clock bucket and within 20% on pre-session volatility.
Two choices constrain what this run can say about Maeda's idea. A historical low gives no evidence that β > 0, and our exits follow fixed rules rather than the optimized band. We also have a known flaw of our own: the annual top-50 ranking may draw on data from later in the traded year, introducing selection look-ahead into the universe. The relevant comparison is the paired gap between support entries and matched controls after the same costs. A weak or flat gap speaks first to our choices. For the selling half of the folklore, see our earlier minute-bar construction.
Skew size sets the value
The theorem keeps entry at L; β_L governs the payoff. If stock supports have pushes near 0.9, the paper's premium of 0.109 to 0.183 of price across regimes A to C at β_L = 0.9 leaves room for costs. If pushes resemble 0.05, the paper's single-support premium of 6.2 basis points at β = 0.05 becomes a cost-accounting problem. A support-versus-control gap that survives the spread and persists across years would show a push large enough to trade.
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.