Filtered Probability Trend Hypothesis (过滤概率趋势假设)

A binary directional classifier for short-horizon equity prices built on the product-measure construction of the Filtration-Indexed Probability Space. The model's output is one of bullish | bearish | undecidable, and the rule is Observation 4 of Achury Arredondo Sanchez Ascanio 2024 Filtration ARIMA Random Walk. This page distils the construction, formula, decision rule, and what is missing for it to be a tradable system.

The filtered probability (Proposition 1)

Let U(\eta_i, \delta/2) = (\eta_i - \delta/2,\ \eta_i + \delta/2) be the i-th random-walk bin and \hat Y_h, \sigma the ARIMA forecast and in-sample RMSE at horizon h. The filtered probability that the price at time h lies in bin i is

P\!\bigl(Y_h \in U(\eta_i, \delta/2)\bigr) = \frac{\displaystyle \Bigl[\Phi\!\Bigl(\tfrac{\eta_i + \delta/2 - \hat Y_h}{\sigma \sqrt h}\Bigr) - \Phi\!\Bigl(\tfrac{\eta_i - \delta/2 - \hat Y_h}{\sigma \sqrt h}\Bigr)\Bigr]\, \tfrac{h!}{i!\,(h-i)!}\, p^i (1-p)^{h-i} }{ \displaystyle \sum_{j=1}^{h+1} \Bigl[\Phi\!\Bigl(\tfrac{\eta_j + \delta/2 - \hat Y_h}{\sigma \sqrt h}\Bigr) - \Phi\!\Bigl(\tfrac{\eta_j - \delta/2 - \hat Y_h}{\sigma \sqrt h}\Bigr)\Bigr]\, \tfrac{h!}{j!\,(h-j)!}\, p^j (1-p)^{h-j} }.

• The left bracket is the ARIMA-implied normal mass on the bin.
• The right factor is the binomial mass of the random walk landing in that bin.
• The denominator renormalises so the bins form a proper partition of (\eta_1 - \delta/2,\ \eta_{h+1} + \delta/2).

Decision rule (Observation 4)

Let P_i denote the filtered probability on bin i, and let U_r, U_s be the two highest-probability bins with P_r \ge P_s. Define the centroid order by the endpoint values: \eta_r > \eta_s means U_r sits to the right of U_s.

1. Common accumulation point + ordered centroids + higher-prob bin is the higher-\eta bin → bullish. \eta_r > \eta_s \;\wedge\; P(U_r) > P(U_s) \;\Rightarrow\; \text{bullish}.
2. Common accumulation point + ordered centroids + higher-prob bin is the lower-\eta bin → bearish. \eta_r > \eta_s \;\wedge\; P(U_r) < P(U_s) \;\Rightarrow\; \text{bearish}.
3. Top-2 bins are not contiguous (no common accumulation point) OR P_r \approx P_s → undecidable.

The "common accumulation point" check is what separates a decided trend from an undecidable one. If the top two bins are separated by a low-probability gap, the model refuses to call the direction.

Visual

flowchart LR
    ARIMA[ARIMA forecast 𝒩(Ŷ_h, σ²h)]
    RW[Random walk binomial mass on h+1 bins]
    ARIMA --> PROD[Product: normal tail-diff × binomial mass]
    RW --> PROD
    PROD --> RENORM[Renormalise over all h+1 bins]
    RENORM --> P[Filtered probability P_i on each bin]
    P --> TOP2[Top two bins P_r ≥ P_s]
    TOP2 --> CONTIG{Share accumulation point?}
    CONTIG -->|no| UNDEC[Undecidable]
    CONTIG -->|yes| COMP{η_r > η_s ?}
    COMP -->|yes & P_r > P_s| BULL[Bullish]
    COMP -->|yes & P_r < P_s| BEAR[Bearish]
    COMP -->|P_r ≈ P_s| UNDEC

Worked example (SAN)

From the source paper's Fig. 4.1: the filtered density for the next 5 closings peaks around the 3.08–3.22 EUR bin, and the binomial mass (green trajectory) puts the modal mass in the same bin. The two top bins are contiguous, the higher-\eta bin has higher probability, so the model returns bullish.

Headline results (full table in the source)

ARIMA-vs-actual fifth-day variation: −11.2% to +13.5% — the model gets the direction right ~62.5% of the time but not the magnitude.

What is missing for trading

• No position sizing. A tradable system needs to convert bullish into a target weight. Achury et al. leave this open; a natural mapping is the filtered probability mass above X_0 as a directional confidence and a Volatility Targeting overlay for the dollar amount.
• No transaction-cost model. The model is binary-event; the question "is the move large enough to cover round-trip costs?" is not asked.
• No regime conditioning. The 5-day horizon is fixed; in a high-vol regime 5 days is too long, in a low-vol regime too short. Ferson-Siegel Optimal Weighting is a candidate for conditioning the binary call on lagged conditioning information.
• No statistical-significance band. A 62.5% hit rate on 24 decisions has a wide 95% binomial CI (roughly 41–81%). No Stationary Bootstrap is applied; the result could be a sample artefact.
• No out-of-sample protocol. Parameters p, q, \delta^\pm are fit on the same 130 observations used to evaluate. A walk-forward split would be the minimum.
• Binary output, not a return distribution. Cannot be plugged into a Kelly / risk-parity allocator directly. Converting to expected return requires the mean and variance of the filtered density — both are computable from the formula above.

Cross-references

• Built on: Filtration-Indexed Probability Space + ARIMA Forecast Model + Random Walk Price Pathwise
• Sister classifier: Time-Series Momentum (uses correlation structure instead of σ-algebra nesting)
• Sister signal-to-weight: Nonlinear Signal Mapping (would take P(\text{bullish}) as input s_t and produce a target weight)
• Sister risk overlay: Volatility Targeting (would take a directional bet and turn it into a sized position)
• Sister statistical test: Stationary Bootstrap (would put a confidence band on the 62.5% headline number)
• Source: Achury Arredondo Sanchez Ascanio 2024 Filtration ARIMA Random Walk