random-walkmarkovbinomialsigma-algebratrend-forecastequity

Random Walk Price Pathwise (参数化随机游走)

A simple random walk over h steps on the asset price, parameterised by the asset's own historical increments. Plays the role of the discrete-landing distribution that lives on the smaller σ-algebra \mathcal{F}_A of the Filtration-Indexed Probability Space. In Achury Arredondo Sanchez Ascanio 2024 Filtration ARIMA Random Walk it supplies the per-bin probability mass that multiplies the ARIMA Forecast Model density to form the trend hypothesis.

Construction (Achury et al. Observation 2)

Start from the last close X_0. For a horizon of h trading days:

  • Step sizes. Use the medians of the positive and negative first differences, not the means: \delta^+ = \tilde x\bigl(\Delta Y_t : \Delta Y_t > 0\bigr), \qquad \delta^- = \tilde x\bigl(\Delta Y_t : \Delta Y_t < 0\bigr), \qquad \delta = \delta^+ + |\delta^-|. Median rather than mean so that one or two outlier jumps do not pull the path off the historical centre.
  • Step probabilities. p = \Delta^+ Y_t / (\Delta^+ Y_t + \Delta^- Y_t) and q = 1 - p, with the count of "no-change" days excluded.
  • Walk. At each step the price moves up by \delta^+ with prob p or down by \delta^- with prob q = 1 - p. The walk is Markov (Definition 7 of the source) and simple (only two outcomes per step).

Endpoint mass

The 2^h paths collapse to h+1 endpoints because the landing price depends only on the count of up- vs down-steps, not on the order:

\eta_i = X_0 + (h - i)\,\delta^+ + i\,\delta^-, \qquad i = 0, 1, \dots, h. The mass on endpoint i is binomial: P_i = \binom{h}{i} p^i (1 - p)^{h - i}.

The endpoints are equally spaced at distance \delta apart. The σ-algebra \mathcal{F}_A is generated by the h+1 open intervals U(\eta_i, \delta/2) of radius \delta/2 around each endpoint.

Worked example (SAN)

  • X_0 = 3.1425, h = 5, p = q = 0.5, \delta^+ = 0.0375, \delta^- = -0.033.
  • 2^5 = 32 paths → h+1 = 6 endpoints: \{3.3300, 3.2595, 3.1890, 3.1185, 3.0480, 2.9775\}.
  • Binomial masses (read off Fig. 3.2 of the source): modal mass at \eta_3 = 3.1890, tapering off to the tails.
  • See Fig. 4.1 of the source for the filtered density, which multiplies this binomial by the ARIMA normal CDF and renormalises — that is the Filtered Probability Trend Hypothesis.

Why a random walk, and why binomial

  • Calibration-free. A simple random walk only needs X_0, \delta^+, \delta^-, p — all directly observable from the close-to-close series. No MLE, no hyperparameter search.
  • Symmetric, low-variance shape. The binomial is unimodal (for h \ge 2 and p \ne 0,1), which makes it easy to reason about top-2-bin order.
  • Markov + exchangeable increments. The future is a function of the present only; the order of up/down steps does not affect the landing distribution.

Where it fails

  • Constant step size. Real markets have heteroscedastic steps; the constant-\delta walk cannot capture volatility clustering. Replacing the binomial with a step-size distribution fit to the empirical histogram is the natural upgrade — and it is essentially what ARIMA Forecast Model adds.
  • Path independence. Only the endpoint matters, not the path. If you need path-dependent statistics (max drawdown over the horizon, time spent above X_0), the binomial mass is no longer enough.
  • No drift-vs-no-drift uncertainty. A simple random walk with p = 0.5 has zero expected drift, but a long-run up-trending asset has p \ne 0.5. Achury et al. let p be the empirical up-frequency, which lets the walk have drift; this is good for the bias, bad when the bias is reversing.
  • Not a model of "fat tails". The binomial gives Gaussian-like tails. Real equity returns are leptokurtic; the random-walk mass under-prices extreme moves and that is exactly where ARIMA's wider normal also under-prices them, so the product measure inherits the problem.

Cross-references