Constructing price-change Markov transition matrices

What the model assumes

A Markov transition model is a forecast construction in which the next classified price, volume, or breadth state depends only on the current state over a stated sampling interval and lookback. Ordered price-change observations can therefore be mapped from one sampling interval to the next.

The worked construction uses three mutually exclusive daily states: up one price unit, unchanged, or down one unit. Those states are recorded over a ten-session lookback, then each prior state is paired with the following state.

Building the one-step transition matrix

A count matrix is filled so each cell records how often a prior-day state was followed by a current-day state. Dividing every count by its row total produces the one-step transition matrix.

A transition matrix is a square table of one-step probabilities. Rows are the prior discrete state, columns are the next state, and the entries in each row sum to one.

Iterating to an equilibrium vector

Multiplying the transition matrix by itself, then repeatedly multiplying each new iterative matrix by the original transition matrix, yields an equilibrium vector. For a regular chain that vector is identical in every row and independent of the starting state.

The equilibrium vector is the stable row of long-run state probabilities obtained by repeating matrix multiplication until the iterative table stops changing.

Choosing state bins

A state bin is the discrete interval, fixed or variable in width, used to classify each observation into a row and a column of the transition matrix. State bins can be ranges rather than single ticks, and the grid can be enlarged.

A five-by-five layout is described as usable. Equal-count variable-width bins that place one-fifth of a sorted sample in each state reduce mid-matrix clustering, but identically placed zeros can keep a chain from reaching a regular equilibrium.

Long-run independence is not a short-horizon forecast

The construction assumes a stable generating distribution. Even when a regular chain's long-run vector ignores the start, short-horizon forecasts remain dependent on the current state.

From probabilities to a risk-of-ruin bound

Transition and equilibrium probabilities are described as inputs for estimating gain and loss magnitudes and for calculating risk of ruin so an exposure decision can be bounded before the model is treated as a standalone trade basis.

Risk of ruin is a pre-trade and in-position filter that turns outcome probabilities and loss magnitudes into a bound on exposure before a position is treated as justified.