AdvancedQuantitative MethodsPython

Run this module

cd "Quantitative Methods - Markov Chains"
python "markov_chains.py"

Quantitative Methods — Markov Chains¶

A Markov chain models a system that hops between a finite set of states where the next state depends only on the current one — never on the full history. That "memoryless" assumption is crude, but it is the backbone of a surprising amount of quant work: market regime models (bull / bear / sideways), credit-rating migration matrices, and toy models of volatility clustering.

Everything you need lives in one object — the transition matrix P, where P[i, j] is the probability of moving from state i to state j in one step. Each row is a probability distribution, so it sums to 1. This module builds P, takes its powers, finds its long-run behaviour, and simulates paths — all from NumPy.

Functions¶

Function	Description
`normalize_rows(matrix)`	Turn observed transition counts into a row-stochastic `P`
`is_stochastic(P)`	Check every row is a valid probability distribution
`n_step(P, n)`	The `n`-step transition matrix `P^n` (Chapman-Kolmogorov)
`stationary_distribution(P)`	Long-run fraction of time in each state (`pi P = pi`)
`simulate(P, start, steps, random_state)`	Generate a sample path of states
`expected_return_time(P, state)`	Mean steps to revisit a state (`1 / pi[state]`)

The stationary distribution¶

Multiply a starting distribution by P over and over and — for a well-behaved (irreducible, aperiodic) chain — it settles to a fixed vector pi that no longer changes: pi P = pi. That pi is the long-run fraction of time the chain spends in each state, independent of where it started. We compute it as the left eigenvector of P for eigenvalue 1.

It answers a genuinely useful question: if today's regime can be any of bull, bear or flat, what fraction of all days are bull days in the long run?

Estimating P from data¶

You rarely know P; you estimate it. Count how many times the market went from each regime to each other regime, then normalise each row:

import numpy as np
from markov_chains import normalize_rows, stationary_distribution

counts = np.array([[820,  30,  60],   # Bull -> Bull/Bear/Flat
                   [ 25, 300,  40],   # Bear -> ...
                   [ 70,  45, 410]])  # Flat -> ...

P = normalize_rows(counts)
print(P.sum(axis=1))               # [1. 1. 1.]
print(stationary_distribution(P))  # long-run regime mix

Example — market regimes¶

import numpy as np
from markov_chains import stationary_distribution, n_step, simulate

P = np.array([[0.90, 0.03, 0.07],
              [0.05, 0.85, 0.10],
              [0.15, 0.10, 0.75]])

print(stationary_distribution(P))  # e.g. [0.58 0.18 0.24]
print(n_step(P, 5)[0])             # where you are 5 days after a Bull day
print(simulate(P, start=0, steps=10, random_state=1))  # a sample path

Practical notes¶

The diagonal is "stickiness." Regimes persist, so real transition matrices have a dominant diagonal — markets stay bull far more often than they flip.
Aperiodic + irreducible guarantees a unique stationary distribution. A chain with an unreachable state, or one that cycles deterministically, breaks that — stationary_distribution will still return a fixed vector, but interpret it with care.
For continuous, mean-reverting state instead of discrete regimes, see Quantitative Methods - Stochastic Processes.
For detecting which regime you are in from returns, see Quantitative Methods - Regime Detection.
The same machinery powers the prediction step in the Quantitative Methods - Kalman Filter.

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