Markov and Martingale Properties in Financial Price Models

Summary

This explanation introduces two properties used in stochastic models of asset prices. The Markov property says that, conditional on the present state, a process’s future distribution does not depend on its earlier states. The article illustrates the idea with coin tosses taking values of positive or negative one and with their cumulative sums: the expected next sum is conditioned on the current sum rather than the full sequence of prior sums.

The martingale property states that the conditional expected future value equals the current value given available information. In the coin-toss example, this corresponds to a fair game in which past outcomes do not create an expected advantage. The article presents these ideas as foundations for defining Brownian motion and later modeling asset paths. These are mathematical assumptions and concepts, not evidence that actual market prices satisfy them; the post does not estimate or test either property on market data.

Key ideas

• A Markov process has a future distribution that depends on its current state rather than its full history.
• The article illustrates the Markov property using coin tosses and cumulative sums.
• A martingale has a conditional expected future value equal to its current value.
• The coin-toss martingale example represents a fair game with no expected gain from past outcomes.
• Markov and martingale properties are introduced as foundations for Brownian motion models, not as empirically tested market facts.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.