The Markov Property and Asset Price History
Summary
The document explains the intuition behind modeling a stock price as a Markov process. Under the stated assumption, the current price captures all relevant past information, so the process’s future behavior depends on its present state rather than requiring the full price history. It connects this idea to a stochastic differential equation for asset returns, though the equation itself is introduced in the question rather than derived in the answer.
The answer’s reasoning relies on the assumption that relevant information accumulates and is represented by the current price. This is an explanatory framing, not proof that real market prices are Markov or that the efficient market hypothesis guarantees the property. In practice, whether price alone is a sufficient state depends on the model and information set; variables such as volatility or other market conditions may also matter.
Key ideas
- A Markov process has future behavior conditioned on its current state rather than its full history.
- The answer explains the property by assuming the current stock price captures relevant past information.
- The explanation does not prove that real asset prices are Markov or that market efficiency guarantees it.
- A model may need state variables beyond price to represent relevant market conditions.
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Full text
# Are changes in the asset price a Markov process?
# Are changes in the asset price a Markov process?
I'm studying the book "The Mathematics of Financial Derivatives - A Student Introduction" and at the start of the second chapter it says that thanks to the effcient market hypothesis (1-the past history is fully reflected in the present price, 2-markets respond immediately to any new information about an asset) the changes in the asset price are a Markov process (full text below). After this, the book introduces the SDE that model the return on the asset: $\frac{dS}{S}=\mu dt + \sigma dW$.
A stochastic process is said to be a Markov process if it satisfies the Markov property: the next state of the process depends solely on the present one, not on the sequence of events that preceded it.
So I'm a bit confused, how can the changes in the price be a Markov process if the present price fully reflects the past history?
Or maybe I misinterpreted it, and the Markov property just says that all past information about the stock price process is incorporated in the current price and therefore only the current price is relevant ?
## Answer by Daneel Olivaw (score 5, accepted)
https://quant.stackexchange.com/a/36676
Let us define a stock process $(S_t)_{t \geq 0}$. We assume the value of the process $S_t$ at $t$ captures all past information which might affect the stock price (its "past history"). Let us consider two times $t_1 < t_2$.
If we make the (realistic) assumption that information accumulates and increases over time then, given the stock price $S_t$ captures all relevant information up to $t$, the stock price $S_{t_2}$ encapsulates all relevant information up to and including $t_2$. Therefore it contains at least the same amount of relevant information than $S_{t_1}$. As this is true for all $t<t_2$, all information affecting the stock process at $t_2$ is contained in $S_{t_2}$ and the history of the process is irrelevant (because it is "encapsulated" in $S_{t_2}$): the stock process is thus a Markov process.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.