# 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 ?

## 1 Answer

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.

• Nice argument, so basically you are saying that 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 ? – sound wave Oct 31 '17 at 17:44
• Indeed. Note that is what your text says: if "the past history is fully reflected in the present price", then any price $S_t$ contains all information up to and including $t$, for any $t$; assuming information accumulates, this implies that the only relevant price is the current one as it contains at least as much information as all other past prices. – Daneel Olivaw Oct 31 '17 at 17:47
• Well, regarding your comment on volatility, we start delving on deeper modeling questions: historically the market has been assumed to be at least mildly efficient as there has been empirical evidence pointing towards it (at least in its weak- and semi-strong forms, check out Wikipedia, I cannot point to specific studies but the expert, Nobel-prize winner in this field is Eugene Fama, he has done a lot of studies on the question), resulting in Markovian modeling to reflect this. – Daneel Olivaw Oct 31 '17 at 17:52
• Of course other modeling approaches are possible and are indeed implemented: if your concern is predicting stock returns in order to deploy an investment strategy, then the Markovian assumption is probably unsufficient; however, if your main concern is the pricing of derivative products and you do not need to be too precise (but rather have a plausible model for stock price movements) then the Markovian assumption is (or at least has been considered so) a fair one. – Daneel Olivaw Oct 31 '17 at 17:56
• Yes it is, as when pricing vanilla derivatives (such as a call option) you only need to know the current price of the stock (instead of its whole price history). – Daneel Olivaw Oct 31 '17 at 21:34