A Markov process is a stochastic process that has the Markov property, which means that the future state of the process depends only on the present state and not on any past states. Mathematically, a Markov process (Xt)t ≥ 0 on a state space S is defined as follows:
$$\begin{aligned}
P(X_{t+s} &= x_{t+s}|X_t=x_t,\cdots,X_0 = x_0) = P(X_{t+s} = x_{t+s}|X_t = x_t)\\
&\phantom{=} \forall s\geq 0, x_i \in S, i=0,\cdots, t+s\end{aligned}$$
The Markov property states that the conditional probability distribution of future states, given the present state, is independent of past states of the process. In other words, knowing the current state of the system is sufficient to determine the probabilities of future states.
Relevance in Quantitative Trading:
Quantitative trading strategies often involve building predictive models that rely on identifying patterns or dependencies in historical market data. The assumption that financial time series follow Markov processes has several implications that are relevant to quantitative trading:
1. **Simplicity**: Since a Markov process depends only on the current state, quantitative models based on Markov processes do not require information about entire historical paths of the asset prices, which simplifies modeling and reduces computational complexity.
2. **Predictability**: If a financial time series is assumed to be a Markov process, then it is possible to make predictions about the future states (asset prices, returns, or volatility) solely based on the present state. This is the foundation of many quantitative trading models like algorithmic trading, statistical arbitrage, and high-frequency trading.
3. **State-Dependent Strategies**: With Markov processes, traders can develop trading strategies that take into account the current state of the market. For example, suppose we have a discrete-time Markov chain with two states representing high and low volatility regimes. A trading strategy can be designed to exploit differences in expected returns across these states, such as entering positions with high expected returns during high volatility states and exiting positions during low volatility states.
4. **Limitation of the Markov Assumption**: While the Markov assumption simplifies modeling and offers a practical approach to quantitative trading, it may not hold true for all financial time series. In some cases, path-dependent or long-memory effects may be present, which would violate the Markov property. It is essential to be aware of these limitations and test the validity of the Markov assumption when developing trading strategies.
One example of how to utilize a Markov process in quantitative trading is predicting stock price movements using a Hidden Markov Model (HMM). HMM can identify hidden states in the financial time series and estimate the probabilities of transitioning between these states, as well as the state-dependent distribution of observed variables (e.g., asset prices or returns). These estimated state transition probabilities and state-dependent distributions can be used to make trading decisions or optimize portfolio allocation.