Simple Markov Chain Model and Application on Securities Market
A Markov Chain is a mathematical model for representing a sequence of events in which the probability of each event depends only on the state of the previous event. In other words, it’s a sequence of events where the future state of the system depends only on the current state, rather than the history of the states that came before it.
A simple Markov Chain can be represented as a state transition diagram, where each state represents an event, and arrows between the states represent the probability of transitioning from one state to another. The probabilities of transitions between states must sum up to 1, indicating that the system is in one of the states at any given time.
An example of a simple Markov Chain could be modeling the weather, where the states are “Sunny”, “Rainy”, and “Cloudy”.
If the current state is “Sunny”, the probability of transitioning to “Rainy” could be 0.2, the probability of transitioning to “Cloudy” could be 0.4, and the probability of staying “Sunny” could be 0.4. These probabilities represent the likelihood of the weather changing from “Sunny” to “Rainy” or “Cloudy” in the next day.
The Markov property states that the future is independent of the past, given the present values of the process, because the future state of the system depends only on the current state and not the previous states. The probability of each event depends only on the state of the previous event.
This property is a fundamental assumption of Markov Chain models and is based on the idea that the current state of the system contains all the relevant information about the history of the system up to that point.
As a result, the future behavior of the system can become predicted based solely on the current state, without the need to consider the entire history of the system.
This assumption can be useful in many applications. As it allows for the modeling of complex systems in a simplified manner. By reducing the amount of information that needs to be taken into account to make predictions.
For example, in weather forecasting, the current temperature, pressure, and other weather variables become utilized to predict the future weather. Without considering the entire history of the weather patterns.
Markov processes are important models of security prices. Because they are often realistic representations of true prices.
Let the filtration F={Ft; t=0,1,…,T} become generated by a stochastic process X={Xt; t=0,1,…,T}. The process takes values in some finite set E, which we call state space.
There usually exists a sample space and a probability measure P. And the information Ft should become thought of as the history of the present and past values of the process X. A stochastic process Xt=j ∈ E, we would say the process is in the state j at time t.
A stochastic process X becomes a Markov Chain if,
P{Xt+1=j | Ft}=P{Xt+1=j | Xt} for all j ∈ E and all t.
It follows that,
P{Xt+s=j | Ft}=P{Xt+s=j | Xt} for all s1.
Thus, a Markov chain is a stochastic process. One where the only information useful for predicting future value is the current state. Moreover, this means that the future state of the system can become determined solely based on the current state and the transition probabilities between states. The past states do not provide any additional information that would affect the future behavior of the system. And therefore, are not considered useful for predicting the future.
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A Markov Chain X is called stationary. If the conditional probability P{Xt+1=j | Ft} does not depend on time t.
We can define the transition probabilities
P(i,j)=P{Xt+1=j | Xt=i}, for i,j ∈ E
To organize them into a transition matrix,
P=[P(i,j)]
It is a square matrix with the sum of the elements in each row of P equals 1. An example would be the process N={Nt; t=1,…,T} represents the number of heads in t flips of a coin that lands “head” with probability p. For the state space, we take {0,1,…,T} and the transition matrix would be
P =
In conclusion, a useful Markov Chain property would be if Y=f(Xt, Xt+1, … , XT) for some function f, thenE[Y | Ft]=E[Y | Xt]. This property basically says that, given the present, the future and the past are conditionally independent. Assume there is a securities market model where the discounted price process is a Markov Chain. It means P{St+1*=j | Ft}=P{St+1*=j | St*} for all j ∈ E and all t. If we want to seek an arbitrage opportunity. Lasltly, we want to know if there exists a risk-neutral probability measure (martingale measure) Q under which S*is a Markov Chain. If there is not a risk-neutral probability measure, then the securities market has an arbitrage opportunity.
Written by Ruizhe Li
