Markov chains are mathematical systems that model transitions between states with fixed probabilities. They're widely used in finance to model market states, option pricing, and risk assessment.
Define the probabilities of moving from one state to another (each row must sum to 1):
| From \ To | State 1 | State 2 |
|---|---|---|
| State 1 | ||
| State 2 |
A Markov chain is a stochastic model describing a sequence of possible events where the probability of each event depends only on the state attained in the previous event.
Key properties:
In finance, Markov chains are used for:
The basic formula for state transitions is:
P(Xn+1 = x | Xn = xn, ..., X1 = x1) = P(Xn+1 = x | Xn = xn)
Where: