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Figure 14.1: Two-state Markov Chain Model
⎛
⎞
A =
⎜
⎝
1 − α α
β
1 − β
⎟
⎠
(14.4)
In order to simulate n steps of a general DSMC model it is possible to define the n-step
transition matrix A(n) as:
A ij (n) := p(X t+n = j | X t = i) (14.5)
It can be easily shown that A(m + n) = A(m)A(n) and thus that A(n) = A(1) n . This
means that n steps of a DSMC model can be simulated simply by repeated multiplication of the
transition matrix with itself.
14.2 Hidden Markov Models
Hidden Markov Models are Markov Models where the states are now "hidden" from view, rather
than being directly observable. Instead there are a set of output observations, related to the
states, which are directly visible. To make this concrete for a quantitative finance example it is
possible to think of the states as hidden "regimes" under which a market might be acting while
the observations are the asset returns that are directly visible.
In a Markov Model it is only necessary to create a joint density function for the observations.
A time-invariant transition matrix was specified allowing full simulation of the model. For
Hidden Markov Models it is necessary to create a set of discrete states z t ∈ {1, . . . , K} (although
for purposes of regime detection it is often only necessary to have K ≤ 3) and to model the
observations with an additional probability model, p(x t | z t ). That is, the conditional probability
of seeing a particular observation (asset return) given that the state (market regime) is currently
equal to z t .
Depending upon the specified state and observation transition probabilities a Hidden Markov