Apr

9

Please correct me if I'm wrong, but am I right in thinking that the particular suitability of Markov models to interest rate derivs derives from the term structure of interest rates vis a vis the section of the wiki entry that states:

"Since the system changes randomly, it is generally impossible to predict the exact state of the system in the future. However, the statistical properties of the system at a great many steps in the future can often be described"

Here is an abstract of a paper that might be of interest.

We introduce a general class of interest rate models in which the value of pure discount bonds can be expressed as a functional of some (low-dimensional) Markov process. At the abstract level this class includes all current models of practical importance. By specifying these models in Markov-functional form, we obtain a specification which is efficient to implement. An additional advantage of Markov-functional models is the fact that the specification of the model can be such that the forward rate distribution implied by market option prices can be fitted exactly, which makes these models particularly suited for derivatives pricing. We give examples of Markov-functional models that are fitted to market prices of caps/floors and swaptions.

Bruno Ombreux writes:

I just checked "Analysis of Financial Times Series", by Tsay. It is a relatively recent book so should be almost state of the art. It looks like Markov chains are used in two areas:

- better time series modeling, with regime switching.

- MCMC as a tool for Bayesian inference, whose main financialapplication at this point seems to be stochastic volatility models.

Practically, that would mean applications in "ever changing cycle" detection and option pricing/hedging.

On the first point, I personally toyed with Hidden Markov models. They work well in hindsight and are able to detect transitions between (high volatily / bear) and (low volatility / bull). But:

- this is all in hindsight. "Past performance is no guarantee…"
- this is low frequency data and I am not sure that the fact they could
detect the handful of past secular changes is that much useful. - a simple volatility threshold might do the job just as well without complication. - what you end up with is a probability transition matrix, which is not very helpful given that you are looking at only a few cycles.

… Maybe they could be more interesting in high frequency. I don't know.

On the second point, my opinion about option pricing is that the price of an OTC option is "as high as the buyer is prepared to pay". Models are an excuse. So I am not sure bayesian stochastic volatility models beat "seat of the pants" marketing.

Phil McDonnell adds:

If we look at the Niederhoffer-Osborne data on p. 899 we can see the number of times the market went from a don tick of various sizes to an uptick state. The reverse is also enumerated. I will take just the -1/8 state and +1/8 to illustrate the following. This matrix is the transition matrix by number of times each prior state led to the subsequent state.

            -1/8       +1/8     total
-1/8      231        777      1008
+1/8      709        236       945

From this we can compute the probabilities of being in each state given the prior state in the previous time period (trade).

Probabilities

           -1/8       +1/8      total
-1/8      .23        .77       1.00
+1/8      .75        .25       1.00

For Markov analysis we can make a prediction by multiplying the probability matrix by the vector which describes the current state. The result is a new vector of probabilities of being in the two states given the initial state. To predict two states ahead we multiply that by the probability matrix yet again. Let us take S as our starting state and P as our probability matrix. Then a prediction k steps ahead is given by:

S(k) = S(0) * P ^ k

Note that * is the matrix multiplication operator and ^k means to multiply P by itself k times.

What often happens is these matrices arrive at some steady state equilibrium after k iterations and we get the happy result that the probabilities are unchanged from iteration to iteration.

Russ Sears writes:

the Society of Actuaries has many research articles using the Markov Chain process. To see them all type in "Markov chain" into their home web search. I believe one of the best papers to give and intuitive understanding of its use and power and limits, is Ms. Christiansen's paper in which she gives a brief intro to many different interest rate generators.


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