<p>You could do a binomial-tree approximation to the process for $z_t$ and then have a different process control the number of steps you take on the tree. This preserves recombining property and it is essentially the method explored in <a href="http://www.jstor.org/stable/2331407" rel="nofollow">On the Computation of  Continuous  Time Option Prices Using Discrete Approximations</a> (Amin (1991)) </p>

<blockquote>
  <p>We  develop  a class of  <strong>discrete, path-independent  models</strong> to 
  compute prices  of American options  within the Black-Scholes  (1973)
  framework,  <strong>including  models in which state variables have 
  time-varying volatility</strong>  functions and models with  multiple  state
  variables. Time-varying volatility  functions are illustrated with 
  applications  to term structure models developed by  Vasicek  (1977) 
  and  Heath, Jarrow,  and Morton  (1988), (1990).  Distinct from 
  previous  work in the  literature,  the multivariate models suggested
  in this  paper are consistent with  arbitrarily large, though
  constant,  covariance functions. Finally,  we compare  and contrast
  the numerical  accuracy  of a  large  number of models with simulation
  results.</p>
</blockquote>

<p>You could then use a <a href="http://www2.hsu-hh.de/mathstat/downloads/Folien_09_12_2.pdf" rel="nofollow">binomial AR(1) process</a> for the number of steps.</p>

CC BY-SA 3.0
2015-01-26T21:58:22.383Z
2015-01-26T21:58:22.383Z
BKay
<p>A common practice when computing solutions to stochastic dynamic optimization problems is to approximate an exogenous forcing process $z_{t+1} = \rho z_t + \sigma \epsilon_{t+1}$ with a finite-state Markov chain, e.g. by Tauchen's or Rouwenhorst's method. What would be a good way to discretize AR(1) process with stochastic volatility?</p>

<p>That is, if the original AR(1)-SV process looks something like this:
$$
\begin{split}
z_{t+1} &amp;= \rho_z z_t + \mathrm{e}^{v_t} \sigma_z \epsilon_{t+1} \\
v_{t+1} &amp;= \rho_v v_t + \sigma_v \eta_{t+1} 
\end{split}
$$
with $\epsilon, \eta$ being independent standard gaussian shocks, my goal is to obtain values $z_i$ and transition probabilities $p_{ij}$ ($i,j=1,\dots,n$) such that the corresponding Markov chain approximates the original continuous-valued process.</p>

<econometrics><time-series><computation><markov-chain><statistics>
How to approximate stochastic volatility model with finite-state Markov chain?
ivansml
# How to approximate stochastic volatility model with finite-state Markov chain?

(Asked by: username_0 on 2015-01-21)

A common practice when computing solutions to stochastic dynamic optimization problems is to approximate an exogenous forcing process $z\_{t+1} = \rho z\_t + \sigma \epsilon\_{t+1}$ with a finite-state Markov chain, e.g. by Tauchen's or Rouwenhorst's method. What would be a good way to discretize AR(1) process with stochastic volatility?

That is, if the original AR(1)-SV process looks something like this:
$$
\begin{split}
z\_{t+1} &= \rho\_z z\_t + \mathrm{e}^{v\_t} \sigma\_z \epsilon\_{t+1} \\
v\_{t+1} &= \rho\_v v\_t + \sigma\_v \eta\_{t+1}
\end{split}
$$
with $\epsilon, \eta$ being independent standard gaussian shocks, my goal is to obtain values $z\_i$ and transition probabilities $p\_{ij}$ ($i,j=1,\dots,n$) such that the corresponding Markov chain approximates the original continuous-valued process.

**Tags:** econometrics, time-series, computation, markov-chain, statistics

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## 1 answers

### Answer by username_1 on 2015-01-26. Score: 2

You could do a binomial-tree approximation to the process for $z\_t$ and then have a different process control the number of steps you take on the tree. This preserves recombining property and it is essentially the method explored in [On the Computation of Continuous Time Option Prices Using Discrete Approximations](http://www.jstor.org/stable/2331407) (Amin (1991))

> We develop a class of **discrete, path-independent models** to
> compute prices of American options within the Black-Scholes (1973)
> framework, **including models in which state variables have
> time-varying volatility** functions and models with multiple state
> variables. Time-varying volatility functions are illustrated with
> applications to term structure models developed by Vasicek (1977)
> and Heath, Jarrow, and Morton (1988), (1990). Distinct from
> previous work in the literature, the multivariate models suggested
> in this paper are consistent with arbitrarily large, though
> constant, covariance functions. Finally, we compare and contrast
> the numerical accuracy of a large number of models with simulation
> results.

You could then use a [binomial AR(1) process](http://www2.hsu-hh.de/mathstat/downloads/Folien_09_12_2.pdf) for the number of steps.


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