1. Hidden Markov Model for Stochastic Volatility
Vasin Suntayodom, Mengyuan Wu
University of Massachusetts, Amherst
Introduction
Recently, Bayesian estimation of stochastic volatility models via Markov chain Monte Carlo (MCMC) have gained some popularity because Bayesian
estimators have become much easier to compute. However, the purpose of Hidden Markov model for stochastic volatility project is
• to show that it provides alternative approach to estimate parameters in stochastic volatility model.
• to compare the goodness of fit between Gaussian Stochastic Volatility model and Non-Gaussian Stochastic Volatility
The key idea is the use of iterated numerical integration. This method involves an approximation to the stochastic volatility likelihood that can be made
arbitrarily accurate.
Numerical Results
The Gaussian stochastic volatility model with-
out leverage was fitted to the daily continuously
compounded returns on the index funds for the
period from the end of 1 May 2006 to the end of
27 May 2016. The starting values are µ0 = 0.7,
φ0 = 0.95 and σ0 = 0.12. Using m = 200 and gt
values from -2.5 to 2.5, we obtain the result as
follows :
β φ σ AIC BIC
vfinx 0.01490 0.99755 0.19384 -16219.70590 -16202.18969
veurx 0.01535 0.99254 0.14358 -14976.17353 -14958.65732
veiex 0.02027 0.99785 0.15170 -14976.24524 -14958.72903
vbltx 0.00616 0.99167 0.07627 -18365.26568 -18347.74947
vbisx 0.00205 0.99905 0.06691 -26627.52034 -26610.00413
vpacx 0.01316 0.98901 0.15359 -15761.37733 -15743.86112
The result obtained in table above lead us to
conclude that the parameter estimated obtained
by HMM stabilizes for m somewhere between
100 and 200. Next we examine Non-Gaussian
Stochastic Volatility with gt values from -7.5 to
7.5, we obtain the result as follows:
β φ σ ν AIC BIC
vfinx 0.00859 0.98457 0.17785 12.91058 -16228.02737 -16204.67242
veurx 0.01118 0.98899 0.13716 14.01139 -14989.97749 -14966.62254
veiex 0.01162 0.98766 0.14411 27.46222 -14981.03284 -14957.67789
vbltx 0.00616 0.99168 0.07627 31058.90450 -18363.26517 -18339.91022
vbisx 0.00106 0.99782 0.04748 7.73066 -26688.34194 -26664.98699
vpacx 0.00990 0.98141 0.16345 28.24942 -15772.48712 -15749.13217
According to the table above, S&P 500 index,
European stock index, Emerging markets index,
Short term bond index and Pacific stock index
under Non-Gaussian Stochastic Volatility model
perform better than the Gaussian Stochastic
Volatility.
On the contrary, the long term bond index is
almost normally distributed and the degree of
freedom we got also support this evidence (ν =
31058.90450). Both AIC and BIC are almost
identical in two models, therefore the models
are indistinguishable. In the case of Short term
bond index, kernal density also indicates that it
deviates from normality assumption.
As a result, the Non-Gaussian stochastic volatil-
ity model performs much better than Gaussian
stochastic volatility model.
Data Descriptions
We will use various major index funds obtained
from Bloomberg from May 2006 until May 2016
on daily basis. The following is the brief descrip-
tion of the data :
1. S&P 500 index (vfinx)
2. European stock index (veurx)
3. Emerging markets stock index (veiex)
4. Long term bond fund (vbltx)
5. Short term bond fund (vbisx)
6. Pacific stock index (vpacx)
The data set covers daily closing price data from
the end of 1 May 2006 to the end of 27 May 2016.
Model Specification
g1 g2 g3 gt gT
r1 r2 r3 rt rT
. . . . . .
The first form of the model which we consider
here and the best known, is Gaussian Stochas-
tic Volatility or Heston Model where the asset
returns rt on the observation equation satisfy
rt = tβ exp (gt/2)
gt+1 = φgt + ηt,
where |φ| < 1 and { t} is Gaussian white noise
sequence with mean 0 and variance 1, {ηt} is also
Gaussian white noise sequence with mean 0 and
variance σ2
ν. Suppose { t} and {νt} are inde-
pendent for Gaussian stochastic volatility model
without leverage effect. Then we improve the ob-
servation equation in the model by
rt = t (β exp (gt/2) + ξ) .
The additional parameter ξ ≥ 0 is persuasive on
the fact that some baseline volatility is always
presented. We also relax normality assumption
and assume that t has t-distribution with ν de-
grees of freedom. The basic model which assume
standard normal distribution for t is a special
case when ν → ∞.
Conclusion
In this work, we introduced the problem of volatility estimation in prices of financial assets. We have
discussed why this is an interesting endeavor to use stochastic volatility models and have introduced
a popular stochastic volatility model to capture some of the statistical features found in real-world
data. It would be interesting to experiment by relaxing assumption that { t} and {νt} have to
be independent for Gaussian stochastic volatility model with leverage effect. Specifically, for all t
t
ηt
∼ N (0, Σ) with Σ =
1 ρσ
ρσ σ2 .
Reference
[1] J. Hull and A. White. The pricing of options on assets with stochastic volatility the Journal of Finance, pp. 42(2):281-300, 1987.
[2] E. Jacquier, N. G. Polson, and P. E. Rossi. Bayesian analysis of stochastic volatility models with fat-tails and correlated errors Journal of Econometric,
122(1):185-212, 2004.
[3] P.Glasserman Monte Carlo methods in financial engineering (2004), Volume 53, Springer
[4] Bartolucci, F.,De Luca, G. Maximum likelihood estimation of a latent variable time-series model.Applied Stochastic Models in Business and Industry, pp. 17:
5-17, 2001
[5] Fridman, M., Harris, L. A maximum likelihood approach for non-Gaussian stochastic volatility models.Journal of Business and Economic Statistics, pp. 16:
284-291, 1998
Acknowledgment
We wish to thank particularly Panit Arunanondchai and Rene Cabrera for his proof reading and very useful comments during the preparation of this paper.