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Nonlinear Filtering of Stochastic
Volatility Aleksandar Zatezalo, University of
Rijeka, Faculty of Philosophy in Rijeka, Rijeka,
Croatia e-mail zatezalo_at_pefri.hr
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Abstract Stochastic volatility models for
increments of logarithms of stock prices are
considered. Given historic data of stock prices
and a state model for volatility, we are
applying nonlinear filtering methods to estimate
conditional probability density function
of stochastic volatility at time
given sigma algebra generated by all stock prices
up to time . Numerical methods for nonlinear
filtering based on Strang splitting scheme are
proposed and numerical simulations are
presented. Method for evaluation and comparison
of different schemes is proposed based on value
at risk (VaR) calculations.
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References
1 W. F. Ames, Numerical Methods for Partial
Differential Equations, Academic Press,
London, 1992. 2 R. Frey and W. J. Runggaldier,
A nonlinear filtering approach to volatility
estimation with a view towards high frequency
data, preprint, to appear in
International Journal of Theoretical and Applied
Finance. 3 A. Kolmogoroff, Zufällige
bewegungen, Annals of Mathematics, Volume 35,
No. 1, 1934. 4 A. N. Shiryaev, Essentials
of Stochastic Finance, World Scientific, New
Jersey, 1999. 5 I. Karatzas and S.E.
Shreve, Methods of Mathematical Finance,
Springer, New York, 1998. 6 J. Ma and
J. Yong, Forward-Backward Stochastic Differential
Equations and Their Applications,
Springer, New York, 1999. 7 J. Cvitanic, R.
Liptser, and B. Rozovskii, Tracking Volatility,
In Proc. 39th IEEE Conf. On Decision and
Control, Sydney, 2000.
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Continuous Model
Let be probability space with
complete filtration ,
Wiener process with respect to the filtration
, and
progressively measurable processes. Let price
process (stock price) satisfies
Black-Scholes type model that allows stochastic
volatility i.e. for we
have (see 7) Further we consider stochastic
process which represents amount of
money investor puts into the stock with price
and let represents interest rate
of the riskless asset
and let wealth process
satisfies (see 6)
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Assumptions
We assume ,
and
which for gives (using
Itôs formula)
(for this model see
4) where
are the so called i.i.d.
random variables. We can consider
as observed data. Considering
discrete observation we have that the change of
wealth is also discrete i.e. we have where
and
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Assumptions (Cont.)
We assume that we do not borrow the money and
that we do not sell shares which we do not own
at the moment i.e. we have constraint Generally
we assume such that
satisfies Itôs differential
equation i.e. we have that is
solution of where is Wiener
process with respect to filtration and
and are suitable functions. We
consider special case of the so called constant
velocity tracking (in radar tracking) i.e. we
have where
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Simulation Examples vs. Real Data
1.25
1.2
Daily stock price in 1995 for ATT stock
1.15
1.1
1.05
1
0.95
0
50
100
150
200
250
300
Simulation
Simulation
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Value at Risk (VaR)
Let be the amount of money which the
agent is willing to risk with probability at
most under observed past data where
represents the sets of measure zero. That is
we are interested in estimating portfolios
such that where
i.e. the problem is to mathematically describe I
f we are able to calculate quantity (2) we should
be able to determine the interval of desirable
portfolios.
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Value at Risk (Cont.)
Let
is
conditional probability density of
given Therefore we have Since
and is strictly decreasing
function on such that
there
exists a unique such that Then the
optimal portfolio is
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Value at risk (Cont.)
The problem is to approximate
i.e. the conditional probability density of
given . The approximation of
will be done at the grid points
i.e. Let where we want to find the maximal
such that
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Predictor for Tracking
Transitional probability density function of the
process defined by (1) is the fundamental
solution of the Fokker-Planck equation Therefore
to give an estimate of the conditional
probability density
defined for any nonnegative or bounded Borel
function by
the following equality we approximate solution
of (5) on interval with
initial condition
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Predictor for Tracking (Cont.)
For model
we have prediction conditional probability
density function given by which is defined by
This is the so called prediction step.
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Correction for Tracking
For correction in tracking Bayes rule can be
applied i.e. Since we
have In calculations instead of
we take Therefore the problem is how to
approximate the solution of (5) as simple and
fast as possible.
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Numerical Schemes
Let and For the exact
solution at we have where
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Numerical Schemes (Cont.)
We have the following two approximations The
approximation in (6) corresponds to Strang scheme
by separately solving in given order the
following PDEs
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Numerical Schemes (Cont.)
Generalization of Strangs scheme is given
by for prescribed we have The
finite difference corresponding to (8) is given
by where
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Numerical Schemes (Cont.)
Let for From the expression for the
fundamental solution of (5) (see 3) we have
that the exact solution of (5), with initial
condition expressed by and is given
by
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Numerical Schemes (Cont.)
Checking Strangs splitting vs. simple operator
splitting. Both schemes converge with increasing
m and refinement of space grid.
vs. Strangs
scheme shows higher accuracy in time variable t.
initial condition
true solution
Strangs scheme
simple operator splitting scheme
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Simulation Example 1
Here we consider stochastic volatility model
given with Stochastic Differential Equation (1)
and Initial conditional probability density
function is given by We consider two
types of estimators maximum probability
estimator and standard conditional probability
estimator (or only prob. estim.) We assume that
10 percent of wealth can be lost with probability
at most 0.1 and we invest according to portfolio
calculated by (3).
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Simulation Example 1 (Cont.)
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Simulation Example 2
For simulated observations we used
where and we
used the tracker from Example 1with
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Real Data (from slide Simulation Examples vs.
Real Data)
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Conclusion
We have the new method of estimating volatility
of stock prices. The method is robust with
respect to the model which governs the
volatility i.e. it can perform well even though
volatility fits better to different model. The
method does not require huge historical data to
estimate the volatility. It gives conditional
probability distribution function of
volatility at any time in future for given sigma
algebra of observation. This gives us possibility
to calculate value at risk (VaR) for any future
investment with better precision and accuracy
depending on how well model corresponds to
reality. The method is satisfactory with respect
to VaR calculated portfolio on given simulation
and real data examples. It is derived and
proposed method for evaluation and comparisons of
different models and methods for estimation of
volatility.
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Further possibilities
Application of the method to continuous SDEs
which are coming from already known discrete
models (e.g. GARCH). Developing suitable
correlation tracking techniques for
accurate portfolio optimization i.e. possible
application of our method to prediction of
time-nonhomogeneous correlation coefficients for
better assessing investments (diversification). P
assively track a stock indexes (DAX). Optimal
control of exchange rates and/or price of a
stock (modeling diffusion of news into
market). Optimal control for portfolio with
transaction costs. Option pricing using partial
differential equations and in more complex
situations stochastic partial differential
equations.