Nonlinear Filtering of
Stochastic Volatility
Aleksandar Zatezalo,
University of Rijeka,
Faculty of Philosophy in Rijeka,
Rijeka, Croatia
e-mail: zatezalo@pefri.hr
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 p x | Fty of
y
F
stochastic volatility at time t given sigma algebra t
generated by all stock prices up to time t . 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.


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. Cvitanić, R. Liptser, and B. Rozovskii, Tracking Volatility, In Proc. 39th
IEEE Conf. On Decision and Control, Sydney, 2000.
Continuous Model
Let , F , P be probability space with complete filtration Ft t 0,
wt , t  0 Wiener process with respect to the filtration Ft t 0,
and rt , t  0 progressively measurable processes. Let price
process St , t  0, (stock price) satisfies Black-Scholes type model
that allows stochastic volatility t , t  0 i.e. for t  0 we
have (see [7])
dSt  St rt dt  t dwt .
Further we consider stochastic process t , t  0, which represents
amount of money investor puts into the stock with price St , t  0,
and let t , tt  0, represents interest rate of the riskless asset


Bt  B0 exp   s ds , t  0, and let wealth process X t , t  0, satisfies

(see [6])  0
t
X t  t
dXt  dSt 
dBt , t  0, X 0  x  0.
St
Bt
Assumptions
n
t2
, t : t n1  tn ,
We assume Bt  1, rt  , t  0, and t 
2
t
S tn1
t  tn , tn1 , which for yn : log
gives (using Itô’s formula)
Stn

yn  nn ,
(for this model see [4])
where n ~ N 0,1, n  0,1,2,, are the so called i.i.d. random
variables. We can consider yn , n  0,1,2,, as observed data.
Considering discrete observation we have that the change of wealth
Stn1  Stn
is also discrete i.e. we have
X n  tn
St n
,
where X n : X tn1  X tn , and Stn1  Stn  e yn  1  en n  1.
St n
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
0  tn  X tn .
Generally we assume t  t , xt   0, such that xt , t  0, satisfies
Itô’s differential equation i.e. we have that xt , t  0, is solution
of
~
dxt  b(t , xt )dt  (t , xt )dwt ,
~ , t  0, is Wiener process with respect to filtration Ft t 0
where w
t
and b(,) and (,) are suitable functions. We consider special
case of the so called constant velocity tracking (in radar tracking)
i.e. we have xt   t , t , t  0, where
~



(1)
dt  dwt ,
d   dt.
t
t
Simulation Examples vs. Real Data
1.25
1.0007
1.0006
1.2
Daily stock price
in 1995
for ATT stock
1.0005
1.0004
1.0003
1.0002
1.0001
1.15
1.1
1.05
1
1
0.9999
0
50
100
150
200
250
0.95
300
0
50
100
150
200
250
300
Prices of Meta-stock for 1995
34
High
Low
Mid
32
Simulation:
(t,  t , t )  t
30
value of prices
0.9998
Simulation:
28
(t ,  t , t )  0.01 et
26
24
22
0
50
100
150
trading days
200
250
300
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
Fny1 : p0 , y0 , y1,..., yn1 A0 where A0 represents
the sets of measure zero.
That is we are interested in estimating portfolios tn such that


P X n   | Fny1  1  ,
where X n  X tn1  X tn , i.e. the problem is to mathematically
describe
( 2)


P X n   | Fny1 .
If we are able to calculate quantity (2) we should be able to
determine the interval of desirable portfolios.
Value at Risk (Cont.)
Let
 : n X tn , tn   n X tn , n ,  n [0,1],
z
2
 x2
F ( z ) :
e dx, z  0,  n ( y), y  R is conditional

0
y
F
probability density of n given n1. Therefore we have
 F  log1  (
n

/  n )  /( y 2 )  n ( y )dy  1  2.


Since 0  n   n  1 and g ()   F  log  /( y 2 )  n ( y )dy  1  2,
R
R
is strictly decreasing function on  [0,1  n ], such that
g (0)  1, g (1  n )  1  2 there exists a unique ~
n such that
g ~
n   1  2. Then the optimal portfolio is
(3)

n 
.
~
1  n 
Value at risk (Cont.)
The problem is to approximate  n ( y), y  R , i.e. the conditional
probability density of n given Fny1.
The approximation ~
 () of () will be done at the
n ( yi )   n ( yi ).
grid points yi i.e. ~
Let
 log  
~
 n ( yi ),
g () :  F  
( 4)

y
2
i
 i


 
~
where we want to find the maximal   0,1   such that

g~()  1  2,   0, ~
.
Xn 
Predictor for Tracking
Transitional probability density function of the process defined
by (1) is the fundamental solution of the Fokker-Planck equation
 )  2  2u(t , ,  )
u(t , ,  )

u
(
t
,

,

 

.
2

t

2

(5)
Therefore to give an estimate of the conditional probability
y
density pt, ,  | Fn1 ,defined for any nonnegative or bounded
 ), ,   R, by the following equality
Borel function g (, 


  g ,  pt
E g tn1 ,  tn1 | Fny1 
R2

 | F y dd ,
,

,

n 1
n 1
we approximate solution of (5) on interval 0, tn1  tn  with initial
y
condition u(0, ,  )  ptn , ,  | Fn .
Predictor for Tracking (Cont.)
 ,  )   , t  0, we have
For model (t , 
t
t
t
prediction conditional probability density function given by

 

p tn1, ,  | Fny  u(tn1  tn , ,  )  u(tn1  tn ,,  ) I 0 ,
which is defined by


  g ,  pt
E g tn1 ,  tn1 | Fny 
R2
This is the so called prediction step.

 | F y dd .
,

,

n 1
n
Correction for Tracking
For correction in tracking Bayes rule can be applied i.e.


p t n 1 , ,  | Fny1 




p yn 1 | ,  p t n 1 , ,  | Fny
  R2.
,

,

p y | ,  p t , ,  | F y dd
 
n 1
n 1
n

R2
Since yn  nn we have


1

p y | ,  
e
2
y2
 2
2
,   0,  ,   R.
In calculations instead of   0,   we take   , ,   0.
Therefore the problem is how to approximate the solution
of (5) as simple and fast as possible.
Numerical Schemes

2
2



Let L1  : 
and L2 :
.
2


2 
For the exact solution at

t  t we have



t L1   L2 

u t  t, ,   e
u t, ,  ,
where

e     : 
t L1   L2
k 0
t k L    L k .
k!
1
2
Numerical Schemes (Cont.)
We have the following two approximations
( 6)
(7 )





tL2
e
e
e e
 O t 3 ,
t L1   L2 
tL1   tL2
e
e
e  O t 2 .
t L1   L2

t
L1 
2
t
L1 
2
 
 
The approximation in (6) corresponds to Strang scheme
by separately solving in given order the following PDE’s
(8)

1 
L1  u ,
2
ut  L2u ,
ut 
ut 

1 
L1  u.
2
Numerical Schemes (Cont.)
Generalization of Strang’s scheme is given by


u t  t , ,   e
 
t
L1 
2m
~
e


t
L2
m
e
 
t
L1 
m
m 1



e
t
L2
m
e
  t 3 
u t , ,   O   .
 m  



t
L1 
2m
~


for prescribed   0 we have m  t /  .
The finite difference corresponding to (8) is given by
1
1

1

t

t




un1   I  A1  I  tA2  I  A1  un ,
2
2




where
y
y

A1u : I yi 0
i
x
ui 1, j  I yi 0

yi
ui 1, j  i ui , j ,
x
x
2
2



A2u :
u

u
 2 ui , j .
2 i , j 1
2 i , j 1
2y
2y
y
2
Numerical Schemes (Cont.)
Let for
 y , x  0, t  0, ,   R,
t  : 2 2y  t 6 2x  t 3   9 2x  2y ,
t  : 6 2x  t 3 ,
3
t
 t  : 2 2x  t 2  2y  .
3
From the expression for the fundamental solution of (5)
(see [3]) we have that the exact solution of (5),    
with initial condition expressed by u 0, ,    e 2 2 / 2x  y ,
and   2 is given by
2
2
x


u t , ,   3e

9t   ( t ) t 2x 
  t  t
2  2x ( t )  ( t )
2
2 2
12  2x  6t 2x  2
2  2x ( t )
 
2

2
2 x
 2
2  2y

2
2
y

/ 2 (t ) , t  0.
Numerical Schemes (Cont.)
Checking Strang’s splitting vs. simple operator splitting.
Both schemes converge with increasing m and refinement of
space grid.
True solution
10
Initial condition
10
initial condition
8
8
6
6
4
4
2
Speed
Speed
2
0
0
-2
-2
-4
-4
true solution
-6
-8
-10
-10
-8
-6
-4
-2
0
Space
2
4
6
8
-6
-8
-10
-10
10
-8
-6
-4
Approximation using Strang scheme
-2
0
Space
2
4
6
8
10
Approximation using simple scheme
10
10
8
Strang’s scheme
6
4
8
6
4
2
0
vs.
-2
-4
Speed
Speed
2
0
-2
-6
-4
-8
-6
-10
-10
-8
-6
-4
-2
0
Space
2
4
6
8
10
simple operator splitting scheme
-8
-10
-10
-8
-6
-4
-2
0
Space
2
Strang’s scheme shows higher accuracy in time variable t.
4
6
8
10
Simulation Example 1
Here we consider stochastic volatility model given with Stochastic
Differential Equation (1) and  (t,t , t )  t , t  0.
Initial conditional probability density function p0 ( x, x) : p(0, x, x | y0 )
y02
x 2
is given by


p0 ( x, x) 
y0
 x  x
2
2
e
2 x 2 2 x 2
.
We consider two types of estimators: maximum probability estimator
| F y 

ˆ n(1) : arg max max
p
t
,

,

n
t

 R
 R
n
and standard conditional probability estimator (or only prob. estim.)
ˆ n(2) : E | Fny .
n
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).
Simulation Example 1 (Cont.)
Wealth
Simulated stock price
10.3
1.3
10.2
1.2
10.1
1.1
10
1
9.9
9.8
0.9
9.7
0.8
9.6
0.7
9.5
0.6
0.5
9.4
9.3
0
50
100
150
trading days
200
250
300
0
50
100
150
trading days
200
250
300
Erors
0.12
0.07
max. prob. error
stand. prob. error
0.06
simulated volatility
max. prob. estimate
standard prob. est.
0.1
0.05
0.08
0.04
0.06
0.03
0.04
0.02
0.01
0
0.02
0
50
100
150
trading days
200
250
300
0
0
50
100
150
200
250
300
Simulation Example 2
For simulated observations we used yn  0.01 n , where
 n ~ N (0,1) and we used the tracker from Example 1with   0.001 .
0.03
0.02
max-prob. error
conditional prob. error
simulated data
max-prob. estimator
conditional estimator
true volatility
0.02
0.015
0.01
0.01
0
0.005
-0.01
0
-0.02
-0.005
-0.03
0
10
20
30
40
50
60
trading days
70
80
90
100
-0.01
0
10
20
30
40
50
60
trading days
70
80
90
100
Real Data (from slide Simulation Examples vs. Real Data)
-3
Initial condition for the Fokker-Planck operator at t=0
x 10
-0.6
0.1
3.5
log(Sn+1/Sn)
maximum probability estimate
conditional expecation estimate
0.08
-0.4
0.06
3
Velocity variable
-0.2
0.04
0.02
2.5
2
0
1.5
0.2
0
1
0.4
-0.02
-0.04
0.5
0.6
0
50
100
150
Trading days
200
250
300
-2
0
Wealth
2
4
6
8
Space variable
10
12
-3
x 10
-4
Conditional PDF at t=32
10.7
0
14
x 10
7
-0.4
10.6
-0.3
6
10.5
-0.2
10.4
Velocity variable
10.3
10.2
10.1
0
4
0.1
3
0.2
10
0.3
9.9
0.4
9.8
5
-0.1
2
1
0.5
0
50
100
150
200
250
300
0.6
0
0.01
0.02
0.03
Space variable
0.04
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.
Further possibilities
Application of the method to continuous SDE’s 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).
Passively 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.