PAPER:
Maximum likelihood estimation of the parameters of a system of
stochastic differential equations that models the returns of the
index of some classes of hedge funds*
Lorella Fatone, Francesca Mariani, Maria Cristina Recchioni, Francesco Zirilli
1.
2.
3.
4.
5.
6.
Abstract
Introduction
Filtering problem Filtering Movies
Estimation problem Estimation Movies
References
FORTRAN code
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*
The numerical experience reported in this paper has been obtained using the computing grid of
ENEA (Roma, Italy). The support and the sponsorship of ENEA are gratefully acknowledged.
1. Abstract
In [1] we study an inverse problem for a parabolic partial differential equation. The parabolic
partial differential equation considered is the Fokker Planck equation associated to a system of
stochastic differential equations and the inverse problem studied consists in finding from suitable
data the values of the parameters that appear in the coefficients of this Fokker Planck equation, that
is of the parameters that define the system of stochastic differential equations. The data used in the
reconstruction of the parameters are observations made at discrete times of the stochastic process
solution of the system of stochastic differential equations. That is, the data of the inverse problem
are a “sample” taken from some of the components of the vector solution of the stochastic
differential equations and not, as usual, observations made on the solution of the parabolic equation.
The choice of the system of stochastic differential equations and of the data used in the inverse
problem are motivated by applications in mathematical finance. The stochastic differential
equations presented can be used to model the dynamics of the log-returns of the index of some
classes of hedge funds, such as, for example, the so called “long short equity” hedge funds and of
some auxiliary variables. The solution of the inverse problem proposed is obtained through the
solution of a filtering and of an estimation problem. The solution of the last two problems is based
on the knowledge of the joint probability density function of the state variables of the model
conditioned to the observations made and to the initial condition. This joint probability density
function is solution of an initial value problem for the Kushner equation that in the circumstances
considered here can be written as a sequence of initial value problems with appropriate initial
conditions for the Fokker Planck equation associated to the system of stochastic differential
equations. An integral representation formula for this probability density function is derived and
used to develop a numerical procedure to solve the estimation problem using the maximum
likelihood method. The computational method proposed has been tested on synthetic data and the
results obtained are presented. This website contains some auxiliary material useful to understand
[1] and contains some animations and the description of some numerical experiments. A more
general reference to the work of the authors and of their coauthors in mathematical finance is the
website: http://www.econ.univpm.it/recchioni/finance.
2. Introduction
Assuming the log-return of the stock prices index, the log-return of the index of a suitable class of
hedge funds (i.e.: for example “long short equity” class) and the stochastic variance of the stock
prices index as state variables and the log returns of the index of the stock prices and of the index
of the hedge funds class observed at discrete times as observations, we study the problem of
forecasting (tracking) the unobserved and time-varying stochastic variance when the model
parameters and initial condition are known using the observations (filtering problem) and the
problem of estimating the model parameters and the unknown initial stochastic variance using the
observations (estimation problem). The solution of this last problem is needed in order to use the
model proposed to forecast in real situations future log returns of the stock prices and hedge funds
indices.
Let xt , zt, and vt be the state variables at time t>0 that is, respectively, the log-return of stock prices
index, the log-return of the index of the hedge funds class and the stochastic variance of the stock
prices index at time t>0. We assume that the stochastic process (xt , zt , vt ), t>0, is solution of the
following system of stochastic differential equations:
1
dx t  (μ̂ - v t )dt  v t dWt1 ,
2
1
dz t   ( μ̂ - v t )dt  v t (  dWt1   dWt3 ),
2
dv t   (θ  v t )dt  ε v t dWt2 ,
t  0,
t  0,
t  0,
(1)
(2)
(3)
where Wt1, Wt2, Wt3 , t>0, are standard Wiener processes such that W01=W02= W03 =0, dWt1, dWt2
, dWt3 , are their stochastic differentials and dWt1dWt2 = 1,2dt, dWt1dWt3 = 1,3dt, dWt2dWt3
= 2,3dt where    denotes the mean of  and 1,2,1,3,2,3[-1,1] are constants known as
correlation coefficients, μ̂ is a real constant related to the behaviour of the mean of the stock prices
index log return, associated to the stochastic process of the stock prices index St , t>0. Note that the
stochastic variance vt, t>0, is assumed to be described by a mean-reverting process with speed 0
and parameters 0 and 0, that is by equation (3). The stochastic differential equations (1), (3)
are known as Heston model [2]. Equation (2) has been introduced to model the log-returns of the
indices of suitable classes of hedge funds (such as, for example, the long short equity class). In fact
equation (2) says that dzt is the sum of dxt and a random disturbance given by  v t dWt3, where
β and γ are real constants. If we interpret zt, t>0, as the log-return at time t of the index of one of
these classes of hedge funds, xt, t>0, as the log-return at time t of the stock prices index, for
example the S&P500 index, and vt, t>0, as the stochastic variance of xt, t>0, equation (2) can be
regarded as a reasonable translation in the continuous time setting of the findings of the time series
analysis presented in [3] and the system (1), (2), (3) can be interpreted as a model to explain the
dynamics of the log-return of the index of one of the classes of hedge funds mentioned previously
that uses the log-return of the S&P500 index and its stochastic variance as explanatory variables.
We remark that the models proposed in [3] are discrete time models.
The equations (1), (2), (3) must be equipped with an initial condition, that is:
x0  ~
x0 ,
~
z 0  z0 ,
v ~
v .
0
(4)
(5)
(6)
0
x 0 , ~z 0 , and ~
v 0 denote random variables concentrated with probability one in a point; for
Here ~
x 0 , ~z 0 , and ~
v 0 . We note that ~
x0 ,
simplicity, we continue to denote these points respectively with ~
~z , that is the log-returns of the stock prices index and of the index of the hedge funds class at t=0,
0
v 0 , that is the stochastic variance at t=0, cannot be observed. The quantities
are observable while ~
μ̂ , , , , ,  and 1,2, 1,3, 2,3 are real constants and are the model parameters that we want to
estimate. Hence, the parameters and unknown initial condition component of the model (1), (2), (3)
v 0 . We assume that 2/2 1, this
that we want to estimate are μ̂ , , , , , , 1,2, 1,3, 2,3 and ~
assumption guarantees that if v 0 is positive with probability one, vt, solution of (3), (5) remains
positive with probability one for t>0.
We suppose that at discrete times 0=t0<t1<t2<…<tn<tn+1 =+, the log-return of the stock prices
index and the log-return of the index of a special class of hedge funds are observed and let Ft=
{( ~
x i , ~zi ) : ti t}, t>0, be the set of the observations up to time t>0 of the log-returns of the stock
prices index xt and of the log-returns of the index of the hedge funds, that is for i=1,2,...,n, let ~
x i be
~
the observation of the log-return of the stock prices index and z i be the observation of the logreturn of the hedge funds index at time t=ti . We assume that the observations are error free, that is
we assume x t i = ~
x i , z t i = ~zi , i=0,1,…,n. Let Θ  (μ̂,  , ε, θ,  ,  , ρ1,2 , ρ1,3 , ρ 2,3 ~
v 0 ) T be the vector of
the model parameters and of the initial stochastic variance, where the superscript T denotes the
x0 , ~
z0 , ~
v 0 )}to simplify some of the
transpose operator. We use the notation t0 =0 and F0={( ~
formulae that follow.
3. Filtering problem
Let us assume the vector  known, we focus on the problem of forecasting the state variables from
the observations previously described. This problem is a filtering problem, where the stochastic
x 0 , ~z 0 and ~
v 0 , at time t0 is the
differential equations (1), (2), (3) represent the “state equations”, ~
~
~
initial condition and, for i=1,2,...,n, ( x i , zi ) at time ti represent the observations of the filtering.
v is known through the
Note that in the filtering problem the initial stochastic variance ~
0
knowledge of  and that since ~
x 0 , ~z0 are observed in the filtering problem the initial condition
~
v 0 is known. The solution of this filtering problem is based on the knowledge of the joint
x 0 , ~z0 , ~
probability density function ~
p(x, z, v, t | Ft , Θ) , (x,z,v)( -,+)  (-,+)  (0,+) , t>0, of the
state variables conditioned to the observations Ft, t>0, and to the initial condition (4), (5), (6).
The joint probability density function ~
p(x, z, v, t | Ft , Θ) (x,z,v) (-,+)  (-,+)  (0, +) , t>0,
of having xt=x, zt=z, vt=v, t>0, conditioned to the observations Ft, t>0, is solution of the following
equations [1] : for i=0,1,...,n:
2
2
2
2
2
~
~
~
~
p 1  2 (v~
p ) 1  2 (ε 2 v~
p ) 1  ((    2 1,3 )vp )  (ε ρ1,2 vp )  (ε ( ρ1,2   ρ 2,3 ) vp )





t 2 x 2
2 v 2
2
xv
zv
z 2

 2 ((    ρ1,3 ) v~
p)
zx

p )  (  (   v/2) ~
p)
 ((μ  v/2) ~
p )  (  (θ  v)~

, (x, z, v)  (-,)  (-,)  (0,),
x
v
z
t i  t  t i 1 ,
i  0,1,..., n
(7)
with initial condition: for i=0:
~
z0 )  (v  ~
x 0 )  (z  ~
v 0 ) , (x,z,v)( -,+)  (-,+)  (0,+) , t=t0
p(x, z, v, t | F0 , Θ) = (x  ~
(8)
and for i=1,2,...,n:
~
p(x, z, v, t | Ft i , Θ) =
δ(x  ~
x i )δz  ~zi )~
p(x, z, v, t i | Ft i 1 , Θ)


 p(x, z, v' , t i
~

0
, (x,z,v)( -,+)  (-,+)  (0,+) ,
| Ft i 1 , Θ)dv'

t=ti.
(9)
~
Once we know the joint probability density p (x,z,v,t|Ft,), (x,z,v)  (-,+)  (-,+)  (0, +),
t>0, we can forecast the values of the random variables (xt, zt ,vt), t>0, conditioned to the
observations using the quantities (x̂ t|Θ , ẑ t|Θ , v̂ t|Θ ), t  0, given by:





0





0


x̂ t|   dx  dz
ẑ t|   dx  dz
v̂ t| 
 x p(x, z, v, t | Ft , )dv ,
 dx  dz


~
 z p(x, z, v, t | Ft , )dv ,
~
t 0,
(10)
t 0,
(11)

 v ~p(x, z, v, t | Ft , )dv ,
t 0.
(12)
0
The quantities x̂ t|Θ , ẑ t|Θ , v̂ t|Θ , t  0, can be regarded as the solution of the filtering problem.
Let G(x,z,v,t,x',z’,v',t'|), (x,z,v), (x',z’,v') (-,+)  (-,+)  (0, +)
fundamental solution of (7) whose expression is given in [1].
, t>t'>0, be the
Using the function G the solution of problem (7), (8), (9) can be written as follows:
~
p(x, z, v, t | F0 , Θ) =G(x,z,v,t, ~x 0 , ~z 0 , ~x 0 , t 0 |  ), (x,z,v)(-,+)  (-,+)  (0,+), t 0 <t  t1 ,
(13)
and for i=1,2,…,n:
~
p(x, z, v, t | Ft , ) 

 G(x, z, v, t, x i , z i , v' , t i | )f i (v' )dv' ,
0
(x, z, v)  (-,)  (-,)  (0,), t i  t  t i 1 ,
(14)
where
f i (v' ) 

~
p(~
x i , ~zi , v' , t i | Ft , )


,
v'  0.
(15)

~
p(~
x i , ~zi , v, t i | Ft , )dv
0
The evaluation of ~p(x, z, v, t | Ft i , Θ) , (x,z,v) (-,+)  (-,+)  (0, +), t>0, for i=1,2,...,n, on a
grid of N3 points in the (x,z,v) variables with a naïve quadrature procedure applied to formula (14)
using N quadrature nodes in each integration variable requires O(N6) complex multiplications
when N goes to infinity. In fact (14) due to the fact that the fundamental solution G is obtained
computing a two dimensional integral can be regarded as a three dimensional integral. In [1] using:
 a wavelet expansion [1], [5] , [6] of the integral kernel appearing in (14) and of the function (15),
 a truncation procedure that exploits the sparsifying properties of the wavelet expansion,
 a suitable expansion in powers of a time step and the Fast Fourier Transform (FFT),
we reduce the computational cost of the previous computation from O(N6) to N1O(N3log N)
complex multiplications when N goes to infinity, where N1 is independent of N. Note that N1 is the
number of the wavelet functions remaining in the expansion of the integral kernel after the use of
the truncation procedure.
Using the estimates (10), (11), (12) we can forecast the values of the log-return of the stock prices
index xt, t>0, of the log-return of the hedge funds class index zt, t>0, and of the stochastic variance
vt, t>0, coherently with the observations
Ft, t0. The continuous update in time of
~
is
the
“trajectory
tracking”
of
(1), (2), (3), (4), (5), (6). The continuous update
(x̂ t|Θ , z t|Θ , v̂ t|Θ ), t  0,
in time of v̂ t|Θ , t  0, is the corresponding “variance tracking”.
The quality of the forecasted values (x̂ t|Θ , ~z t|Θ , v̂ t|Θ ), t  0, depends on the variance of the random
variables xt, zt vt, t>0, conditioned to the observations, that is we can estimate the quality of the
estimates (10), (11), (12) computing respectively the quantities:

( x t | Ft , ) 
(z t | Ft , ) 
(v t | Ft , ) 


 dx  dz  (x  x̂ t| )
2~
t  0,
(16)
p(x, z, v, t | Ft , )dv,
t  0,
(17)
2
 dx  dz  (v  v̂ t| ) ~p(x, z, v, t | Ft , )dv,
t  0,
(18)


0





0



 dx  dz  (z  ẑ t| )


p(x, z, v, t | Ft , )dv,
2~
0
The estimates (10), (11), (12) are good, when the variances (16), (17), (18) are small.
Click here to download a digital movie showing the trajectory tracking.
Click here to download a digital movie showing the evolution in time of the integral with respect
to x of the joint probability density function conditioned to the observations obtained solving the
filtering problem . That is the digital movie shows the evolution in time of the function D(z,v,t),
5 

(z, v)  (-,)  (0, ) ,
t  0,
,
where
D
is
defined
as
 252.5 

5 

D(z,v,t)=  ~p(x, z, v, t | Ft , )dx, (z, v)  (-,)  (0, ) , t  0,
.
 252.5 
-
In the trajectory tracking shown in Figure 1 and in the digital movie showing the trajectory tracking
the solid line represents the true trajectory made of the values (xt, zt vt), t[0,24/252.5], obtained
computing numerically for t>0 with Euler method one trajectory of the stochastic differential
equations (1), (2), (3) with the initial condition (4), (5), (6) with ~x 0  0, ~z 0  0, and ~v 0  0.5 ; the
dashed line represents the trajectory followed by the forecasted values (x̂ t|Θ , ẑ t|Θ , v̂ t|Θ ),
t[0,24/252.5], obtained solving the filtering problem computing the integrals (10), (11), (12) and
using the data shown in Table 1. The squares and the asterisks indicate, respectively, the values of
(xt, zt vt) and of (x̂ t|Θ , ẑ t|Θ , v̂ t|Θ ), at the observation times t=ti, i=0,1,...,5. We have chosen  =
(0.026, 5.94, 0.306, 0.01159,1,0.1, -0.576,0,0, 0.5)T. We note that , , 1,2, 1,3, 2,3 are
adimensional quantities and that μ̂ , , , , ~v 0 dimensionally are 1/time. We assume that the
components μ̂ , , , , ~v 0 of the parameter vector  are expressed in annualized unit, that is in the
unit 1/year. We use a “year” made of the average number of trading days per year, that is made of
252.5 days, so that for example t=24/252.5 corresponds to the end of the trading day number 24 of
the regular year.
i
0
1
2
3
~
xi
0
0
4/252.5 0.1046324981
8/252.5 -0.0389636566
12/252.5 -0.1106878449
ti
~z
i
0
-0.09423814923
-0.03170475035
-0.09375359522
4 16/252.5 0.01355863752 0.036249189791
5 20/252.5 0.04209650394 0.053437008036
Table 1. The data of the filtering problem
Below we show two figures (Figure 1 and Figure 2) that help the understanding of the quality of the
estimates obtained solving the filtering problem. We remind that Figure 1 shows the trajectory
tracking obtained as solution of the filtering problem. In particular the solid line represents the
true trajectory made of the values (xt, zt vt), t[0,24/252.5], obtained computing with Euler
method one trajectory of the initial value problem (1), (2), (3), (4), (5), (6), the dashed line
represents the trajectory followed by the forecasted values of x̂ t|Θ , ẑ t|Θ , v̂ t|Θ , , t[0,24/252.5].
Figure 2 shows (xt | Ft, ), (zt | Ft, ), (vt | Ft, ) versus time when t[0,24/252.5]. In the
figures the time is expressed in days that is ~t  252.5t .
~
Figure 1. Tracking of xt, zt, vt versus t
~
(vt|Ft ,) versus t
Figure 2. (xt|Ft ,), (zt|Ft ,),
4. Estimation problem
In Section 3 we have assumed  known to solve the filtering problem but in real situations when
the model (1), (2), (3) is used  is not known and it must be estimated from the knowledge of the
observations. The dynamics of the stochastic process (xt, zt vt), t>0, and of the estimates
(x̂ t|Θ , ẑ t|Θ , v̂ t|Θ ), t>0, obtained from the solution of the filtering problem, depend on the vector
Θ  (μ, χ, ε, θ , β , γ, ρ1,2 , ρ1,3 , ρ 2,3 ~
v0 )T .
Let
M  { :   0,   0, θ  0,   0,   0, 2  θ /  2  1, 1  1, 2 , 1,3 ,  2,3  1} be the set of the admissible
vectors .
We solve the problem of estimating the vector  from the observations using the maximum
likelihood method. That is we search the vector M that maximizes the following (log-)likelihood
function:
n 1

F(Θ)   log  ~
p(x i1 , z i1 , v, t - i1 | Ft i , Θ)dv,
i 0
Θ M.
(19)
0
Note that (19) represents the sum of the logarithms of the integrals of the transition probability
~
densities p (xi+1, zi+1,v,ti+1|Fti ,), i=0,1,...,n-1. The function F() represents the (log-)likelihood of
the parameter vector  given the observations Ft , t>0, and the initial condition ~x 0 , ~z 0 , ~v 0 at
time t=t0.
In order to maximize the (log-)likelihood function shown in (19), we use an iterative optimization
procedure that searches the maximum likelihood estimate * solution of the maximization
problem:
max F (),
M
(20)
starting from an initial guess 0M.
The iterative procedure that solves the optimization problem (20) using 0 as initial guess finds
during the iterations the admissible vectors k,, k=1,2,...,iter, where iter is the maximum number of
iterations allowed. The vectors k, k=0,1,…,iter, are such that F(k+1)F(k), k=0,1,…,iter-1.
In the first numerical experiment we use as data of the estimation problem the same data used for
the filtering problem that is the data shown in Table 1. These are synthetic data that were generated
using =true = (0.026, 5.94, 0.306, 0.01159,1,0.1 -0.576,0,0, 0.5)T. We choose as initial guess of
the optimization procedure the vector 0=true(1+0.8), where  has been sampled from 10dimensional standard Gaussian variable with independent components. We assume 0M if this is
not the case we discard it and we sample the 10 dimensional Gaussian variable again.
We fix the maximum number of iterations allowed iter=100, we use the variable metric steepest
ascent method described in [1] as optimization method to solve problem (20) and we use the
following stopping criterion:
k 1
| F(Θ )  F(Θ ) |
  1 , k  10
k 1
| F(Θ ) |
k
(21)
where 1 is a given tolerance. Choosing 1 =0.001 the variable metric steepest ascent procedure
stops at the iteration k=19, with 19=(0.025,4.765,0.252, 0.011,0.96,0.106, -0.521, 0.071,0.019,0.492)T and we have F(19)  23.54, F(0)  12.91, and F(true)  23.10 .
Click here to download a digital movie showing the trajectory tracking for
 = k M ,
k=0,1,2,…,19, obtained during the iterative optimization procedure and the true trajectory
corresponding to the parameter vector =true obtained integrating numerically (1), (2), (3), (4),
(5), (6).
The second experiment consists in analyzing a series of daily observations during a period of two
years, that is a series made of 505 observation times. Remind that we have considered a “year”
made of 252.5 trading days. We consider 505 observation times corresponding to 1010 observations
~
x i , ~z i , i=0,1,…,504, where we have chosen t0=0, x0= ~x 0 =0, z0= ~z 0 =0, v0= ~v 0 =0.5. The data are
generated integrating numerically with Euler method one trajectory of (1), (2), (3), (4), (5), (6) for
the two years period. We choose the vector  of the stochastic differential system that generates
the data to be equal to the vector   1 used in the previous experiment in the first year of data
(i.e. when t=ti , i=0,1,…,252) and to the vector    2 in the second year of data (i.e. when t=ti ,
i=253,254,…,504) where 1 and  2 are given by: 1 =(0.026,5.94,0.306,0.01159,1,0.1,0.576,0,0,0.5) T,  2 =(0.4,2,0.01,0.01,1,0.01,0.5,0,0,0.012) T.
Click on the names to download the file containing the observations
(samplezz.txt), i=0,1,…,504.
~
x i (samplexx.txt) , ~z i
We solve problem (20) using a time window made of 9 consecutive observation times
corresponding to 18 observations. In particular we divide the time interval [0,504/252.5] in 63
consecutive disjoint time windows containing 8 consecutive observation times that is for
j=0,1,…,62 the time values t=ti,j=t8j+i, i=1,2,…,8 belong to the same time interval. Note that when
considering the time window j (j1) we use the notation t0,j=t8j that is t0,j is the last observation time
of the time window j-1. Note that 863=504 and that the observation time t=t0 is not considered in
the time windows considered above. The observation time t=t 0 is associated to the first time
window (j=0) and we use the notation t0,0=t0. We solve problem (20) using 9 consecutive
observations that is ~x i , ~z i , at time t=ti,j i=0,1,…,8, that is for j=0,1,…,62 the time values ti,j=t8j+i,
i=0,1,2,…,8 belong to the same time window. We solve 63 problem (20) starting from different
initial guesses. In Figure 3 the diamond represents the vector 1 and the square represents the
vector  2 . The unit length in Figure 3 is || 1 || . The origin is the mid point of the segment that
joins the points 1 and  2 and the length of the segment that joins the diamond and the square is
proportional to the Euclidean norm of the vector 1 -  2 . The vector determined solving problem
(20) using the optimization procedure are represented by stars. A point P= is plotted around 1
||    2 ||
||   1 ||
when the quantity
is smaller than the quantity
, otherwise it is plotted around
||  2 ||
|| 1 ||
 2 . The point P is plotted on the left or on the right of the point that represents 1 or  2
according with the sign of  8 , the eighth component of the vector  , that is the estimated
correlation coefficient 1,3 . Finally the difference of the abscissae of the point 1 and a point P=
of its cluster is || 1   || . A similar statement holds for  2 and a point P on the cluster associated
to  2 . We remind that in Figure 3 the unit length is || 1 || .
Figure 3: Reconstruction of the parameter vectors 1 ,  2
Click here to download a digital movie showing the points determined solving the 63 problem
(20) described above as a function of the time window of the observations.
Figure 3 and the digital movie that shows the solution of the 63 problem (20) show that the points
obtained solving problem (20) are concentrated on two disjoint segments one on the left and one on
the right of the origin and that they form two disjoint clusters, that is, the solution of the 63
problems (20) corresponding to the time windows described previously shows that two sets of
parameters seem to generate the data studied. This is really the case. Moreover as expected the
points of the cluster around 2 are those obtained by the analysis of the observations made in the
second year.
5. References
[1]
L.Fatone, F.Mariani, M.C.Recchioni, F.Zirilli Maximum likelihood estimation of the
parameters of
a system of stochastic differential equations that models the returns of the index of some
classes of hedge funds, submitted to Journal of Inverse and Ill Posed Problems.
[2]
S.L.Heston, A closed-form solution for options with stochastic volatility with applications to
bond and currency options , Review of Financial Studies, 6 (1993), 327-343.
[3]
P. Pillonel, L. Solanet, Predictability in hedge fund index returns and its application in fund
of hedge funds style allocation, Master's Thesis in Banking and Finance at Université de
Lausanne, Hautes Etudes Commerciales (HEC), (2006).
[4]
L.Fatone, M.C.Recchioni, F.Zirilli New wavelet bases made of piecewise polynomial
functions: approximation theory, quadrature rules and applications to kernel sparsification
and image compression, submitted to SIAM Journal on Scientific Computing.
[5]
L.Fatone, G.Rao, M.C.Recchioni, F.Zirilli, High performance algorithms based on a new
wavelet expansion for time dependent acoustic obstacle scattering, to appear in
Communications in Computational Physics.
[6]
F.Mariani, G.Pacelli, F.Zirilli, Maximum likelihood estimation of the Heston stochastic
volatility model using asset and option prices: an application of nonlinear filtering theory,
to appear in Optimization Letters.
6. FORTRAN Code
The FORTRAN code distributed in this website computes the solution of the estimation problem
(20) using the formulae outlined in Sections 3 and 4 and derived in [1] . The data of the estimation
problem are those described in Section 4 and must be provided by the user. The FORTRAN code
consists of two programs.
The first program preproc.f reads and writes on appropriate files the data given by the user,
initializes the parameter vector 0 and computes the number of wavelet coefficients needed after
the truncation procedure in order to approximate satisfactorily the integral kernel contained in (14)
associated to the initial guess 0.
click here to download preproc.f
The second program estimate.f starting from the initial guess 0 generates a sequence of
admissible vectors k, k=1,2,…,iter, using a variable metric steepest ascent optimization procedure.
click here to download estimate.f
Entry n.