PAPER:
The calibration of the Heston stochastic volatility model using filtering and maximum likelihood
methods *
Lorella Fatone, Francesca Mariani, Maria Cristina Recchioni, Francesco Zirilli
1.
2.
3.
4.
5.
Abstract
Introduction
Estimation problem using synthetic data Estimation Movie 1, Estimation Movie 2
Estimation problem and forecasting experiments using real data Forecasting Movie 1, Forecasting Movie 2
References
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1. Abstract
In [1] we study the problem of obtaining accurate estimates of the parameters, of the initial stochastic variance and of the risk premium parameter
of the risk neutral measure of the Heston stochastic volatility model from the observation at discrete times of the stock log-returns and of the prices
of a European call option on the stock. This problem is an inverse problem known in the literature as calibration problem. As a byproduct of the
solution of the calibration problem we develop a tracking procedure that can be used to forecast the stochastic variance and the stock price. From a
mathematical point of view the problem considered is formulated as a constrained optimization problem where the objective function is the
logarithm of the likelihood associated to the parameter, the initial stochastic variance and the risk premium parameter values given the observed
stock log-returns, option prices and observation times, this function is called (log-)likelihood function. The evaluation of the (log-)likelihood
function associated to a given choice of the parameter, of the initial stochastic variance and of the risk premium parameter values requires the
solution of a filtering problem for the Heston model. An accurate and computationally efficient solution of this filtering problem is necessary for a
satisfactory solution of the calibration problem for the Heston model. A similar problem has been considered in [2] and [3]. The aim of this paper is
to extend and improve the results obtained in [2] with respect to the formulation of the problem, the accuracy of the solution obtained and the
computational efficiency of the solution method. In particular in comparison with [2] we reformulate the problem adding to the quantities that must
be estimated the risk premium parameter and to the observations the option price at the initial time. The addition of the risk premium parameter to
the quantities that must be estimated makes the problem considered more realistic and makes interesting the analysis of time series of real financial
data using the method proposed. The addition of the option price at the initial time to the data used in the solution of the problem is natural and
improves significantly the estimate of the initial stochastic variance. Moreover we simplify the expression of some formulae used in [2] in the
computation of the (log-)likelihood function and we improve the optimization method employed to solve the maximum likelihood problem
introducing some ad hoc preliminary optimization steps. Finally a new, easy to compute, formula that gives the ``Heston" option price is derived.
The use of this new formula reduces substantially the computational cost of evaluating the (log-)likelihood function when compared to the cost of
the (log-)likelihood function evaluation in [2]. Some numerical examples of the calibration problem using synthetic and real data are presented. As
real data we use the 2005 historical data (precisely the data of the period Jan. 3, 2005, May 11, 2005) of the US S&P500 index and of the
corresponding option prices. Very good results are obtained forecasting future values of the S&P500 index and of the corresponding option price
using the solution of the calibration problem and the forecasting and tracking procedure mentioned above. In this website some auxiliary material
useful in the understanding of [1] including some animations and some numerical experiments is shown. 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
The problem that we are considering is the calibration of the Heston stochastic volatility model including the estimation of the initial stochastic
variance and of the risk premium of volatility implicit in option prices. The main differences between the work contained in [1] and the work
presented in [2] are: i) the addition to the quantities that must be estimated of the risk premium parameter contained in the risk neutral measure of
the Heston model and the addition to the observations used in the estimation problem of the observation at the initial time of the option price, ii) the
reduction of the formula used to compute the ``Heston" option price from being a three dimensional integral that must be evaluated numerically to
being a one dimensional integral that must be evaluated numerically, iii) an improved optimization method that includes some ad hoc preliminary
steps that is used to maximize the likelihood function. These differences lead to a substantial improvement of the accuracy of the estimates obtained
of the model parameters, of the initial stochastic variance and of the risk premium parameter and to great savings in the computational cost of
solving the calibration problem. Note that the addition of the risk neutral parameter to the quantities that must be estimated makes the problem
considered more relevant from the practical point of view and together with the improved computational efficiency in the solution of the problem
makes possible the use of the method proposed to analyze time series of real financial data. The work presented in [2] corresponds to choose the risk
premium parameter equal to zero. In particular we show that the addition of the initial option price to the data and the improved optimization
method determine a substantial improvement of the quality of the estimate of the initial stochastic variance obtained and, consequently, an
improvement of the quality of the estimates of all the remaining quantities. In fact the numerical experience shows that the accuracy of the estimate
of the initial stochastic variance is crucial to obtain a good solution of the calibration problem and satisfactory forecasted values of the stock and
option prices. On the other hand the numerical evaluation of a one dimensional integral in order to compute the option prices instead than the three
dimensional integral evaluated numerically in [2] is responsible for the substantial reduction of the computational time needed to solve the
calibration problem. The improved numerical efficiency makes practical the use of observation samples bigger than those used in [2] and this gives
an extra possibility of improving the quality of the results obtained.
Let us give some details. Assuming the stock prices and their stochastic volatilities or variances as state variables and the stock prices and option
prices observed at discrete times as observations, we study the problem of forecasting (tracking) the unobserved and time-varying stochastic
volatility or variance when the model parameters, initial condition and risk premium parameter are known using the observations (filtering problem)
and the problem of estimating the model parameters, the unknown initial stochastic volatility or variance and the risk premium parameter using the
observations of the stock price and of the price of a European call option on the stock (estimation problem).
Let St and vt be the state variables at time t>0, that is, respectively, the stock price and the stochastic variance at time t>0, the Heston model
prescribes the following dynamics for the state variables (St,vt), t>0 (see [4]):
dS t  μ S t dt  v t S t dWt1 ,
t  0,
(1)
dv t  γ(θ  v t )dt  ε v t dWt2 ,
t  0,
(2)
where Wt1, Wt2, t>0, are standard Wiener processes such that W01=W02=0, dWt1, dWt2 are their stochastic differentials and dWt1dWt2 = dt
where    denotes the mean of  and [-1,1] is a constant known as correlation parameter;  is a real constant related to the behaviour of the
mean of the stock log returns, associated to the stochastic process 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 (2). The equations (1), (2) must be equipped with an initial
condition, that is:
~
S0  S0 ,
v ~
v .
0
(3)
(4)
0
~
Here S0 and
~v
0 denote random variables concentrated with probability one in a point; for simplicity, we continue to denote these points
~
~
~
~
respectively with S0 and v 0 . We note that S0 , the stock price at t=0, is observable while v 0 , the stochastic variance at t=0, cannot be observed.
The quantities , , ,  and  are real constants and are the model parameters that we want to estimate. Hence, the parameters and unknown initial
condition component of the Heston model that we want to estimate are , , , , , ~v 0 , moreover we want to estimate the risk premium parameter
of the risk neutral measure . The meaning of the parameter  is explained below. We assume that 2/2 1, this assumption guarantees that if
~v is positive with probability one, vt, solution of (1), (2), remains positive with probability one for t>0.
0
We suppose to have the observations of the prices of the stock and the prices of a European call option of known maturity and strike price on the
~ ~
underlying whose price St, t>0, satisfies (1), (2), (3), (4) at prescribed discrete times t0= 0<t1<t2<…<tn<tn+1 =+, let Ft= { ( Si , C i ) : ti t}, t>0, be
~
the set of the observations up to time t>0 of the stock prices St and of the option prices Ct that is for i=1,2,...,n let Si be the observation of the stock
~
price and C be the observation of the option price at time ti and let Θ  (  ,  , ε, θ, ρ, ~
v ,  ) be the vector made of the model parameters, of the
i
0
initial stochastic variance and of the risk premium parameter. We note that since we consider option prices as data and we evaluate the option prices
under the risk neutral measure we add to the parameters and to the initial condition component of the Heston model (1), (2), (3), (4) to be estimated
the risk premium parameter  [4] that appears in the risk neutral measure used to price the options. Note that as shown in [4] the risk neutral
measure is obtained from the statistical measure associated to (1), (2), (3), (4) with a simple explicit transformation. We use the notation t0 =0 and
~ ~
F0={ (S0 , C 0 ) }, this notation simplifies some of the formulae that follow. The option prices considered refer to the prices of an European call option
on the underlying whose prices are described by St, t>0, with known strike price E and maturity time T such that T>t n, we denote with
C(S,v,t;E,T,) the price under the risk neutral measure of this option in the Heston stochastic volatility model when 0t<T, St=S and vt=v. We
assume that the option prices observed are given by the theoretical option prices in the Heston model affected by a Gaussian error, that is: for
~
~
i=1,2,...,n, C i =C( Si , ~
vi ,ti; E,T,)+ui, where ui is an additive Gaussian error term; that is we assume that ui is sampled from a Gaussian variable
~
with mean zero and known variance  i , and C( Si , ~
vi ,ti; E,T,) is the option price in the Heston model assuming known the (unobserved) variance
~
~
~
vi and the (observed) stock price Si at time ti. Note that we assume Si and ~
vi known without error.
Let us briefly describe how we get the forecasted values of the state variables S t , v t , t>0 (see [1], [2] for further details). Let us introduce the stock
log returns:
~
x t  log(S t / S0 ),
(5)
t  0,
~
without loss of generality, we can assume S0= S0 =1. From (1), (2), (3), (4) and (5) it is easy to see that (xt,vt), t>0, satisfy the following equations:
1
v t )dt  v t dWt1 ,
2
dv t  γ(θ  v t )dt  ε v t dWt2 ,
dx t  (μ 
t  0,
(6)
t  0,
(7)
with initial condition:
x0  ~
x0  0 ,
(8)
v0  ~
v0 .
(9)
Given Θ  (  ,  , ε, θ, ρ, ~
p(x, v, t | Ft , Θ), (x,v)(-,+)  (0, +) , t>0, of the state
v 0 ,  ) , we can compute the joint probability density function ~
variables x t , v t that is of the stock log return and of the stochastic variance, conditioned to the observations Ft, t>0, and to the initial condition (8),
(9) as solution of the equation presented in [1], [2] , that is: for i=0,1,...,n:
~
p 1  2 (v~
p) 1  2 (ε 2 v~
p)  2 (ε ρ v~
p)  2 ((μ  v/2) ~
p) (γ (θ  v) ~
p)





, (x, v)  (-,)  (0,), t i  t  t i 1 , (10)
2
2
t 2 x
2 v
xv
x
v
with initial condition: for i=0:
~
p(x, v, t 0 | Ft 0 , Θ)  δ(x  ~
x 0 ) (v  ~
v 0 ),
and for i=1,2,...,n:
(x, v)  (-,)  (0,),
(11)

~
~
 C
(x, v, t i ; E, T, Θ)  C i

exp 

2i

~
p (x, v, t i | Ft i , Θ) 
2i

2




δ(x  ~
x i )~
p (x, v, t i | Ft i 1 , Θ)

~
~
 C
(x, v, t i ; E, T, Θ)  C i
1

 exp  
2i
2i 0



2


~
p(x, v' , t i | Ft i 1 , Θ)dv'



(x,v)(-,+)  ( 0,+) .
,
(12)
~
where for i=1,2,...,n: ~
x i =log(Si/ S0 ) and C (x,v,ti;E,T, ) is the value in the Heston model at time t=ti (see [1] Section 3) when x t i  x , v t i  v of
~
a call option on the stock whose price is St= S0 exp(xt), t>0, with strike price E and maturity time T.
Once we know the joint probability density ~p (x,v,t|Ft,), (x,v)  (-,+)  (0, +), t>0, solution of (10), (11), (12) we can forecast the values of
the random variables (xt,vt), t>0, using the formulae:
x̂ t|Θ 
v̂ t|Θ 



0



0
~
 dx  xp(x, v, t | Ft , Θ)dv,
 dx  vp(x, v, t | F , Θ)dv,
~
t
t  0,
(13)
t  0.
(14)
~
The quantities x̂ t|Θ , v̂ t|Θ , t  0, are used to forecast the stock prices i.e. we forecast S t with S0 exp( x̂ t|Θ ) , t>0, and the stochastic variance i.e. we
forecast v t with v̂ t|Θ , t>0. Once known the forecasted values of the stock price and of the stochastic variance, of the Heston parameters and of the
risk premium parameter we can forecast the value of the option price using a formula derived in [1] Section 3 . This formula involves only the
numerical evaluation of a one dimensional integral.
~ ~
In order to reconstruct the vector Θ  (  ,  , ε, θ, ρ, ~
v 0 ,  ) from a set of observations ( Si , C i ) made at time t=ti, i=0,1,…,n, we use the maximum
likelihood method that consists in searching the vector M that maximizes the following (log-)likelihood function:
n 1

F(Θ)   log  ~
p(~
x i 1 , v, t i 1 | Ft i , Θ)π1 (~
x i 1 , v, t i | Θ)dv  log π1 (~
x0 , ~
v 0 , t 0 | Θ) ,
i 0
ΘM,
(15)
0
where M  { :   0,   0, θ  0, 2θ /  2  1, 1    1} is the set of the admissible vectors  and π1 (x, v, t i | Θ) , i=0,1,…,n-1, is the following
function:
~ 2
 C(x, v, t ; E, T, Θ)  C
i
i

exp  


2i

 , (x, v)  (-,)  (0,), i  0,1,..., n  1 . (16)
π1 (x, v, t i | Θ) 
2i
We note that (15) represents the sum of the logarithms of the integrals of the product between the transition probability densities
~
p (~
x i 1 , v, t i 1 | Ft i , Θ) and the weight functions π1 (~
x0, ~
v0 , t 0 | Θ) . The use of these weight
x i1, v, t i | Θ) , i=0,1,...,n-1, and of the logarithm of π1 (~
~
functions in the expression of the likelihood function (15) has the purpose of exploiting the fact that also the option prices C i , i=0,1,...,n, are
~
observed, assigning bigger weights to the values of the v variable that make more likely the observation of C i at time t=ti, i=0,1,...,n. The function
~
x 0 , C 0 at time t=t0.
F() represents the (log-)likelihood of the parameter vector  given the observations Ft, t>0, and ~


A solution of the estimation problem (i.e. the problem of calibrating the parameters of Heston model (1), (2), (3), (4), the stochastic initial variance
~v and the risk premium parameter ) can be obtained solving the following problem:
0
max F (),
M
(17)
using an iterative method (such as, for example, a variable metric steepest ascent method) starting from an initial guess 0M. Problem (17) is the
maximum (log)-likelihood problem that has been announced.
Further details can be found in the website http://www.econ.univpm.it/pacelli/mariani/finance/w1
3. Estimation problem using synthetic data
In this Section we describe some numerical experiments obtained applying the numerical method proposed in [1] and summarized in Section 2 to
reconstruct the vector  on some test problems. We remind that once known , we can forecast the stock prices, the stochastic variance and the
option prices solving the filtering problem, that is using formulae (13), (14) and the appropriate formula for the option price.
The first numerical experiment consists in the calibration of the Heston model through the solution of problem (17) . We use synthetic data. The
time unit is a ``year" made of 252.5 trading days. We consider a temporal interval of 36 consecutive trading days and n=9 observation times in this
time period. Note that we choose t0=0, and the observation times ti equally spaced, that is ti=ti-1+, i=1,2,...,9, where =4/252.5. The choice of the
vector  that generates the trajectory of the initial value problem (6), (7), (8), (9) used
to generate the data is:
~
~
-4
~
Θ  (  ,  , ε, θ, ρ, v 0 ,  ) = Θ  (0.026, 5.94, 0.306,0.01159,-0.576,0.5,0) . Moreover we choose x 0 =0 and  i =10 , i=0,1,…,9. We remind that the
~
vector Θ  Θ is expressed in annualized unit. Similarly all the remaining choices of the vector Θ made in the numerical experiments are
expressed in annualized unit. Table 1 shows the synthetic data generated integrating numerically one trajectory of (6), (7), (8), (9) and using the
appropriate change of variable and option price formula. Starting from Θ 0  (0.5,8.22,0.15,0.0067,-0.634,0.3,0.01)  M , we solve the optimization
~
0
problem (17) associated to the data of Table 1. Note that we have: Θ  Θ  2.3 .
i
ti
0
1
2
3
4
5
6
7
8
9
0
4/252.5
8/252.5
12/252.5
16/252.5
20/252.5
24/252.5
28/252.5
32/252.5
36/252.5
~
xi
0
1.2355e-1
6.8935e-2
-2.9809e-2
1.2049e-2
4.4983e-2
-7.7662e-4
-1.3411e-2
-1.2461e-1
-7.1041e-2
~
Ci
0.5099
0.6410
0.5804
0.4791
0.5201
0.5538
0.5065
0.4936
0.3894
0.4377
Table 1. The synthetic data of the first experiment
The numerical procedure evaluates the objective function (15) in a point Θ  M solving the filtering problem (10), (11), (12) and evaluates the
gradient of the objective function (15) in a point Θ  M using elementary finite differences formulae so that to compute the function (15) and its
seven dimensional gradient in a point Θ  M it is necessary to solve eight filtering problems.
The iterative procedure used to solve the optimization problem (17) is a variable metric steepest ascent algorithm that generates a sequence of
k
vectors Θ , k=1,2,..,itmax, and in our numerical experience we fix the maximum number of iterations itmax=200. We use the following stopping
criterion:
| F(Θ )  F(Θ
k
| F(Θ
k 1
k 1
)|
)|
 δ1 ,
(18)
where δ1 is a given tolerance. In this experiment we choose δ1 =10-6 and the maximization procedure that starts at Θ  Θ stops at the k=92-th
~
k
~
|| Θ  Θ ||
k
iteration. In Table 2 we show the values of F( Θ ), and of
, when k=0, 92 and we show the value of F( Θ ). Note that the maximum
~
|| Θ ||
~
*
92
likelihood procedure finds as estimated value of the vector Θ the vector Θ   .
0
~
k
|| Θ  Θ ||
~
|| Θ ||
0
-929.14
0.392
92
43.17
0.041
~
k
Table 2. Value of the (log)-likelihood function and relative distance between Θ and Θ when Θ   , k=0, 92
k
k
F( Θ )
0
92
Figure 1 shows the tracking of the stochastic variance obtained with formula (14) (dashed dotted line) when Θ = Θ and when Θ = Θ , the (blue)
~
solid line represents the “true trajectory” obtained integrating numerically one trajectory of (6), (7), (8), (9) with Θ   . The substantial
improvement obtained in the tracking of the stochastic variance passing from Θ   to Θ  
is a valid formulation of the calibration problem.
0
92
shows that the maximum likelihood problem (17)
Figure 1. Tracking of the stochastic variance
Estimation Movie 1, click here to download a digital movie showing the tracking of the stochastic variance trajectory when  = k M ,
k=0,1,…,92, remind that these  are vectors obtained during the iterative optimization procedure used to solve problem (17) with the data of
~
Table 1. Moreover the movie shows the true trajectory of the stochastic variance corresponding to the parameter vector =  obtained integrating
numerically one trajectory of (6), (7), (8), (9), this trajectory is the one used to generate the data of Table 1.
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 , C i , t=ti=i/252.5, i=0,1,…,504, where we have chosen t0=0, x0= ~x 0 =0, v0= ~v 0 =0.5, T=2.5, E=0.5, i  10 4 , i=0,1,2,…,504. The data
are generated integrating numerically with Euler method one trajectory of (6), (7), (8), (9) for the two years period. We choose the vector 
associated to the stochastic differential system that generates the data to be equal to the vector   1 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,-0.576,0.5,0),and  2 =(0.4,2,0.01,0.01,0.5,0.012,0.7) .
~
Click on the names to download the file containing the observations ~x i (samplex.txt) , C i (sampleC.txt), made at time ti=i/252.2, i=0,1,…,504 used
in the numerical experiment.
~
We solve problem (17) using 8 consecutive observations that is the observations ~x i , C i , made at time t=ti,j i=0,1,…,7 when j=0,1,…,62, that is for
j=0,1,…,62 the time values ti,j=t8j+i, i=0,1,2,…,7 belong to the same time window j. We solve 63 problem (17) corresponding to the data associated
to the data to the 63 time windows described above starting from different initial guesses. In Figure 2 the diamond represents the vector 1 and the
square represents the vector  2 . The unit length in Figure 2 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 the Euclidean norm of the vector 1 -  2 . The vector determined solving
problem (17) using the optimization procedure are represented by stars. A point P= is plotted around 1 when the quantity
|| Θ  Θ1 ||
is smaller
|| Θ1 ||
|| Θ  Θ 2 ||
, otherwise it is plotted around  2 . The point P belonging to the cluster associated to 1 is plotted on the left or on the
|| Θ 2 ||
right of the point that represents 1 according to the sign of the first component of 1   . A similar statement holds for  2 and for a point P= 
belonging to the cluster associated to  2 .
than the quantity
Figure 2. Reconstruction of the parameter vectors
1 and  2
Estimation Movie 2, click here to download a digital movie showing the points determined solving the 63 problem (17) described above as a
function of the time window of the observations.
Figure 2 and the Estimation Movie 2 show, as expected, that the points obtained solving problem (17) 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 (17) corresponding
to the time windows described previously shows that two sets of parameters seem to generate the data studied. Moreover as expected the points
obtained using data contained in the first year of observation cluster around   1 and the points obtained using data contained in the second year
of observation cluster around    2 .
4. Estimation problem and forecasting experiments using real data
In this section we show some numerical results obtained applying the calibration procedure to real data.We analyze the daily closing values of the
U.S. S&P 500 index and the corresponding bid prices of a European call option on the U.S. S&P 500 index with maturity date December 16, 2005
and strike price E=1200 during the period of about four months going from January 3, 2005 to May 11, 2005. Due to the number of data actually
available in 2005 the time unit is a “year” made of 253 trading days. Note that we have selected an option that during the period considered is at the
money and that this option has a significant volume traded. We have chosen as initial time t=t0=0 the day January 3, 2005 and we have considered
81 windows of 10 daily consecutive observation times. In particular the observation times are chosen as ti,j=(i+j)/253, i=0,1,…,9, j=0,1,…,80 that is
in each window we consider daily observations and when we move from a window to the next one we discard the observations made in the first day
of the window and we add the next daily observations as last observations to build the new window. In the observation times t=ti,j , i=0,1,…,9,
j=0,1,…,80 the index j labels the window and the index i labels the observation dates in the window j. In each window we solve problem (17)
choosing δ1 =5.10-4 and we choose the starting point taking into account the implied volatility associated to the option prices, the standard deviation
and the mean value of the historical log-returns data. The calibration procedure, applied to the eighty-one windows considered provides eighty-one
parameter vectors. We use the parameter vector obtained using the data contained in the j-th window (i.e. observation times ti,j =(i+j)/253,
i=0,1,…,9) to forecast the value of the S&P500 index and the bid price of the option the day next to the last observation time of the j-th window
(see Figures 3 and 4), that is we forecast St, C at time t=(j+10)/253. This is done for the j-th window, j=0,1,…,80. The forecasts are made using
formula (13) and a similar formula for the option price (see [1] Section 3 and Section 4 ). We note that in the eighty-one calibration and forecasting
problems solved the mean relative error made in the forecasts of the “next day” S&P500 index value and option bid price are 0.0063 and 0.0785
respectively. We note that in order to forecast the option price we use the value of the stochastic variance obtained with the tracking procedure, that
is obtained with formula (14).
Figure 3. Tracking of the U.S. S&P 500 index
Figure 4. Tracking of the bid option price on the U.S. S&P 500 index
Figures 3 and 4 show the forecasted and the observed values of the index and of the bid price of the option versus time (expressed in days)
respectively. The solid line and the square represent the “true” values provided by the historical data while the dotted line and the stars are the
forecasted values obtained using the procedure described previously.
The forecasts shown in Figure 3 and Figure 4 are “next day” forecasts. We consider also forecasts two, three, four and five days in the future
obtained with the vectors Θ estimated previously. In Table 2 we show the mean relative errors made in the forecast of the S&P500 index value
eindex and of the bid price of the option eoption obtained forecasting up to five days in the future. The mean relative errors are computed averaging the
errors made in the solution of the eighty-one minus the number of days in the future of the forecasts problems considered. For example when we
make “next day” forecasts the average is made on eighty values, when we made “two days” in the future forecasts the average is made on seventynine values and so on. The errors are computed comparing the forecasts with the historical data.
Days in the
eindex
eoption
future
1
0.0063
0.0785
2
0.0083
0.1010
3
0.0097
0.1181
4
0.0109
0.1296
5
0.0118
0.1435
Table 3. Mean relative errors on forecasted values of the SP&500 and of the corresponding option price
Forecasting Movie 1, click here to download a digital movie showing the one and four days in the future forecasted values of the
SP&500 index as function of time. The blue boxes are the “true” (historical) values, the red stars are the one day in the future
forecasted values and the green stars are the four days in the future forecasted values.
Figure 5 shows the true bid option prices on the US S&P500 index provided by the historical data (blue solid line and squares) and the bid prices of
the option forecasted as explained below.
Figure 5. Daily estimates of bid option prices on the US S&P 500 index
Forecasting Movie 2, click here to download a digital movie showing the bid option prices forecasted as explained below as a
function of time. The blue boxes are the “true” (historical) values and the red stars are the forecasted values.
Finally Figure 5 and the Forecasting Movie 2 show the bid prices of the option computed using the value of the stock price actually observed and
the value of the stochastic variance obtained as one day in the future forecast derived from the parameters obtained solving the calibration problem
with the data observed in the previous ten days. In this case mean relative error made is e1option=0.0298.
5. References
[1]
L. Fatone, F. Mariani, M.C. Recchioni, F. Zirilli: “The calibration of the Heston stochastic volatility model using filtering and maximum
likelihood methods”, submitted to Proceedings of Dynamic Systems and Applications (2007).
[2]
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 (http://www.econ.univpm.it/pacelli/mariani/finance/w1 contains
downloadable software).
[3]
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", Journal of Inverse and Ill-Posed Problems, 15, 329-362 (2007).
(http://www.econ.univpm.it/recchioni/finance/w5 contains downloadable software).
[4]
S.L. Heston, “A closed-form solution for options with stochastic volatility with applications to bond and currency options”, Review of
Financial Studies, 6, 327-343 (1993).
Entry n.