Lorella Fatone, Francesca Mariani, Maria Cristina Recchioni, Francesco Zirilli
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We use filtering and maximum likelihood methods to solve a calibration problem for a multiscale stochastic volatility model. The multiscale
stochastic volatility model considered has been introduced in [4],
[5] and describes the dynamics of the asset price using
as auxiliary variables two stochastic variances varying on two different time
scales. Our aim is to estimate the
parameters of this multiscale model
(including the risk premium parameters when necessary) and its two initial
stochastic variances from the knowledge, at discrete times, of the asset
price and, eventually, of the prices of
call and/or put European options on the asset. This problem is translated in a
maximum likelihood problem with the likelihood function defined through the
solution of a filtering problem. The estimated values of the parameters and of
the two initial stochastic variances are characterized as being a constrained maximizer
of a likelihood function. Furthermore we
develop a tracking procedure that is
able to track the asset price and the values of its two stochastic
variances for time values where there are no data
available. The solution of the calibration problem and the tracking procedure
are used to do the analysis of data time
series. Numerical examples of the
solution of the calibration problem and of the performance of the tracking
procedure using synthetic and real data
are presented in [3].
The synthetic data time series is
analyzed using data samples made of observations done with a given time frequency.
We consider observation frequencies ranging from “high” to “low” and we study the performance of the calibration
procedure as a function of the observation frequency. The real data studied are
two time series of electric power price data taken from the
Acknowledgments:
· It is a pleasure
to thank V. Kholodnyi and R. Whaley of Platts Analytics Inc. (
· The numerical
experience and data analysis reported in this website have been obtained using the
computing resources of CASPUR (
· The research reported in this website is partially supported by
MUR – Ministero Università e Ricerca (
Keywords. Multiscale
stochastic volatility models, filtering problem, calibration model, option
pricing.
AMS (MOS) Subject
Classification. 62F10, 35R30, 91B70.
M.S.C. classification. 34M50, 60H10, 91B28.
J.E.L. classification. C53, G12, C61.
2. The multiscale stochastic volatility model, the filtering and
calibration problems
For the convenience of the
reader we summarize some results contained in [5]. Let R
and R^{+} be the sets of real and of positive real numbers respectively
and t be a real variable that denotes time. Let S_{t}, t ≥0, be a
stochastic process describing the asset (stock, commodity, index) price at time
t≥0 and x_{t} = log(S_{t}/S_{0}), t ≥ 0, be
the corresponding logreturn. We associate to the asset price two stochastic
variances described by the stochastic processes v_{1,t}, v_{2,t},
t ≥ 0, one fluctuating on a fast time scale and the other fluctuating on
a long time scale. The dynamics of the stochastic process x_{t}, v_{1,t},
v_{2,t}, t > 0, is defined by
the following system of stochastic differential equations:
_{}
where the quantities _{}, _{} a_{i}, b_{i},
i = 1,2, are real constants satisfying the following conditions: c_{i} ³ 0, e_{i} ³ 0, q_{i} ³ 0, b_{i}
≥ 0, i = 1,2. These conditions are due to the financial meaning of
the constants.
The state variables
v_{1,t}, v_{2,t}^{ }, t>0, are the stochastic
variances associated to x_{t}, t>0,
W_{t}^{0,1}, W_{t}^{0,2 },W_{t}^{1},
W_{t}^{2}, t>0, are standard Wiener processes such that W_{0}^{0,1}=
W_{0}^{0,2}=W_{0}^{1}=W_{0}^{2}=0, dW_{t}^{0,1}, dW_{t}^{0,2
},dW_{t}^{1}, dW_{t}^{2} are their stochastic
differentials and we assume that <dW_{t}^{0,1}dW_{t}^{0,2}
>= 0, <dW_{t}^{0,1}dW_{t}^{1}
>= r_{0,1}dt, <dW_{t}^{0,2}dW_{t}^{2}
>= r_{0,2}dt, <dW_{t}^{1}dW_{t}^{2}>=0 , where < × > denotes the mean
of × and r_{0,1,} r_{0,2}Î[1,1] are constants
known as correlation coefficients.
The fact that the model is a two scale stochastic volatility model is
translated in the assumption that _{}. The equations (1), (2), (3)
must be equipped with an initial condition, that is:
_{}
The random variables _{}, _{}, _{} are assumed to be
concentrated in a point with probability one and we choose _{}=0. We assume that _{} _{}, _{}>0, i=1,2 and later in the multiscale model we will choose
a_{i}=1/2, b_{i}=1, i=1,2.
The model (1), (2), (3) generalizes the Heston stochastic volatility
model. In fact choosing a_{1} = 1/2, a_{2} = 0, b_{1}
= 1, b_{2} = 0 the equations (1), (2) reduce to the Heston model and
they become decoupled from equation (3). Remind that the stochastic variances and in particular the initial
stochastic variances appearing in (5), (6) cannot be observed in real markets.
As already mentioned in all the numerical experiments
presented here we use the multiscale model model and we choose
in equations (1), (2), (3) a_{1}
= a_{2} = ^{1}/_{2} and b_{1} = b_{2} =
1. This choice translates in the
multiscale model the usual relation between asset price, asset logreturn and
stochastic variance known from the Heston model. Remember that when the option
prices are used as data of the calibration problem the discount rate r , that
appears in the option price formulae, becomes a parameter that must be
determined in the calibration, and in that case we will choose _{}in the option price formulae. More general problems where r
is not necessarily equal to _{} and a_{i}, b_{i}, i = 1,2, are
parameters to be determined in the calibration can be studied. We will not
consider them here.
Let us introduce the vector Q =(e_{1},q_{1},r_{0,1},c_{1},_{}, _{},l_{1},e_{2},q_{2},r_{0,2},c_{2}, _{},l_{2}) Î R^{13} where the parameters _{}, i=1,2, are the risk premium parameters that must be used
when we consider option prices. The set M Ì R^{13} of the feasible vectors is defined as follows:
_{}
The constraints that define M translate some elementary properties required to model (1), (2),
(3). Moreover, let t_{i}, i =
0,1,¼,n, be time values
such that t_{i}
< t_{i+1},
i = 0,1,¼,n1, without loss
of generality we choose t_{0} = 0, and for later convenience we define
t_{n+1} = +¥. Let _{} be respectively the
logreturn of the asset price S_{t} and the prices of European vanilla
call and put options on the asset having maturity time T_{i} (T_{i} > t_{i}) and strike
price K_{i}, observed at time t = t_{i}, i = 0,1,¼,n. We suppose that the option
prices observations _{}, _{} are affected by a Gaussian error with mean
zero and known variance f_{i}, i = 0,1,¼,n, and that the asset logreturns _{}, i = 0,1,¼,n, are observed without error. The filtering problem that we consider
is the following:
Given the vector Q Î M find
the probability density function p(x,v_{1},v_{2},tF_{t},
Q), (x,v_{1},v_{2})
Î R×R^{+}×R^{+},
t
> 0, of the random variables x_{t}, v_{1,t}
and v_{2,t}, t > 0, solution of (1), (2), (3), (4), (5), (6),
conditioned to the observations F_{t} = {_{} : t_{i} £ t, i > 0}, t > 0, and
forecast the values of the state variables x_{t}, v_{1,t}, v_{2,t}
for t
> 0 and in particular for t ¹ t_{i},
i = 0,1,¼,n and for t > t_{n}.
For t = t_{0} = 0 we define _{} = {_{} = 0 }. Note that in
the filtering problem stated above we do not make use of the option prices _{}, _{} observed at time t = t_{0}. That is in
the filtering problem at time t = t_{0} = 0, we use as datum only x_{0}
=_{} =
Let us
define the following functions:


to be the joint probability
density functions of the random variables x_{t}, v_{1,t}, v_{2,t},
solution (1), (2), (3), (4), (5), (6), when t_{i} < t < t_{i+1} conditioned to
the observations _{} made up to time
t = t_{i}, i = 0,1,¼,n. The functions p_{i}(x,v_{1},v_{2},t Q), (x,v_{1},v_{2}) Î R×R^{+}×R^{+},
t_{i} < t < t_{i+1}, i = 0,1,¼,n can be computed as
solutions of a set of initial value problems for the Fokker Planck equation (see
[5] and the reference therein) with suitable initial
conditions that involve the knowledge of the option prices.
Using the functions p_{i}(x,v_{1},v_{2},t Q), (x,v_{1},v_{2}) Î R×R^{+}×R^{+},
t_{i} < t < t_{i+1}, i = 0,1,¼, n, we can forecast the
values of the state variables of the model x_{t}, v_{1,t}, v_{2,t},
t > 0, respectively as
the expected values_{}, _{} , _{}, t
> 0,
conditioned to the observations contained in F_{t}, t > 0, of the random variables x_{t},
v_{1,t}, v_{2,t}, t > 0, that is:
_{}
_{}
where E(·  +) denotes the expected
value of · conditioned to + and the functions f_{i}(v_{1},v_{2};Q) are defined
through the knowledge of the conditioned probability densities (8) (see [3] for further details) and are given as twodimensional
integrals of explicitly known integrands. The formulae (9), (10) can be used to track
the random variables x_{t}, v_{1,t}, v_{2,t}, for t > 0, and in particular for t ¹ t_{i}, i = 0,1,¼,n and for t > t_{n} and they
constitute the tracking procedure announced in the Abstract.
We note that the model (1), (2), (3) together with the associated
option price formulae are parameterized by 13 real quantities, that is: the
model parameters _{}, c_{i}, q_{i}, e_{i}, i = 1,2, the risk
premium parameters l_{i}, i = 1,2, the
correlation coefficients r_{0,i}, i = 1,2, and the
initial stochastic variances _{}, i = 1,2. That is, when we consider the calibration problem using as data asset and
option prices we use the vector Q =(e_{1},q_{1},r_{0,1},c_{1},_{}, _{},l_{1},e_{2},q_{2},r_{0,2},c_{2}, _{},l_{2}) Î R^{13} and the set M Ì R^{13} defined in
(10).
The vector Q Î R^{13} is the unknown of the calibration problem considered
here when we use as data asset and option prices and the set M defines
the set of the “feasible” vectors, that is the set of vectors satisfying the
constraints. Later in the formulation of the calibration problem we will assume
Q Î M. When we consider the filtering and calibration problems using as
data only asset prices the risk premium parameters l_{i}, i = 1,2, that
appear in the option price formulae, must be removed from the vector Q and the definition of the
vector Q and of the set M
must be changed consequently. To fix the ideas we discuss in detail only the situation where
asset prices and option prices are used as data in the filtering and in the
calibration problem. We leave to the reader to work out the obvious changes
that are needed when simpler problems are considered.
The calibration problem consists in
determining the value of the vector Q Î M that makes most likely the observations _{}, made at time t = t_{i}, i = 0,1,¼,n. In the calibration problem
we add to the data of the filtering problem the call and put option prices at
time t = t_{0} and this fact improves greatly the quality of the
estimates of the initial stochastic variances resulting from the solution of
the calibration problem. The improvement of the quality of the estimated
initial stochastic variances determines an improvement in the estimates of all
the remaining parameters contained in the vector Q.
As suggested in [6], [10] given the observations _{} at time t = t_{i},
i = 0,1,¼,n, we measure the (log)likelihood of a vector Q Î M through the following
(log) likelihood function:
_{}
where we
have defined:
_{}
and p_{i}(x,v_{1},v_{2},t_{i}^{} Q) = _{}^{ }p_{i}(x,v_{1},v_{2},t Q), _{}^{ } means left limit for t that goes to t_{i},
i = 1,2,¼,n and C and P are
the formulae derived in [5] for the price of European
vanilla call and put options in the multiscale model (1), (2), (3) respectively
. The calibration problem that we
consider is the following one:
Given the observations _{} at time t = t_{i},
i = 0,1,¼,n, determine the vector Q^{*} Î M Ì R^{13}
solution of the following nonlinear constrained optimization problem:

(13) 
Problem (23) is known as maximum (log)likelihood problem since the
vector Q^{*} Î M solution of (13) is the vector that makes “more likely” the
observations actually made. The vector Q^{*} Î M is the solution of the calibration problem. Note that problem (13)
is only one possible way of using the maximum likelihood method to formulate
the calibration problem that we are considering. Many other formulations of the
calibration problem are possible and legitimate.
3 Calibration problem and forecasting experiments using real data
In this section we show two
experiments that use real data. In these two experiments daily data are used.
The first experiment consists in the analysis of two time series of electric
power price data, the first time series does not contain spikes while the
second one contains spikes. No option prices are used as data in this
experiment. The electric power price data studied are taken from the
The first numerical experiment considers two time
series of electric power price data. The first time series consists of 365
daily price observations _{} i = 0,1,¼,364, that is it is a year,
made of 365 days, of daily observations. The second time series consists of 765
daily price observations _{} , i = 0,1,¼764. In this experiment no
option price data are used. Figure 1 shows the daily logreturn increment _{} =_{}, i = 1,2,¼,364, of the
electric power price data _{}, i = 0,1,¼,364. The time
series shown in Figure 1 does not contain spikes. Figures 2 and 3 show the
daily logreturn increment of the second time series of electric power price
data that is _{}=_{}, i=1,2,…,764. Figure 2 shows the daily logreturn increment of the
second time series of electric power price data in the first year of
observation and Figure 3 shows the daily logreturn increment in the second
year of observation. The time series shown in Figures 2 and 3 contains spikes.
Figure 1: Logreturn increment log(S_{t}/S_{t1})
of the electric power prices versus time t (days) (time series with no spikes _{}, i=0,1,…,364,
daily data).
Figure 2: Logreturn increment log(S_{t}/S_{t1})
of the electric power prices versus time t (days) (time series with spikes _{}, i=0,1,2,…,364, first year of observation, daily data).
Figure 3: Logreturn increment log(S_{t}/S_{t1})
of the electric power prices versus time t (days) (time series with spikes _{},
i=365,366,…,764, second year of observation, daily data).
Figure 4: Estimated parameter values obtained
solving 340 calibration problems using as data a data window of the electric power prices
(time series with no spikes, Figure 1) versus the calibration problem number i_{1}.
Figure 5: Estimated parameter values obtained
solving 340 calibration problems using as data a data window of the electric
power prices (time series with spikes, Figure 2) versus the calibration problem
number i_{1}.
Figure 6: Estimated parameter values obtained
solving 740 calibration problems using as data a data window of the electric
power prices (time series with spikes, Figures 2, 3) versus the calibration
problem number i_{1}.
We begin the analysis of these time series calibrating
the multiscale model using a data window
made of 26 consecutive daily observations and we move this window along the
time series substituting the first observation of the window with the next
observation after the window. Using this procedure in the case of the time
series shown in Figure 1 we solve 340 (340=36526+1) calibration problems
numbered with the index i_{1}, i_{1} = 1,2,¼,340 and in the case of the
time series shown in Figures 2 and 3 we solve 740(740=76526+1) calibration
problems numbered with the index i_{1}, i_{1} = 1,2,¼, 740. The index i_{1}
is the value of the index i associated to the first day contained in the data
window used in the calibration. Figures 4 and 5 show the results obtained
solving the calibration problems associated to the two time series of data as a
function of the index i_{1} for i_{1} = 1,2,¼,340. Figure 4 shows the
results obtained processing the time series of electric power price data with
no spikes (shown in Figure 1). Remind that this time series is made of one year
of daily data. Figure 5 shows the results obtained processing the first year of
the time series of electric power price data with spikes (shown in Figure 2).
We focus our attention on Figures 4, 5 to point out the different behaviour of
the estimated parameter values in the two cases. In absence of spikes (Figure 4)
the parameters e_{i}, q_{i}, c_{i}, i = 1,2, are
stable, that is they are approximately constants as a function of i_{1},
and c_{1} and c_{2} are of the same
order of magnitude (i.e.: c_{1} » 1, c_{2} » 4). In presence of spikes
(Figure 5) the parameters e_{i}, q_{i}, c_{i}, i = 1,2, have a
jump in correspondence of the first spike. However comparing Figure 2 and
Figure 5 we can see that the calibration procedure produces two values of the
ratio c_{1}/c_{2}, in particular
after the spike (i.e. i_{1} > 100) Figure 5 shows a jump in the ratio c_{1}/c_{2} and we have c_{1} << c_{2}.
Figure 6 shows the estimated values of the parameters e_{i}, q_{i}, c_{i}, i = 1,2, during
the entire observation period covered by the data shown in Figures 2 and 3
(time series with spikes). We note the presence of a jump in the estimated
parameter values around i_{1} = 395. Note that when we solve the
calibration problem corresponding to i_{1} = 395 we use as data the 26
consecutive daily observations starting with the 395^{th} observation
day. That is, this calibration problem takes as input data one of the spikes
(see Figure 3) contained in the time series. The fact that Figures 4, 5, 6 show
that the model parameters as a function of the index i_{1} are
approximately piecewise constant functions suggests that the model and the
calibration procedure proposed are well suited to establish a stable
relationship between the data and the unknown of the calibration problem.
Moreover using formulae (9), (10), that is using the
tracking procedure, we forecast the
electric power prices using as data of the calibration problems the data
corresponding to the time series containing the spikes (shown in Figures 2, 3).
That is we take a data window of 26 consecutive days in the time series of
electric power price data we solve the corresponding calibration problem, we
use the (approximate) solution of the calibration problem Q = Q^{*} and formulae (9),
(10) to forecast the values of the state variables in the future. The future
consists in the time period after the last day whose observation is contained
in the data window used in the calibration problem. The data window used as
input data of the calibration problem is rolled through the time series using
the procedure already described.
The results obtained in this forecasting exercise are illustrated
in Table 1, Figures 7, 8, Animation 1, Animation 2, Animation 3.
Table 1 establishes the quality of the forecasted
values of the electric power logreturns when we do forecasts up to fifteen days
in the future and we process the data corresponding to the time series shown in
Figures 2, 3. That is Table 1 contains the mean value of the relative error, _{}, committed approximating the observed logreturn with the
forecasted logreturn, the column “trend” of Table 1 contains (in percent) the
number of times that the forecasted logreturn moves in the correct direction
with respect to the logreturn of the first day contained in the data used in
the calibration, that is increases when the historical logreturn increases and
decreases when the historical logreturn decreases. The last column_{} of Table 1 contains
the same quantity than _{} computed on the
logreturns observed before the first spike, that is for i_{1} £
Finally Figures 7, 8 show the forecasted values of the logreturn
(one day in the future) and the time series of the logreturns corresponding to
the time series shown in Figures 2, 3 as a function of time t (days). Figure 8 reproduces
on a different scale the part of Figure 7 corresponding to 300 £ t £ 500. This is the region where
severe spikes occur (see Figures 2, 3). Figures 7 and 8 show the very high
quality of the (one day in the future) forecasted logreturns in a more
convincing way than Table 1 where we consider highly aggregated performance
indices. The Animations 1, 2,
3 show respectively the quality of the forecasted logreturns one day, five
days and fifteen days in the future.
of days in the future 
_{} 
trend (%) 
_{} 
1 
0.0485 
90.53% 
0.0167 
2 
0.0786 
82.27% 
0.0268 
3 
0.0968 
77.54% 
0.0310 
4 
0.1078 
73.88% 
0.0324 
5 
0.1131 
72.94% 
0.0322 
15 
0.1773 
63.06% 
0.0474 
Table 1: Mean relative errors and “trend” of
the forecasted values of the electric power logreturns of the time series with
spikes when compared to the logreturns actually observed.
Figure 7: Forecasted values (one day in the
future) (stars) and observed values (squares) of the logreturns of electric
power prices (time series with spikes, Figures 2, 3) versus time t (days).
Figure 8: Forecasted values (one day in the
future) (stars) and observed values (squares) of the logreturns of electric
power prices (time series with spikes, Figures 2,3) versus time t (days).
In all the animations the square is the observed value of the logreturn
and the star is the forecasted value.
Animation 1: Forecasted logreturn one day in
the future (data time series with spikes Figures 2, 3) click here, reduced animation on a different
scale click here
Animation 2: Forecasted logreturn five days in
the future data time series with spikes Figures 2, 3) click here, reduced animation on a different
scale click here
Animation 3: Forecasted logreturn fifteen days
in the future data time series with spikes Figures 2, 3) click here, reduced animation on a
different scale click here
The second numerical experiment consists in the
analysis of the daily closing values of the S&P 500 index and of the daily
bid prices of the European call and put options on the S&P 500 index with
maturity date December 16, 2005 and strike price K = 1200 during the period of
about eleven months that goes from January 3, 2005 to November 28, 2005. Due to
the number of trading days in the year 2005 the time unit is a “year" made
of 253 days. Note that the call and the put options mentioned above whose
prices have been used as data in this experiment during the period January,
November, 2005 had a significant volume traded. To avoid excessive numerical
work we analyze in detail the market data (i.e. S&P 500 index and the
European vanilla call and put option prices on this index) only in the months
of January, February, May, June, October and November, 2005. The results
presented are representative of the results obtainable from an exhaustive
analysis of the eleven months of data available. We have solved the calibration
problem (13) using the data contained in a window made of fifteen consecutive
observation days, that is fifteen daily observations of the logreturn of the
S& P 500 index and of the call and put option prices on S& P 500 index
having maturity time December 16, 2005 and strike price K =
We solve three calibration problems, the first one
uses as data the data relative to the first fifteen observation days of January
2005, the second one uses as data the data relative to the last twelve
observation days of May 2005 and the first three observation days of June 2005,
and the third one uses as data the data relative to the last twelve observation
days of October 2005 and the first three observation days of November 2005.
These three calibration problems have been chosen in order to be able to
compare the results presented here with those presented in [5].
The three sets of estimated parameter
values obtained solving these three calibration problems are used to forecast
the value of the S&P 500 and the prices of its call and put European
vanilla options one day, one week and one month in the future (that is one day,
one week, one month after the day of the last observation contained in the data
used in the calibration problem). The quality of these forecasts is established
by comparison with the corresponding historical data. Moreover a comparison of
the results obtained using the calibration procedure proposed here with the
results obtained using the least squares formulation of the calibration problem
proposed in [5] is presented.
The forecast of the S& P 500 index is done with
the tracking procedure derived from (9), (10) already discussed. The forecast
of the option prices is done using formulae derived in [5] and
recalled in [3] starting from the model parameters,
including the correlation coefficients, the risk premium parameters and the
initial stochastic variances, estimated solving the calibration problem. We use
option formulae with two different
choices of asset price, that is: the price actually observed in the day of the
forecast or the asset price forecasted using formula (10). The first choice is
considered in order to compare the results obtained in [3] and in this website
with those presented in [5] where the asset price actually observed the
day of the forecast was used. We note that in the option price formulae we use
always the values of the stochastic variances obtained using formula (10). That
is, given the stochastic variances v_{1,t}, v_{2,t}, at time t
= t_{0} = _{} where _{} is the time of the first observation used in
the calibration problem (that is given the initial stochastic variances _{}, i = 1,2 that are estimated solving the calibration problem)
we forecast v_{1,t}, v_{2,t}, t > t_{0}, using formula (10) where the vector Q is chosen to be the vector Q^{*} obtained as
solution of the calibration problem. Starting from the solution of the first
calibration problem (data corresponding to the first fifteen observation days
of January 2005) we forecast the index value and the option prices (in some
specified days of January and February 2005). In Table 2 we show the relative
errors made in the forecast of the S&P 500 index value e_{index}
and of the bid prices of the European vanilla call option, e_{call option},
and put option, e_{put option}, having strike price K = 1200 and
maturity time T given by December 16, 2005 obtained forecasting up to a month
in the future. In Table 2 the option prices are forecasted using as asset price
the forecasted values of the S&P 500 index in the days of the forecasts.
The errors are computed comparing the forecasted values with the historical
data. The results shown in Table 2 relative to some specified days of January
and February, 2005 are somehow representative of the results obtained with a
more exhaustive analysis of the data available. Note that the forecasted values
of the S&P 500 index are really satisfactory and of much higher quality
than the forecasted values of the electric power prices (compare Table 1 and
Table 2). This difference is due to two facts: the first one is the use of the
option prices in the calibration procedure when we study the S& P 500 data.
The option prices improve the quality of the solution of the calibration
problem and of the conditioned probability densities. No option prices are used
in the solution of the calibration problem for the electric power price data.
The second one is the huge historical volatility of the time series of the
electric power price data in comparison with the historical volatility of the
S&P 500 index in 2005. This last fact makes the forecasting problem for the
electric power price data substantially more difficult than the corresponding
problem for the S&P 500 index in 2005.
of days
in the
future 
e_{index} 
e_{call option} 
e_{put option} 
1 
7.522·10^{5} 
0.0659 
0.0407 
2 
1.303·10^{4} 
0.0559 
0.0737 
3 
1.7509·10^{4} 
0.0893 
0.0952 
4 
2.6962·10^{4} 
0.0528 
0.0793 
5 
2.9114·10^{4} 
0.0496 
0.0822 
15 
3.7106·10^{4} 
0.0268 
0.1810 
30 
3.5094·10^{4} 
0.0717 
0.1099 
Table 2:
Relative errors of the forecasted values of the SP&500 and of the
corresponding call and put option prices in some specified days of January and February 2005 when compared to the
prices actually observed. The model parameters employed have been obtained
solving the calibration problem using as data the data relative to the first
fifteen observation days of January 2005.
Finally we use the estimated model parameter vectors
obtained as solution of the calibration problems to forecast the prices of all
the European call options on the S&P 500 index traded in a given day. The
days that we consider are January 28, 2005, June 7 and 28, 2005 and November 7,
14 and 28, 2005. Note that in the year 2005 we had approximately three hundreds
European call options on the S&P 500 index traded. The forecast of the
option prices of January 28, 2005 is done using the vector Q obtained solving the
calibration problem using as data the first fifteen observation days of January
2005. The forecast of the option prices of June 7 and 28, 2005 and November 7,
14 and 28, 2005 are done using as data the first three observation days of the
current month (June and November respectively) and the last twelve observation
days of the previous month (May and October respectively). We consider the days
January 28, June 7 and 28, November 7, 14 and 28, 2005 because we have at our
disposal the results presented in [5] relative to the
forecasts of the call option prices of these days made using the model parameter vectors
obtained as solution of the calibration problem formulated with the least
squares procedure discussed in [5]. We compare the call
option prices obtained solving the calibration problem with the maximum
likelihood procedure presented in this paper with those obtained solving the
calibration problem with the least squares procedure introduced in [5]. Let us introduce some notation necessary to illustrate
the comparison. Let n_{obs} be the number of call option prices
available at time _{}and let _{}, _{} and _{} denote respectively
the (observed) prices at time _{} of European call options with strike price K_{i}
and maturity time T_{i}, the prices generated using the appropriate
formula (see [3], [5]) with the
parameter vector obtained solving the calibration problem with the least
squares procedure (LS) and with the parameter vector obtained solving the
calibration problem using the maximum likelihood procedure (ML). The index i
spans all the European call options traded at time _{}. Note that n_{obs} may depend on _{}, we omit this dependence for simplicity. Let us define the absolute error:
_{}
obtained using the two
calibration procedures (A = LS (least squares procedure), ML (maximum
likelihood procedure)), in [3] we consider _{} as a function of the
moneyness K/S_{0} = K_{i}/S_{0}, i = 1,2,¼,n_{obs}, where S_{0}
=_{} is the value of the
S&P 500 index at the transaction day _{} where the option prices are forecasted and as
a function of the time to maturity t_{i} = T_{i}_{}, i = 1,2,¼, n_{obs}. Note that the
sets of the values assumed by the moneyness and by the time to maturity may
contain repeated values. In Table 3 we show the mean absolute error:
_{}
Date t 
e^{t}_{mean, ML} 
e^{t}_{mean,LS} 
January 28, 2005 
4.69 ·10^{3} 
2.84·10^{3} 
June 7, 2005 
5.99 ·10^{3} 
1.75·10^{3} 
June 28, 2005 
6.87 ·10^{3} 
2.56·10^{3} 
November 7, 2005 
3.68·10^{3} 
3.04·10^{3} 
November 14, 2005 
3.07·10^{3} 
2.21·10^{3} 
November 28, 2005 
3.21·10^{3} 
2.41·10^{3} 
Table 3: Quality of the
forecasted values of the option prices established comparing the forecasted
prices with the prices actually observed when the maximum likelihood (ML) or
the least squares (LS) method are used in the calibration of the multiscale
model.
Note that the absolute errors shown in Table 3 are referred to quantities roughly
speaking of order 0.1.
We
conclude this Section making some comments on the results obtained doing the
forecasts with the least squares (LS) or with the maximum likelihood (ML)
procedures.
We recall that in [5] we have
solved three calibration problems using the least squares formulation and the
call option price data of January 3, 2005, June 3, 2005 and November 3, 2005.
Note that the least squares procedure to solve the calibration problem of
January 3, 2005 uses n_{obs} = 281 call option prices. Similarly the
calibration problem of June 3, 2005 uses n_{obs} = 281 call option
prices and the calibration problem of November 3, 2005 uses the prices of n_{obs}
= 303 call options prices. We use the parameter estimated using the call option
prices of January 3, 2005 (n_{obs} = 281) to forecast the option prices
of January 28, 2005 (n_{obs} =
258) and similarly we use the parameters estimated using the call option prices
of June 3, 2005 (n_{obs} = 281) to forecast the option prices of June 7,
2005 (n_{obs}= 288) and June 28,
2005 (n_{obs} = 278) and the parameters estimated using the call option
prices of November 3, 2005 (n_{obs} = 303) to forecast the option
prices of November 7, 2005 (n_{obs} = 303), November 14, 2005 (n_{obs}
= 305) and November 28, 2005 (n_{obs} = 292).
We proceed similarly with the parameter values
estimated solving the three calibration problems mentioned above with the maximum likelihood procedure. In fact
we use the parameters estimated using the first fifteen daily observations of
January 2005 of the S& P 500 index and of the prices of the European call
and put options with K = 1200 and T given by December 16, 2005 to forecast the
prices of all the call options traded on January 28, 2005. That is we forecast
the prices of January 28, 2005 of n_{obs} = 258 call options. Moreover
we use the parameters estimated using the last twelve observation days of May
2005 and the first three observation days of June 2005 of the S& P 500
index and of the prices of the European call and put options with K = 1200 and
T given by December 16, 2005 to forecast the prices of all the call options
traded on June 7 and June 28, 2005. That
is we forecast the prices of June 7of n_{obs}= 288 call options and the
prices of June 28, 2005 of n_{obs}
= 278 call options. Finally, we use the parameters estimated using the last
twelve observation days of October 2005 and the first three observation days of
November 2005 of the S& P 500 index and of the price of the European call
and put options with K = 1200 and T given by December 16, 2005 to forecast the
prices of all the call options traded on November 7, 14, 28, 2005. That is we
forecast the prices of November 7, 2005 of n_{obs} = 303 call options,
the prices of November 14, 2005 of n_{obs} = 305 call options and the
prices of November 28, 2005 of n_{obs} = 292 call options.
Table 3 shows the results obtained. Remind that in the forecasts considered in
Table 3 we use as value of the S&P 500 index the actual value of the index
the day of the forecast and not its forecasted value. The results presented in
Table 3 show that the option prices forecasted with the two methods are
approximately of the same quality. We can conclude that the maximum likelihood
approach to the calibration problem has two advantages when compared to the
least squares approach presented in [5]. The first one
is that a relatively small number of data is used to calibrate the model. In
fact only one European call and one European put option prices and the
corresponding value of the index are considered for each day of data and a time
window of fifteen consecutive trading days is used as input data in the
calibration problem. This corresponds to fortyfive data. The least squares procedure
uses as data all the call options traded in a given day that is approximately
three hundred data. The second one is the fact that the maximum likelihood
approach thank to the tracking procedure discussed previously allows to
forecast the value of the asset, that is the value of the S&P 500 index.
This is a consequence of the solution of the filtering problem. This makes
possible to “forecast” option prices without knowing the value of the S&P
500 index the day of the forecast. Finally we note that going back to Table 2 we can say that the parameter
values obtained using the maximum likelihood approach give satisfactory results
when used to forecast the option prices up to a month in the future.
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