Home > PAPER: European option prices under an explicitly solvable multiscale stochastic volatility model and the analysis of the implied values of
European option
prices under an explicitly solvable multiscale stochastic volatility
model and the analysis of the implied values of the parameters, of the
correlation coefficients and of the initial stochastic variances of
the model ^{*}
Lorella Fatone, Francesca Mariani, Maria Cristina Recchioni, Francesco Zirilli
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1. Abstract
In [3] we study an explicitly solvable multiscale
stochastic volatility model that generalizes the Heston model. The model
describes the dynamics of an asset price and of its two stochastic variances
using a system of three Ito stochastic differential equations. The two
stochastic variances vary on two different time scales and can be regarded
as auxiliary variables introduced to model the dynamics of the asset
price. Under some assumptions the transition probability density function
of the stochastic process solution of the model is represented as a
one dimensional integral of an explicitly known integrand. In this sense
the model is explicitly solvable. In mathematical finance multiscale
stochastic volatility models are used for several purposes such as,
for example: i) the study of commodity prices, in fact usually commodity
prices are characterized by spikes that can be modeled using a fast
time scale volatility together with a standard time scale volatility,
ii) the study of financial products that live for long time periods
(such as life insurance contracts) that can be modeled using a long
time scale volatility together with a standard time scale volatility.
In alternative to multiscale stochastic volatility models the stochastic
jump models, such as, for example, L��vy's processes with stochastic
volatility, can be used. However this last type of models is explicitly
solvable only in very special circumstances and in general the transition
probability density functions associated to them are defined as solutions
of integrodifferential equations that must be solved numerically. So
that the use of jump models in practical situations is, in general,
cumbersome when compared to the use of the model proposed here. In order
to use the multiscale stochastic volatility model considered to compute
option prices we define its risk neutral measure and the associated
risk premium parameters. In [3] we derive a formula for the price
of the European vanilla (call and put) options in the multiscale stochastic
volatility model. The formulae obtained are given as one dimensional
integrals of explicitly known integrands and in a special case reduce
to the corresponding formulae in the Heston model
[7]. This last
formula in the case of a European vanilla call option improves the formula
given in Heston [7] formula (10) in a sense made clear
later. We use the option price formulae obtained to study the values
of the model parameters, of the correlation coefficients of the Wiener
processes defining the model and of the initial stochastic variances
implied by the ``observed" option prices using both synthetic and
real data. The real data analyzed are those relative to the S&P
500 index in the year 2005. That is we generalize to the model proposed
here the so called term structure analysis of the implied volatility
used in the Black Scholes model. This analysis is translated in the
solution of a constrained nonlinear optimization problem with an objective
function containing an integral that must be evaluated numerically.
The real data analysis presented shows that the multiscale stochastic
volatility model gives a satisfactory explanation of the observed option
prices including those of the out of the money options and that the
implied values of the model parameters, of the correlation coefficients
and of the initial stochastic variances can be used to obtain high quality
forecasts of the option prices. This website contains some auxiliary
material including some animations that helps the understanding of [3]. 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
We present a multiscale stochastic volatility model in mathematical finance. Multiscale modeling is used in a variety of scientific and technical fields and there are many problems that can be described satisfactorily within a multiscale framework (such as, for example, problems in turbulent flow, porous media, weather forecasting, material science, wave propagation, microstructure simulation and biological cell behaviour). However only recently multiscale models have attracted the attention of the mathematical finance community. In [3] we consider the use of multiscale models in the study of the stochastic volatility associated to an asset price. The idea of using stochastic volatility models to describe the dynamics of an asset price derives from the empirical evidence that the asset price dynamics is driven by processes with non constant volatility. In fact when the constant volatility Black and Scholes model [1] is used to interpret financial data two phenomena are observed: (i) the volatility ��smile�� that appears in the volatilities implied by the observed option prices, (ii) the presence of skewness and kurtosis in the asset price probability density function deduced from the empirical data. These phenomena show that the Black Scholes model fails to describe correctly the market behaviour. Several alternatives have been proposed to improve the market description, among them we consider specifically the stochastic volatility models. In these models the stochastic volatility or the stochastic variance of the asset price is defined as the solution of a stochastic differential equation that is coupled with the asset price stochastic differential equation. Models of this type are, for example, the Heston model [7], that has received special attention since in several circumstances it is able to describe satisfactorily the price dynamics of relevant assets. Furthermore the Heston stochastic volatility model is an explicitly solvable model in the sense that in the Heston model the joint probability density function associated to the asset price and to its stochastic variance can be written as a one dimensional integral of an explicitly known integrand. Heston in [7] derived a formula for the price of a European vanilla call option in the Heston model given by a one dimensional integral. Later Lipton [8] pagg. 602605 derived for the Heston model a representation formula for the joint probability density function of the asset price and of the associated stochastic variance given by a one dimensional oscillatory integral of an explicitly known integrand and he used this formula to price forward starting options. Using the ideas presented in [8] pagg. 602605 we have derived in [4], in the Heston model framework, an integral representation formula for the prices of European vanilla options that involves only a one dimensional integral of an explicitly known integrand. In a sense explained in [3] and summarized here at the end of this Section the formulae derived in [9] and refined in [4], are different from that derived in [7] and are an improvement of the formula derived in [7].
In the construction of the multiscale stochastic volatility model proposed we start from the fact that several empirical studies of real data have shown that the term structure of the implied volatility of the price of several underlyings (such as, for example, market indices, commodities,....) seems to be driven by two different factors: one fluctuating on a fast time scale and the other fluctuating on a long time scale. Moreover it has been observed that the financial derivatives with long residual life are not priced satisfactorily by one factor stochastic volatility models such as the Heston model. That is there is empirical evidence of the fact that in the circumstances mentioned above one factor stochastic volatility models are inadequate to fully capture the volatility smile and the volatility dynamics. In the last two decades several authors have developed models able to deal satisfactorily with at least one of the market features mentioned previously. In particular for this purpose two different classes of models have been developed, that is: the multiscale stochastic volatility models (see for example [6], [3]), and the jump models (see for example [2]). Usually the first kind of models are used to price financial derivatives having medium or long residual life while the second kind of models are used to price derivatives with short residual life on underlyings whose prices have spikes. In particular we mention the jump models containing exponential L��vy processes, these models have been widely used to describe market fluctuations and to price derivatives (see for example [2]). In general when exponential L��vy processes are used option prices are obtained as solutions of suitable integrodifferential equations and ad hoc numerical methods must be used to compute them (see [2], Chapter 12). However in some one factor L��vy process model the characteristic function of the probability density function of the logreturn of the underlying asset price can be expressed as an elementary function and the corresponding European vanilla option prices can be computed as a double integral having as integration variables the logreturn of the underlying asset price and its conjugate variable in the Fourier transform.
Since empirical studies (see [3] and the reference therein) have shown that the volatility dynamics of the price of many important assets is governed by two factors, a highly persistent factor and a quickly mean reverting factor, it is natural to believe that in these circumstances multiscale stochastic volatility models can describe satisfactorily the volatility dynamics. For these reasons we advocate the idea of using multiscale volatility models instead than jump models to describe the asset price dynamics.
We use multiscale stochastic volatility models in their simplest form, that is we use a two scale stochastic volatility model to study mediumlong time scale phenomena and fastmedium time scale phenomena. We propose a multiscale stochastic volatility model (i.e. a three factor stochastic volatility model) that generalizes the Heston model and that, under some hypotheses specified later, is explicitly solvable in the sense that it is possible to write a formula for its transition probability density function given by a one dimensional integral of an explicitly known integrand. Starting from this representation formula for the transition probability density function we derive two formulae given by one dimensional integrals of explicitly known integrands to price European vanilla (call and put) options on an underlying asset whose price dynamics is described by the multiscale stochastic volatility model.
Let us go into details. Let t be a real variable that denotes time, we consider the (vector valued real) stochastic process (x_{t},v_{1,t}, v_{2,t}), t0, solution of the following system of stochastic differential equations:
where the asset logreturn state variable x_{t} is related with the asset price S_{t} , t>0, via the following relation:
(4)
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 <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 quantities , , i=1,2 are real constants.
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 have =0. Later we will assume , >0, i=1,2 and when we use the multiscale model we will choose a_{i}=1/2, b_{i}=1, i=1,2.
We conclude
this Section noting that when we evaluate the option prices we use
the no arbitrage pricing theory, that is we compute the option price
as expected value of a discounted payoff with respect to an equivalent
martingale measure, known as riskneutral measure. The risk neutral
measure is a probability measure equivalent to the statistical measure
associated to the process (1), (2), (3) that guarantees that the discounted
value of the underlying is a martingale. When we assume that the volatility
is constant, the risk neutral measure is unique. In the case of one
factor stochastic volatility models, such as the Heston model, this
uniqueness does not hold anymore and we have a family of risk neutral
measures parameterized by a real parameter called risk premium parameter.
It is easy to see that in the case of the multiscale stochastic volatility
model (1), (2), (3) we have a family of risk neutral measures parameterized
by two risk premium parameters _{1} and _{2}. This family of measures
is given by the statistical measures associated to the following dynamical
systems:
equipped with
the initial conditions (5), (6), (7), where _{i}^{*}(_{i})
= _{i}+_{i}, ^{*}_{i}(_{i})
= _{i}_{i}/(_{i}+_{i}),
i = 1,2, where _{i}, _{i},
i=1,2 are the constants appearing in the stochastic differential system
(1), (2), (3) and the real parameters _{i}, i = 1,2 are the risk
premium parameters. We impose ^{*}_{i}(_{i})
0, ^{*}_{i}(_{i})
0, i =1,2, that is we impose _{i}
_{i},
i = 1,2.
In [3] we use the risk neutral measure defined through (8), (9), (10) to derive formulae involving one dimensional integrals of explicitly known integrands to price European call and put vanilla options. These formulae when we choose a_{1}=0, a_{2}=1/2, b_{1}=0, b_{2}=1 price European vanilla options under the Heston model. The formula derived in [3] for the price of a European vanilla call option under the Heston model improves the well known formula derived in [7] to price options under the Heston model (see formula (10) of [7]). In fact the formula derived in [7] does not generalize naturally the Black Scholes formula since it is not the expected value of the discounted payoff computed using the risk neutral measure induced by the fundamental solution of the Fokker Planck equation associated to the Heston dynamical system. This is due to the fact that the measure used by Heston in [7] is not a probability density in the two variables logreturn and stochastic variance at the initial time t=0 and, as a consequence, it is not a probability density function at time t, t0. The formula derived in [3] instead is truly analogous to the original Black Scholes formula in the context of the stochastic volatility models considered.
The calibration problem that we solve consists in determining the values of the model parameters, of the correlation coefficients and of the initial stochastic variances that give the best approximation of the option prices used as data. We consider the approximation in the least squares sense so that we reformulate the calibration problem as a nonlinear constrained least squares problem.
In the numerical experiments proposed here and in Section 4 we choose in equations (1), (2), (3) a_{1} = a_{2} = ^{1}/_{2} and b_{1} = b_{2} = 1. Let us denote with = (_{1},_{1},_{0,1},_{1},, ,_{1},_{2},_{2},_{0,2},_{2}, ,_{2}) the vector of the parameters of the multiscale model (including the risk premium parameters), of the correlation coefficients and of the initial stochastic variances and let m be a positive integer, we denote with R^{+} the price of the asset at time t, t 0, with C^{t}(,T_{i},K_{i}), i = 1,2,,m, the data, that is the prices at time t of the European vanilla call options having maturity time T_{i} and strike price K_{i}, i = 1,2,,m, and with C_{M}^{t,}^{}(,T_{i},K_{i}), i = 1,2,,m, the prices of the same European vanilla call options obtained using formula (59) of [3] that prices European vanilla call options in the multiscale stochastic volatility model considered taking as maturity time = T_{i}t, i = 1,2,,m, and as asset price S_{0} = . When necessary we denote with C_{H}^{t, }^{}(,T_{i},K_{i}), i = 1,2,,m, and with C_{B}^{t, }^{}(,T_{i},K_{i}), i = 1,2,,m, the prices of the same options obtained using the Heston formula, that is the formula deduced from the multiscale formula when we choose a_{1} = 0 , b_{1} = 0, a_{2} = 1/2 and b_{2} = 1, and the Black Scholes formula respectively.
Note that C_{H}^{t, }^{} and C_{B}^{t, }^{} depend only on some of the components of the vector . Please note that in the experiments presented in this website and in [3] we use as data only the prices of call options. More general sets of data will be considered elsewhere.
Let R^{13} be the 13dimensional
real Euclidean space and let M be the set of the admissible vectors ,
that is:
The constraints that define M translate some elementary properties of the model (1), (2), (3).
At time t, t 0, the calibration problem considered is translated into the following nonlinear least squares problem:

(12) 
where the objective function
L_{t}( ) is defined as follows:
When needed we solve problem (12) replacing C_{M}^{t,}^{} with C_{H}^{t,}^{} or with C_{B}^{t,}^{}. The formulation of the calibration problem through (11), (12), (13) is only one formulation among many other possible formulations. It is known that the objective function (13) may have flat regions where the objective function changes slightly in correspondence of significant changes of the independent variables. So that special attention must be devoted to the choice of the initial guess of the iterative procedure used to solve the minimization problem (12). In fact in order to choose the initial guess of the minimization procedure we first explore the feasible region M taking a set of (feasible) random points and evaluating the objective function on this set of points. From these points we select a subset of points to be used as initial guesses. We solve the optimization problem (12) using a projected steepest descent method. This method is an iterative procedure that, starting from an initial feasible vector ^{0}, generates a sequence {^{n}}, n = 0,1, of feasible vectors, ^{n} M, n = 0,1,, moving along a descent direction obtained via a suitable projection on the active constraints of minus the gradient with respect to of L_{t}. The optimization procedure stops when the vector ^{n} generated satisfies for the first time the following criterion:

(14) 
Let us present a numerical experiment using synthetic data (i.e.: synthetic option prices). This experiment shows that the formulation of the calibration problem (11), (12), (13) is able to determine the model used to generate the synthetic option prices. We generate three sets of synthetic data, that is three sets of synthetic option prices. The first set is obtained generating the option prices with the Black Scholes formula choosing the volatility equal to and the risk free interest rate r equal to 0.03, the second one is obtained generating the option prices with the Heston model formula obtained using the formula derived in [4], [3] and choosing the parameters a_{1} = 0, b_{1} = 0, a_{2} =0.5, b_{2} = 1 and , _{2} = 0.5, _{2} = 9.9, _{2} = 0.06, _{2} = 0.4, _{2} =0.1, _{2,0} = 0.5. The third one is obtained generating the option prices using the multiscale model formula contained in [3] choosing , _{1} = 0.8, _{2} = 0.5, _{1} = 1.5, _{2} = 9.9, _{1} = 0.04, _{2} = 0.06, _{1} = 0.05, _{2} = 0.1, _{1} = 0.4, _{2} = 0.2, _{1,0} = 0.8, _{2,0} = 0.5. In the generation of the Heston and of the multiscale option prices we choose the risk free interest rate . The synthetic data generated with the Black Scholes formula, with the Heston formula and with the multiscale formula have been perturbed with an additive stochastic noise uniformly distributed of magnitude in absolute value at most equal to 1% of the absolute value of the datum perturbed.
We calibrate the multiscale model solving the constrained least squares optimization problem (12) with the variable metric steepest descent procedure announced previously choosing in (14) e_{tol} = 0.03 and n_{max} = 10000.
Figures 1, 2, 3 show the trajectories of the slow varying variance v_{1,t}, t0, and of the fast varying variance v_{2,t}, t0, obtained calibrating the model on the previous three sets of data and the ��true�� trajectories of the stochastic variances v_{1,t}, v_{2,t}, t0, obtained integrating the Black Scholes model, the Heston model and the multiscale model with the ��true�� parameters, correlation coefficients and stochastic initial variances. Note that the ��true�� trajectories of the stochastic variances together with the corresponding ��true�� trajectories of the asset price are the trajectories used to generate the synthetic data and that when we use the asset price at time t>0 in (13) the quantities coming from the solution of the calibration problem (11), (12), (13) must be interpreted as the stochastic variances at time t>0.
Figures 1,
2 show the results obtained using as data the option prices generated
by the Black Scholes and the Heston models respectively. These Figures
show that the paths reconstructed with the calibration procedure of
the stochastic variance v_{1,t}, t0, are substantially constant and that
the estimated values of v_{1,t}, t0, are small. While the paths obtained
solving the calibration problem using the third set of data, that is
the data generated using the multiscale model, show that the calibration
procedure works correctly (Figure 3). Furthermore in the case of the
first set of data that is the data generated using the Black Scholes
model (Figure 1) the reconstructed stochastic variance v_{2,t},
t0,
is substantially constant. This is in agreement with the well known
properties of the Black Scholes model.
Figure
1: Trajectories of the stochastic variances v_{1,t} (red
dotted line corresponding to a reconstructed _{1} approximately equal to
9.96·10^{4}), v_{2,t} (blue dotted line corresponding
to a reconstructed _{2} approximately equal to
7.71) obtained solving the calibration problem using as data the option
prices generated with the Black Scholes model and trajectory of the
``true" Black Scholes variance (green line) versus time t
Figure
2: Trajectories of the stochastic variances v_{1,t} (red
dotted line corresponding to a reconstructed _{1} approximately equal to
7.93·10^{3}), v_{2,t} (blue dotted line corresponding
to a reconstructed _{2} approximately equal to
7.68) obtained solving the calibration problem using as data the option
prices generated with the Heston model and trajectory of the ``true"
Heston variance (green line) versus time t
Figure
3: Trajectories of the stochastic variances v_{1,t} (red
dotted line corresponding to a reconstructed _{1} approximately equal to
8.84·10^{1}), v_{2,t} (blue dotted line corresponding
to a reconstructed _{2} approximately equal to
5.5) obtained solving the calibration problem using as data the option
prices generated with the multiscale model and trajectory of the ``true"
stochastic variances of the multiscale model (green lines) versus time
t
For
the three models considered (Black Scholes, Heston, multiscale [3]) the following movies show a sample
trajectory of the logreturn x_{t} obtained using the
vector obtained solving the calibration problem (12) and the same sample
trajectory obtained using the ��true�� vector as function
of time. This ��true�� vector is the one used to generate the data
of the calibration problem.
Black Scholes model  Click here to see a digital movie 
Heston model  Click here to see a digital movie 
Multiscale model  Click here to see a digital movie 
In this Section we propose some experiments that use real data. We consider three models that is the Black Scholes model, the Heston model and the multiscale model (1),(2),(3). We study the values of the model parameters (including the risk premium parameters), of the correlation coefficients and of the initial stochastic variances implied by the observed prices (available to us) of the European vanilla call options on the S&P 500 index of the New York Stock Exchange in the year 2005 and we show that the multiscale stochastic volatility model captures satisfactorily the ``smile" effect. To avoid excessive numerical work we analyze the market data (i.e. S&P 500 index and the corresponding option prices) only in the months of January, June and November, 2005. The results presented relative to these months are representative of the results obtainable from an exhaustive analysis of the 2005 data. We have solved the calibration problems relative to the Black Scholes model, to the Heston model and to the multiscale model, using the call option prices of January 3, 2005, or of June 3, 2005 and or of November 3, 2005. The formulation of the calibration problem for the Black and Scholes model and for the Heston model is analogous to the formulation (11), (12), (13) used for the multiscale model. Note that for each model (Black Scholes, Heston, multiscale) we have solved the calibration problem three times (using as data the prices of January 3, June 3, November 3, 2005 respectively) and that since we use all the call option prices available the calibration procedure works simultaneously on out of money, at the money and in the money options. In the stopping criterion (14) of the optimization procedure we use e_{tol}=0.05 and n_{max} = 10000. We note that in some cases the optimization procedure stops because we have nn_{max}, in these cases the smallest value of e_{tol} reached is approximately 0.07.
Once solved
the calibration problems, we use the calibrated models to forecast the
option prices in future days using the Black Scholes formula or
the Heston formula or the multiscale model formula (see [3] for the explicit form of these formulae).
In these formulae the asset price value used is the asset price value
actually observed in the day of the forecast. We forecast the stochastic
variances v_{1,t}, v_{2,t}, t0, using the mean values _{1,t}_{}_{ }_{}, _{
2,t}_{}_{ }_{}, tt_{0}, conditioned to the estimates
made at time t =t_{0} of the random variables v_{1,t},
v_{2,t}, tt_{0}, that we denote with
F_{0} (i.e.:F_{0} = (_{1,0}, _{
2,0})) (see [3], [4] for further details), that is using
the formulae:
where E( · ) is the expected value of · , and , i = 1,2 are the initial stochastic variances at time t=t_{0} obtained solving the calibration problem. Note that in the analysis of the real data t=t_{0} corresponds to January 3, or to June 3, or to November 3, 2005, and that we can translate the origin of the time axis to t_{0}.
Let n_{obs}
be a positive integer that is the number of call option prices at time
t available in a given day, and let C^{t}(,T_{i},K_{i}),
C_{B}^{t, }^{}(,T_{i},K_{i}),
C_{H}^{t, }^{}(,T_{i},K_{i}), C_{M}^{t, }^{}(,T_{i},K_{i})
denote respectively the prices of European call options with strike
price K_{i} and maturity time T_{i} (that corresponds
to time to maturity _{i} = T_{i}t) observed,
generated by Black Scholes formula, generated by Heston formula
and generated by multiscale formula, i=1,2,��, n_{obs}. The
following figures (i.e. Figures 4, 5, ��,12) show the absolute error:
obtained using the three models (L = B(Black Scholes), H(Heston), M(Multiscale)) as a function of the moneyness K/S_{0} = K_{i}/S_{0}, i = 1,2,, n_{obs}, when S_{0} = is the value of the S&P 500 index at the transaction day t considered and as a function of the time to maturity _{i} = T_{i}t, i = 1,2,, n_{obs}.
We use the parameters estimated on January 3, 2005 (n_{obs} = 281) to forecast the option prices on January 7 (n_{obs}=282), January 14 (n_{obs}=283), January 28 (n_{obs}=258), 2005 (see Figures 4, 5, 6). We use the parameters estimated on June 3, 2005 (n_{obs} = 281) to forecast the option prices on June 7 (n_{obs}= 288), June 14 (n_{obs} = 291), June 28 (n_{obs} = 278), 2005 (see Figures 7, 8, 9). We use the parameters estimated on November 3, 2005 (n_{obs} = 303) to forecast the option prices on November 7 (n_{obs} = 303), November 14 (n_{obs} = 305), November 28 (n_{obs} = 292), 2005 (see Figures 10, 11, 12)
Figures 4, 5,...,12 and the corresponding movies show the behaviour of the absolute error committed forecasting the values of the option prices. We note that the bigger the time to maturity of the option considered is, the larger is the difference among the behaviour of the three models and the advantage of using the multiscale model when compared to the use of the Heston model or of the Black Scholes model.
The numerical experiment presented shows that the solutions of the calibration problem obtained using the data of January 3, 2005, June 3, 2005 and November 3, 2005 work well when used to forecast the prices of the options in January 7, 14, 28, June 7, 14, 28 and November 7, 14, 28, 2005 respectively.
Figure 4: January 7, 2005: Absolute error committed on the observed call option prices using the prices forecasted by the Black Scholes model, the Heston model and the multiscale model versus moneyness K/S_{0} 
Click here to see the corresponding digital movie 
Figure 5: January 14, 2005: Absolute error committed on the observed call option prices using the prices forecasted by the Black Scholes model, the Heston model and the multiscale model versus moneyness K/S_{0} 
Click here to see the corresponding digital movie 
Figure 6: January 28, 2005: Absolute error committed on the observed call option prices using the prices forecasted by the Black Scholes model, the Heston model and the multiscale model versus moneyness K/S_{0} 
Click here to see the corresponding digital movie 
Figure 7: June 7, 2005: Absolute error committed on the observed call option prices using the prices forecasted by the Black Scholes model, the Heston model and the multiscale model versus moneyness K/S_{0} 
Click here to see the corresponding digital movie 
Figure 8: June 14, 2005: Absolute error committed on the observed call option prices using the prices forecasted by the Black Scholes model, the Heston model and the multiscale model versus moneyness K/S_{0} 
Click here to see the corresponding digital movie 
Figure 9: June 28, 2005: Absolute error committed on the observed call option prices using the prices forecasted by the Black Scholes model, the Heston model and the multiscale model versus moneyness K/S_{0} 
Click here to see the corresponding digital movie 
Figure 10: November 7, 2005: Absolute error committed on the observed call option prices using the prices forecasted by the Black Scholes model, the Heston model and the multiscale model versus moneyness K/S_{0} 
Click here to see the corresponding digital movie 
Figure 11: November 14, 2005: Absolute error committed on the observed call option prices using the prices forecasted by the Black Scholes model, the Heston model and the multiscale model versus moneyness K/S_{0} 
Click here to see the corresponding digital movie 
Figure 12: November 28, 2005: Absolute error committed on the observed call option prices using the prices forecasted by the Black Scholes model, the Heston model and the multiscale model versus moneyness K/S_{0} 
Click here to see the corresponding digital movie 
5.
References
[1]
F. Black, M. Scholes, ��The pricing of options and corporate liabilities��,
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[2]
R. Cont, P. Tankov, Financial modelling with jump processes,
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[3]
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[4]
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[5]
L. Fatone, F. Mariani, M.C. Recchioni, F. Zirilli: ��Maximum likelihood
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equations that models the returns of the index of some classes of hedge
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329362, (http://www.econ.univpm.it/recchioni/finance/w5 contains downloadable software).
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Entry n.
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