The Use of Statistical Tests to Calibrate the Black-Scholes Asset Dynamics Model Applied to Pricing Options with Uncertain Volatility

1 Dipartimento di Matematica e Informatica, Università di Camerino, Via Madonna delle Carceri 9, 62032 Camerino, Italy 2 CERI-Centro di Ricerca “Previsione, Prevenzione e Controllo dei Rischi Geologici”, Università di Roma “La Sapienza”, Palazzo Doria Pamphilj, Piazza Umberto Pilozzi 9, Valmontone 00038 Roma, Italy 3 Dipartimento di Scienze Sociali “D. Serrani”, Università Politecnica delle Marche, Piazza Martelli 8, 60121 Ancona, Italy 4 Dipartimento di Matematica “G. Castelnuovo”, Università di Roma “La Sapienza”, Piazzale Aldo Moro 2, 00185 Roma, Italy


Introduction
The Black-Scholes formulae 1 used to price European call and put options are based on an asset price dynamics model.This model is a stochastic dynamical system that may be written as a stochastic differential equation.The solution to this model takes the form of a stochastic process called geometric Brownian motion.The model contains two real parameters: volatility and drift.The volatility and drift parameter values are necessary to apply the model to asset and option price forecasting; therefore, in practice, the values of these parameters must be determined prior to using the Black-Scholes dynamics model and the option pricing formulae derived from it.The problem of estimating the asset price dynamics model parameters must be considered based on the available data.This problem is a calibration problem and is an inverse problem for a stochastic dynamical system defined by a stochastic differential equation.The estimated parameter values obtained by solving the calibration problem may be used to forecast asset prices at future time points and to evaluate the option pricing formulae.The "accuracy" and reliability of the estimated parameter values determine the accuracy and reliability of the forecasted asset prices and the computed and/or forecasted option prices.
Note that in recent years, the validity of the asset price dynamics model proposed by Black and Scholes in 1973 1 has been disputed in the mathematical finance literature, and several other refined models, such as the Heston model 2 , have been introduced to describe asset price dynamics.Nevertheless, the Black-Scholes asset dynamics model and, particularly, the option pricing formulae derived from it remain widely used in financial market practice.We will see that the solution to the Black-Scholes model calibration problem, using statistical tests, and an investigation of effects on the option pricing problem, are easily derived using elementary mathematics.For this reason, the Black-Scholes asset dynamics model is the natural choice to begin our study of solutions to the problems associated with calibrating stochastic dynamical systems using statistical tests.The ideas introduced in this paper are rather general and are not limited to the study of the Black-Scholes model.
The data used in the calibration problem include observations of the asset prices over a finite set of known equispaced discrete time values.We show how elementary statistical tests i.e., the Student's t and the χ 2 tests may be used to estimate the drift and volatility parameters of the Black-Scholes model with statistical significance.Recall that the first step in formulating a hypothesis testing problem consists of defining the null hypothesis, H 0 , as well as an alternative hypothesis, H 1 .When the goal is to establish an assertion about a probability distribution parameter based on support from a data set, the assertion is usually taken to be the null hypothesis H 0 , and the negation of the assertion itself is taken to be the alternative hypothesis H 1 or vice versa .The null hypothesis H 0 considered later is the hypothesis that a parameter of a probability distribution belongs to a given interval.In a statistical test, we must additionally consider the statistical significance.The statistical significance α, 0 < α < 1, is the maximum probability of rejecting the null hypothesis H 0 when the hypothesis is true.Defining a type I error as the error associated with rejecting the null hypothesis H 0 when H 0 is true, the statistical significance α, 0 < α < 1, is the maximum probability of making a type I error.The result of the test is a decision to accept or reject the null hypothesis H 0 with a significance level α, 0 < α < 1.
Let us describe the content of the paper in greater detail.As mentioned, the solution to the Black-Scholes asset price dynamics model is a stochastic process called geometric Brownian motion, which depends on two parameters: the drift and the volatility.From this fact, it follows that the asset price at any given time may be modeled as a random variable with a log-normal distribution and, therefore, the log return of the asset price is normally distributed.It is easy to see that the asset price log return increments associated with observations of the asset price over a finite set of equispaced time values comprise a sequence of values sampled from a set of independent identically distributed Gaussian random variables.The mean and variance of these Gaussian random variables can be expressed using elementary formulae as a function of the drift and volatility parameters of the Black-Scholes asset price dynamics model and of the time interval between observations.Starting from a sample of observations of the log return increments over a finite set of equispaced known time values, elementary statistical tests i.e., the Student's t and χ 2 tests may be used to estimate, with a given statistical significance, the mean and variance of the Gaussian random variables associated with the log return increments.In fact, the Student's t-test 3, 4 is the test used to establish, with statistical significance, whether the mean of a normally distributed population of independent samples has a certain value or belongs to an interval specified in a "null" hypothesis.Similarly, the χ 2 test 4, 5 is the test used to establish, with statistical significance, whether the variance of a normally distributed population of independent samples has a value or belongs to an interval specified in a "null" hypothesis.Knowledge of the Gaussian random variable parameters associated with the log return increments derived from the statistical tests permits recovery of the corresponding parameters of the Black-Scholes model and the associated statistical significances.
The use of statistical tests to solve the problem associated with calibrating stochastic dynamical systems, such as the Black-Scholes model, is an interesting approach to inverse problems used elsewhere in mathematical finance as well as in application contexts other than mathematical finance.The significance levels obtained for the parameter values intervals in the Black-Scholes model are relevant in many practical situations.In Section 3, we consider the problem associated with pricing an option with an uncertain volatility.This problem has been considered by several authors in the scientific literature.For simplicity, we refer the reader to 6, 7 , in which the uncertain volatility in the Black-Scholes framework is assumed to belong to a known interval and the corresponding price intervals for the European vanilla option prices may be determined using the Black-Scholes-Barenblatt equations.Thanks to our methodology, statistical significance levels may be attributed to the option price intervals determined using the Black-Scholes-Barenblatt equations.
Finally, the approach used to calibrate the Black-Scholes model is applied to the study of synthetic and real data.The synthetic data considered were generated by numerically integrating the stochastic differential equation that defines the asset price dynamics in the Black-Scholes model, for several choices of the parameter values.The real data studied were the time series of the daily closing values of the S&P500 index and the associated European vanilla option prices during the year 2005.The numerical results obtained were computationally simple and statistically convincing.
In mathematical finance, the problem associated with estimating the volatility of asset prices, starting from a time series of observed data asset and/or option prices , has received significant attention.The methods most commonly used in the literature include the implied and historical or realized volatility methods.The approach proposed here for solving this problem is distinct.Unlike the implied volatility method, we do not consider the volatility implied by the option prices observed in the market.That is, we do not estimate the volatility parameter of the asset price using the prices of various options to the asset with different strikes and expiration dates.By contrast, our method estimates the volatility based on the asset prices observed over a finite set of discrete time values, similar to the approach used in the historical volatility method.Our method improves on the historical volatility method by associating a significance level to the volatility estimate.In some sense, the transition from the historical volatility method to the method proposed here corresponds to a transition from the use of a volatility estimate for "sample volatility" to the use of a statistical test i.e., the χ 2 test tailored to the random variable which depends on the parameter to be estimated.
Roughly speaking, the transition is from a method of descriptive statistics to a method of mathematical statistics.
The work presented here has several valuable features that are worth noting.First, the idea proposed here is very simple.Although the approach is introduced in the context of the Black-Scholes model, it can be applied to other stochastic dynamical systems used in mathematical finance to describe asset price dynamics, for example, the Heston model see 2, 8, 9 , or some of its variants, such as the models introduced in 10-13 to study specific problems.In 8-13 , the models were calibrated using the "implied volatility method" or the maximum likelihood method.These studies did not implement statistical tests to solve the calibration problems or estimate the parameters; therefore, no statistical significance could be associated with the parameter estimates.Note that in calibrating these models or generic stochastic dynamical system models in general, no elementary statistical tests such as the Student's t or χ 2 tests could be used.Extending the method suggested here to studies of more general dynamical systems will depend on the development of new ad hoc statistical tests.The application of these new statistical tests to a sample data will most likely require substantial use of numerical methods.The website http://www.econ.univpm.it/recchioni/finance/w11/makes available auxiliary material, including animations, to assist the reader in understanding the discussion present here.References to the authors' more general studies in the field of mathematical finance are available at the website http://www.econ.univpm.it/recchioni/finance/.
The remainder of the paper is organized as follows.Section 2 formulates and solves the calibration problem for the Black-Scholes model.Section 3 addresses the problem of pricing options with statistical significance when the volatility value is uncertain and lies in a specified range.Section 4 tests the proposed method on time series data, and the results obtained from studies of the synthetic and real data are discussed.Section 5 presents some concluding remarks and a few possible extensions of this work.

The Calibration Problem for the Black-Scholes Model
Let S t > 0 denote the asset price at time t ≥ 0. The Black-Scholes model 1 assumes that S t , t > 0, is a stochastic process, the dynamics of which are governed by the following stochastic differential equation: with the initial condition: where μ and σ are real parameters, μ is the drift, σ > 0 is the volatility, W t , t > 0, is a standard Wiener process, W 0 0, dW t , t > 0 is its stochastic differential, and the initial condition S 0 > 0 is a given random variable.For simplicity, we assume that S 0 is a random variable concentrated at a point with probability one.Abusing the notation, we denote this point as S 0 > 0. The parameters μ and σ are the unknowns of the calibration problem.
The stochastic differential 2.1 defines the so-called geometric Brownian motion.In fact, 2.1 can be rewritten as dS t S t μdt σdW t , t > 0.

2.3
Let ln • denote the logarithm of • for t ≥ 0. The quantity G t ln S t / S 0 is the log return at time t of the asset with a price S t .Using 2.3 and It ô's Lemma see 14 , it follows that the process G t , t > 0 satisfies the following stochastic differential equation: with initial conditions that follow from 2.2 , that is, G 0 0. Equation 2.4 implies that G t ln S t / S 0 , t > 0, is a generalized Wiener process with a constant drift μ − σ 2 /2 and a constant volatility σ > 0. Therefore, for t ≥ 0, τ > 0, the increment in G t ln S t / S 0 occurring between time t and time t τ is a Gaussian random variable with mean μ − σ 2 /2 τ and variance σ 2 τ.That is where, for M and V real constants, V / 0, N M, V 2 denotes the Gaussian distribution with mean M and variance V 2 .From 2.5 , it follows that the Black-Scholes asset price log return increments associated with a discrete set of equispaced time values form a sequence of independent identically distributed Gaussian random variables.Let Δt > 0 be a time increment, and let t i iΔt, i 0, 1, . . ., n, be a discrete set of equispaced time values that later will be chosen as the set of observation times.We define X t i , the asset price log return increment as t increases from t i−1 to t i , as It is easy to see that the random variables X t i , i 1, 2, . . ., n are independent identically distributed Gaussian random variables with mean M and variance V 2 , where We then have Δt, σ 2 Δt , i 1, 2, . . ., n.

2.8
Consider the following calibration problem: given a time increment Δt > 0, a statistical significance level α, 0 < α < 1, and the asset price S i observed at time t t i iΔt, i 0, 1, . . ., n, determine two intervals in which the parameters μ and σ of the model 2.1 , respectively, fall with the given significance level α.
Note that in order to apply the calibration problem solution to the problem of pricing options with uncertain volatility see Section 3 , point estimates of the tested parameters are not useful.Instead, an interval of variability for the tested parameter i.e., with a given significance level α is necessary.Studies of the problem presented in Section 3, in particular, require the identification of an interval of variability for the volatility σ.That is, we would like to determine the interval within which the parameter σ may be found, and to this interval we would like to associate a significance level α.The option prices in the Black-Scholes model are independent of the drift parameter μ and depend on the risk-free interest rate.Usually, the risk-free interest rate is known for a given problem; however, in the numerical experiments presented in Section 4, a risk-free interest rate was selected, depending on the interval of variability of μ determined in the calibration problem.For more details see Section 4.
The observations S i of the asset price at time t t i , i 0, 1, . . ., n, are nonnegative real numbers that are assumed to be unaffected by errors.These n 1 observations of the asset price and the corresponding time values are the data used in the calibration problem.The corresponding observed log return increments x i ln S i / S i−1 , i 1, 2, . . ., n, are a sample of n observations taken, respectively, from the random variables X t i , i 1, 2, . . ., n, that is, taken from a set of independent identically distributed Gaussian random variables.Using this sample data, the Student's t-test and χ 2 test see 3-5 , we identified two intervals over which the mean M and variance V 2 of the Gaussian random variables were determined to within a given significance level α.From knowledge of the intervals determined for M and V 2 , the corresponding intervals for μ and σ could be recovered by inverting the relations 2.7 .
In greater detail, given a significance level α, 0 < α < 1, we can perform statistical tests on the variance V 2 and on the mean M of the random variables X t i ln S t i /S t i−1 , i 1, 2, . . ., n starting from the sample data x i ln S i / S i−1 , i 1, 2, . . ., n, using the χ 2 and Student's t tests, respectively.This implies that for a given α, 0 < α < 1, and the relations 2.7 , we can accept or reject, with a significance level α, the following hypotheses regarding the parameters of the Black-Scholes asset price dynamics model: 2.9 and where and the quantities that define the corresponding hypotheses on M and V , respectively.This is performed simply by translating the results on V 2 and M obtained using the statistical tests to σ and μ.We can proceed as follows.Given a sample data, 12 with M and V 2 defined in 2.7 , it is possible first to estimate V 2 and, therefore, σ 2 using the χ 2 test along with 2.11 , and, subsequently, to estimate M and, therefore, μ, thanks to knowledge of σ 2 acquired using the χ 2 test using the Student's t-test along with 2.11 .
It should be noted that in many circumstances, it is more practical to try to determine an interval of variability for the drift μ and volatility σ parameters in the Black-Scholes model than to try to determine their "exact" values.
To estimate V 2 from the log return increments 2.12 , we must choose the hypotheses that we would like to test.This can be done in many ways.The analysis of data time series in Section 4 uses the following procedure.
Procedure 1.Given the sample data x i , i 1, 2, . . ., n, fix a statistical significance level α, 0 < α < 1. Choose a sufficiently large interval and apply the χ 2 test to test the hypothesis V 2 ∈ I 0 i , i 1, 2, . . ., m.We restrict our attention to the subinterval s which the composite hypothesis: 13 may be accepted with a significance level α, 0 < α < 1.If the hypothesis 2.13 cannot be accepted over any subintervals I 0 i * , the choice of I 0 and/or m may be changed.If the subinterval I 0 i * is unique, we set I 1 a 1 , b 1 I 0 i * , otherwise, we set I 1 a 1 , b 1 equal to the union of the intervals over which 2.13 is accepted with a significance level α.In both cases, the procedure described above is repeated, in the first case with the division of I 1 , and in the second case, with the shrinking of I 1 .In this way, a sequence of subintervals I k a k , b k , k 1, 2, . . . is constructed such that the hypothesis: is accepted with a significance level α, 0 < α < 1.This procedure stops when b k − a k < tol, where tol is a given tolerance.We take the last set constructed using this procedure, over which the formulated hypothesis may be accepted, as a final estimate of an interval within which V 2 falls with a significance level α.
Procedure 2. A similar procedure is used to estimate an interval within which M falls with a significance level α, based on the data sample x i , i 1, 2, . . ., n, using the Student's t test.
The synthetic and real data are analyzed in Section 4 using Procedures 1 and 2 to identify intervals within which the parameters σ and μ fall with a significance level α.
Let us briefly discuss the design of Procedures 1 and 2. We focus on the motivations for the design of Procedure 1, which are not substantially different from the motivations for the design of Procedure 2. In Procedure 1, the statistical tests performed during the calibration process were chosen as hypotheses with the goal of balancing two needs.On the one hand, a careful and conservative procedure is desirable, which means that a sufficiently large interval surrounding the volatility σ must be specified in the null hypothesis.On the other hand, the interval surrounding the volatility σ should be small enough to provide a meaningful range of variability around the corresponding option prices.Too large an interval of variability around the volatility σ generates a large interval of variability around the corresponding option prices, making the information about the option prices useless.To satisfy these contrasting needs, we initially select a large interval for the volatility σ, such that we trust that the estimated parameter lies within this interval.All scenarios, even the most extreme cases, may be considered systematically based in this choice.We then use an ad hoc procedure i.e., Procedure 1 to gradually reduce the size of the chosen interval based on the idea that if a given hypothesis has already been accepted, it should be possible to refine it.The suggested procedures are iterative.At each step, the null hypothesis H 0 of the considered test is modified in light of what has been discovered in the preceding steps.This leads to an iterative hypothesis testing procedure.Note that the only purpose underlying this iterative approach is to identify a satisfactory formulation of the test's null hypothesis.The sample data used in the tests do not vary with the iterative procedure.Type I errors that occur during each test are calculated as if the other tests were not performed.This iterative procedure is conceptually different from a multiple testing procedure 15 and from an analytic induction procedure 16 .In fact, in a multiple testing procedure see 15 , a set of statistical inferences is considered simultaneously i.e., each test has his own sample data , and the type I errors increase as the number of comparisons increases, unless the tests are perfectly dependent.In an analytic induction procedure see 16 , the type I errors decrease because the size of the sample data set considered increases during the induction procedure.
An alternative approach to determining the range within which the volatility, σ, varies involves using an elementary method based on elementary statistics.For example, the sample data is used to obtain a point estimate of the parameter σ i.e., compute the volatility of the sample data , then a confidence interval is constructed around this point estimate to quantify the uncertainty.Similarly, elementary statistics may be used to determine the range of variability of the drift μ.
For the purposes pursued here, the method described in Procedures 1 and 2 is preferable to methods based on elementary statistics.Elementary statistics methods risk yielding dubious results in the event that a low-quality point estimate is selected.Also, the construction of large intervals of variability for the volatility can spoil any predictions regarding the corresponding option prices.
Finally, we note that in this paper, the idea of calibrating stochastic dynamical systems using statistical tests is suggested and implemented in the context of the Black-Scholes model.Procedures 1 and 2 are only heuristic procedures for selecting test hypotheses.Many other methods are available for estimating the model parameters using statistical tests that are plausible, such as methods based on elementary statistics, as mentioned previously.

Option Prices with Uncertain Volatility and Statistical Significance
Consider the Black-Scholes asset price dynamics model 2.1 and the problem of pricing options with uncertain volatility in the Black-Scholes model with a given statistical significance α, 0 < α < 1.We assume that the volatility value σ is not known exactly, but it is known that the volatility lies within a specified range, say, σ 1 ≤ σ ≤ σ 2 , where σ 1 and σ 2 are constants and 0 < σ 1 < σ 2 , with a given significance level α.The χ 2 test may be applied to the data used in the calibration problem, as described in Section 2, to decide whether to accept or reject the statement: with significance level α. 3.1 We limit our attention to European vanilla call and put options.
The question that we want to answer is given a significance level α, 0 < α < 1, and assuming that the hypothesis H 0 : σ 1 ≤ σ ≤ σ 2 is accepted with a significance level α, determine the corresponding range within which the value of a European vanilla option lies with a significance level α .
The answer to this question follows from the work of Avellaneda et al. 6 and of Lyons 7 .These authors proposed a method for estimating price options in the Black-Scholes model if the volatility σ is not known exactly but it is known that: 3.2 In 6, 7 , significance levels were not considered.We remark that the bounds σ 1 and σ 2 in 3.1 and 3.2 were determined in different ways.In 3.1 , these bounds correspond to parameters that can be chosen together with a significance level in such a way that the χ 2 test will accept the hypothesis H 0 : σ 1 ≤ σ ≤ σ 2 over the sample data considered with a significance level α.Alternatively, these bounds are parameters determined from the sample data through an iterative procedure, such as Procedure 1 described in Section 2. In 3.2 , these bounds are assigned using common sense assumptions, or they are determined either by looking at extreme values of the volatility implied by the observed option prices or by looking at the low and high peak values of the historical volatility.
Let r be the risk-free interest rate, t be the time variable, S be the asset price, T > 0, g S , S > 0, be the expiration date, and the pay-off function of the option to be priced.References 6, 7 showed that in the Black-Scholes framework, when the volatility satisfies 3.2 , there exists an interval V 1 , V 2 , depending on S and t, such that the price V V S, t , S > 0, 0 < t ≤ T, of the option lies within this interval.That is, 3.2 implies that

3.3
The worst case option value V 1 S, t , S > 0, 0 < t ≤ T satisfies the following nonlinear partial differential equation: where

3.7
Similarly, the best-case option value V 2 S, t , S > 0, 0 < t ≤ T satisfies the following nonlinear partial differential equation: where

3.11
For example, for a European vanilla call option, we have g S max S−K, 0 , S > 0, and for a European vanilla put option, we have g S max K − S, 0 , S > 0, where K is the strike price of the option.To ensure the existence of a unique solution for the European vanilla call option, the following boundary conditions must be added to 3.4 and 3.5 and to 3.8 and 3.9 : V S, t −→ 0 as S −→ 0, 0 < t < T, 3.12 V S, t ∼ S − Ke −r T−t as S −→ ∞, 0 < t < T.

3.13
Similar boundary conditions must be added to 3.4 and 3.5 and to 3.8 and 3.9 when considering a European vanilla put option.Equations 3.4 and 3.8 are known as the Black-Scholes-Barenblatt equations, and they reduce to the usual Black-Scholes equation in the absence of uncertainty about the volatility value i.e., in the case of σ 1 σ 2 .For a general pay-off function, these equations do not have a closed-form solution and must be solved numerically; however, the properties of the Black-Scholes-Barenblatt 3.4 and 3.8 or, more generally, of the maximum principle of parabolic partial differential equations, ensure that when a call or a put option is considered, the convexity of the corresponding payoff functions g S , S > 0 imply that the functions ∂ 2 V 1 /∂S 2 and ∂ 2 V 2 /∂S 2 do not change sign for S > 0, 0 < t < T. That is, when S > 0, 0 < t < T, the functions ∂ 2 V 1 /∂S 2 and ∂ 2 V 2 /∂S 2 retain the value of their sign at t T; therefore, in the context of a call or a put option, 3.4 and 3.8 reduce to the Black-Scholes equation.Note that when t T, we have V i S, T g S , S > 0, i 1, 2, and in the context of the payoff functions g of the call and put European vanilla options, the corresponding ∂ 2 g/∂S 2 are Dirac delta functions, which requires that the sign of ∂ 2 g/∂S 2 must be interpreted in the context of the distributions.The Black-Scholes equation is linear.Under simple final conditions, such as the conditions relevant to the call and put options, this equation can be solved explicitly to yield the Black-Scholes formulae.
For example, consider the situation in which the worst-case option value V 1 is the solution to the following problem relating to a European call option: with boundary conditions 3.12 and 3.13 .Similarly, the best-case call option value V 2 satisfies V 2 S, T max S − K, 0 , S > 0, 3.17 with boundary conditions 3.12 and 3.13 .The explicit solutions of 3.14 and 3.15 and of 3.16 and 3.17 , in the context of the boundary conditions 3.12 and 3.13 are respectively 1 : where

3.22
A similar analysis can be used to determine the worst-value and best-value European vanilla put options.Note that the formulae 3.18 and 3.19 are the Black-Scholes formulae.
We are now in a position to address the question posed at the beginning of this Section.First, let us assume that a "true" value of the volatility σ exists, even if it is unknown.In the Black-Scholes model, the price V of an option is a monotonically increasing function of the volatility σ.We can therefore conclude that when 3.1 holds, we have: where V 1 S, t , V 2 S, t , S > 0, 0 < t < T, are the solutions to the appropriate Black-Scholes-Barenblatt equations.Note that if we consider European vanilla call and put options V 1 S, t , V 2 S, t , S > 0, 0 < t < T, can be determined explicitly see formulae 3.18 and 3.19 , and similar formulae that can be deduced for put options .The meaning of 3.23 can be restated as follows: if we assume that a "true" value of the volatility σ exists and that an analysis of sample data permits us to accept the hypothesis that this true value lies in the range σ 1 , σ 2 with a significance level α, that is, σ 1 ≤ σ ≤ σ 2 with probability 1 − α, it follows that the corresponding "true" value of the option price V lies in the range V 1 , V 2 with probability 1 − α, that is, "V 1 S, t ≤ V S, t ≤ V 2 S, t , S > 0, 0 < t < T, with significance level α", where V 1 and V 2 are the solutions to the appropriate Black-Scholes-Barenblatt equations.

Numerical Experiments
Several numerical experiments were conducted to demonstrate the principles described above.As a first example, we consider a numerical experiment that involves solving the calibration problem discussed in Section 2 using synthetic data.This experiment features an analysis of the time series corresponding to daily data of asset prices over a period of ten years.We assume that one year is composed of 253 trading days.The time series studied comprises 253 • 10 1 2531 daily asset price data points, that is, it comprises the asset price S i observed at time t t i iΔt, i 0, 1, . . ., 2530, Δt 1/253.The synthetic data were obtained by computing one trajectory using the stochastic differential equation 2.1 for several choices of the parameter values, then examining the computed trajectory at time t t i iΔt, i 0, 1, . . ., 2530.We choose as the initial conditions, time t t 0 0, S 0 S 0 1200.We choose μ μ 1 0.01 and σ σ 1 0.1 in the first five years i.e., for t t i , i 0, 1, . . ., 1264 , μ μ 2 0.06 and σ σ 2 0.4 in the sixth and seventh years i.e., for t t i , i 1265, 1266, . . ., 1770 , and μ μ 3 0.03 and σ σ 3 0.2 in the last three years i.e., for t t i , i 1771, 1772 . . ., 2530 .The synthetic data were generated such that the last data point of the fifth year was the initial data point of the sixth year.Similar statements hold for the initial data point values of the last three years.The daily log return increments of the synthetic asset prices generated in this way are shown in Figure 1.The fact that the data are generated using three different parameters can be readily seen by inspection of Figure 1.
Consider the following calibration problem: given Δt 1/253, α 0.1, and the asset price observation S i at time t t i iΔt, i 0, 1, . . ., 2530, determine intervals within which the parameters μ and σ of model 2.1 fall with a significance level α 0.1.
We solve this calibration problem by applying Procedures 1 and 2, described in Section 2, to the data associated with a time window comprising 253 consecutive observations, that is, the observations corresponding to 253 consecutive trading days one year .This window is moved stepwise across the ten years of data, discarding the datum corresponding to the first observation time within the window and inserting the datum corresponding to the next observation time after the window.The calibration problem is solved for each data time window by applying Procedures 1 and 2 described in Section 2. In other words, the problem is solved 2530 − 253 1 2278 times, identifying two intervals for each calibration problem solved within which the volatility and drift parameters fall with a significance level α 0.1.Referring to Procedures 1 and 2 described in Section 2, we choose m 2, tol 2 • 10 −4 and appropriate initial intervals I I 0 to determine an interval within which the parameters σ and μ fall.The parameter reconstructions obtained from moving the window along the data are shown in Figure 2. The abscissa in Figure 2 corresponds to the data window used to reconstruct the model parameters.The data windows are numbered in ascending order, beginning with one, according to the first day within the window being considered.Figure 2 shows that the intervals containing the parameters μ and σ, chosen by Procedures 1 and 2, and the times at which the parameter values changed were reconstructed satisfactorily.
The second numerical experiment involved the use of real data.The real data studied were the 2005 daily values of the U.S. S&P 500 index see Figure 3  option prices.We limit our study to the call and put prices corresponding to options with a positive volume i.e., a positive number of contracts and a positive bid price, traded on the day corresponding to the price considered.This data set included 46 823 options prices.Because there are 253 trading days in the year 2005, we define a "year" as 253 consecutively ordered trading days.The time t t 0 0 was assigned to the day of January  This data set were interpreted using the Black-Scholes model, as described in Section 2. We begin by studying the variance and drift in the Black-Scholes model applying the S&P 500 index over the year 2005.The S&P 500 daily log return increments x i ln S i / S i−1 , i 1, 2, . . ., 252, were analyzed using the Black-Scholes model.The data were considered as a sample see Figure 4 of 252 observations taken from a set of independent identically distributed Gaussian random variables: where the mean M and the variance V 2 are defined in 2.7 .Procedures 1 and 2, described in Section 2, were used to estimate V 2 and, therefore, σ 2 using the χ 2 test.Subsequently, M and, therefore, μ was estimated using the Student's t-test.
Given the sample data comprising the S&P 500 daily log return increments x i , i 1, 2, . . ., 252, the significance level α 0.1, m 2, tol 10 −4 , and the appropriate initial intervals I I 0 , we initiated Procedures 1 and 2 to find that the hypotheses: 2.5297 are accepted with a significance level α 0.1.We next perform a type of stability analysis over the intervals 4.2 and 4.3 determined from the statistical tests.To do this, we fixed α 0.1 and applied Procedures 1 and 2 to determine the intervals within which σ 2 and μ lay with the significance level α, starting from a window of 70 consecutive observations corresponding to 70 consecutive observation times i.e., 70 consecutive trading days .The data window was shifted stepwise over the data time series, discarding the datum corresponding to the first observation time of the window and inserting the datum corresponding to the next observation time after the window.This procedure generated 252−70 1 data windows over the data time series.For each window, the corresponding calibration problem was solved.We found 252−70 1 pairs of intervals within which the volatility and drift parameters lay with a significance level of α 0.1.Figure 5 shows that as the data window was shifted, the intervals determined through Procedures 1 and 2 remained stable.The abscissa in Figure 5 represents the data windows used to reconstruct the model parameters, numbered in ascending sequential order, beginning with one.Figure 5 shows the intervals determined by Procedures 1 and 2 of Section 2 corresponding to each data window.Finally, we considered the 46.823S&P 500 European call and put option prices during the year 2005, and we attempted to interpret these values using the method developed in Section 3. The estimates 2.9 , 2.10 established with a significance level α 0.1 permitted us to determine the corresponding option price intervals using the Black-Scholes-Barenblatt equations.The worst-case option values V 1 S, t , S > 0, 0 < t ≤ T, and the best-case option values V 2 S, t , S > 0, 0 < t ≤ T could thereby be determined.Note that V 1 and V 2 could be determined using the Black-Scholes formula using 2.9 .We selected r μ 1 μ 2 /2, with μ 1 and μ 2 , as defined in 2.10 .It should be noted that the value of r was not relevant in determining V 1 and V 2 , and varying r over a reasonable range did not substantially change V 1 or V 2 .Moreover, the time to maturity within the Black-Scholes framework may be computed by considering a year composed of 365 days.We compute the percentage of European call % call and put % put option prices on the S&P 500, observed over the year 2005, that verified 3.23 , assuming 2.9 .The results obtained are shown in Tables 1-3.In these tables, N call and N put denote respectively the number of call prices and put prices corresponding to options with the characteristics described in the caption of the table.The quantities I call and I put denote respectively the average relative amplitude of the call price intervals and pricing problem were considered.The pricing problem for European call and put options with uncertain volatility in the Black-Scholes framework was reinterpreted by associating a statistical significance to the option price intervals determined using the Black-Scholes-Barenblatt equations.The proposed method was tested on synthetic and real data.The real data considered were S&P 500 values and the corresponding European call and put option prices over the year 2005.The calibration problem for the Black-Scholes model was solved based on the S&P 500 data, and the S&P 500 call and put option price data were interpreted in the framework of option prices with uncertain volatility and significance level.In the money and at the money S&P 500 option prices could be modeled with a high degree of accuracy.The out of the money S&P 500 option prices were not explained well by this model.
The methodology discussed in this paper was introduced in the context of the Black-Scholes asset dynamics model to take advantage of the model simplicity and to offer an alternative approach to the calibration techniques in widespread use by practitioners in the financial markets.The calibration problem may be solved using statistical tests that can affect the call, put option pricing, and these effects may be studied using elementary mathematics.Several extensions of our work may be considered.For example, the skew-normal models see, e.g. 17 are commonly used to interpret financial data.Many financial data sets exhibit asymmetric distributions, and several models, including the skew-normal models, have been proposed to capture this asymmetry and repair any inadequacies of the Black-Scholes model.In these models, a solution to the calibration problem that uses statistical tests should not differ too significantly from the solution proposed here.The extension of our methodology to the study of more general stochastic dynamical models, such as the Heston model or some of its variants, may require the development of new ad hoc statistical tests.Any application of new statistical tests to a sample data set should rely on the use of numerical methods.Another interesting idea to explore involves attempting to simplify the new statistical tests developed, solve the calibration problem, and approximate the relevant probability density functions using an asymptotic expansion within some meaningful limit.The intelligent use of these asymptotic expansions can provide significant computational advantages.

Figure 1 :
Figure 1: The daily log return increments synthetic data .