ON AN IMPLICIT ASSESSMENT OF FUZZY VOLATILITY IN THE BLACK AND

SCHOLES ENVIRONMENT

Andrea CAPOTORTI Gianna FIGÀ-TALAMANCA

Quaderno n. 106 — Ottobre 2012

QUADERNI DEL DIPARTIMENTO DI ECONOMIA, FINANZA

E STATISTICA

ISSN 1825-0211

On an implicit assessment of fuzzy volatility inthe Black and Scholes environment∗

Andrea Capotorti - Gianna Figa-TalamancaDip. Matematica e Informatica - Dip. Economia, Finanza e Statistica

Universita degli Studi di Perugia - Italy

Abstract

In this work we suggest a methodology to obtain the membershipof a non observable parameter through implicit information. To thisaim we profit from the interpretation of membership functions as co-herent conditional probabilities. We develop full details for the wellknown Black and Scholes pricing model where the membership of thevolatility parameter is obtained from a sample of either asset pricesor market prices for options written on that asset.Keywords: Fuzzy membership elicitation, Implicit Information, Co-herent Conditional Probability Assessments and Extension, Probability-Possibility Transformation

1 Motivation and assumptions

Most mathematical models rely on unknown quantities (parameters) whichare usually estimated through sampling techniques. One of the main issues inthe application of such models is the additional problem of non-observabilityof some of these parameters. In this case direct sampling is not possible and itis compulsory to rely on indirect information. A typical example in financialapplications is the derivation of the volatility of a risky asset. In particular,in the context of Black and Scholes model [4] (BS model, hereafter), it is

∗Manuscript submitted to Fuzzy Sets and Systems

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assumed that the price St of the risky asset follows a geometric Brownianmotion:

dStSt

= µdt+ σdWt, (1)

where σ is the volatility of the asset, t is the time index, µ the mean assetreturn and Wt a standard Wiener process. Under this dynamics assumption,a closed formula can be derived for the evaluation of European-style deriva-tives, e.g. the price at time t for a European Call option with strike price Kand expiration time T is:

C(St) = Φ(d1)St − Φ(d2)K e−r(T−t), (2)

with

d1,2 =ln(St

K) + (r ± σ2

2(T − t))

σ√T − t

, (3)

r the continuously compounded riskless interest rate, and Φ(·) the cdf of thestandard normal distribution.

Note that the pricing formula (2) only depends on the rate r and onthe volatility parameter σ. Since r is indeed independent of the stock price,σ is the most relevant parameter in (1). As a consequence, its estimationis crucial and has inspired a huge literature on estimation procedures (see,among others, [3],[30],[33]). Besides, generalizations of BS model have beenintroduced in which σ is assumed to be random; seminal papers in thisdirection are [26, 34, 39] and [24].

However, even in the simple constant case, the estimation of volatilityis debatable. In fact, while volatility is not directly observable, indirectinformation is available through associated quantities such as the price St ofthe risky asset itself as well as the price of some of its derivative claims. Purestatisticians usually base their evaluations on a sample of stock prices andestimate σ by the standard deviation of sample log-returns.

On the contrary, market practitioners mostly rely on indirect informationinferred from a sample of market option prices, which they claim to be moreinformative about the current market beliefs than asset prices themselves.There is anyhow a general agreement that both kinds of available informationcan only produce vague statements about the value of σ.

Recently, several authors proposed to explicitly include imprecision inparameters estimation by grading their admissibility through membership

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functions. Within the BS environment, in [40] it is assumed that the stockprice at time t as well as the parameters r and σ are triangular fuzzy-numbers.This assumption is relaxed in [23] where other shapes for the membershipsof the fuzzy values are adopted. By applying different methodologies bothcontributions derive a fuzzy price for Call and Put options with crisp matu-rities and strike prices. We stress that the support, the core and the shape offuzzy variables in the applications suggested therein are assumed as known(or preliminarily assigned). In [9] the authors apply BS model to price realoptions; they assume both the stock price and the strike of the option tobe modelled as trapezoidal fuzzy numbers St and K. The Call option priceis obtained as a fuzzy number by applying a hybrid version of formula (2),where the value in (3) is computed by replacing St and K with the possi-bilistic mean value of their fuzzified version, while the volatility parameter σis replaced by the square-root of the possibilistic variance of the fuzzy stockprice St. Again, the support and the shape of fuzzy values St and K areinitially assigned. A different approach is adopted in [36] where the authorsassume that volatility is described through a fuzzy number σ. They makeuse of the Heston stochastic volatility model [24], where the probability den-sity function for the instantaneous variance is explicitly given. The authorsprofit from this density function and from a probability/possibility transfor-mation in order to obtain the membership function for the fuzzy volatilityσ. Once again, in their example, parameters for the known distribution aregiven values. Similar remarks apply to other fuzzy generalizations of the BSenvironment (see, among others, [25] and [41]).

In this paper, we agree on introducing vagueness-imprecision for thevolatility parameter in BS model; our contribution is to propose a method-ology for a proper elicitation of σ which goes further the pre-assignmentadopted in the quoted papers. This is done by using indirect informationobtained from either asset or derivatives prices. In addition, contrarily towhat commonly done in literature, the volatility parameter and its estimateare treated as two distinct entities. Nevertheless, we are able to infer onσ by observing the behavior of each selected estimator θ through an inter-mediate pseudo-membership which permits the flow of information from theestimator θ to the parameter σ.

In order to elicit such pseudo-memberships we profit from the interpreta-tion of membership functions as coherent conditional probabilities providedin [11, 14, 15] and from their possible extensions. In fact, coherent condi-tional probability can be looked at as a general non-additive “uncertainty

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measure” µ(·) = P (E|·) of the conditioning events. This gives rise to a clear,precise and mathematical frame, which allows to define fuzzy subsets and tobound memberships of other fuzzy sets. Moreover, this interpretation per-mits the use of Bayesian methodologies to get a proper membership functionby joining different pieces of probabilistic information.

Of course, our approach could be straightforwardly extended to any otherfield where a parametric model is involved and where the evaluation of anunknown and unobservable parameter is affected both from the availablesources of indirect information and from sample variability. Potential appli-cations could be in engineering (e.g. [6]) or in models for climate changes(see, among others, [29]).

The paper is organized as follows: in Section 2 we describe the theoreticalframework where the membership elicitation procedure can be embedded: inSubsec. 2.1 we briefly recall the main notions of coherence while in Subsec. 2.2we formalize the procedure to obtain the searched membership µσ(x) throughscenarios identification, prior and likelihoods elicitation, pseudo-membershipdetermination and the eventual probability/possibility transformation. InSection 3 we describe the two sources of information to be considered in thenumerical application which is fully developed in the subsequent Section 4.Section 5 concludes and gives some hints for future developments.

2 Obtaining the membership via Bayes Rule

or coherent extension

Let us sketch the aforementioned interpretation of memberships as coherentconditional assessments [11, 14, 15]. The arguments usually brought for-ward to distinguish grades of membership from probabilities often refer to arestrictive interpretation of event and probability. On the contrary we inter-pret probability as a measure of belief in given propositions. In particulara conditional probability P (E|H) can be directly introduced through coher-ence without resorting to the usual ratio between unconditional probabilitiesP (E ∧H) and P (H), ratio which needs P (H) to be positive.

Going into details, the formal interpretation of fuzzy subset given in [14]through coherent conditional probability is:

Definition 1 Given a random quantity X with range CX and a related prop-erty ϕ, a fuzzy subset E∗ϕ of CX is the pair (Eϕ, µEϕ) with

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• Eϕ is the proposition “An assessor claims ϕ”,

• µEϕ(x) = P (Eϕ|X = x) is a coherent conditional probability, looked onas a real function of the argument x ∈ CX .

A coherent conditional probability P (Eϕ|X = x) is, hence, a measureof how much an assessor, given the event (X = x), is willing to claim theproperty ϕ, and it plays the role of the membership function of the fuzzysubset E∗ϕ. To let the paper be self-contained, we briefly introduce the basicconcepts of the conditional coherence paradigm.

2.1 Coherence

Coherence for partial conditional probabilities can be reduced to the propertyof compatibility with the well established mathematical model of the so calledfull conditional probabilities, as introduced by [18] and in line also with [16,28, 32]. Full conditional probabilities are characterized by the following setof axioms:

Definition 2 Given a Boolean algebra B, a full conditional probability onB × H0 (with H ⊆ B closed with respect to logical sums, and putting H0 =H \ ∅) is a function P : B ×H0 → [0, 1] such that(i) P (·|H) is a finitely additive probability on B for any given H in H0;(ii) P (H|H) = 1 for all H ∈ H0;(iii)P (A ∧B|C) = P (B|C)P (A|B ∧C) for every A,B ∈ B, C,B ∧C ∈ H0,where ∧ denotes the usual logical conjunction.

In discrete settings, the Boolean algebra B is usually the power set P(Ω)of some sample space Ω. What is peculiarly relevant is the possibility toassess and handle conditional probabilities given on domains without anyparticular structure. In fact we have:

Definition 3 If E = [A1|H1, . . . , An|Hn] is an arbitrary set of conditionalevents, an assessment P (·|·) on E is said to be coherent if there exists afull conditional probability P ′(·|·) defined on P(Ω) × P(Ω)0 (with P(Ω) thepower set of sample space Ω spanned by the events A1, H1, . . . , An, Hn) whichcoincides with P (·|·) on E.

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Whereas, for infinite domains E , coherence is equivalent to coherence ofP (·|·) on any finite subset F ⊂ E ([13, §11.3]).

Contrary to usual approaches, within this view conditional probabilitiescan be given directly without the use of joint and marginal evaluations. Anoperational check of coherence is anyway possible thanks to the followingcharacterization theorem ([10, 11, 12, 13]):

Theorem 4 Let E be an arbitrary finite family of conditional events and Ωdenote the set of atoms ωr generated by the events A1, H1, . . . , An, Hn. For areal function P on E the following two statements are equivalent:(a) P is a coherent conditional probability assessment on E;(b) there exists (at least) a class of probabilities P0, P1, . . . , Pk, each prob-ability Pα being defined on a suitable subset Ωα ⊆ Ω, such that for anyAi|Hi ∈ E there is a unique Pα with

∑ωr|=Hi

Pα(ωr) > 0 P (Ai|Hi) =

∑ωr|=Ai∧Hi

Pα(ωr)∑ωr|=Hi

Pα(ωr);

moreover, Ωα′ ⊂ Ωα′′ for α′ > α′′ and Pα′′(ωr) = 0 if ωr ∈ Ωα′.

Any class P0, P1, . . . , Pk singled-out by condition (b) is said to agreewith the conditional probability P .

With regard to our purposes, what is crucial is the possibility to makeinference from a coherent assessment to any new target conditional event. Infact the following theorem, essentially due to [16], holds:

Theorem 5 Let P (·|·) be an assessment on an arbitrary family E; then thereexists a (possibly not unique) coherent extension of P to any family K ⊃ E ifand only if P (·|·) is coherent on E.

Thanks to the characterization given by the aforementioned Theorem4, it is possible to obtain such a coherent extension through specific linearoptimization problems based on the classes P0, P1, . . . , Pk (for details refere.g. to [8, 11]).

The role of coherence is simply that of allowing not structured domain. Asubsequent step is to identify an effective method for its evaluation, as donein what follows for our purposes.

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2.2 Coherent membership elicitation

Going back to our original problem, we have to look at an operational wayto assess a membership function for the volatility parameter σ. To followclosely Def.1, we deal with some vague statement about σ, e.g. “σ is arounda specific value”, and we have to grade the willingness of an assessor to claimit, conditionally to the actual, but unknown, value of the parameter. Morespecifically, we should look for

µHs(x) = P (σ “is claimed to be a value around ” σs|σ = x), (4)

where different values of σs are representatives of distinct market behaviorscenarios Hs, s = 1, . . . , ns, while x is the precise value attained by theparameter σ. Given the non-observability of the parameter, beliefs about σmust be based on some estimator so what we can actually elicit is

µHs(θ) = P (Hs|Infoθ), (5)

where Infoθ is any information gathered from an estimator θ of σ. We profitfrom pseudo-memberships (5) to select the most plausible scenario(s) for thevolatility parameter σ, which will serve the final elicitation of membership(4) as detailed afterwards. As already remarked, the volatility parameterand its estimate are seen as two distinct entities. However, by observing thebehavior of θ, we are able to infer on the true parameter σ.

Let us assume, for computational purposes, that any empirical or simu-lated distribution for θ is discretized into nb classes θb, named ”bins”. With alittle abuse we set θb ≡ (θ ∈ θb) so that values for the pseudo-memberships (5)

are computed ”bin by bin” with Infoθ ≡ θb. By using the notation introduced

in Subsec. 2.1, the input domain E is given byHs, θb|Hs

s=1,...,ns;b=1,...,nb

i.e.

by the unconditional scenarios ranges and by the realization of the estimatorinside one of its possible bins, conditioned to different market scenarios.

As a first step we need the pseudo-memberships (5); they can be de-rived either via Bayes rule, whenever scenarios Hs, s = 1, . . . , ns, come froman exhaustive partition, or via coherent extension, whenever scenarios comefrom partial knowledge, i.e. they overlap, or they do not cover all the pos-sibilities or there is some logical constraint among them and possible valuesof the parameter. In any case, the computation of (5) is based on likeli-hoods P (θb|Hs) and priors P (Hs). In the numerical application we obtain

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the priors by experts evaluations about scenarios based on historical data,whereas likelihoods are derived through simulation. More precisely, we as-sume that, by randomly generating values of σ from a specific distributionπs for each scenario Hs, we are able to obtain the distribution for θ condi-tioned to Hs, s = 1, 2, ..., ns. The latter step relies on the availability of anexplicit relation between σ and θ. Obviously, different approaches may leadto different assessments.

Once likelihoods P (θb|Hs) are obtained, it is possible to infer on theprobabilities P (θb). If we have full information, i.e. the scenarios form apartition and there is not any logical constraint among the Hs and the θb,then the available assessment P (·|·) on E is surely coherent [37, Prop.1]. Inthis case we obtain the P (θb) through the usual disintegration formula

P (θb) =ns∑s=1

P (θb|Hs)P (Hs), (6)

and the pseudo-membership P (Hs|θb) by Bayes rule

P (Hs|θb) =P (θb|Hs)P (Hs)

P (θb). (7)

Otherwise, whenever information is partial, overall conditional coherenceof the assessment P (·|·) on E must be checked and, once it has been ensured,we can extend it by the procedures detailed in [8, 11] to Hs|θb, obtainingcoherent intervals

[P∗(Hs|θb), P ∗(Hs|θb)]. (8)

Given the incompatibility of the various bins θb, Theorem 2 in [14] guaran-tees the coherence of any value inside the intervals (8) . Hence, we obtaina set of plausible pseudo-memberships instead of a single one, so that wehave to deal with interval type-2 [22] pseudo-memberships (see e.g. Fig.1).

The further step is to consider the current (observed) value θobs of the pa-

rameter estimator. On the base of the bin θb including θobs we can select

most plausible scenarios Hs by maximizing P (Hs|θb). Tails among type-1 orinterval incomparability among type-2 pseudo-memberships can induce morethan one plausible scenario; in this case, any of such scenarios can be a validcandidate so we may take the disjunction of them. Note that we assume theset of pseudo-memberships µHs(θ), s = 1, 2, ..., ns to hold for a suitable timeperiod, hence its computation is performed once in a while.

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Figure 1: Example of interval type-2 membership function.

At this point it is possible to elicit the searched membership µσ(x) bytransforming the simulating distributions πs associated with the selectedscenario(s). At first, membership µHs

(x) is obtained through a probabil-ity/possibility transformation [20] of the simulating distribution for each se-lected scenario. In particular, for the sake of computational simplicity, weadopt the following:

µHs(x) =

2Fπs(x) if x ≤ σMπs2(1− Fπs(x)) if x > σMπs

, (9)

where Fπs(x) and σMπs denote, respectively, the cumulative distribution func-tion and the median of the simulating distribution πs associated to Hs. Notethat such probability-possibility transformation is, among those proposed in[21], that one induced by confidence intervals around the median.

In case a single most plausible scenario Hs is selected, we obtain a fuzzynumber and µσ(x) ≡ µHs

(x), otherwise, µσ(x) may be obtained through areasonable t-conorm of the memberships associated to each selected scenario.

3 Indirect information: the choice of input

data

If we believe in the efficiency of financial markets, then the price of an assetrepresents the agents opinion, i.e. the total information available on thatasset. Stock and derivative assets are traded in different financial markets

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which provide different sources of indirect information on the volatility ofthe asset. As already mentioned, we can infer the volatility σ as the samplestandard deviation of asset returns; in this case indirect information is basedon a sample of stock prices. Alternatively, we can obtain the volatility asimplied by the price of some derivative assets written on that stock e.g.,given the price of a traded option, we may use the so-called BS impliedvolatility, obtained by numerical inversion of the BS option pricing formula(2) with respect to the volatility parameter σ. As well, we may profit, whenavailable, from the value of a volatility index, usually obtained by consideringas input a set of suitably selected traded Call and Put options. The two latterprocedures both rely on a sample of derivatives prices.

In our numerical application we aim at the elicitation of a membershipfunction for the volatility of the S&P500 Index. Hence, St is the value attime t of the Index and as alternative estimators of σ we consider eitherthe sample standard deviation of the Index log-returns or the VIX volatilityindex, as defined in what follows. The former is based on a time series ofthe S&P500 returns while the latter is based on a sample of Call and Putoptions prices.

3.1 Historical data of the underlying

In the first approach we estimate the volatility σ by means of the historicalvolatility, which is computed, given a time series xi, i = 1, 2, ..., n of theS&P500 Index returns, as:

σ =

√∑ni=1(xi − x)2

n− 1, (10)

where x is the sample mean. In order to elicit the priors on scenariosHs, s = 1, 2, ..., ns we use the empirical distribution of σ obtained by thelargest available time series. More precisely, we compute σ on m (overlap-ping) moving windows with n observations each. As remarked in Subsec.2.2, the empirical distribution of σ is discretized into nb classes σb and rep-resented through the associated histogram (see Fig.2 (a)). On the base ofsuch empirical distribution, an expert can choose the number ns of scenariosand their relative ranges. The easiest way to do it is on the base of specificquantiles for σ. Then, for each scenario Hs, we count the number ms of

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Figure 2: (a):S&P500 historical volatility empirical distribution elicitedfor moving windows of length n=45. (January 1950- October 2010) -(b):VIX/100 empirical distribution (January 1990- October 2010)

values falling in that scenario and we estimate P (Hs) as

P (Hs) =ms

m.

We stress that if Hs, s = 1, 2, ..., ns is a partition then∑ns

s=1ms = m so that∑ns

s=1 P (Hs) = 1.Furthermore, once scenarios Hs, s = 1, 2, ..., ns are determined, we com-

pute the associated likelihoods for σ|Hs as follows:

• we randomly generate N values for the volatility in the scenario Hs

from a known distribution πs, namely σi, i = 1, 2, ..., N ;

• for each of these values σi, we simulate, through the BS model (1), a

sample x(i)1 , x

(i)2 , ..., x

(i)n of n values for the Index return;

• for each sample of returns we then compute the corresponding value ofσ(i), for i = 1, 2, ..., N, as

σ(i) =

√∑nt=1(x

(i)t − x(i))2n− 1

.

Note that the N values for σ(i) are obtained from distribution πs condi-tionally on the scenario Hs selected at the beginning of the procedure; thedependence on s has been omitted to simplify the notation. Such N val-ues are also summarized in nb classes and the likelihoods discretized intohistogram densities P (σb|Hs), b = 1, . . . , nb.

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Once such likelihoods are obtained, we can infer on pseudo-membershipsP (Hs|σb) as in (7) or at least compute their coherent bounds (8). On thebase of the actual value of the estimator σobs , it is possible, by comparison,to find the most plausible scenarios and obtain the searched membershipsµσ(x) as described in Subsec. 2.2.

3.2 Market option prices and the VIX

The alternative setting for the volatility estimator relies on a time seriesfor the VIX, which is a volatility index obtained through a set of prices foroptions written on the S&P500 Index (SPX options). Let us define

V 2T =

2

T

∑i

∆Ki

K2i

eRTQ(Ki)−1

T

[F

K0

− 1

]2, (11)

where T is the time to maturity, F is the forward index level (derived fromindex option prices), K0 is the first strike below F , Ki is the strike price ofthe i-th out-of-the-money option and ∆Ki is the interval between strikes. Ris the risk free rate while Q(Ki) is the midpoint of the bid-ask values foroption with strike Ki.

The selected options are out-of-the-money SPX Calls and Puts centeredaround an at-the-money strike price K0. Only SPX options quoted withnon-zero bid prices are used in the VIX calculation. The squared VIX valueis finally obtained as a weighted mean of V 2

T1and V 2

T2with T1 and T2 cor-

responding to near and to next term maturities. For more details on VIXcomputation see [38].

Note that the VIX is expressed as a percentage; we adopt ν = VIX/100

as a proxy for the volatility (i.e. θ = ν) so that the distribution of interest issimply given by the empirical distribution of ν.

Assume we are given a time series of length m for ν; similarly to theprevious subsection, we can discretize its empirical distribution into nb bins(see Fig.2 (b)). On the base of this distribution experts can grasp ns relevantscenarios and consequently elicit the priors as P (Hs) = ms

m. Afterwards,

likelihoods are obtained as below:

• we randomly generate N values for the volatility in the scenario Hs

from a known distribution πs, namely σi, i = 1, 2, ..., N ;

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• for each simulated σi, we compute, according to BS option pricingformula, the Call and Put option prices for the two maturities and forthe selected Strikes needed for the calculation of the VIX;

• we compute the corresponding value of the VIX index to get ν(i), fori = 1, 2, ..., N .

• the proportion of ν(i) falling in each of the nb classes can be adoptedas a discrete density approximation P (νb|Hs).

Once again, through the current value νobs, we can obtain the searchedmembership µσ(x) by the probability/possibility transformation of the simu-lating functions associated to the scenariosHs with highest pseudo-membershipsupport P (Hs|νobs) (see Subsec. 2.2).

4 Numerical Illustration

In this section we describe how to elicit the membership function for thevolatility parameter in practice. We apply either σ or ν, introduced in theprevious section, and we consider the largest time series available for theS&P500 Index as well as for the VIX Index; the former starts on January 3,1950, the latter on January 2, 1990 and both end on October 10, 20101. The(discretized) distributions of σ and ν, from which we derive priors P (Hs),are shown in Fig.2 and compared in Fig.3.

Figure 3: Prior distribution comparison for σ (solid line) and ν (dashed line)estimators.

1Data are downloaded from the Yahoo Finance web-site.

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We take into account three or five volatility scenarios; namely ns = 3 cor-responds to scenarios of low, medium and high volatility and ns = 5 to sce-narios of low, medium-low, medium, medium-high and high volatility. Thesescenarios are associated to intervals of the real line, which may be disjointas well as overlapping, obtained by dividing the range of the correspondingestimator according to its percentiles.

Three different parametric functions πs are chosen to simulate the volatil-ity in order to obtain the conditional likelihoods. At first, the Uniform distri-bution is adopted as representative of minimal knowledge; as an alternativethe Log-normal distribution is considered since, as pointed out in [35], it re-sembles the density function of the integrated volatility in the Heston model(see [24]). Furthermore, the Gamma distribution is selected to obtain thesimulated variance since it describes well the invariant distribution of thelocal variance of the stock in the Garch diffusion setting (see [1]). As pro-totypes, we fully describe here three cases as reported in Table 1, where qarepresents the a-th percentile of θ.

scenarioCase ns L ML M MH H Simul.D.1 3 [q0, q25] - [q25, q75] - [q75, q100] U2 3 [q0, q30] [q20, q80] [q70, q100] LN3 5 [q0, q20] [q20, q40] [q40, q60] [q60, q80] [q80, q100] G

Table 1: Several scenarios detection based on θ estimator’s quantiles, scenariolabels (L=low, ML=medium-low, M=medium, MH=medium-high, H=high)and simulating distributions (U=uniform, LN=log-normal, G=gamma).

It is possible to analyze each case study following the proposed method-ology.

In our exercise the VIX time series provides itself the m = 5244 ob-servations for ν while we obtain a sample of m = 15254 values for σ byconsidering moving windows of length n = 45 from the time series of theS&P500 log-returns. Concerning the derivation of likelihoods, N values forthe parameter σ are generated through the density function πs (Uniform,Log-normal or Gamma) with specific parameters for each scenario Hs; inCase 1 N = m while in Case 2 and Case 3 N = 100000 to guarantee ob-servations in the tails. The number of bins is fixed at nb = 50 for all thedistributions, while the effective ranges will depend on the estimator at hand.

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It is worth noticing that, although the parameter σ is obtained by simulationfrom πs, the corresponding estimators are derived as described in Subsecs.3.1, 3.2 and depend on the implicit relation between the parameter and itsestimates. The distributions of the estimators are different from the originaldistributions πs for σ. For example in Case 1, even starting from disjointuniform πs, we get empirical distributions of σ and ν which overlaps and aremostly not uniform (see Fig.4).

Figure 4: Empirical distributions of σ (a) and ν (b) for Case 1.

In Cases 1 and 3 it is possible to apply plain Bayes rule (7) to obtain thepseudo-memberships (Figs. 5-6), since the scenarios are partitions.

Figure 5: pseudo-memberships P (Hs|θ), s = L,M,H for case 1 with θ ≡ σ(a) and θ ≡ ν (b).

On the other hand, in Case 2 the scenarios overlap, hence the best wecan obtain is the interval type-2 pseudo-memberships as plotted in Fig. 7.Let us assume that the current values for the estimators are σobs = 0.16 andνobs = 0.19, highlighted with a vertical dotted line in all the aforementioned

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Figure 6: pseudo-memberships P (Hs|θ), s = L,ML,M,MH,H θobs for Case3 with θ ≡ σ (a) and θ ≡ ν (b).

Figure 7: Interval type-2 pseudo-memberships P (Hs|θ), s = L,M,H for Case2 with θ ≡ σ (a) and θ ≡ ν (b).

figures. The most plausible scenarios are uniquely determined for Cases 1and 3, both for σ and ν. In Case 1 we select the ”medium” (s = M)scenario and, by the probability-possibility transformation (9) of the Uniformsimulating distribution πM , we get the memberships for σ to be “around”0.124 and “around” 0.1923, respectively(see Fig.8). Similarly, in Case 3we get the ”medium-high” (s = MH) scenario and, by the probability-possibility transformation (9) of the associated distribution πMH , we get σ“around” 0.147 and 0.23, respectively (see Fig.9).

Note that in Case 1 the current value σobs is not included in the supportof the fuzzy number elicited for σ; similarly, in Case 3 we get a negligiblemembership value for νobs. If, on one side, this could result counterintuitive,on the other hand it emphasizes the difference between the real parameter σ

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Figure 8: Triangular fuzzy numbers for σ expressing the membershipsµσ|θobs(s) for Case 1 deriving from σobs = 0.16 (a) and νobs = 0.19 (b) andmedium volatility scenario with uniform simulated likelihood.

and its estimators σ and ν. In fact, the elicited memberships concern possiblevalues for σ based on the observed values σobs and νobs of the estimators whichare two representatives of indirect information.

For Case 2, and in particular for σobs = 0.16, we get two overlapping plau-sibility intervals. This produces two most plausible scenarios to be selectedso the more general approach is to take both the ”medium” and the ”high”scenarios as supported by the current observation. Hence, we transform thetwo simulating distributions into two different plausible memberships whichare respectively “around” 0.1182 and “around” 0.2024 (see Fig.10(a)). Sev-eral choices are possible, among known t-conorms of the two memberships(see. e.g. [19]), to get the final membership for σ. Differently, on the base ofthe VIX estimator we select a single scenario and get an evaluation of σ tobe “around” 0.1980 (see Fig.10(b)).

5 Conclusive remarks and future directions

In this work we give a contribution to research in stochastic models withfuzzy parameters. The novelty of our approach is to avoid a pre-assignmentof memberships and to suggest a methodology to elicite them.

To this aim we profit from the interpretation of membership functions ascoherent conditional probabilities and from their possible extensions. Withthis approach we embed the elicitation procedure into a rigorous mathemati-cal frame which also allows to join different pieces of probabilistic information

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Figure 9: Fuzzy numbers for σ expressing the memberships µσ|θobs(s) forCase 3 deriving from σobs = 0.16 (a) and νobs = 0.19 (b) and medium-highvolatility scenario with Gamma simulated likelihood.

by applying generalized Bayesian methodologies.In our application we focus on the elicitation of a proper membership func-

tion µσ(x) for the volatility parameter in Black and Scholes model. Moreover,since volatility is not-observable, we base our elicitation procedure on indi-rect information obtained either from a sample of asset prices or from a setof observed derivatives values.

A peculiarity of our contribution is that the volatility parameter and itsestimates are treated distincly. Though, by observing the behavior of eachselected estimator, we are able to infer on the original volatility parameter,by introducing an intermediate pseudo-membership which permits the flowof information from one to the other. Our approach could be extended toother parametric models as well as to different application fields.

The proposed numerical exercise shows the feasibility of the method andemphasizes the fact that different sources of information can be adopted.Indirect information obtained respectively through the historical volatilityor through the VIX volatility Index lead to different conclusions. To jointlyprofit from them, an aggregation or merging of the two is compulsory. Specif-ically, there are two stages where this integration might be performed: at thepseudo-memberships level (P (Hs|σ) and P (Hs|ν)) or at the final member-ships one (µσ|σobs(s) and µσ|νobs(s)). Within the former choice we might adopta specific connective depending on the semantic properties one wants to rep-resent (see e.g. [5, 19]) or, on the contrary, to apply a more general approachby considering all possible coherent extensions of the two (as depicted in[14]). On the other side, working directly with the final memberships, we

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Figure 10: Fuzzy numbers for σ deriving from medium or high volatilityscenarios associated to σobs = 0.16 (a) and medium scenario associated toνobs = 0.19 (b) and with Log-Normal simulated likelihood.

could average the two by computing a fuzzy mean (see e.g. [17]) or mergethem by some specific pseudo-distance minimization (see e.g. [7]). Of course,appropriateness of such a choice will depend on practical consequences forthe specific application. Since in this paper we do not face this issue, we donot develop any further analysis on this aspect that anyhow will deserve ourattention in the next future.

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