
Silvia MUZZIOLI
Professore Ordinario Dipartimento di Economia "Marco Biagi"

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2023
 Assessing skewness in financial markets
[Articolo su rivista]
Campisi, G.; La Rocca, L.; Muzzioli, S.
abstract
It is a matter of common observation that investors value substantial gains but are averse to heavy losses. Obvious as it may sound, this translates into an interesting preference for rightskewed return distributions, whose right tails are heavier than their left tails. Skewness is thus not only a way to describe the shape of a distribution, but also a tool for risk measurement. We review the statistical literature on skewness and provide a comprehensive framework for its assessment. Then, we present a new measure of skewness, based on the decomposition of variance in its upward and downward components. We argue that this measure fills a gap in the literature and show in a simulation study that it strikes a good balance between robustness and sensitivity.
2023
 Financial innovation, FinTech, and implications for financial markets
[Capitolo/Saggio]
Gambarelli, Luca; Muzzioli, Silvia
abstract
2023
 Googling Investor Sentiment around Europe
[Working paper]
Gambarelli, L.; Muzzioli, S.
abstract
2023
 Hedging effectiveness of cryptocurrencies in the European stock market
[Articolo su rivista]
Gambarelli, L.; Marchi, G.; Muzzioli, S.
abstract
2023
 The LGBT plus University Inclusion Index and its application to Italian universities
[Articolo su rivista]
Russo, T; Addabbo, T; Muzzioli, S; Damiani, F; De Baets, B
abstract
This paper outlines an index of LGBT+ inclusion in universities and provides an initial ranking of Italian public universities based on this index. The LGBT+ University Inclusion Index incorporates an extensive set of indicators based on existing LGBT+ inclusion indices and expert opinions and is based on a fuzzy rulebased system of the MamdaniAssilian type to measure inclusiveness. The index has a twofold aim: to help tertiary education institutions to assess their degree of inclusiveness, and to assess the level of LGBT+ inclusion in tertiary education by the National Agency for the Assessment of Higher Education and Research. Best practices in LGBT+ inclusion, implemented by Italian universities and identified in this study, are highlighted with the aim of suggesting and recommending guidelines helpful to fighting homobitransphobic discrimination in higher education institutions.
2023
 Understanding the SKEW Index: the relationship with sentiment and returns
[Working paper]
Elyasiani, E.; Gambarelli, L.; Muzzioli, S.
abstract
2022
 An OWA Analysis of the VSTOXX volatility index
[Working paper]
Gambarelli., L.; Muzzioli, S.; De Baets, B.
abstract
In this paper, we analyze the information value of the VSTOXX (volatility) index as a measure of risk for the EU stock market. Employing daily data from 2007 to 2017, we inspect and contrast the properties of the VSTOXX index under various market conditions and in high and lowvolatility periods. Moreover, to evaluate the contribution of each countryspecific index to the VSTOXX index, we employ the Ordered Weighted Averaging (OWA) operator, which provides a flexible aggregation procedure ranging between the minimum and the maximum of the input values. We obtain a number of useful insights. The correlation between the VSTOXX index and the volatility indices is high during the entire period only for France and Germany. Moreover, the VSTOXX index acts more like an ORlike measure than as an ANDlike measure of volatility for the EU stock markets and acts as an average only during periods of extreme volatility.
2022
 Asymmetric SemiVolatility Spillover in a Nonlinear Model of Interacting Markets
[Articolo su rivista]
Campisi, Giovanni; Muzzioli, Silvia
abstract
This paper develops an heterogeneous agents model with fundamentalists and chartists trading in two different speculative markets. It examines whether investors’ behaviour is related to the volatility and its dynamics. We find that investors’ heterogeneity in price trends and trading strategies can significantly explain asymmetry in semivolatility transmission.
2022
 Asymmetric correlations and hedging effectiveness of cryptocurrencies for the European stock market
[Working paper]
Gambarelli., L.; Marchi, G.; Muzzioli, S.
abstract
The aim of the paper is twofold: first, to examine the hedging effectiveness of cryptocurrencies and cryptocurrency portfolios for European equities in bearish and bullish market conditions, and second, to contrast cryptocurrencies with gold as a safe haven asset. To this end, daily data from 2018 to 2021 were employed in a linear and nonlinear Autoregressive Distributed Lag (ARDL) framework.
The findings have significant implications for investors, financial intermediaries and regulators. First, none of the cryptocurrencies under investigation acts as a safe haven for the European stock market. Second, an asymmetric relationship was found between Bitcoin / Ethereum returns on the one hand and stock market returns on the other, indicating the risk of large joint losses during periods of market turmoil. Third, cryptocurrency portfolios appear to perform better than Bitcoin and Ethereum for diversification purposes. Fourth, among cryptocurrency portfolios, the portfolio made up of the top ten cryptocurrencies appear to be the best in terms of diversification benefits and the riskreturn profile. Finally, during the 2020 bear market conditions, not even gold acted as a safe haven for European stocks, highlighting the need to investigate alternative safe haven assets to mitigate portfolio risks.
2022
 Asymmetric semivolatility spillover in a nonlinear model of interacting markets
[Working paper]
Campisi, G.; Muzzioli, S.
abstract
This paper develops an heterogeneous agents model with fundamentalists and chartists trading in two different speculative markets. It examines whether investors’ behaviour is related to the volatility and its dynamics. We find that investors’ heterogeneity in price trends and trading strategies can significantly explain asymmetry in semivolatility transmission.
2022
 Forecasting returns in the US market through fuzzy rulebased classification systems
[Working paper]
Campisi, G.; De Baets, B.; Gambarelli, L.; Muzzioli, S.
abstract
The paper aims to investigate the forecasting ability of fuzzy rulebased classification systems (FRBCS) on future direction of the S&P500 index. To this end, we apply four FRBCS methods. Moreover, we compare both the forecasting accuracy and the interpretability of the results of FRBCS with the recently used machine learning techniques. Overall, among the two approaches, we prefer the FRBCS methods, since they allow a good balance between accuracy and interpretability, and provide sharper results than the machine learning techniques.
2022
 News Sentiment indicators and the Cross‐Section of Stock Returns in the European Stock Market
[Working paper]
Gambarelli., L.; Muzzioli, S.
abstract
This paper investigates whether the Bloomberg investor sentiment index can provide valuable information for investors and fund managers for the purposes of stock picking and portfolio selection. The dataset consists of all the listed companies in the Euro area for the period from 2010 to 2021. By exploiting portfolio sorting strategies, the paper evaluates to what extent and how long investor sentiment can affect stock returns. Moreover, it considers whether additional factors can affect the relationship between sentiment and returns, casting light on the asymmetric effect related to positive and negative news.
The findings are as follows. First, high (low) sentiment stocks exhibit high (low) returns on average. The average return of the portfolio that takes a long position in the stocks with very high sentiment and a short position in stocks with very low sentiment is statistically and economically significant and is robust to the inclusion of commonly used risk factors. Second, the predictability of stock returns using the sentiment indicator declines fast after one month. Third, evidence is found of the profitability of a longshort strategy that invests in stocks with low capitalization: profitability declines with the duration of the investment period. Finally, it is found that positive news is factored into the stock price more slowly than negative news, especially for stocks with low market capitalization.
2022
 Regional innovation in southern Europe: a posetbased analysis
[Working paper]
Damiani, F.; Muzzioli, S.; De Baets, B
abstract
This paper examines the performance of regional innovation across the 60 southern European regions of Greece, Italy, Portugal and Spain. A posetbased analysis is carried out in two phases. The first phase establishes a ranking of the clusters in which regions are grouped to identify patterns of comparable regions. The second phase focuses on the country level, where the regions of each of the four countries are ranked into five different performance levels. The outcomes of the two phases are compared with the results described in the Regional Innovation Scoreboard 2019, with a view to providing insights for policymakers.
2022
 The power of deterministic optionimplied trees in pricing European options
[Articolo su rivista]
Elyasiani, E.; Muzzioli, S.
abstract
The aims of the current article are threefold. First, to investigate the power of deterministic optionimplied trees, constructed either by forward or by backward induction, in pricing European options, in order to assess the proper representation of the smile. Second, to investigate and contrast the power of deterministic optionimplied trees during tranquil and volatile market conditions. Last, to assess the correctness of the representation of the smile in different parts of the riskneutral distribution. Three main results are obtained. First, the pricing performance of the Enhanced Derman and Kani model (EDK), based on forward induction, is superior to that of the Rubinstein model, based on backward induction. Second, the EDK model produces better results (smaller errors) on the left tail of the distribution, i.e. it is better in pricing outofthemoney put options. Third, it performs better in turmoil periods where correct pricing a challenge, and accuracy is of greater importance than in tranquil periods. Diebold and Mariano test of equal predictive accuracy confirms the superiority of the EDK model in both subperiods.
2021
 A TOPSIS analysis of regional competitiveness at European level
[Working paper]
Ferrarini, F.; Muzzioli, S.; De Baets, B.
abstract
.
2021
 A comparison of machine learning methods for predicting stock returns in the US market
[Working paper]
Muzzioli, S.; Campisi, G.; De Baets, B.
abstract
2021
 A posetbased analysis of regional innovation at European level
[Working paper]
Damiani, F.; Muzzioli, S.; De Baets, B.
abstract
This paper examines the performance of regional innovation across 220 European regions. First, a cluster analysis is performed in order to detect patterns of comparable regions. Subsequently, a posetbased approach is adopted to obtain a ranking of the different clusters of European regions. The outcome is compared with the results described in the Regional Innovation Scoreboard 2019. Useful insights for policymakers are obtained.
2021
 Designing volatility indices for Austria, Finland and Spain
[Articolo su rivista]
Campisi, G.; Muzzioli, S.
abstract
2021
 Existence of a fundamental solution of partial differential equations associated to Asian options
[Articolo su rivista]
Anceschi, Francesca; Muzzioli, Silvia; Polidoro, Sergio
abstract
We prove the existence and uniqueness of the fundamental solution for Kolmogorov operators associated to some stochastic processes, that arise in the Black & Scholes setting for the pricing problem relevant to path dependent options. We improve previous results in that we provide a closed form expression for the solution of the Cauchy problem under weak regularity assumptions on the coefficients of the differential operator. Our method is based on a limiting procedure, whose convergence relies on some barrier arguments and uniform a priori estimates recently discovered.
2021
 Nonlinear dynamics in asset pricing: the role of a sentiment index
[Articolo su rivista]
Campisi, G.; Muzzioli, S.; Zaffaroni, A.
abstract
This paper aims to contribute to the literature on the role of sentiment indices in heterogeneous asset pricing models. A new sentiment index in financial markets is proposed in which transactions take place between two groups of fundamentalists with divergent perceptions of fundamental value. It is assumed that the proportion of fundamentalists in the two groups depends on the sentiment index. After examining the analytical properties of the deterministic discrete dynamical system, stochastic components are added to the expectations of fundamentalists. First, the study measures the performance of the model in reproducing the stylized facts of financial data relying on the S&P 500 index. Second, the forecasting power of the model to predict the daily prices of the S&P 500 index is examined. For this purpose, the forecasting accuracy of the proposed dynamical model, where the sentiment index is explicitly modelled, is compared with a model where the sentiment index is not taken into account. In this case, the predictions are obtained by means of a machine learning technique (lasso regression). The results show that the sentiment index is important in explaining the stylized facts of financial returns and in forecasting prices.
2021
 The skewness index: uncovering the relation with volatility and market returns
[Articolo su rivista]
Elyasiani, 2. E.; Gambarelli, L.; Muzzioli, S.
abstract
2021
 The skewness index: uncovering the relationship with volatility and market returns
[Articolo su rivista]
Elyasiani, E.; Gambarelli, L.; Muzzioli, S.
abstract
The SKEW index of the Chicago Board Options Exchange (CBOE), launched in February 2011, measures the tail risk not fully captured by the VIX index. In this paper we introduce, for the first time, a skewness index for the Italian stock market (ITSKEW) and investigate the pairwise and trilateral relations of this index with volatility and market returns. The results are compared with those of the US market. Data for the period 3 January 2011 to 29 December 2017 are used and three main results are found. First, in both the US and the Italian markets, the skewness index acts as a measure of market greed, as opposed to market fear, in the sense that it captures investor excitement to a larger extent than investor fear. Second, increases in the skewness index are related to returns with high significance in the Granger causality test, while the reverse is not the case. Last, volatility and skewness may give conflicting signals. When skewness and volatility indices move in the same direction, investors should rely on volatility because it has a stronger influence on market returns. The implications for investors and policymakers are outlined.
2021
 Tools and practices for LGBT+ inclusion in tertiary education: the development of the LGBT+ University Inclusion Index and its application to Italian universities
[Working paper]
Russo, T.; Addabbo, T.; Muzzioli, S.; De Baets, B.
abstract
The literature provides evidence that lesbian, gay, bisexual and trans learners ore often victims of homobitransphobic discrimination in educational environments.
This paper outlines an index of LGBT+ inclusion for Universities. The index has a twofold aim: to help tertiary education institutions to assess their degree of inclusivenesss, and to detect the level of LGBT+ inclusion in tertiary education. Starting from the existing LGBT+ inclusion indices like the Campus Pride Index and LGBT+ Inclusive Education Index, the LGBT+
inclusive university index refers to an extended set of indicators and is based on fuzzy logic techniques to measure inclusiveness (Zadeh, 1965, 1988).
Best practices in LGBTI+ inclusion, implemented by Italian universities and identified in this study, are highlighted with the aim of suggesting and recommending guidelines helpful to fighting homobitransphobic discrimination in higher education institutions.
2021
 Towards new volatility measures for the EU stock market
[Relazione in Atti di Convegno]
Gambarelli, L.; Muzzioli, S.; De Baets, B.
abstract
This paper analyzes the role of the VSTOXX volatility index as a measure of risk for the EU stock market. Employing daily data from 2007 to 2017, we study and contrast the properties of the VSTOXX index in various market conditions. Moreover, to investigate the information content of each countryspecific index for the VSTOXX, we exploit the Ordered Weighted Averaging (OWA) operator, which provides a flexible aggregation procedure ranging between the minimum and the maximum of the input values. The VSTOXX index can correctly measure the volatility risk only for France and Germany, while the results depend on the period under investigation for the other countries. Moreover, VSTOXX acted more like an ORlike measure than an ANDlike measure of volatility for the EU stock markets and represented an average for the EU volatility only during periods of extreme volatility.
2021
 Uncertainty about fundamental and pessimistic traders: a piecewiselinear maps approach
[Working paper]
Campisi, G.; Muzzioli, S.; Tramontana, F.
abstract
2021
 Uncertainty about fundamental, pessimistic and overconfident traders: a piecewiselinear maps approach
[Articolo su rivista]
Campisi, G.; Muzzioli, S.; Tramontana, F.
abstract
We analyze a financial market model with heterogeneous interacting agents where fundamentalists and chartists are considered. We assume that fundamentalists are homogeneous in their trading strategy but heterogeneous in their belief about the asset’s fundamental value. On the other hand, we consider that chartists, when they are optimistic become overconfident and they trade more than when they are pessimistic. Consequently, our model dynamics are driven by a onedimensional piecewiselinear continuous map with three linear branches. We investigate the bifurcation structures in the map’s parameter space and describe the endogenous fear and greed market dynamics arising from our assetpricing model.
2020
 Assessing skewness in financial markets
[Working paper]
Campisi, G.; La Rocca, L.; Muzzioli, S.
abstract
It is common knowledge that investors like large gains and dislike large losses.
This translates into a preference for rightskewed return distributions, with right tails heavier than left tails.
Skewness is thus interesting not only as a way to describe the shape of a distribution, but also for risk measurement.
We review the statistical literature on skewness and provide a comprehensive framework for its assessment.
We present a new measure of skewness, based on a relative comparison between above average and below average returns.
We show that this measure represents a valid complement to the state of the art.
2020
 Fundamentalists heterogeneity and the role of the sentiment indicator
[Working paper]
Campisi, G.; Muzzioli, S.
abstract
This paper is a contribution to the literature on the role of the sentiment indices in heterogeneous asset pricing models. We propose a new sentiment index in a financial market where we assume that transactions take place between two groups of fundamentalists that differentiate on the perception of the fundamental value. We assume that the fraction of fundamentalists in the two groups depends on the sentiment index. After studying the analytical properties of the deterministic discrete dynamical system we compare the new index with a previous index introduced in financial literature. For this purpose, by adding stochastic components to the fundamentalist' demands, we measure the performance of our model under different sentiment indices and we test its explanatory power to reproduce the stylized facts of financial data relying on the S&P500 index.
2020
 Investor sentiment and trading behavior
[Articolo su rivista]
Campisi, G.; Muzzioli, S.
abstract
The aim of this paper is to model trading decisions of financial investors based on a sentiment index. For this purpose, we analyze a dynamical model, which includes the sentiment index in the agents' trading behavior. We consider the setup of a discrete dynamical system, assuming that in financial markets, transactions take place between two groups of fundamentalists that differ in their perception of fundamental value. This assumption is motivated by a degree of uncertainty about the true fundamental value. The proportion of fundamentalists in the two groups is assumed to depend on the sentiment index. The sentiment index used is related to the risk asymmetry index, enabling us to consider both the variance and the asymmetry of the prediction error between the two groups of fundamentalists. We identify the equilibria of the model and conduct a numerical analysis in order to capture stylized facts documented empirically in the financial literature.
2020
 Investor sentiment and trading behavior
[Working paper]
Campisi, G.; Muzzioli, S.
abstract
The aim of this paper is to model trading decisions of financial investors based on a sentiment index. For this purpose, we analyse a dynamical model which includes the sentiment index in the agents' trading behavior. We consider the set up of a Discrete Dynamical System, assuming that in financial markets transactions take place between two groups of fundamentalists that differ in their perception of fundamental value. The proportion of fundamentalists in the two groups is assumed to depend on the sentiment index. The sentiment index used is related to the risk asymmetry index (RAX) enabling us to consider both the variance and the asymmetry of the prediction error between the two groups of fundamentalists.
We identify the equilibria of the model and conduct a numerical analysis in order to capture stylized facts documented empirically in the financial literature.
2020
 Moment risk premia and the crosssection of stock returns in the European stock market
[Articolo su rivista]
Elyasiani, Elyas; Gambarelli, Luca; Muzzioli, Silvia
abstract
This article investigates whether volatility, skewness, and kurtosis risks are priced in the European stock market and assess the signs and the magnitudes of the corresponding risk premia. To this end, we adopt two approaches: a modelfree approach based on swap contracts, and a modelbased approach built on portfoliosorting techniques. A number of results are obtained. First, stocks with high exposure to innovations in implied market volatility (skewness) exhibit low (high) returns on average. Second, the estimated premium for bearing market volatility (skewness) risk is negative (positive), robust to the two approaches employed, and statistically and economically significant. Third, in contrast with studies on the US stock market, we identify the existence of a size premium in the European stock market: small capitalization stocks earn higher returns than high capitalization stocks.
2020
 Option implied moments obtained through fuzzy
regression
[Articolo su rivista]
Muzzioli, Silvia; Gambarelli, Luca; De Baets, Bernard
abstract
The aim of this paper is to investigate the potential of fuzzy regression methods for
computing more reliable estimates of higherorder moments of the riskneutral distribution.
We improve upon the formula of Bakshi et al. (RFS 16(1):101–143, 2003),
which is used for the computation ofmarket volatility and skewness indices (such as the
VIX and the SKEW indices traded on the Chicago Board Options Exchange), through
the use of fuzzy regression methods. In particular, we use the possibilistic regression
method of Tanaka, Uejima and Asai, the least squares fuzzy regression method of
Savic and Pedrycz and the hybrid method of Ishibuchi and Nii.We compare the fuzzy
moments with those obtained by the standard methodology, based on the Bakshi et al.
(2003) formula, which relies on an exante choice of the option prices to be used and
cubic spline interpolation.We evaluate the quality of the obtained moments by assessing
their forecasting power on future realized moments. We compare the competing
forecasts by using both the Model Confidence Set and Mincer–Zarnowitz regressions.
We find that the forecasts for skewness and kurtosis obtained using fuzzy regression
methods are closer to the subsequently realized moments than those provided by the
standard methodology. In particular, the lower bound of the fuzzy moments obtained
using the Savic and Pedrycz method is the best ones. The results are important for
investors and policy makers who can rely on fuzzy regression methods to get a more
reliable forecast for skewness and kurtosis.
2020
 The use of option prices to assess the skewness risk premium
[Articolo su rivista]
Elyasiani, E.; Gambarelli, L.; Muzzioli, S.
abstract
The aims of this study are twofold. First, to determine the sign and magnitude of the skewness risk premium (SRP) in the Italian index option market using two procedures: (i) skewness swap contracts, (ii) option trading strategies consisting of positions in options and their underlying assets. Second, to investigate the term structure of the SRP for 30, 60 and 90day maturities to provide investors with a proper time horizon for profitable skewness trading strategies. Several results are obtained. First, the SRP, defined as the difference between the physical and the riskneutral skewness, is positive and statistically and economically significant. These findings indicate that the SRP does exist, it is positive in sign, and it can be quantified. Second, the SRP is higher in magnitude for shortterm maturity (€35 for the 30day maturity) and lower for 60day and 90day maturities (both about €27). Third, skewness trading strategies confirm our finding of a positive and economically significant SRP. Fourth, a strategy that sells outofthemoney puts is more profitable for mediumterm maturities compared to shortterm maturities. A strategy that takes a long position on outofthemoney calls, and a short position on outofthemoney puts, yields a higher return, if nearterm options are used.
2019
 Construction and properties of volatility indices for Austria, Finland and Spain
[Working paper]
Campisi, Giovanni; Muzzioli, Silvia
abstract
The volatility index of the Chicago Board Options Exchange (VIX) is the first to have been introduced and it has attracted international imitators worldwide since it is considered as a barometer of investor fear. The aim of the paper is threefold. First, by following the VIX methodology, we construct a volatility index for three European countries (Austria, Finland and Spain) that do not have yet that piece of market information for investors. Second, we investigate the properties of the new volatility indices. In particular, we test their ability to act as fear indicators and as predictors of future returns. Moreover, we shed light on the term structure of the proposed volatility indices, by computing spot and forward implied volatility indices for different time to maturities (30, 60 and 90 days). Our results indicate that volatility indices are useful not only for investors to improve their trading decisions, but also for policy makers to choose the appropriate economic measure to guarantee stability in the market.
2019
 How do normalization schemes affect net spillovers? A replication of the Diebold and Yilmaz (2012) study
[Articolo su rivista]
Giuseppe Caloia, Francesco; Cipollini, Andrea; Muzzioli, Silvia
abstract
This paper replicates the Diebold and Yilmaz (2012) study on the connectedness ofthe commodity market and three other financial markets: the stock market, the bond market, and the FX market, based on the Generalized Forecast Error Variance Decomposition, GEFVD. We show that the net spillover indices (of directional connectedness), used to assess the net contribution of one market to overall risk in the system, are sensitive to the normalization scheme applied to the GEFVD. We show that, considering data generating processes characterized by different degrees of persistence and covariance, a scalarbased normalization of the Generalized Forecast Error Variance Decomposition is preferable to the row normalization suggested by Diebold and Yilmaz since it yields net spillovers free of sign and ranking errors.
2019
 Riskasymmetry indices in Europe
[Working paper]
Gambarelli, Luca; Muzzioli, Silvia
abstract
The objectives of this study are threefold. First, we introduce for the first time a skewness index (SKEW) for each European country. Second, we compute an alternative measure of asymmetry (RAX) based on corridor implied volatilities to assess whether it outperforms the standard skewness index in measuring tail risk. Third, we investigate the properties of the proposed indices by uncovering the contemporaneous linear relationship among skewness, volatility, and returns and the information content of skewness on future returns, which is highly debated in the literature. Last, we propose two aggregate indices of asymmetry to monitor the risk of the EU financial market as a whole. To deal with the limited availability of optionbased data for European countries, that represent the main obstacle for the construction of such indices in the EU, we delineate a countryspecific procedure.
Several results are obtained. First, all the asymmetry indices are on average higher than 100, indicating that the riskneutral distribution is in general leftskew for the 12 EU countries under investigation. Second, the relation between changes in asymmetry indices and contemporaneous market returns in positive, indicating that asymmetry indices are not able to capture the same fear effect captured by volatility indices. Third, the results for the relationship between asymmetry and volatility (future returns) are mixed both in terms of magnitude and significance and do not allow us to delineate general conclusions. Last, the aggregate asymmetry index based on the RAX methodology is the only one able to forecast future negative returns for all the EU countries in our dataset when it reaches very high levels.
2019
 The Future of Fuzzy Sets in Finance: New Challenges in Machine Learning and Explainable AI
[Capitolo/Saggio]
Muzzioli, Silvia
abstract
Traditional statistical analysis is oriented towards finding linear relationships between the variables under investigation, often accompanied by strict assumptions about the problem and data distributions. Moreover, traditional analysis endorses data reduction as much as possible before modeling, and, as a result, part of the original information is lost. On the other hand, machine learning does not impose rigid preassumptions about the problem and data distributions since the underlying ratio is to “learn from data”, without the need for data reduction or a priori knowledge before the learning. For these reasons machine learning has experienced a rapid dissemination in a large number of sectors including healthcare, finance, transportation, retail and social media services industry. Machine learning is the core technology of the new age of AI applications. Machine learning methods offer tremendous benefits, but are limited by their opaqueness, nonintuitiveness and difficulty to understand.
In finance in particular, machine learning methods have played a crucial role in improving the forecasting ability of financial models and trading systems, due to their ability to process a large amount of data and the peculiarity of capturing also nonlinear relationships between variables. In recent years, the availability of sample data at very high frequencies (intraday or tick by tick) resulted in a fertile domain for their application, especially in the coding of indicators and patterns of technical analysis. Deep learning systems are the most advanced form of machine learning. They can match humans in recognizing images or driving a car, but why they come up with the solutions remains difficult to tell exactly. Businesses would have used machine learning more widely if they could understand how machines come up with their recommendations on trading, fraud detection, insurance and banking.
A challenge for AI in finance is the need to analyze and aggregate a large amount of information obtained from different sources. In the financial literature, the use of artificial intelligence (AI) and machine learning techniques is often limited to the coding of technical analysis indicators (such as moving averages or the flag pattern) for trading strategy purposes. As pointed out in [1], most of the contributions investigating machine learning methods in financial markets propose trading strategies that rely mainly on technical analysis and focus on a single stock market or market index. Moreover, financial markets are generally treated as compartmentalized and most of the contributions investigate only a single market or a specific asset [2]. On the other hand, investors and regulators need comprehensive risk measures able to aggregate and synthesize different types of information in a single indicator.
Another challenge is represented by the increasing dominance of computerized trading, which may cause more volatility during market downturns. The rising frequency of 'flash crashes' across many major markets, the increasing incidents of volatility such as the VIX spike on Feb. 5, 2018, the 10year Treasury bond on Oct. 15, 2014, and the British pound on Oct. 6, 2016, are an important early warning sign that machines have to be closely supervised and understood. New measures and tools to control the volatility of financial markets [3,4,5] should be developed.
The semantic properties of linguistic fuzzy sets, their good coverage even in the case of lack of data, their management of the uncertainty, especially in Big Data, make them a very interesting tool for nowadays applications, especially when the practitioner need to understand why a given decision has been made.
2019
 Towards a fuzzy index of skewness
[Capitolo/Saggio]
Muzzioli, Silvia; Gambarelli, L.; De baets, Bernard
abstract
The aim of this paper is to investigate the potential of fuzzy regression methods for computing a measure of skewness for the market. A quadratic version of the Ishibuchi and Nii hybrid fuzzy regression method is used to estimate the third order moment. The obtained fuzzy estimates are compared with the one provided by standard market practice. The proposed approach allows us to cope with the limited availability of data and to use all the information that is present in the market.
In the Italian market, the results suggest that the fuzzyregression based skewness measure is closer to the subsequently realized measure of skewness than the one provided by the standard methodology. In particular, the upper bound of the Ishibuchi and Nii method provides the best forecast. The results are important for investors and policy makers who can rely on fuzzy regression methods to get a more reliable forecast of skewness.
2018
 Asymmetric semivolatility spillover effects in EMU stock markets
[Articolo su rivista]
Giuseppe Caloia, Francesco; Cipollini, Andrea; Muzzioli, Silvia
abstract
The aim of this paper is to quantify the strength and the direction of semivolatility spillovers between five EMU stock markets over the 20002016 period. We use upside and downside semivolatilities as proxies for downside risk and upside opportunities. In this way, we aim to complement the literature, which has focused mainly on the contemporaneous correlation between positive and negative returns, with the evidence of asymmetry also in semivolatility transmission. For this purpose, we apply the Diebold and Yilmaz (2012) methodology, based on a generalized forecast error variance decomposition, to downside and upside realized semivolatility series. While the analysis of Diebold and Yilmaz (2012) is based on a stationary VAR, we take into account the longmemory behaviour of the series, by using the multivariate extension of the HAR model (named VHAR model). Moreover, we cast light on how the choice of the normalization scheme can bias the netspillover computation in a full sample as well as in a rolling sample analysis.
2018
 INDICES FOR FINANCIAL MARKET VOLATILITY OBTAINED THROUGH FUZZY REGRESSION
[Articolo su rivista]
Muzzioli, Silvia; Gambarelli, Luca; De Baets, Bernard
abstract
The measurement of volatility is of fundamental importance in finance. Standard market practice adopted for volatility estimation from option prices leads to a considerable loss of information and the introduction of an element of arbitrariness in the volatility index computation. We propose to resort to fuzzy regression methods in order to include all the available information from option prices and obtain an informative volatility index. In fact, the obtained fuzzy volatility indices do not only offer a most possible value, but also a lower and an upper bound for the interval of possible values, providing investors with an additional source of information. We also propose a defuzzification procedure in order to select a representative value within this interval. Moreover, we investigate the occurrence of truncation and discretization errors in the volatility index computation by resorting to an interpolationextrapolation method. We also test the forecasting power of each volatility index on future realized volatility.
2018
 On the financial connectedness of the commodity market: a replication of the Diebold and Yilmaz (2012) study
[Working paper]
Caloia, F.; Cipollini, A.; Muzzioli, S.
abstract
In this paper we replicate the Diebold and Yilmaz (2012) study on the connectedness of the Commodity market and three other financial markets: the stock market, the bond market, and the FX market. We show that both the row and the column normalization schemes of the Generalized Forecast Error Variance Decomposition, suggested by the authors, lead to inaccurate measures of net contribution to risk transmission, in terms of ranking and sign. We show that, considering data generating processes characterized by different degrees of comovement and persistence, a scalar based normalization of the Generalized Forecast Error Variance Decomposition yields consistent (free of sign and ranking errors) net spillovers.
2018
 Risk aversion connectedness in five European countries
[Articolo su rivista]
Cipollini, Andrea; Lo Cascio, Iolanda; Muzzioli, Silvia
abstract
In this paper we compute an aggregate index of risk aversion and indices of vulnerability and the contribution to systemic risk aversion for five European countries. The variance risk premium proxies risk aversion. The contribution to the literature is twofold. First, this is the first study estimating not only the common component, but also indices of directional connectedness among variance risk premia. Second, it is the first to estimate the interconnections by means of a FIVAR model, in order to account for long memory. Our analysis indicates measures of total and directional connectedness unlike those that would be obtained with the use of a short memory VAR. These differences arise when the focus is on market turmoil periods and on forecast horizons of thirty days. Future research evaluating spillovers among long memory series can benefit from our results. Policymakers should take these interconnections into account when adopting effective macroeconomic policies.
2018
 The RiskAsymmetry Index as a New Measure of Risk
[Articolo su rivista]
Elyasiani, E.; Gambarelli, L.; Muzzioli, S.
abstract
The aim of this paper is to propose a simple and unique measure of risk that subsumes the
conflicting information contained in volatility and skewness indices and overcomes the
limitations of these indices in accurately measuring future fear or greed in the market. To this
end, the concept of upside and downside corridor implied volatility, which accounts for the
asymmetry in the riskneutral distribution, is exploited. The riskasymmetry index is intended to
capture the investors’ pricing asymmetry towards upside gains and downside losses. The results
show that the proposed riskasymmetry index can play a crucial role in predicting future returns,
at various forecast horizons, since it subsumes the information embedded in both the volatility
and skewness indices. Furthermore, the riskasymmetry index is the only index that, at very high
values, possesses the ability to clearly highlight a risky situation for the aggregate stock market.
2018
 The properties of a skewness index and its relation with volatility and returns
[Working paper]
Elyasiani, E.; Gambarelli, L.; Muzzioli, Silvia
abstract
The objective of this study is threefold. First, we investigate the properties of a skewness index in order to determine whether it captures fear (fear of losing money), or greed in the market (fear of losing opportunities). Second, we uncover the combined relationship among skewness, volatility and returns. Third, we provide further evidence and possible explanations for the relationship between skewness and future returns, which is highly debated in the literature. The stock market investigated is the Italian one, for which a skewness index is not traded yet. The methodology proposed for the construction of the Italian skewness index can be adopted for other European and nonEuropean countries characterized by a limited number of option prices traded.
Several results are obtained. First, we find that in the Italian market the skewness index acts as measures of market greed, as opposed to market fear. Second, for almost 70% of the daily observations, the implied volatility and the skewness index move together but in opposite directions. Increases (decreases) in volatility and decreases (increases) in the skewness index are associated with negative (positive) returns. Last, we find strong evidence that positive returns are reflected both in a decrease in the implied volatility index and in an increase in the skewness index the following day. Implications for investors and policy makers are drawn.
2018
 The use of option prices in order to evaluate the skewness risk premium
[Working paper]
Elyasiani, E.; Gambarelli, L.; Muzzioli, Silvia
abstract
2017
 Fuzzy approaches to option price modelling
[Articolo su rivista]
Muzzioli, Silvia; De Baets, Bernard
abstract
The aim of this paper is to review the literature that has addressed direct and inverse problems in option pricing in a fuzzy setting. In a direct problem, the stochastic process for the underlying asset is assumed and the option prices are derived by noarbitrage or equilibrium conditions. In an inverse problem, the option prices are taken as given and used to infer the underlying asset process. Models are divided into discretetime and continuoustime ones. Special attention is paid to real options, a particular class of nonfinancial options that are used to evaluate real investments. Directions for future research are outlined. In particular in inverse problems, there is still room for promising research, both in discrete time and in continuous time. Moreover, given that many proposed methods remain difficult to use in practice, there is mainly the need to apply the fuzzy models obtained on real market data and to compare the results with nonfuzzy techniques in order to assess the usefulness and the improvements in the modeling of imprecise data with fuzzy sets and fuzzy random variables.
2017
 The information content of corridor volatility measures during calm and turmoil periods
[Articolo su rivista]
Elyasiani, E.; Gambarelli, L.; Muzzioli, S.
abstract
Measurement of volatility is of paramount importance in finance because of the effects on risk measurement and risk management. Corridor implied volatility measures allow us to disentangle the volatility of positive returns from that of negative returns, providing investors with additional information beyond standard market volatility. The aim of the paper is twofold. First, to propose different types of corridor implied volatility and some combinations of them as risk indicators, in order to provide useful information about investors’ sentiment and future market returns. Second, to investigate their usefulness in prediction of market returns under different market conditions (with a particular focus on the subprime crisis and the European debt crisis). The data set consists of daily index options traded on the Italian market and covers the 20052014 period. We find that upside corridor implied volatility measure embeds the highest information content about contemporaneous market returns, claiming the superiority of call options in measuring current sentiment in the market. Moreover, both upside and downside volatilities can be considered as barometers of investors’ fear. The volatility measures proposed have forecasting power on future returns only during high volatility periods and in particular during the European debt crisis. The explanatory power on future market returns improves when two of the proposed volatility measures are combined together in the same model.
2017
 Towards a Fuzzy Volatility Index for the Italian Market
[Relazione in Atti di Convegno]
Muzzioli, Silvia; Gambarelli, Luca; De Baets, Bernard
abstract
The measurement of volatility is of fundamental
importance in finance. The standard market practice adopted
for the computation of a volatility index imposes to discard
some option prices quoted in the market, resulting in a
considerable loss of information. To overcome this drawback,
we propose to resort to fuzzy regression methods in order to
include all the available information and obtain an informative
volatility index for the Italian stock market.
2016
 A note on normalization schemes: The case of generalized forecast error variance decompositions
[Working paper]
Caloia, F. G.; Cipollini, A.; Muzzioli, S.
abstract
The aim of this paper is to propose new normalization schemes for the values obtained from the generalized forecast error variance decomposition, in order to obtain more reliable net spillover measures. We provide a review of various matrix normalization schemes used in different application domains. The intention is to contribute to the financial econometrics literature aimed at building a bridge between different approaches able to detect spillover effects, such as spatial regressions and network analyses. Considering DGPs characterized by different degrees of correlation and persistence, we show that the popular row normalization scheme proposed by Diebold and Yilmaz (2012), as well as the alternative column normalization scheme, may lead to inaccurate measures of net contributions (NET spillovers) in terms of risk transmission. Results are based on simulations and show that the number of errors increases as the correlation between the variable increases. The normalization schemes we suggest overcome these limits.
2016
 Fear or greed? What does a skewness index measure?
[Working paper]
Elyasiani, E.; Gambarelli, L.; Muzzioli, S.
abstract
The ℎ ℎ (CBOE) SKEW index is designed to capture investors’ fear in the US stock market. In this paper we pursue two objectives. First, we investigate the properties of the CBOE SKEW index in order to assess whether it captures fear or greed in the market. Second, we introduce and compare three measures of asymmetry of the Italian index options return distribution. These measures include: (i) the CBOE SKEW index adapted to the Italian market (we call it ITSKEW) and (ii) two modelfree measures of skewness based on comparison of a bear and a bull index. Finally, we explore the existence and sign of the skewness risk premium. Several results are obtained. First, the Italian skewness index (ITSKEW) presents many advantages compared to the two modelfree measures: it has a significant contemporaneous relation with market index returns and with modelfree implied volatility. Both the ITSKEW and the CBOE SKEW indices act as measures of market greed (the opposite of market fear), since returns react positively to an increase in the skewness indices. Trading strategies show that the Italian market is characterized by a significant positive skewness risk premium.
2016
 Forecasting and pricing powers of optionimplied tree models: Tranquil and volatile market conditions
[Working paper]
Elyasiani, E.; Muzzioli, S.; Ruggieri, A.
abstract
The aims of this paper are twofold. First, to investigate the accuracy of different optionimplied trees in pricing European options in order to assess the power of implied trees in replicating the market information. Second, to compare deterministic volatility implied trees and stochastic implied volatility models (Bakshi et al. (2003)) in assessing the forecasting power of implied moments on subsequently realised moments, and ascertaining the existence, magnitude and sign of variance, skewness, and kurtosis riskpremia. The analysis is carried out using the Italian daily market data covering the period 20052014. This enables us to contrast the pricing performance of implied trees and to assess the magnitude and sign of risk premia in both a tranquil and a turmoil period. The findings are as follows. First, the pricing performance of the Enhanced Derman and Kani (EDK, Moriggia et al. 2009) model is superior to that of the Rubinstein (1994) model. This superiority is stronger especially in the high volatility period due to a better estimation of the left tail of the distribution describing bad market conditions. Second, the Bakshi et al. (2003) formula is the most accurate for forecasting skewness and kurtosis, while for variance it yields upwardly biased forecasts. All models agree on the signs of the risk premia: negative for variance and kurtosis, and positive for skewness, but differ in magnitude. Overall, the results suggest that selling (buying) variance and kurtosis (skewness) is profitable in both high and low volatility periods.
2016
 Moment Risk Premia and the CrossSection of Stock Returns
[Working paper]
Elyasiani, E.; Gambarelli, L.; Muzzioli, S.
abstract
The aim of this paper is to assess the existence and the sign of moment risk premia. To this end, we use methodologies ranging from swap contracts to portfolio sorting techniques in order to obtain robust estimates. We provide empirical evidence for the European stock market for the 20082015 time period. Evidence is found of a negative volatility risk premium and a positive skewness risk premium, which are robust to the different techniques and cannot be explained by common riskfactors such as market excess return, size, booktomarket and momentum. Kurtosis risk is not priced in our dataset. Furthermore, we find evidence of a positive risk premium in relation to the firm’s size.
2016
 The RiskAsymmetry Index
[Working paper]
Elyasiany, E.; Gambarelli, L.; Muzzioli, S.
abstract
The aim of this paper is to propose a simple and unique measure of risk, that subsumes the conflicting information in volatility and skewness indices and overcomes the limits of these indices in correctly measuring future fear or greed in the market. To this end, we exploit the concept of upside and downside corridor implied volatility, which accounts for the asymmetry in riskneutral distribution, i.e. the fact that investors like positive spikes in returns, while they dislike negative ones. We combine upside and downside implied volatilities in a single asymmetry index called the riskasymmetry index (ܴܣ .(ܺThe riskasymmetry index ሺܴܣܺሻ plays a crucial role in predicting future returns, since it subsumes all the information embedded in both the Italian skewness index ܫܵܶܧܭ ܹand the Italian volatility index (ܫܸܶܫ .(ܺThe ܴܣ ܺindex is the only index that is able to indicate (when reaching very high values) a clearly risky situation for the aggregate stock market, which is detected neither by the ܫܸܶܫ ܺindex nor by the ܫܵܶܧܭ ܹindex.
2015
 A comparison of fuzzy regression methods for the estimation of the implied volatility smile function
[Articolo su rivista]
Muzzioli, Silvia; Ruggieri, A.; De Baets, B.
abstract
The information content of option prices on the underlying asset has a special importance in finance. In particular, with the use of option implied trees, market participants may price other derivatives, estimate and forecast volatility (see e.g. the volatility index VIX), or higher moments of the underlying asset distribution. A crucial input of option implied trees is the estimation of the smile (implied volatility as a function of the strike price), which boils down to fitting a function to a limited number of existing knots. However, standard techniques require a onetoone mapping between volatility and strike price, which is not met in the reality of financial markets, where, to a given strike price, two different implied volatilities are usually associated (coming from different types of options: call and put).
In this paper we compare the widely used methodology of discarding some implied volatilities and interpolating the remaining knots with cubic splines, to a fuzzy regression approach which does not require an apriori choice of implied volatilities. To this end, we first extend some linear fuzzy regression methods to a polynomial form and we apply them to the financial problem. The fuzzy regression methods used range from the possibilistic regression method of Tanaka et al.[28], to theleast squares fuzzy regression method of Savic and Pedrycz [27]and to the hybrid method of Ishibuchi and Nii[11].
2015
 Financial connectedness among European volatility risk premia
[Working paper]
Cipollini, A.; Lo Cascio, I.; Muzzioli, S.
abstract
In this paper we use the Diebold Yilmaz (2009 and 2012) methodology to estimate the contribution and the vulnerability to systemic risk of volatility risk premia for five European stock markets: France, Germany, UK, Switzerland and the Netherlands. The volatility risk premium, which is a proxy of risk aversion, is measured by the difference between the implied volatility and expected realized volatility of the stock market for next month. While Diebold and Yilmaz focus is on the forecast error variance decomposition of stock returns or range based volatilities employing a stationary VAR in levels, we account for the (locally) long memory stationary properties of the levels of volatility risk premia series. Therefore, we estimate and invert a Fractionally Integrated VAR model to compute the cross forecast error variance shares necessary to obtain the index of total and directional connectedness.
2015
 The optimal corridor for implied volatility: From periods of calm to turmoil
[Articolo su rivista]
Muzzioli, Silvia
abstract
Corridor implied volatility is obtained from modelfree impliedvolatility by truncating the integration domain between two barriers. Empirical evidence on volatility forecasting in various marketspoints to the utility of trimming the riskneutral distribution ofthe underlying stock price, in order to obtain unbiased measuresof future realized volatility. The aim of this paper is to investigatethe optimal corridor of strike prices for volatility forecasting in theItalian market, by analyzing numerous symmetric and asymmetric corridors in a dataset for the years 2005–2010 spanning both arelatively calm period and a period of turmoil. The results indicatethat put prices, providing information on the probability of a downturn of the underlying asset, provide the best indication of futurerealized volatility, particularly in a period of turmoil.
2015
 Towards a skewness index for the Italian stock market
[Working paper]
Elyasiani, E.; Gambarelli, L.; Muzzioli, S.
abstract
The present paper is a first attempt of computing a skewness index for the Italian stock market. We compare and contrast different measures of asymmetry of the distribution: an index computed with the CBOE SKEW index formula and two other asymmetry indexes, the SIX indexes, as proposed in Faff and Liu (2014). We analyze the properties of the skewness indexes, by investigating their relationship with modelfree implied volatility and the returns on the underlying stock index. Moreover, we assess the profitability of skewness trades and disentangle the contribution of the left and the right part of the risk neutral distribution to the profitability of the latter strategies. The data set consists of FTSE MIB index options data and covers the years 20112014, allowing us to address the behavior of skewness measures both in bullish and bearish market periods.
We find that the Italian SKEW index presents many advantages with respect to other asymmetry measures: it has a significant contemporaneous relation with both returns, modelfree implied volatility and has explanatory power on returns, after controlling for volatility. We find a negative relation between volatility changes and changes in the Italian SKEW index: an increase in modelfree implied volatility is associated with a decrease in the Italian SKEW index. Moreover, the SKEW index acts as a measure of market greed, since returns react more negatively to a decrease in the SKEW index (increase in risk neutral skewness) than they react positively to an increase of the latter (decrease in risk neutral skewness).
The results of the paper point to the existence of a skewness risk premium in the Italian market. This emerges both from the fact that implied skewness is more negative than physical one in the sample period and from the profitability of skewness trading strategies. In addition, the higher performance of the portfolio composed by only put options indicates that the mispricing of options is mainly focused on the left part of the distribution.
2015
 Volatility comovements: a timescale decomposition analysis
[Articolo su rivista]
Cipollini, Andrea; Lo Cascio, Iolanda; Muzzioli, Silvia
abstract
In this paper we are interested in detecting contagion from US to European stock market volatilities in the period immediately after the Lehman Brothers collapse. The analysis is based on a factor decomposition of the covariance matrix, in the time and frequency domain, using wavelets. The analysis aims to disentangle two components of volatility contagion (anticipated and unanticipated by the market). Once we control for heteroskedasticity, the results show no evidence of contagion (from the US) in market expectations (coming from implied volatility) and evidence of unanticipated contagion (coming from the volatility risk premium) for almost any European country. Finally, the estimation of a threefactor model specification shows that a European common shock plays an important role in determining volatility comovements mainly in the tranquil period, while in the period of financial turmoil the US common shock is the main driver of volatility comovements.
2014
 Volatility risk premia and financial connectedness
[Working paper]
Cipollini, A.; Lo Cascio, I.; Muzzioli, S.
abstract
In this paper we use the Diebold Yilmaz (2009 and 2012) methodology to construct an index of
connectedness among five European stock markets: France, Germany, UK, Switzerland and the
Netherlands, by using volatility risk premia. The volatility risk premium, which is a proxy of risk
aversion, is measured by the difference between the implied volatility and expected realized
volatility of the stock market for next month. While Diebold and Yilmaz focus is on the forecast
error variance decomposition of stock returns or range based volatilities employing a stationary
VAR in levels, we account for the (locally) long memory stationary properties of the levels of
volatility risk premia series. Therefore, we estimate and invert a Fractionally Integrated VAR
model to compute the cross forecast error variance shares necessary to obtain the index of total
connectedness and the net contribution of each series to total connectedness.
The results show that, over January 2000August 2013, the index of total connectedness among
volatility risk premia has been relatively stable with an increasing role played by France and with a
positive (but decreasing) role played by Germany and the Netherlands. Non EMU countries such as
the UK and Switzerland are negative net contributors to the index.
2014
 Volatility risk premia and financial connectedness
[Working paper]
Cipollini, A.; Lo Cascio, I.; Muzzioli, S.
abstract
In this paper we use the Diebold Yilmaz (2009 and 2012) methodology to construct an index of
connectedness among five European stock markets: France, Germany, UK, Switzerland and the
Netherlands, by using volatility risk premia. The volatility risk premium, which is a proxy of risk
aversion, is measured by the difference between the implied volatility and expected realized
volatility of the stock market for next month. While Diebold and Yilmaz focus is on the forecast
error variance decomposition of stock returns or range based volatilities employing a stationary
VAR in levels, we account for the (locally) long memory stationary properties of the levels of
volatility risk premia series. Therefore, we estimate and invert a Fractionally Integrated VAR
model to compute the cross forecast error variance shares necessary to obtain the index of total
connectedness and the net contribution of each series to total connectedness.
The results show that, over January 2000August 2013, the index of total connectedness among
volatility risk premia has been relatively stable with an increasing role played by France and with a
positive (but decreasing) role played by Germany and the Netherlands. Non EMU countries such as
the UK and Switzerland are negative net contributors to the index.
2013
 A comparative assessment of different fuzzy regression methods for volatility forecasting
[Articolo su rivista]
Muzzioli, Silvia; B., De Baets
abstract
The aim of this paper is to compare different fuzzy regression methods in the assessment of the information content on future realised volatility of optionbased volatility forecasts. These methods offer a suitable tool to handle both imprecision of measurements and fuzziness of the relationship among variables. Therefore, they are particularly useful for volatility forecasting, since the variable of interest (realised volatility) is unobservable and a proxy for it is used. Moreover, measurement errors in both realised volatility and volatility forecasts may affect the regression results.
We compare both the possibilistic regression method of Tanaka, Uejima and Asai (1982) and the least squares fuzzy regression method of Savic and Pedrycz (1991). In our case study, based on intradaily data of the DAXindex options market, both methods have proved to have advantages and disadvantages. Overall, among the two methods, we prefer the Savic and Pedricz (1991) method, since it contains as special case (the central line) the ordinary least squares regression, is robust to the analysis of the variables in logarithmic terms or in levels, and provides sharper results than the Tanaka, Uejima and Asai (1982) method.
2013
 A comparison of fuzzy regression methods for the estimation of the implied volatility smile function
[Working paper]
Muzzioli, S.; Ruggieri, A.; De Baets, B.
abstract
The information content of option prices on the underlying asset has a special importance in finance. In particular, with the use of option implied trees, market participants may price other derivatives, estimate and forecast volatility (see e.g. the volatility index VIX), or higher moments of the underlying asset distribution. A crucial input of option implied trees is the estimation of the smile (implied volatility as a function of the strike price), which boils down to fitting a function to a limited number of existing knots. However, standard techniques require a onetoone mapping between volatility and strike price, which is not met in the reality of financial markets, where, to a given strike price, two different implied volatilities are usually associated (coming from different types of options: call and put). In this paper we compare the widely used methodology of discarding some implied volatilities and interpolating the remaining knots with cubic splines, to a fuzzy regression approach which does not require an apriori choice of implied volatilities. To this end, we first extend some linear fuzzy regression methods to a polynomial form and we apply them to the financial problem. The fuzzy regression methods used range from the possibilistic regression method of Tanaka, Uejima and Asai [14], the least squares fuzzy regression method of Savic and Pedrycz [13] and the hybrid method of Ishibuchi and Nii [4].
2013
 Option implied trees and implied moments
[Working paper]
Muzzioli, S.; Ruggieri, A.
abstract
Implied trees are simple nonparametric discretizations of one or twodimension diffusions, aimed at introducing nonconstant volatility in an option pricing model. The aim of the paper is twofold. First we investigate the ability of different option implied trees in pricing European options. Second, we compare the implied moments obtained with the use of option implied trees with the risk–neutral moments obtained with the use of Bakshi et al. (2003) formula and with realised physical moments. The comparison is pursued in the Italian market by analysing a data set which covers the years 20052009 and span both a relatively tranquil and a turmoil period. Keywords:
2013
 The forecasting performance of corridor implied volatility in the Italian market
[Articolo su rivista]
Muzzioli, Silvia
abstract
Corridor implied volatility introduced in Carr and Madan (1998) and recently implemented in Andersen and Bondarenko (2007) is obtained from modelfree implied volatility by truncating the integration domain between two barriers. Corridor implied volatility is implicitly linked with the concept that the tails of the riskneutral distribution are estimated with less precision than central values, due to the lack of liquid options for very high and very low strikes. However, there is no golden choice for the barrier levels, which are likely to change depending on the underlying asset risk neutral distribution. The latter feature renders its forecasting performance mainly an empirical question.
The aim of the paper is to investigate the forecasting performance of corridor implied volatility by choosing different corridors with symmetric and asymmetric cuts, and compare the results with the preliminary findings in Muzzioli (2010b). Moreover, we shed light on the information content of different parts of the risk neutral distribution of the stock price, by using a modelindependent approach based on corridor measures. To this end we compute both realized and modelfree variance measures accounting for both falls and increases in the underlying asset price.
The forecasting performance of volatility measures is evaluated both in a statistical and an economic setting. The economic significance is assessed by employing trading strategies based on deltaneutral straddles. The comparison is pursued by using intraday synchronous prices between the options and the underlying asset.
2013
 The information content of option based forecasts of volatility: evidence from the Italian stock market
[Articolo su rivista]
Muzzioli, Silvia
abstract
The aim of this paper is to comprehensively compare optionbased measures of volatility, with the ultimate plan of devising a new volatility index for the Italian stock market. The performance of the different implied volatility measures in forecasting future volatility is evaluated both in a statistical and in an economic setting. The properties of the implied volatility measures are also explored, by looking at both the contemporaneous relationship between implied volatility changes and market returns and the usefulness of the proposed index in forecasting future market returns.
The results of the paper are of practical importance for both policymakers and investors. The volatility index, based on corridor measures, could be used to forecast market volatility, for Value at Risk purposes, in order to determine trading strategies on the underlying index and as an early warning for future market conditions.
2013
 The optimal corridor for implied volatility: from calm to turmoil periods
[Working paper]
Muzzioli, S.
abstract
Corridor implied volatility is obtained from modelfree implied volatility by truncating the integration domain between two barriers. Empirical evidence on volatility forecasting, in various markets, points to the utility of trimming the riskneutral distribution of the underlying stock price, in order to obtain unbiased measures of future realised volatility (see e.g. [9], [3]). The aim of the paper is to investigate, both in a statistical and in an economic setting, the optimal corridor of strike prices to use for volatility forecasting in the Italian market, by analysing a data set which covers the years 20052010 and span both a relatively tranquil and a turmoil period.
2013
 Volatility comovements: a time scale decomposition analysis
[Working paper]
Cipollini, A.; Lo Cascio, I.; Muzzioli, S.
abstract
In this paper we investigate shortrun comovements before and after the Lehman Brothers’ collapse among the volatility series of US and a number of European countries. The series under investigation (implied and realized volatility) exhibit longmemory and, in order to avoid missspecification errors related to the parameterization of a long memory multivariate model, we rely on wavelet analysis.
More specifically, we retrieve the time series of wavelet coefficients for each volatility series for high frequency scales, using the Maximal Overlapping Discrete Wavelet transform and we apply Maximum Likelihood for a factor decomposition of the shortrun covariance matrix.
The empirical evidence shows an increased interdependence in the postbreak period and points at an increasing (decreasing) role of the common shock underlying the dynamics of the implied (realized) volatility series, once we move from the 24 days investment time horizon to the 816 days. Moreover, there is evidence of contagion from the US to Europe immediately after the Lehman Brothers’ collapse, only for realized volatilities over an investment time horizon between 8 and 16 days.
2012
 Corridor implied volatility and the variance risk premium in the Italian market
[Relazione in Atti di Convegno]
Muzzioli, Silvia
abstract
Corridor implied volatility introduced in Carr and Madan (1998) and recently implemented in Andersen and Bondarenko (2007) is obtained from modelfree implied volatility by truncating the integration domain between two barriers. Corridor implied volatility is implicitly linked with the concept that the tails of the riskneutral distribution are estimated with less precision than central values, due to the lack of liquid options for very high and very low strikes. However, there is no golden choice for the barriers levels’, which will probably change depending on the underlying asset risk neutral distribution. The latter feature renders its forecasting performance mainly an empirical question.The aim of the paper is twofold. First we investigate the forecasting performance of corridor implied volatility by choosing different corridors with symmetric and asymmetric cuts, and compare the results with the preliminary findings in Muzzioli (2010b). Second, we examine the nature of the variance risk premium and shed light on the information content of different parts of the risk neutral distribution of the stock price, by using a modelindependent approach based on corridor measures. To this end we compute both realised and modelfree variance measures which accounts for drops versus increases in the underlying asset price. The comparison is pursued by using intradaily synchronous prices between the options and the underlying asset.
2012
 PutCall Parity and options’ forecasting power
[Articolo su rivista]
Muzzioli, Silvia
abstract
Numerous papers have investigated the forecasting power of BlackScholes volatility versus a time series volatility forecast (see e.g. Poon (2005)). However, as far as we know, little is the evidence about the different information content of implied volatilities obtained from options with different type (call or put). Even if theoretically call and put options with the very same strike price and expiration date should yield the same implied volatility due to no arbitrage considerations, empirically there are many reasons that may cause call and put implied volatilities to differ (see e.g. Hentshle (2003), Buraschi and Jackwerth (2001)).
The aim of this paper is twofold: to investigate how the information content of implied volatility varies according to option type and to compare the latter option based forecasts with historical volatility in order to see if they subsume all the information contained in the latter. Two hypotheses are tested: unbiasedness and efficiency of the different volatility forecasts w.r.t. historical volatility. The investigation is pursued in the Dax index options market. Differently from previous studies, that use settlement prices, we are using the more informative synchronous prices, matched in a one minute interval. This is very important to stress, since our implied volatilities are real “prices”, as determined by synchronous noarbitrage relations.
2011
 Assessing the information content
of optionbased volatility forecasts using
fuzzy regression methods
[Working paper]
Muzzioli, S.; De Baets, Bernard
abstract
Volatility is a key variable for portfolio selection models, option pricing models and risk management techniques. Volatility can be estimated and forecasted by using either historical information or option prices. The present paper focuses on optionbased volatility forecasts for three main reasons. First, for the forward looking nature of optionbased forecasts (as opposed to the backward looking nature of historical information); second, for the average superiority, documented in the literature, of optionbased estimates in forecasting future realized volatility; third, for the widespread use of option prices in the computation of the most important market volatility indexes (see e.g. the VIX index for the Chicago Board Options Exchange). The aim of this paper is to assess the information content on future realised volatility of different optionbased volatility forecasts, through the use of fuzzy regression methods. The latter methods offer a suitable tool to handle both imprecision in measurements and fuzziness of the relationship among variables. Therefore, they are particularly useful for volatility forecasting, since the variable of interest (realised volatility) is unobservable and a proxy for it is used. Moreover, measurement errors in both realised volatility and volatility forecasts may affect the regression results. Fuzzy regression methods have not yet been used in volatility forecasting. Our case study is based on intradaily data on the DAXindex options market.
2011
 Corridor Implied Volatility and the Variance Risk Premium in the Italian Market
[Working paper]
Muzzioli, S.
abstract
Corridor implied volatility introduced in Carr and Madan (1998) and recently implemented in Andersen and Bondarenko (2007) is obtained from modelfree implied volatility by truncating the integration domain between two barriers. Corridor implied volatility is implicitly linked with the concept that the tails of the riskneutral distribution are estimated with less precision than central values, due to the lack of liquid options for very high and very low strikes. However, there is no golden choice for the barriers levels’, which will probably change depending on the underlying asset risk neutral distribution. The latter feature renders its forecasting performance mainly an empirical question.The aim of the paper is twofold. First we investigate the forecasting performance of corridor implied volatility by choosing different corridors with symmetric and asymmetric cuts, and compare the results with the preliminary findings in Muzzioli (2010b). Second, we examine the nature of the variance risk premium and shed light on the information content of different parts of the risk neutral distribution of the stock price, by using a modelindependent approach based on corridor measures. To this end we compute both realised and modelfree variance measures which accounts for drops versus increases in the underlying asset price. The comparison is pursued by using intradaily synchronous prices between the options and the underlying asset.
2011
 The Skew Pattern of Implied Volatility in the DAX Index Options Market
[Articolo su rivista]
Muzzioli, Silvia
abstract
The aim of this paper is twofold: to investigate how the information content of implied volatility varies according to moneyness and option type and to compare optionbased forecasts with historical volatility. The different information content of implied volatility is examined for the most liquid atthemoney and outofthemoney options: put (call) options for strikes below (above) the current underlying asset price. Two hypotheses are tested: unbiasedness and efficiency of the different volatility forecasts. The investigation is pursued in the Dax index options market, by using synchronous prices matched in a oneminute interval. It was found that the information content of implied volatility has a humped shape, with outofthemoney options being less informative than atthemoney ones. Overall, the best forecast is atthemoney put implied volatility: it is unbiased (after a constant adjustment) and efficient, in that it subsumes all the information contained in historical volatility.
2011
 Towards a volatility index for the Italian stock market
[Relazione in Atti di Convegno]
Muzzioli, Silvia
abstract
The aim of this paper is to analyse and empirically test how to unlock volatility information from option prices. The information content of three option based forecasts of volatility: BlackScholes implied volatility, modelfree implied volatility and corridor implied volatility is addressed, with the ultimate plan of proposing a new volatility index for the Italian stock market. As for modelfree implied volatility, two different extrapolation techniques are implemented. As for corridor implied volatility, five different corridors are compared.Our results, which point to a better performance of corridor implied volatilities with respect to both BlackScholes implied volatility and modelfree implied volatility, are in favour of narrow corridors. The volatility index proposed is obtained with an overall 50% cut of the risk neutral distribution. The properties of the volatility index are explored by analysing both the contemporaneous relationship between implied volatility changes and market returns and the usefulness of the proposed index in forecasting future market returns.
2010
 Optionbased forecasts of volatility: An empirical study in the DAXindex options market
[Articolo su rivista]
Muzzioli, Silvia
abstract
Volatility estimation and forecasting are essential for both the pricing and the risk management of derivative securities. Volatility forecasting methods can be divided into optionbased ones, that use prices of traded options in order to unlock volatility expectations, and time series volatility models that use historical information in order to predict future volatility. Among optionbased volatility forecasts we distinguish between the “modeldependent” BlackScholes implied volatility and the “modelfree” implied volatility proposed by BrittenJones and Neuberger (2000) that does not rely on a particular option pricing model.The aim of this paper is to investigate the unbiasedness and efficiency, with respect to past realised volatility, of the two optionbased volatility forecasts. The comparison is pursued by using intradaily data on the DAXindex options market. Our results suggest that BlackScholes implied volatility subsumes all the information contained in past realised volatility and is a better predictor for future realised volatility than modelfree implied volatility.
2010
 The relation between implied and realised volatility in the DAX index options market
[Capitolo/Saggio]
Muzzioli, Silvia
abstract
The aim of this paper is to investigate the relation between implied volatility, historical volatility and realised volatility in the DAX index options market. Since implied volatility varies across option type (call versus put) we run a horse race of different implied volatility estimates: implied call and implied put. Two hypotheses are tested in the DAX index options market: unbiasedness and efficiency of the different volatility forecasts. Our results suggest that both implied volatility forecasts are unbiased (after a constant adjustment) and efficient forecasts of future realised volatility in that they subsume all the information contained in historical volatility.
2010
 Towards a volatility index for the Italian stock market
[Working paper]
Muzzioli, S.
abstract
The aim of this paper is to analyse and empirically test how to unlock volatility information from option prices. The information content of three option based forecasts of volatility: BlackScholes implied volatility, modelfree implied volatility and corridor implied volatility is addressed, with the ultimate plan of proposing a new volatility index for the Italian stock market. As for modelfree implied volatility, two different extrapolation techniques are implemented. As for corridor implied volatility, five different corridors are compared. Our results, which point to a better performance of corridor implied volatilities with respect to both BlackScholes implied volatility and modelfree implied volatility, are in favour of narrow corridors. The volatility index proposed is obtained with an overall 50% cut of the risk neutral distribution. The properties of the volatility index are explored by analysing both the contemporaneous relationship between implied volatility changes and market returns and the usefulness of the proposed index in forecasting future market returns
2009
 On the no arbitrage condition in option implied trees
[Articolo su rivista]
V., Moriggia; Muzzioli, Silvia; Torricelli, Costanza
abstract
The aim of this paper is to discuss the noarbitrage condition in option implied trees based on forward induction and to propose a noarbitrage test that rules out the negative probabilities problem and hence enhances the pricing performance. The noarbitrage condition takes into account two main features: the position of the node in the tree and the relation between the dividend yield and the riskfree rate. The proposed methodology is tested in and out of sample with Italian index options data and findings support a good pricing performance.
2009
 The skew pattern of implied volatility in the DAX index options market
[Working paper]
Muzzioli, S.
abstract
The aim of this paper is twofold: to investigate how the information content of implied volatility varies according to moneyness and option type, and to compare optionbased forecasts with historical volatility in order to see if they subsume all the information contained in historical volatility. The different information content of implied volatility is examined for the most liquid atthemoney and outofthemoney options: put (call) options for strikes below (above) the current underlying asset price, i.e. the ones that are usually used as inputs for the computation of the smile function. In particular, since atthemoney implied volatilities are usually inserted in the smile function by computing some average of both call and put implied ones, we investigate the performance of a weighted average of atthemoney call and put implied volatilities with weights proportional to trading volume. Two hypotheses are tested: unbiasedness and efficiency of the different volatility forecasts. The investigation is pursued in the Dax index options market, by using synchronous prices matched in a oneminute interval. It was found that the information content of implied volatility has a humped shape, with outofthemoney options being less informative than atthemoney ones. Overall, the best forecast is atthemoney put implied volatility: it is unbiased (after a constant adjustment) and efficient, in that it subsumes all the information contained in historical volatility.
2009
 The skew pattern of implied volatility in the DAXindex options market
[Working paper]
Muzzioli, Silvia
abstract
The aim of this paper is twofold: to investigate how the information content of implied volatility varies according to moneyness and option type, and to compare optionbased forecasts with historical volatility in order to see if they subsume all the information contained in historical volatility. The different information content of implied volatility is examined for the most liquid atthemoney and outofthemoney options: put (call) options for strikes below (above) the current underlying asset price, i.e. the ones that are usually used as inputs for the computation of the smile function. In particular, since atthemoney implied volatilities are usually inserted in the smile function by computing some average of both call and put implied ones, we investigate the performance of a weighted average of atthemoney call and put implied volatilities with weights proportional to trading volume. Two hypotheses are tested: unbiasedness and efficiency of the different volatility forecasts. The investigation is pursued in the Dax index options market, by using synchronous prices matched in a oneminute interval. It was found that the information content of implied volatility has a humped shape, with outofthemoney options being less informative than atthemoney ones. Overall, the best forecast is atthemoney put implied volatility: it is unbiased (after a constant adjustment) and efficient, in that it subsumes all the information contained in historical volatility.
2008
 American Option Pricing with Imprecise RiskNeutral Probabilities
[Articolo su rivista]
Muzzioli, Silvia; H., Reynaerts
abstract
The aim of this paper is to price an American option in a multiperiod binomial model,when there is uncertainty on the volatility of the underlying asset.American option valuation is usually performed, under the riskneutralvaluation paradigm, by using numerical procedures such as the binomialoption pricing model of Cox, Ross, Rubinstein (1979). A key input of themultiperiod binomial model is the volatility of the underlying asset,that is an unobservable parameter.As it is hard to give a precise estimate forthe volatility, in this paper we use a possibility distribution in order to modelthe uncertainty on the volatility. Possibility distributions are one of the mostpopular mathematical tools for modelling uncertainty. The standard riskneutralvaluation paradigm requires the derivation of the riskneutral probabilities, thatin a one period binomial model boils down to the solution of a linear system ofequations. As a consequence of the uncertainty in the volatility, we obtain apossibility distribution on the riskneutral probabilities. Under these measures,we perform the riskneutral valuation of the American option.
2008
 Option based forecasts of volatility: An empirical study in the DAX index options market
[Working paper]
Muzzioli, S.
abstract
Option based volatility forecasts can be divided into “model dependent” forecast, such as implied volatility, that is obtained by inverting the Black and Scholes formula, and “model free” forecasts, such as model free volatility, proposed by BrittenJones and Neuberger (2000), that do not rely on a particular option pricing model. The aim of this paper is to investigate the unbiasedness and efficiency in predicting future realized volatility of the two option based volatility forecasts: implied volatility and model free volatility. The comparison is pursued by using intradaily data on the Daxindex options market. Our results suggest that BlackScholes volatility subsumes all the information contained in historical volatility and is a better predictor than model free volatility.
2007
 American option pricing with imprecise risk neutral probabilities: from plain intervals to fuzzy sets
[Working paper]
Muzzioli, Silvia; H., Reynaerts
abstract
The aim f this paper is to price an American style option when there is uncertainty on the underlying asset volatility.
2007
 Call and put implied volatilities and the derivation of option implied trees
[Articolo su rivista]
Moriggia, V; Muzzioli, Silvia; Torricelli, Costanza
abstract
The aim of this paper is to discuss the noarbitrage condition in option implied trees based on forward induction and to propose a noarbitrage test that rules out the negative probabilities problem and hence enhances the pricing performance. The noarbitrage condition takes into account two main features: the position of the node in the tree and the relation between the dividend yield and the riskfree rate. The proposed methodology is tested in and out of sample with Italian index options data and findings support a good pricing performance.
2007
 Option pricing in the presence of uncertainty
[Capitolo/Saggio]
Muzzioli, Silvia; H., Reynaerts
abstract
In this chapter we investigate the derivation of the European option price in the CoxRossRubinstein binomial model in the presence of uncertainty on the volatility of the underlying asset. We propose two different approaches to the issue that concentrate on the fuzzification of one or both the two jump factors.
2007
 Solving parametric fuzzy linear systems by a non linear programming method
[Articolo su rivista]
Muzzioli, Silvia; H., Reynaerts
abstract
Linear systems of equations, with uncertainty on the parameters,play a major role in various problems in economics and finance. In this paperparametric fuzzy linear systems of the general formA_1x+b_1=A_2x+b_2, with A_1, A_2, b_1 and b_2 matrices with fuzzy elements,are solved by means of a nonlinear programming method.The relationbetween this methodology and the algorithm proposed in Muzzioli and Reynaerts (2006a)is highlighted.The methodology is finally applied to an economic and afinancial problem.
2007
 The relation between implied and realised volatility: are call options more informative than put options? Evidence from the DAX index options market
[Working paper]
Muzzioli, S.
abstract
The aim of this paper is to investigate the relation between implied volatility, historical volatility and realised volatility in the Dax index options market. Since implied volatility varies across option type (call versus put) we run a horse race of different implied volatility estimates: implied call, implied put and average implied that is a weighted average of call and put implied volatility with weights proportional to traded volume. Two hypotheses are tested in the Dax index options market: unbiasedness and efficiency of the different volatility forecasts. Our results suggest that all the three implied volatility forecasts are unbiased (after a constant adjustment) and efficient forecasts of future realised volatility in that they subsume all the information contained in historical volatility.
2007
 The solution of fuzzy linear systems by nonlinear programming: a financial application
[Articolo su rivista]
Muzzioli, Silvia; H., Reynaerts
abstract
Fuzzy linear systems of equations play a major role in various financial applications. In this paper we analyse a particular fuzzy linear system: the derivation of the risk neutral probabilities in a fuzzy binary tree. This system has previously been investigated and different solutions to different forms of the same system have been proposed. The aim of this paper is twofold. First, we highlight that the different solutions proposed, arise from different forms of the same system. Second, in order to find a unique vector solution for the system, we propose a practical algorithm that boils down to the solution of a nonlinear optimization problem. (c) 2005 Elsevier B.V. All rights reserved.
2006
 Fuzzy binary tree model for European options
[Capitolo/Saggio]
Muzzioli, Silvia; H., Reynaerts
abstract
The derivation of the riskneutral probabilities in a binary tree, in the presence of uncertainty on the underlying asset moves, boils down to the solution of dual fuzzy linear systems. The issue has previously been addressed and different solutions to the dual systems have been found. The aim of this paper is to apply a methodology which leads to a unique solution for the dual systems.
2006
 Fuzzy linear systems of the form A(1)x plus b(1) = A(2)x plus b(2)
[Articolo su rivista]
Muzzioli, Silvia; H., Reynaerts
abstract
Linear systems of equations, with uncertainty on the parameters, play a major role in several applications in various areas such as economics, finance, engineering and physics. This paper investigates fuzzy linear systems of the form A(1) x + b(1) = A(2)x + b(2) with A(1), A(2) square matrices of fuzzy coefficients and b(1), b(2) fuzzy number vectors. The aim of this paper is twofold. First, we clarify the link between interval linear systems and fuzzy linear systems. Second, a generalization of the vector solution of Buckley and Qu [Solving systems of linear fuzzy equations, Fuzzy Sets and Systems 43 (1991) 3343] to the fuzzy system A(1) x + b(1) = A(2)x + b(2) is provided. In particular, we give the conditions under which the system has a vector solution and we show that the linear systems Ax = b and A(1)x + b(1) = A(2)x + b(2), with A = A(1)  A(2) and b = b(2)  b(1), have the same vector solutions. Moreover, in order to find the vector solution, a simple algorithm is proposed.
2005
 Fuzzy up and down probabilities in a financial problem
[Working paper]
Muzzioli, Silvia; H., Reynaerts
abstract
The aim of this paper is to solve a fuzzy linear system of equation that arises in the computation of the risk neutral probabilities.
2005
 Solving fuzzy systems of linear equations by a nonlinear programming method
[Relazione in Atti di Convegno]
H., Reynaerts; Muzzioli, Silvia
abstract
Linear systems of equations with uncertainty on the parameters play a major role in various problems in economics and finance. In this paper fuzzy linear systems of the most general form A_1x+b_1=A_2x+b_2, with A_1, A_2, b_1, and b_2 matrices with fuzzy elements are solved by means of a nonlinear programming method. The methodology is finally applied to an economic and a financial problem.
2005
 The no Arbitrage Condition in Option Implied Trees: Evidence from the Italian Index Options Market
[Working paper]
Moriggia, V.; Muzzioli, S.; Torricelli, C.
abstract
2005
 The pricing of options on an interval binomial tree. An application to the DAXindex option market
[Articolo su rivista]
Muzzioli, Silvia; Torricelli, Costanza
abstract
This paper implements a model setup in Muzzioli and Torricelli [Int. J. Intell. Syst. 17 (6) (2002) 577594] for deriving implied trees and pricing options when the putcall parity is not fulfilled. The model basically extends Derman and Kani´s [Risk 7 (2) (1994) 3239], whereby call (put) prices are also used in the lower (upper) part of the tree thus exploiting the information content of both call and put prices. The DAXindex option market is chosen for this application because it is a relatively new European market where shortselling restrictions may induce putcall parity violations and the nature of the option (European) and of the underlying (dividends reinvested in the index) avoid some estimation problems. In order to test the pricing fit of the model, a nonlinear optimisation procedure is proposed to estimate a unique implied tree which allows a comparison between the model prices, Derman and Kani´s and market prices. The results suggest that the MT model improves the pricing.
2004
 A multiperiod binomial model for pricing options in a vague world
[Articolo su rivista]
Muzzioli, Silvia; Torricelli, Costanza
abstract
The aim of this paper is the pricing of European options in a multiperiod binomial model characterised by illdefined states of the world. The pricing methodology is still the riskneutral valuation approach. However, the vagueness in the stock price movements implies that both the riskneutral probabilities and the stock price are weighted intervals. An empirical validation of the model with DAXindex option data is also provided.
2004
 Fuzzy binary tree model for European style vanilla options
[Relazione in Atti di Convegno]
Muzzioli, Silvia; H., Reynaerts
abstract
The derivation of the risk neutral probabilities in a binary tree, in the presence of uncertainty on the underlying asset moves, boils down to the solution of dual fuzzy linear systems. The issue has previously been addressed and different solutions to the dual systems have been found. The aim of this paper is to apply a methodology which leads to a unique solution for the dual systems.
2003
 A note on fuzzy linear systems
[Working paper]
Muzzioli, Silvia
abstract
The aim of this paper is to analyse the solution of a fuzzy system when the classical solution based on standard fuzzy mathematics fails to exist. In particular we analyse the solution of the system Ax=b with A squared matrix with positive fuzzy coefficients and y crisp vector of positive elements. This system is particularly important for financial applications. We propose two different solution methods that are based respectively on the work of Buckley et al. (2002) and Friedman, Ming and Kandel (1998). An application to an important financial problem, the derivation of the artificial probabilities in a lattice framework, is provided.
2003
 OPTION IMPLIED TREES WHEN THE PUTCALL PARITY IS NOT FULFILLED
[Working paper]
Moriggia, V.; Muzzioli, S.; Torricelli, C.
abstract
2002
 Implied trees in illiquid markets: A Choquet pricing approach
[Articolo su rivista]
Muzzioli, Silvia; Torricelli, Costanza
abstract
Implied trees are necessary to implement the risk neutral valuation approach, and standard methodologies for their derivation are based on the validity of the put call parity. However, in illiquid markets the put call parity fails to hold, and the uniqueness of the artificial probabilities leaves room for an interval. The contribution of this article is twofold. First we propose a methodology for the derivation of implied trees in illiquid markets. Such a methodology, by contrast with standard ones, takes into account the information stemming both from call and put prices. Second, we set up a framework for pricing derivatives written on an underlying asset traded on an illiquid market. To this end we have extended the Choquet integral definition to account for interval payoffs of the underlying asset. The price interval we obtain may be interpreted as a bidask price quoted by the intermediary issuing the derivative security.
2001
 A Multiperiod Binomial Model for Pricing Options in an Uncertain World
[Relazione in Atti di Convegno]
Muzzioli, Silvia; Torricelli, Costanza
abstract
The aim of this paper is to price an option in a multiperiod binomial model, when there is uncertainty on the states of the world at each node of the tree. As a consequence, also the stock price at each state takes imprecise values. Possibility distributions are used to handle this type of problems. The pricing methodology is still based on a risk neutral valuation approach, but, as a consequence of the uncertainty on the two jumps of the stock, we obtain weighted intervals for riskneutral probabilities. The distinctive feature of our model is that it tracks back the arising of these probability intervals to the imprecision of the value of the stock price in the up and down states. This paper provides a generalization of the standard binomial option pricing model. We obtain an expected value interval for the option price within which it is possible to find a crisp representative value and an index of the uncertainty present in the model.
2001
 A model for pricing an option with a fuzzy payoff
[Articolo su rivista]
Muzzioli, Silvia; Torricelli, Costanza
abstract
This paper sets up a one period model for pricing an option with a fuzzy payoff. The option is written on an underlying asset that has a fuzzy price at the end of the period, modelled by means of triangular fuzzy numbers. The pricing methodology used is the standard one for pricing derivatives, i.e. the so called risk neutral valuation. Combining the standard Binomial Option Pricing Model with a fuzzy representation of the option payoff offers some advantages. First it provides an intuitive way of looking at the future price of an asset. Second it includes the results of the Standard Binomial Model, allowing the market to have different levels of information.
2001
 A multiperiod binomial model for ricing options in an uncertain world
[Abstract in Atti di Convegno]
Muzzioli, Silvia; Torricelli, Costanza
abstract
The aim of this paper is to price an option in a multiperiod binomial tree, when there is uncertainty on the states of the world at each node of the tree.
2001
 Implied Trees in Illiquid Markets: a Choquet Pricing Approach
[Working paper]
Muzzioli, S.; Torricelli, C.
abstract
2000
 Combining the Theory of Evidence with Fuzzy Sets
for Binomial Option Pricing
[Working paper]
Muzzioli, S.; Torricelli, C.
abstract
1999
 Pricing options on a vague asset
[Relazione in Atti di Convegno]
Muzzioli, Silvia; Torricelli, Costanza
abstract
This paper deals with the problem of pricing an option in a oneperiod model when the price of the underlyng asset is vague. The vagueness is modelled by the use of triangular fuzzy numbers and the pricing methodlogy is based on the noarbitrage principle. A comparison with the corresponding binomial option pricing model is provided, in particular we show that it can be viewed as a special case of our model.
1998
 A Model for Pricing an Option with a Fuzzy Payoff
[Working paper]
Muzzioli, S.; Torricelli, C.
abstract
1998
 Note on ranking fuzzy triangular numbers
[Articolo su rivista]
Facchinetti, Gisella; R., Ghiselli Ricci; Muzzioli, Silvia
abstract
In this paper we present a new method for ranking fuzzy numbers
1997
 New Methods For Ranking Triangular Fuzzy Numbers: An Investment Choice
[Working paper]
Facchinetti, Gisella; Ghiselli Ricci, Roberto; Muzzioli, Silvia
abstract