Silvia Muzzioli

Note on ranking fuzzy triangular numbers

This paper deals with the problem of ranking a set of alternatives, represented by triangular fuzzy numbers, in decision‐making situations. Three new methods are proposed, and a notion of preference between alternatives is suggested. A comparison with other methods is provided in the concluding table.

A multiperiod binomial model for pricing options in a vague world

Abstract The aim of this paper is the pricing of European options in a multiperiod binomial model characterised by ill-defined states of the world. The pricing methodology is still the risk-neutral valuation approach. However, the vagueness in the stock price movements implies that both the risk-neutral probabilities and the stock price are weighted intervals. An empirical validation of the model with DAX-index option data is also provided.

On the no-arbitrage condition in option implied trees

The aim of this paper is to discuss the no-arbitrage condition in option implied trees based on forward induction and to propose a no-arbitrage test that rules out the negative probabilities problem and hence enhances the pricing performance. The no-arbitrage condition takes into account two main features: the position of the node in the tree and the relation between the dividend yield and the risk-free rate. The proposed methodology is tested in and out of sample with Italian index options data and findings support a good pricing performance.

The solution of fuzzy linear systems by non-linear programming: a financial application

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 non-linear optimization problem.

Option-based forecasts of volatility: an empirical study in the DAX-index options market

Volatility estimation and forecasting are essential for both the pricing and the risk management of derivative securities. Volatility forecasting methods can be divided into option-based ones, which use prices of traded options in order to unlock volatility expectations, and time series volatility models, which use historical information in order to predict future volatility. Among option-based volatility forecasts, we distinguish between the ‘model-dependent’ Black–Scholes implied volatility and the ‘model-free’ implied volatility, proposed by Britten-Jones and Neuberger [Option prices, implied price processes and stochastic volatility. Journal of Finance 55: 839–66], 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 option-based volatility forecasts. The comparison is pursued by using intra-daily data on the DAX-index options market. Our results suggest that Black–Scholes implied volatility subsumes all the information contained in past realised volatility and is a better predictor for future realised volatility than model-free implied volatility.

American option pricing with imprecise risk-neutral probabilities

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 risk-neutral valuation paradigm, by using numerical procedures such as the binomial option pricing model of Cox et al. [J.C. Cox, S.A. Ross, S. Rubinstein, Option pricing, a simplified approach, Journal of Financial Economics 7 (1979) 229-263]. A key input of the multiperiod binomial model is the volatility of the underlying asset, that is an unobservable parameter. As it is hard to give a precise estimate for the volatility, in this paper we use a possibility distribution in order to model the uncertainty on the volatility. Possibility distributions are one of the most popular mathematical tools for modelling uncertainty. The standard risk-neutral valuation paradigm requires the derivation of the risk-neutral probabilities, that in a one-period binomial model boils down to the solution of a linear system of equations. As a consequence of the uncertainty in the volatility, we obtain a possibility distribution on the risk-neutral probabilities. Under these measures, we perform the risk-neutral valuation of the American option.

A comparison of fuzzy regression methods for the estimation of the implied volatility smile function

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 one-to-one 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 a-priori 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 the least squares fuzzy regression method of Savic and Pedrycz 27] and to the hybrid method of Ishibuchi and Nii 11].

Implied trees in illiquid markets: A Choquet pricing approach

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 bid‐ask price quoted by the intermediary issuing the derivative security.

The Information Content of Option-Based Forecasts of Volatility: Evidence from the Italian Stock Market

The aim of this paper is to comprehensively compare option-based 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 policy-makers 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.

A comparative assessment of different fuzzy regression methods for volatility forecasting

The aim of this paper is to compare different fuzzy regression methods in the assessment of the information content on future realised volatility of option-based 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 et al. (IEEE Trans Syst Man Cybern 12:903–907, 1982) and the least squares fuzzy regression method of Savic and Pedrycz (Fuzzy Sets Syst 39:51–63, 1991). In our case study, based on intra-daily data of the DAX-index options market, both methods have proved to have advantages and disadvantages. Overall, among the two methods, we prefer the Savic and Pedrycz (Fuzzy Sets Syst 39:51–63, 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 et al. (IEEE Trans Syst Man Cybern 12:903–907, 1982) method.