Gianna Figà-Talamanca

Runs tests for assessing volatility forecastability in financial time series

In this work we refine a nonparametric methodology firstly applied in Christoffersen and Diebold [Review of Economics and Statistics 82 (2000) 12] for assessing volatility forecastability in financial time series based on discretization and on the use of runs tests. Empirical results are provided for SP500 and MIB30 indexes that lead naturally to a discretized one-period Markov chain. The results are confirmed with other persistence measures and their robustness is studied via numerical simulation.

Conditional tail behaviour and Value at Risk

In this paper we study the tail behaviour of eight major market indexes stratifying data according to the violation of a high threshold on the previous day. The distributional differences found can be exploited to improve VaR calculations in several settings, giving rise to what we call ‘MCVaR’. We compare the performance of MCVaR with unconditioned VaR calculation methods and with GARCH VaR by means of several back-testing techniques that take into account not only the number of violations but also their magnitude and clustering.

DETECTING AND MODELING TAIL DEPENDENCE

The aim of this work is to develop a nonparametric tool for detecting dependence in the tails of financial data. We provide a simple method to locate and measure serial dependence in the tails, based on runs tests. Our empirical investigations on many financial time series reveal a strong departure from independence for daily logreturns, which is not filtered out by usual Garch models.

Towards a Coherent Volatility Pricing Model: An Empirical Comparison

Many empirical analysis suggest that market prices dynamics are not well captured by Black and Scholes model. A valid generalization is attained by allowing volatility to change randomly and different approaches have been proposed in literature since the pioneering model by Hull and White [9].

Fuzzy Option Value with Stochastic Volatility Models

Uncertainty and vagueness play a central role in financial models and fuzzy numbers can be a profitable way to manage them. In this paper we generalize the Black and Scholes option valuation model (with constant volatility) to the framework of a volatility supposed to vary in a stochastic way. The models we take under consideration belongs to the main classes of stochastic volatility models: the endogenous and the exogenous source of risk. Fuzzy calculus for financial applications requires massive computations and when a good parametric representation for fuzzy numbers is adopted, then the arithmetic operations and fuzzy calculus can be efficiently managed. Good in this context means that the shape of the resulting fuzzy numbers can be observed and studied in order to state fundamental properties of the model.

Smart Fuzzy Weighted Averages of Information Elicited through Fuzzy Numbers

We illustrate a preliminary proposal of weighted fuzzy averages between two membership functions. Conflicts, as well as agreements, between the different sources of information in the two new operators are endogenously embedded inside the average weights. The proposal is motivated by the practical problem of assessing the fuzzy volatility parameter in the Black and Scholes environment via alternative estimators.

A Continuous Time Model for Bitcoin Price Dynamics

This chapter illustrates a continuous time model for the dynamics of Bitcoin price, which depends on an attention or sentiment factor. The model is proven arbitrage-free under mild conditions and a quasi-closed pricing formula for European style derivatives is provided.

A Generalized SMART Fuzzy Disjunction of Volatility Indicators Applied to Option Pricing in a Binomial Model

In this paper we extend our previous contributions on the elicitation of the fuzzy volatility membership function in option pricing models. More specifically we generalize the SMART disjunction for a multi-model volatility behavior (Uniform, LogNormal, Gamma, ...) and within a double-source (direct vs. indirect) information set. The whole procedure is then applied to the Cox-Ross-Rubinstein framework for option pricing on the S&P500 Index where the historical volatility, computed from the Index returns’ time series, and the VIX Index observed data are respectively considered as the direct and indirect sources of knowledge. A suitable distance among the resulting fuzzy option prices and the market bid-ask spread make us appreciate the proposed procedure against the classical fuzzy mean.

A Statistical Test for the Heston Model

We introduce a formal test to detect whether a times series of financial log-returns is consistent with the Heston stochastic volatility model as data generating process. The test is based on the auto-covariance structure of the integrated volatility, which is available in closed form for the model under investigation. The test suggested in this contribution also relies on the outcomes of a companion paper where we prove asymptotic results for the distribution of sample moments of the squared log-returns in the fully-specified Heston model.