# Calibrating the Italian Smile with Time-Varying Volatility and Heavy-Tailed Models

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## Abstract

In this paper, we consider several time-varying volatility and/or heavy-tailed models to explain the dynamics of return time series and to fit the volatility smile for exchange-traded options where the underlying is the main Italian stock index. Given observed prices for the time period we investigate, we calibrate both continuous-time and discrete-time models. First, we estimate the models from a time-series perspective (i.e. under the historical probability measure) by investigating more than 10 years of daily index price log-returns. Then, we explore the risk-neutral measure by fitting the values of the implied volatility for numerous strikes and maturities during the highly volatile period from April 1, 2007 (prior to the subprime mortgage crisis in the US) to March 30, 2012. We assess the extent to which time-varying volatility and heavy-tailed distributions are needed to explain the behavior of the most important stock index of the Italian market.

## Keywords

Volatility smile Stochastic volatility models GARCH model Non-Gaussian Ornstein-Uhlenbeck processes Lévy processes Tempered stable processes and distributions## Mathematics Subject Classification

60E07 60G51 91G20 91G60## References

- Albrecher, H., Mayer, P., Schoutens, W., & Tistaert, J. (2007).
*The little Heston trap*. London: Wilmott Magazine.Google Scholar - Barndorff-Nielsen, O. E., & Shephard, N. (2001). Non-Gaussian Ornstein-Uhlenbeck-based models and some of their uses in financial economics.
*Journal of the Royal Statistical Society: Series B (Statistical Methodology)*,*63*(2), 167–241.CrossRefGoogle Scholar - Barone Adesi, G., Engle, R. F., & Mancini, L. (2008). A GARCH option pricing model with filtered historical simulation.
*Review of Financial Studies*,*21*(3), 1223–1258.CrossRefGoogle Scholar - Bhar, R. (2010).
*Stochastic filtering with applications in finance*. Singapore: World Scientific.CrossRefGoogle Scholar - Bianchi, M. L., & Fabozzi, F. J. (2015). Investigating the performance of non-Gaussian stochastic intensity models in the calibration of credit default swap spread.
*Computational Economics*,*46*(2), 243–273.CrossRefGoogle Scholar - Bianchi, M.L., Rachev, S.T., Fabozzi, F.J. (2016). Tempered stable Ornstein-Uhlenbeck processes: A practical view.
*Communications in Statistics: Simulation and Computation*.Google Scholar - Brigo, D., & Mercurio, F. (2006).
*Interest rate models: Theory and practice: With smile, inflation, and credit*. New York: Springer.Google Scholar - Carr, P., & Madan, D. (1999). Option valuation using the fast fourier transform.
*Journal of Computational Finance*,*2*(4), 61–73.CrossRefGoogle Scholar - Centanni, S., Ongaro, A. (2011). Computing option values by pricing kernel with a stochastic volatility model.
*Working paper*.Google Scholar - Christoffersen, P., Heston, S., & Jacobs, K. (2013). Capturing option anomalies with a variance-dependent pricing kernel.
*Review of Financial Studies*,*26*(8), 1963–2006.CrossRefGoogle Scholar - Christoffersen, P., Jacobs, K., & Mimouni, K. (2010). Volatility dynamics for the S&P500: Evidence from realized volatility, daily returns, and option prices.
*Review of Financial Studies*,*23*(8), 3141–3189.CrossRefGoogle Scholar - Ciccone, E., Giordano, L., Grasso, R. (2011). Why are derivative warrants more expensive than options? The Italian case.
*Discussion papers, Consob*, n. 2.Google Scholar - Cont, R., & Tankov, P. (2004).
*Financial modelling with jump processes*. London: Chapman & Hall.Google Scholar - Cui, C., Zhang, K. (2011). Equity implied volatility surface.
*Quantitative Research and Development, Bloomberg*, Version 3.5.Google Scholar - Douc, R., Cappé, O. (2005). Comparison of resampling schemes for particle filtering. In
*Proceedings of the IEEE 4th international symposium on image and signal processing and analysis*(pp. 64–69).Google Scholar - Duan, J. C. (1995). The GARCH option pricing model.
*Mathematical Finance*,*5*(1), 13–32.CrossRefGoogle Scholar - Duan, J. C., & Simonato, J. G. (1998). Empirical martingale simulation for asset prices.
*Management Science*,*44*(9), 1218–1233.CrossRefGoogle Scholar - Engle, R. F., & Ng, V. K. (1993). Measuring and testing the impact of news on volatility.
*The Journal of Finance*,*48*(5), 1749–1778.CrossRefGoogle Scholar - Fang, F., Jönsson, H., Oosterlee, K., & Schoutens, W. (2010). Fast valuation and calibration of credit default swaps under Lévy dynamics.
*Journal of Computational Finance*,*14*(2), 1–30.CrossRefGoogle Scholar - Geman, H., Madan, D. B., & Yor, M. (2001). Time changes for Lévy processes.
*Mathematical Finance*,*11*(1), 79–96.CrossRefGoogle Scholar - Geman, H., Madan, D. B., & Yor, M. (2002). Stochastic volatility, jumps and hidden time changes.
*Finance and Stochastics*,*6*(1), 63–90.CrossRefGoogle Scholar - Guillaume, F. (2012). Sato two-factor models for multivariate option pricing.
*Journal of Computational Finance*,*15*(4), 159–192.CrossRefGoogle Scholar - Guillaume, F., & Schoutens, W. (2012). Calibration risk: Illustrating the impact of calibration risk under the Heston model.
*Review of Derivatives Research*,*15*(1), 57–79.CrossRefGoogle Scholar - Heston, S. L. (1993). A closed-form solution for options with stochastic volatility with applications to bond and currency options.
*Review of Financial Studies*,*6*(2), 327–343.CrossRefGoogle Scholar - Heston, S. L., & Nandi, S. (2000). A closed-form GARCH option valuation model.
*Review of Financial Studies*,*13*(3), 585–625.CrossRefGoogle Scholar - Hull, J. (2002).
*Options, futures and other derivatives*(5th ed.). Upper Saddle River: Prentice Hall.Google Scholar - Hurst, S. R., Platen, E., & Rachev, S. T. (1997). Subordinated market index models: A comparison.
*Asia-Pacific Financial Markets*,*4*(2), 97–124.Google Scholar - Hurst, S. R., Platen, E., & Rachev, S. T. (1999). Option pricing for a logstable asset price model.
*Mathematical and Computer Modelling*,*29*(10), 105–119.CrossRefGoogle Scholar - Kim, Y. S., & Lee, J. H. (2007). The relative entropy in CGMY processes and its applications to finance.
*Mathematical Methods of Operations Research*,*66*(2), 327–338.CrossRefGoogle Scholar - Kim, Y. S., Rachev, S. T., Bianchi, M. L., & Fabozzi, F. J. (2010). Tempered stable and tempered infinitely divisible GARCH models.
*Journal of Banking & Finance*,*34*(9), 2096–2109.CrossRefGoogle Scholar - Lando, D. (2004).
*Credit risk modeling: Theory and applications*. Princeton: Princeton University Press.Google Scholar - Lehar, A., Scheicher, M., & Schittenkopf, C. (2002). Garch vs. stochastic volatility: Option pricing and risk management.
*Journal of Banking & Finance*,*26*(2), 323–345.CrossRefGoogle Scholar - Li, J. (2011). Sequential bayesian analysis of time-changed infinite activity derivatives pricing models.
*Journal of Business and Economic Statistics*,*29*(4), 468–480.CrossRefGoogle Scholar - Lopes, H. F., & Tsay, R. S. (2010). Particle filters and Bayesian inference in financial econometrics.
*Journal of Forecasting*,*30*, 168–209.CrossRefGoogle Scholar - Malik, S., Pitt, M.K. (2011). Modelling stochastic volatility with leverage and jumps: A simulated maximum likelihood approach via particle filtering.
*Banque de France, Working paper No. 318*.Google Scholar - Mandelbrot, B. (1963). The variation of certain speculative prices.
*Journal of Business*,*36*(4), 394–419.CrossRefGoogle Scholar - Muzzioli, S. (2011). Corridor implied volatility and the variance risk premium in the Italian market.
*Working paper*.Google Scholar - Nicolato, E., & Venardos, E. (2003). Option pricing in stochastic volatility models of the Ornstein-Uhlenbeck type.
*Mathematical Finance*,*13*(4), 445–466.CrossRefGoogle Scholar - Rachev, S., & Mittnik, S. (2000).
*Stable Paretian models in finance*. New York: Wiley.Google Scholar - Rachev, S. T., Kim, Y. S., Bianchi, M. L., & Fabozzi, F. J. (2011).
*Financial models with Lévy processes and volatility clustering*. New York: Wiley.CrossRefGoogle Scholar - Sato, K. I. (1999).
*Lévy processes and infinitely divisible distributions*. Cambridge: Cambridge University Press.Google Scholar - Scherer, M., Rachev, S. T., Kim, Y. S., & Fabozzi, F. J. (2012). Approximation of skewed and leptokurtic return distributions.
*Applied Financial Economics*,*22*(16), 1305–1316.CrossRefGoogle Scholar - Schoutens, W. (2003).
*Lévy processes in finance: Pricing financial derivatives*. New York: Wiley.CrossRefGoogle Scholar - Tassinari, G. L., & Bianchi, M. L. (2014). Calibrating the smile with multivariate time-changed Brownian motion and the Esscher transform.
*International Journal of Theoretical and Applied Finance*,*17*(4).Google Scholar - Wu, L. (2008). Modeling financial security returns using Lévy processes. In J. R. Birge & V. Linetsky (Eds.),
*Handbooks in operations research and management science, volume 15, financial engineering*(pp. 117–162). Amsterdam: Elsevier.Google Scholar - Yu, C. L., Li, H., & Wells, M. T. (2011). MCMC estimation of Lévy jumps models using stock and option prices.
*Mathematical Finance*,*21*(3), 383–422.CrossRefGoogle Scholar