PAPER:

The Stein-Stein stochastic volatility model: transition probability density function, moment formulae, option pricing, implied volatility, calibration

Lorella Fatone, Francesca Mariani, Maria Cristina Recchioni, Francesco Zirilli

 

  1. Abstract
  2. The Stein-Stein model and the option pricing formulae
  3. The moment formulae of the log-price variable: an interactive application and an app (interactive application, app).
  4. Model calibration using European option prices and option price forecasts (movie 1)
  5. Model calibration using implied volatilities and option price forecasts (movie 2)
  6. References

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1.   Abstract

In [1]  we give an integral representation formula of the transition probability density function  of the Stein-Stein stochastic volatility model. Using this formula we obtain explicit expressions for the moments of the log-price and of the stochastic volatility, we price European options,  we define implied volatility and we formulate  two calibration problems. Examples of the calibration problems   involving real data are solved numerically. The real data considered are  time series of daily data of the U.S.A. S&P 500 index  and of the prices of some of its European options or of the corresponding implied volatilities. The validity of the Stein-Stein model and of the previous calibration problems is established  comparing the option prices obtained with the calibrated models with the observed option prices. A general reference to the work of the authors and of their coauthors in mathematical finance is the website: http://www.econ.univpm.it/recchioni/finance.

 

Interactive Application

Movie 1

Movie 2

App

UNDER CONSTRUCTION

6 References  

[1] Fatone, L., Mariani, F., Recchioni, M.C., Zirilli, F: The Stein-Stein stochastic volatility model: transition probability density function, moment formulae, option pricing, implied volatility, calibration, submitted (2015).

 

[2] Stein, E., Stein, J.:  Stock price distributions with stochastic volatility: an analytic approach,- The Review of Financial Studies,  4(4),  (1991),  727-752.           

 

[3] Fatone, L., Mariani, F., Recchioni, M.C., Zirilli, F.: The use of statistical tests to calibrate the normal SABR model, Journal of Inverse   

and Ill Posed Problems, 21(1),  (2013), 59-84, http://www.econ.univpm.it/recchioni/finance/w15/ .

 

[4] Fatone, L., Mariani, F., Recchioni, M.C., Zirilli, F.: Closed form moment formulae for  the lognormal SABR model and applications

to calibration problems, Open Journal of Applied Sciences, 3, (2013) , 345-359, http://www.econ.univpm.it/recchioni/finance/w16/ .

 

[5] Berestycky, H., Busca, J., Florent I.:  Computing the implied volatility in stochastic volatility models, Communications on Pure and Applied Mathematics, Vol. LVII, (2004), 1352-1373.

 

[6]  Fatone, L.,  Mariani, F. Recchioni, M.C., Zirilli, F.: Some explicit formulae for the Hull and White stochastic volatility model, International Journal of Modern Nonlinear Theory and Application  2, (2013), 14-33, http://www.econ.univpm.it/recchioni/finance/w17/

 

 

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