Preprint B28/2009
Modelos de Volatilidade Estocástica para o Índice Ibovespa: Reversão Rápida à Média e Análise Assintótica

Cassio Alves

**Keywords: **
Stochastic Volatility. Fast Mean Reversion. Asymptotics Analysis. Implied Volatility Surface. IBOVESPA.

In 1973, Black & Scholes and Merton derived a pricing formula for options,
nowadays known as the Black & Scholes formula. This formula became
widely used in financial markets worldwide. However, there are empirical
evidences, such as the non-normality of the returns and the smile effect,
that show inconsistencies in some of the hypothesis postulated by Black
& Scholes. The most controversial one being the assumption of constant
volatility. In recent works, J-P. Fouque, G. Papanicolaou & R. Sircar,
consider a model with stochastic volatility, that reverts to a mean under the
assumption of fast mean-reversion. They derived an asymptotic correction
for the Black & Scholes price, which is independent of the volatility driving
process state. They verified that such regime, is indeed present in the
S&P500. The calibration of the correction is a non-parametric one, through
the use of the market implied volatility surface. In this work, it is verified
that the main Brazilian stock index, IBOVESPA, displays a mean reversion
behaviour. For the calibration of the correction, due to lack of liquidity of
the Brazilian option market, an appropriate interpolation of the sparse data
is necessary to generate an implied volatility surface. This is done using a
method derived by N. Kahale that guarantees that there is no arbitrage in
the interpolated surface.