Preprint B41/2010
Volatilidade Estocástica Multiescala: Modelagem, Estimação e Aplicação com preços da Petrobras.

Guillermo Gómez

**Keywords: **
Multiscale Analysis. Stochastic Volatility. Estimation. Finance Modeling. Mathematic Method in Finance.

This work is developed using a stochastic volatility model proposed by Fouque, Papanicolau,Sircar & Solna (2003) that incorporates two different scales to drive the volatility process.
The asymptotic analysis developed by the authors needs certain conditions the parameters associated to the model must satisfy. Fouque, Papanicolau & Sircar (2000) proposed
a non-parametric estimation method for the simplified model that considers only a fast scale.
Given that in this work we are considering a more general model, we need to validate the hypotheses of a reversal on a large scale. This is why we use the method developed by
Fouque, Papanicolau & Sircar (2000). Moreover, we describe a new form to estimate the parameters which is more difficult to implement because it needs a data series that is not observed in the market.
With this in mind we use the Fourier method proposed by Malliavin & Mancino (2002).
Then, we estimate the parameters using the parametric estimation method of Maximum Likelihood proposed by Bibby & Sorensen (1995). The data used are Petrobras prices
traded on the American market in high frequency to estimate a fast scale and also on a daily basis to estimate a long scale.
Once the necessary suppositions for the model are confirmed, we use an asymptotic analysis to estimate an approximation of the implied volatility surface associated to the call
option prices of Petrobras in this model with two scales. Finally, we compare our approximations with the implied volatility surface obtained using the Kahalé method (2005).