Abstract

This paper aims to provide a framework for modeling and estimating the implied volatilities of stock prices using stochastic processes. The CIR (Cox-Ingersoll-Ross) model is proposed to capture the mean reversion characteristic as shown in the movement of the daily implied volatilities of the S$\&$P 500 Index and Nikkei 225 Index. The maximum likelihood procedure is applied to estimate the parameters appearing in the model, where both analytical and approximation techniques are sought to handle the difficulties arising from the corresponding optimization problem. The procedure is validated with varying sampling methods by setting different time horizons and observation intervals. Results from numerical experiments show that the stochastic volatility model proposed in this paper functions well in both the US and Japan markets. As one of the applications, our approach is tested to be effective in detecting jumps in likelihood ratios, hence useful for forecasting stock market shocks and crisis.

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