subsequent local volatility surface is never considered. In this article, we develop a deep learning approach for interpolation … performance. A novel component is the use of the Dupire formula to enforce bounds on the local volatility associated with option …

, especially on the BS implied volatility. Implied binomial trees (IBT) models capture the variations of the implied volatility … known as volatility smile. They provide a discrete approximation to the continuous risk neutral process for the underlying … Barle and Cakici (BC). After the formation of IBT we can estimate the implied local volatility and the state price density …

volatility of the S&P 500 and FTSE 100 indexes. As a proxy for daily volatility, we consider a consistent and unbiased estimator … of the integrated volatility that is computed from high frequency intra-day returns. We also consider a simple algorithm …

stochastic volatility model and Brent's iterative root-finding method for the calculation of implied volatilities. The numerical …