Local Volatility and Implied Volatility : A Two-Step Numerical Approach to Pricing European Options - Cenizario Domínguez, Julio Jonathan (2025)
We research the process of finding the European option surface and the volatility surface for the S&P 500 options, where the arbitrage-free condition must be fulfilled. We propose a methodology as follows: (1) calculate the market implied volatility using the Black-Scholes equation, then (2) calculate the total implied volatility using the SVI model, then (3) calculate the local volatility via implied volatility and finally (4) obtain the option pricing using numerical methods to solve the partial differential equations. Understanding how to price European options is a key factor since even more complex financial derivatives are based on this one, for example, analysts use European options to price American options, Asian options, Barrier options and others complex ones for which may not exist yet an analytic solution form and only numerical methods are applied. On the other hand, since the volatility is not given by the market, we need to estimate it previous at all calculations of the options, furthermore, not matters what type of option we want to price, anyone of them is depending on a volatility, then it should be carefully estimated with high reliability and at the least time interval possible, since if there is any issue in the volatility, all the option price may be useless.