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I have 2 questions:

  1. What is the most commonly used equity option pricing model? I learned jump diffusion at school, read about Hensen and a few other models online.

  2. I am actually only calibrating the surface for easy retrieval of historical data. Right now to answer questions like "what is the implied vol of an option with strike = 105% of forward price and time = 3 month, for each day for the past 10 years", I literally have to interpolate on 3650 different option chains. Since each option chain is a 2000 * 4 grid, the process is slow. I am hoping that after calibrating, I only have to look at the parameters, instead of the entire surface.

But since calibrating loses accuracy, my second question is, is it viable to increase accuracy by cutting the surface into a few segments, and then calibrating each segment separately? I am honestly only looking for a function that fits through the surface, and I have no problem making that function piece-wise since any fitting attempt is better than my current method of using the entire surface.

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The most used equity volatility models in the industry are the Black-Scholes model (including its time dependent version) and the local volatility model. It always come along with stochastic rates, discrete dividends and quanto effects (a must-have when pricing even simple payoffs) so the calibration/pricing process is much more involved than what you might expect. Of course stochastic volatility are used but it all boils down to the payoffs you are selling to clients and the subsequent risks you are managing. Jumps are less common to my knowledge but might be used in some specific cases (exotic variance products).

As for your question if you are just computing implied volatilities for the sake of storing data then you are not using any model (pricing any other payoff) so I assume this is not related to your first question. What you might want to do is however to fit a parametrical implied volatility surface to your data set. Have a look at Gatheral's SVI/SSVI for example. This is basically a parametrical function that can fit many market shapes (so you only have to store some parameters per maturity) but most importantly can lead to arbitrage free surfaces.

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