BS price as the first term of option price expansion

Question

I recently saw someone write, on a generally non-technical platform, that the Black-Merton-Scholes vanilla option price is the first term of an expansion of the price of a vanilla option.

I get that in the context of stochastic volatility models by making use of the Hull and White mixing formula. And thus also for the specific case of stochastic volatility with (Poisson) jumps in the underlying asset.

For more general processes, can someone derive or point to a paper where it is derived that the BSM price is indeed the first term of price expansion? I would very much like to see what the general series expansion looks like.

This question was prompted by this comment:

which reacted to the perceived suggestion that the BSM model is useless.

Answer 1

The closest thing I can find to a general expansion of an option price in terms of BS price + correction terms is the following paper by Merino and Vives. It basically shows this using three methods, namely Ito calculus, Functional Ito calculus, and Malliavin calculus. The 'stochastic volatility' in the title of the paper actually includes local stochastic volatility models as well.

Merino and Vives, A generic decomposition formula for pricing vanilla options under stochastic volatility