Black–Scholes as a Base Term in Option Price Expansions

Summary

The document asks whether the Black–Merton–Scholes (BMS) vanilla option price can serve as the leading term in a broader expansion for option prices under general processes. It notes that this interpretation is already familiar for stochastic volatility models through the Hull–White mixing formula, including cases with Poisson jumps in the underlying.

The response points to work by Merino and Vives on a generic decomposition for vanilla options under stochastic volatility. It says the paper derives a BMS price plus correction terms using Itô calculus, functional Itô calculus, and Malliavin calculus, and that its scope includes local stochastic volatility models. The document does not give the expansion itself, quantify its accuracy, or establish a general result for every process; it offers a relevant reference rather than a full derivation.

Key ideas

• The BMS vanilla option price can be used as a base term with additional corrections in stochastic volatility settings.
• The Hull–White mixing formula provides one route to this interpretation, including models with Poisson jumps.
• Merino and Vives present a generic decomposition using three calculus-based methods.
• The cited decomposition covers local stochastic volatility models as well as stochastic volatility models.

Tags

# BS price as the first term of option price expansion


# BS price as the first term of option price expansion












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 by Frido (score 1)

https://quant.stackexchange.com/a/75585

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

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.