Martingale Conditions in the Rough Bergomi Model

Summary

The post raises a theoretical question about the rough Bergomi stochastic volatility model and the martingale property of its price process under a risk-neutral measure. It contrasts the change of measure used in an early model derivation with later work examining whether the process is only a local martingale or is a true martingale under particular conditions.

The central issue is that a local martingale need not be a true martingale, so the risk-neutral pricing interpretation requires attention to integrability and model conditions. The document refers to work by Bayer and coauthors and a later analysis by Gassiat, but includes no equations or details from either source. It poses the tension rather than resolving it: the excerpts described by the author appear to make different claims, and the post does not specify the parameter conditions needed to reconcile them. Readers need the cited papers’ precise assumptions to determine when the model price is a true martingale.

Key ideas

• The post questions whether the rough Bergomi price process is a true martingale under the risk-neutral measure.
• A local martingale is not necessarily a true martingale, which matters for pricing interpretations.
• The author compares an early change-of-measure derivation with later work on martingale conditions.
• The document supplies no equations or assumptions, so it does not resolve when the true martingale property holds.

Tags

# Rough Volatility and Change of Measure


# Rough Volatility and Change of Measure












When deriving the rough Bergomi model, Bayer et al in "Pricing Under Rough Volatility" (2015) perform a change of measure to ensure the price process is a martingale as shown in the screenshot below:

However, there is recent academic work, such as the "On the martingale property in the rough Bergomi model" by Gassiat in 2019, showing that the process is a local martingale and a true martingale under certain conditions as shown in the screenshot below:

I don't understand how, under the risk neutral measure, the price process is a martingale and yet current work says it's an unanswered question.

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