Why Jump-Diffusion Markets Can Be Incomplete

Summary

The note explains why a jump-diffusion model may not have a unique risk-neutral measure, contrasting it with a one-stock geometric Brownian motion model. Its central intuition counts the independent sources of risk against the traded assets: Brownian shocks and jump processes each introduce risks that must be priced, while stocks provide equations for determining those prices. When the number of independent risks exceeds the number of stocks, the market is incomplete and the risk-neutral measure need not be unique.

This is a useful introductory heuristic, but it is not a full mathematical criterion. Completeness depends on the structure of the assets and the risks they span, not only a raw count; jump specifications and attainable payoffs also matter. The document gives no derivation or empirical example, and its statement about geometric Brownian motion assumes a suitable one-factor setting with a traded asset spanning the Brownian risk.

Key ideas

• Market incompleteness arises when traded assets cannot span all sources of risk.
• A one-factor geometric Brownian motion with a suitable traded stock can have a unique risk-neutral measure.
• Jump-diffusion models add jump risks alongside diffusion risks.
• Comparing the number of independent risks with traded assets offers an initial intuition, not a complete test.

Tags

# Unique risk neutral measure for jumps or incomplete markets for jumps


# Unique risk neutral measure for jumps or incomplete markets for jumps












I wanted to understand why the market is incomplete in jump-diffusion models. whereas if we have a model following geometric Brownian motion then we can get a risk-neutral measure and hence a complete market.

## Answer by user78189 (score 0)

https://quant.stackexchange.com/a/81148

A market is incomplete if you have more unknown parameters then number of equations (number of stocks). Geometric Brownian Motion has only one Brownian Motion with one market price of risk to determine for one stock. If there would be more Brownian motions, the market could also be incomlete. In Jump Diffusion Model you have for example D Brownian motions and M Poisson processes, so D+M unknown. The market is incomplete if the number of stocks n<D+M. Even if you have one Brownian Motion and one Poisson process, if you only have one stock the market is incomplete.

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