Why Jump-Diffusion Models Are Incomplete

Summary

The document gives a simple argument for incompleteness in a jump-diffusion market. A market is incomplete when more than one equivalent probability measure makes traded asset prices martingales. In the example, an asset has continuous Brownian risk and jumps whose size is observable, while the jump arrival rate is represented by a parameter that cannot be known with certainty from a finite observed path.

Changing that arrival rate produces alternative equivalent martingale measures while preserving the model’s pricing condition, yielding multiple candidate measures and therefore incompleteness. The explanation contrasts the observability of volatility and jump size with the uncertainty around intensity. It is a schematic argument under stated simplifying assumptions, including zero interest rates and a particular compensated jump process. It does not develop the full measure-change conditions or a general proof for every jump-diffusion specification, so those assumptions matter when applying the reasoning elsewhere.

Key ideas

• A market is incomplete when multiple equivalent martingale measures are consistent with its traded assets.
• In the example, changing the jump arrival intensity creates alternative martingale measures.
• Observed jump times can inform an intensity estimate without determining the intensity with certainty.
• The argument relies on a simplified jump-diffusion setup and does not cover every model specification.

Tags

# Formal proof market incompleteness under jump diffusion


# Formal proof market incompleteness under jump diffusion












Does anyone have formal proof of markets incompleteness under jump diffusion ?

I am familiar with the intuitive approach as mentioned in Tankov (delta), yet I am looking for a formal approach and some reference paper :)

Thank you in advance ! :)

## Answer by Andrea (score 1)

https://quant.stackexchange.com/a/81174

A model is incomplete if it allows more than 1 equivalent measure under which all tradable assets are martingales.

In a jump diffusion model, it is enough to change the jump intensity to a different value, so you can actually produce infinite martingale measures.

Assume 0 interest rates and

$dS_t = \sigma S_t dW_t + \gamma S_t (d N_t - \lambda dt)$ under some measure $\mathcal P$: $W$ is a BM and the intensity of $N$ is $\lambda$ and $N_t - \lambda t$ is a martingale.

A change of intensity (to $\eta$) produces an equivalent martingale measure where the SDE becomes.

$dS_t = \sigma S_t dW_t + \gamma S_t (d N_t - \eta dt)$

Since there are no exogenous constraints to chose the intensity, the model is incomplete.

The intuition behind it is: you only live on a single trajectory of the SDE, and anything which is not 100% observable could come from a different (but equivalent) martingale measure.

$\sigma$ is indeed observable (you need to continuous trajectory, but it is observable).

The jump size is easily observable.

But the intensity is not: if I give you the jump times, you can have only an idea of the intensity, but will never be completely certain about it.

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