Continuous-Time Limits for Multi-Asset Price Processes

Summary

The document asks whether Merton’s result for the continuous-time limit of a single asset price process extends to multiple assets. Under specified assumptions, the referenced result characterizes the limiting process as an Itô process when rare events are absent or restricted, and as a jump-diffusion when a broader class of rare events is allowed.

The author distinguishes this general limit result from the familiar binomial-model convergence to diffusion, which relies on a particular model form. The question seeks a multi-asset generalization but includes no answer or derivation. It therefore serves as a prompt about the assumptions and scope of continuous-time modeling, rather than a complete method. Any extension would need to address joint behavior across assets, including dependence and shared sources of jumps; the document does not specify how these are handled.

Key ideas

• The referenced result describes continuous-time limits for a single asset under stated assumptions.
• Permitted rare-event behavior determines whether the limit is an Itô process or a jump-diffusion.
• The author asks how the result generalizes to a system of multiple assets.
• Binomial convergence to diffusion is presented as less general because it assumes a particular model form.
• The document poses the multi-asset question without providing a proof or answer.

Tags

# The continuous-time limit of asset price processes where there is more than one asset


# The continuous-time limit of asset price processes where there is more than one asset












I've read Merton's article "On the Mathematics and Economics Assumptions of Continuous-Time Models" (Reprinted in Continuous-time Finance, Chapter 3), where Merton proved that the price of an asset follows an Itô process (no "rare events", or only those of type I/II) or a jump-diffusion process (with type III "rare events") under certain assumptions when taking the continuous-time limit.

However, in Merton's article he only considered the case of one asset. Are there any generalizations of this result to $n\geqslant 2$ assets? (I assume such generalizations exist, but I can't find any)

Here is a link of the article mentioned above: https://www.dropbox.com/scl/fi/of0rmatcimlrrqn2aucxu/On-the-Mathematics-and-Economics-Assumptions-of-Continuous-time-Models.pdf?rlkey=t498s5p91f8fo1x93smflc3m1&dl=0

Edit: There is a well-known result of prices in the binomial model (Cox-Ross-Rubinstein model) approaching a diffusion process in continuous time. However, the result in Merton's article is much stronger, since it does not specify a functional form.

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