How No-Arbitrage Constrains HJM Term-Structure Dynamics

Summary

The document asks whether no-arbitrage in term-structure models constrains only bond prices and yields at a given time or also their evolution through time. The answer uses a simplified description of an HJM setup: forward rates move over a time step according to specified volatilities and pairwise correlations, with the construction described as arbitrage-free within that step. It also says successive time steps are generated independently, presenting this as protection against arbitrage arising from the simulated dynamics.

The exchange gives only a brief, model-specific explanation and does not derive the no-arbitrage conditions or clarify their assumptions. In particular, it should not be read as a general statement that independent time steps are what makes HJM arbitrage-free; practical HJM models impose drift restrictions tied to volatility and the risk-neutral measure. Further detail is needed to distinguish cross-sectional consistency from dynamic arbitrage restrictions.

Key ideas

• The question distinguishes cross-sectional no-arbitrage from restrictions on bond and yield evolution over time.
• The answer describes HJM forward rates evolving with specified volatility and correlation structure.
• It claims independent time-step generation avoids arbitrage through the model’s dynamics.
• The short answer does not derive HJM’s formal drift restrictions or their assumptions.

Tags

# No-arbitrage in term-structure models


# No-arbitrage in term-structure models












I am a bit confused about what the implication of "no-arbitrage" in popular term struchture models (such as affine term struchtre models or HJM models) are?

Is it solely a restriction on the cross-section of bonds/yields in the sense that at time $t$ arbitrage oppurtunities are excluded or does it also provide a restriction on the time series dimension of bonds/yields? I am confused since e.g. the HJM model provides a dynamic equation for the evolution of forward rates through time and I am unsure if this only implies that for each point in time $t$ oppurtunities are excluded or does it also imply that the dynamic evolution of bonds/yields cohere such that arbitrage oppurtunities are excluded?

## Answer by dm63 (score 1)

https://quant.stackexchange.com/a/42766

In the typical HJM model, during a single time step forward rates evolve according to their individual volatilities and according to pairwise correlations which can be specified. That arrangement is arbitrage free within the time step. In addition, different time steps are independently generated. This latter feature ensures that the model does not generate arbitrage possibilities through its "dynamic evolution", as you say. Does that address your question?

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

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