Skip to content
All library documents

Modeling Alpha and Non-Martingale Views in Equilibrium Finance

Article Quant Q&A · Author: athos
Read the original

Summary

The document asks whether quantitative finance can model markets without assuming martingale behavior. It frames the question around the view that martingales capture unpredictability and are associated with the efficient-market hypothesis, while exploitable inefficiencies may exist temporarily before being discovered and traded away.

The answer points to equilibrium models that allow alpha, or returns beyond those explained by market risk, and suggests CAPM with heterogeneous investor beliefs as a theoretical example. In that setup, investors whose beliefs differ from the market’s average can perceive opportunities for alpha. The cited account says that over the long run alpha diminishes, bringing the outcome in line with efficient-market and homogeneous-belief CAPM results. This is a conceptual illustration, not a developed model specification or empirical demonstration; the post does not explain how to estimate alpha or test the assumptions in trading data.

Key ideas

  • The post asks about finance models that allow non-martingale opportunities.
  • Equilibrium models can represent alpha as an opportunity beyond compensation for market risk.
  • CAPM with heterogeneous beliefs is offered as a theoretical example of perceived alpha.
  • Investors whose beliefs diverge from the market average may see different opportunities.
  • The cited account says alpha diminishes over the long run, aligning outcomes with standard CAPM and efficient-market results.

Tags

StatisticsFactor investingPortfolio construction
Full text
# Assumptions based on non-martingale?


# Assumptions based on non-martingale?












Quantitative finance formular are mostly based on martingales, Poisson jump, GBM, CEV, etc.. The logic behind it is that martingale means the future could not be predicted, or, EMH (Efficient-market hypothesis).

fair enough. however, EMH is only a hypothesis now, the other side of the story, that the market is inefficient, could also stands -- provided the inefficiency is always found and exploited then disappeared.

so, is there some theory / math model, that is based on non-martingale analysis?

## Answer by Alexey Kalmykov (score 1, accepted)

https://quant.stackexchange.com/a/3063

In the equilibrium models you can assume that there exists so called Alpha, i.e. an opportunity that can be exploited. Most of the buy side models (i.e. asset allocation, portfolio construction) are based on this idea.

As a theoretical model, you can consider CAPM with heterogeneous beliefs:

> Hedge funds claim to generate the “Alpha”, i.e., excess returns that cannot be explained by market risk. ... The assumption of homogeneous beliefs was always under scrutiny among finance theorists. Removing it we can model the Alpha within the CAPM, i.e., as a property of financial market equilibria! We show that in a CAPM with heterogeneous beliefs every investor who holds beliefs different to the average market belief, sees some Alpha.

from Financial Economics: A Concise Introduction to Classical and Behavioral Finance

In the long-run the outcome of this model is consistent with the efficient market hypothesis and also with the CAPM based on homogeneous beliefs, i.e. Alpha diminishes.

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.