2025 XJTLU Derivatives and FinTech Workshop�Suzhou, 2025/5/17

Belief Dispersion in the Market for Event Risk

Dan Luo and Zhentao Zou

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Discussant：Zhuo Chen

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Motivation and Overview

• This paper introduces a dynamic GE model with heterogeneous beliefs about rare jumps of assets’ cash flows, and then provide closed-form solutions for belief weights, stock and option prices, event-risk insurance premium, and the size of the insurance market, all of which are sufficiently characterized by two statistics: the time-varying average belief and belief dispersion.
• Findings:
• Amplified jump risk: belief dispersion amplifies event risk in cash flows, leading to a larger jump in the stock price and self-exciting jump risk in option pricing.
• Countercyclical Risk Premium: higher in bad states (after crashes) when pessimistic beliefs dominate, and lower in good states when optimists hold more wealth.
• Inverted U-shaped relationship between the degree of belief dispersion and the size (open interest) of the event-risk insurance market.
• The continuum of beliefs plays an important role in generating those dynamics as a comparable two-investor model cannot reproduce some effects.
• Contribution:
• A model with closed-form solution for several quantities of interest can be summarized by two sufficient statistics.
• The model aligns with existing empirical regularities, and also provides testable implications for future empirical work

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Comment 1: Extension on the assumption of jump intensity

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• The paper assume the heterogeneous beliefs of the jump intensity λ, another possible investor belief heterogeneity is about the relative jump size L, which measures the economic magnitude on the cash flow when a rare event happens).
• It is possible that the jump intensity (or relative jump size) is endogenously determined by the distribution of investor beliefs, e.g., an increasing function of the fraction of optimistic investors. What would be the model implications?
• Empirically, can we disentangle whether investors’ heterogeneity lies in jump intensity, relative jump size, or a combination of the two?

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Comment 2: Short selling constraints and time-varying probability of rare events

• The impact of short selling constraints
• The model assumes that optimists and pessimists can trade frictionlessly, fully reflecting their views in market prices.
• But what if short-selling constraints prevent pessimists from acting on their outlook (see,. e.g., Shen et al., 2017; Stambaugh et al., 2012, 2015; Liu et al., 2020), how might the model implications change as we introduce such constraints (or add additional costs for short sellers)?
• The probability of rare event can also be time-varying
• In reality, rare events happen when almost all investors hold the same belief and trade on the same direction (“Chaos is not dangerous until it begins to look orderly”).
• Probability of rare events is also likely to depend on the wealth of optimistic investors.
• After a series of rare events, will the probability decrease as the average belief increases, i.e., investors update their beliefs after observing the events.
• Exploring a learning mechanism or Bayesian updating may provide some interesting implications.
• Empirical tests to support model predictions
• The model’s predictions hinge on two endogenous moments—average belief and belief dispersion. Could the authors suggest empirical proxies for each? The greater challenge will likely be devising a robust measure of belief dispersion.

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