Density Square Integrability, Finite Variance, and Tail Risk

Summary

The document asks whether stock-price densities under models such as Heston, SABR, Black–Scholes, and Variance-Gamma are square-integrable. Its answers disagree: one emphasizes that a probability density need not generally be square-integrable and notes that option pricing often uses a characteristic function; another asserts square-integrability when variance exists and raises concerns about heavy-tailed financial returns.

The exchange points to a useful distinction between properties of a density and the moments of the modeled distribution, but it does not establish a general result for the listed models. Finite variance alone is not presented with a proof sufficient to settle the question, and the blanket claim should not be taken as a verified conclusion. The risk observation is that models with finite-variance returns may understate tail risk if actual returns are heavier-tailed. The excerpt offers no empirical comparison or model-specific analysis.

Key ideas

• A probability density is not necessarily square-integrable in general.
• The responses disagree about whether finite variance is enough to ensure square-integrability.
• Characteristic functions may be used in option pricing without directly working from a density.
• Finite-variance models may understate risk if actual returns have heavier tails.
• The excerpt does not prove the claim for each named model or settle the disagreement.

Tags

# Are densities used in finance square integrable?


# Are densities used in finance square integrable?












Let $f$ be the density of the stock asset under some model (Heston, SABR, Black Scholes, Variance-Gamma, etc).

Is $f$ square-integrable in these models?

## Answer by user39119 (score 2)

https://quant.stackexchange.com/a/50280

I've never heard about the density function of the stock price in Heston model. For pricing, one uses the characteristic function which can be derived from the Heston-PDE. In general, a density function is not always square-integrable. Check this post https://math.stackexchange.com/questions/756540/is-a-probability-density-function-necessarily-a-l2-function

## Answer by A. Angeli (score 1)

https://quant.stackexchange.com/a/50277

Yes, $f$ is square-integrate. This follows from the fact that the variance exists.

Note that in general for most models this is the case but it is debatable given that financial returns are so heavy tailed. Thus, by modeling finance with models that output returns with finite variance we might be heavily underestimating the risks.

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