Heston Return Distributions: Filtering and Calibration Limits
Summary
The document discusses attempts to describe stock return distributions under the Heston stochastic volatility model, including a proposed unconditional density and later statistical comparisons with a lognormal distribution. It raises the concern that an unconditional density removes the latent volatility path and can discard information contained in volatility transitions. When mean reversion is weak, robust identification from the stationary distribution may require a long, clean sample.
The responses favor approaches that use conditional dynamics, such as filtering latent volatility, and mention joint estimation using implied volatilities. They also frame some apparent model problems as calibration difficulties: the model's dependence on parameters need not produce a convex optimization objective, which can make fitting unstable and interpretations ambiguous. These are qualitative suggestions and references, not a worked estimation recipe or evidence that one technique reliably outperforms another. The document supplies no new closed-form solution and emphasizes the difficulty of estimating unobserved volatility.
Key ideas
- An unconditional return density can discard information in the volatility transition process.
- Weak volatility mean reversion can make parameter identification from stationary data difficult.
- Filtering methods use conditional dynamics to infer latent volatility.
- Joint estimation with implied volatility data is presented as another possible approach.
- Nonconvex parameter calibration can produce unstable fits and ambiguous interpretations.
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Full text
# Stock Returns Distribution in Heston Model # Stock Returns Distribution in Heston Model There is a paper by Dragulescu and Yakovenko (DY) in 2002 proposing a pdf for the stock returns in the Heston model. However, in a paper by Daniel, Bree and Joseph, they actually perform statistical tests on DY's pdf and show it is not really any better than a log normal pdf. Is anyone aware of more recent attempts at a closed-form solution for the distribution of returns under the Heston model? ## Answer by Kiwiakos (score 2) https://quant.stackexchange.com/a/17150 The issue I have with these approaches is that they use the unconditional distribution to eliminate the latent volatility. However, when the volatility process has very weak mean reversion one would need a very long and clean sample to make robust parameter identification from the unconditional density. They just throw away all the information from the transition dynamics. My preference is a filtering approach. There have been some older papers that did that, google for Heston together with Ghysels, Gallant, Renault, Chernov, Tauchen, Pan, Bates, Shephard, MCMC, unscented Kalman filter and you will get some references. It is still ugly, since volatility is unobserved, but at least you are looking at conditional transitions rather than the stationary distribution. Even better, you can implied vols and perform joint estimation. Some of the references above do that too. ## Answer by zuiqo (score 1) https://quant.stackexchange.com/a/16755 Gatheral (Amazon) has a quite extensive discussion on that, and dives into calibration issues. In summary, what you describe appears to be less of a modeling issue, and more of a calibration problem. This is primarily because the model functions (such as the Heston model) are not by nature convex in their input parameters. This is simply result of the fact that they are designed with intuitive understanding of parameters in mind, and not functional properties. You can intuitively understand what variance of variance implies, but there's simply not technical reason it should be convex. This results in highly unstable optimizations, and thus ambiguous interpretations, such as described in the sources you give.
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