Revisiting the Excess Volatility Puzzle Through the Lens of the Chiarella Model

We amend and extend the Chiarella model of financial markets to deal with arbitrary long-term value drifts in a consistent way. This allows us to improve upon existing calibration schemes, opening the possibility of calibrating individual monthly time series instead of classes of time series. The technique is employed on spot prices of four asset classes from ca. 1800 onward (stock indices, bonds, commodities, currencies). The so-called fundamental value is a direct output of the calibration, which allows us to (a) quantify the amount of excess volatility in these markets, which we find to be large (e.g. a factor $\approx$ 4 for stock indices) and consistent with previous estimates; and (b) determine the distribution of mispricings (i.e. the difference between market price and value), which we find in many cases to be bimodal. Both findings are strongly at odds with the Efficient Market Hypothesis. We also study in detail the ‘sloppiness’ of the calibration, that is, the directions in parameter space that are weakly constrained by data. The main conclusions of our study are remarkably consistent across different asset classes, and reinforce the hypothesis that the medium-term fate of financial markets is determined by a tug-of-war between trend followers and fundamentalists.

💡 Research Summary

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This paper revisits the classic excess‑volatility puzzle by updating the Chiarella heterogeneous‑agent model to accommodate arbitrary long‑term drifts of the fundamental value. The authors introduce a linear price‑impact equation (following Kyle) and decompose total market demand into three representative agent groups: fundamentalists, trend‑followers, and noise traders. Fundamentalists submit demand proportional to the mispricing (V_t-P_t) with strength (\kappa); trend‑followers react to an exponentially weighted moving‑average of excess returns, generating a saturated demand (\beta\tanh(\gamma M_t)); noise traders contribute a Brownian term with volatility (\sigma_N). Crucially, the trend signal (M_t) is defined so that the long‑run drift (g_t) of the fundamental value does not contaminate it, thereby decoupling the deterministic fixed point from the drift.

The resulting stochastic differential system (Eq. 3) is analytically tractable. Linearising around the fixed point ((\delta,M)=(0,0)) (where (\delta=P-V) is the mispricing) yields a Jacobian whose eigenvalues determine stability. The fixed point is stable when (\kappa > \alpha(\beta\gamma-1)); in this regime price spirals toward the fundamental value. When the inequality reverses, the fixed point loses stability via a Hopf bifurcation at (\alpha^* = \kappa\beta\gamma-1), and a stable limit cycle emerges, producing persistent oscillations of price around value. This bifurcation condition matches that of earlier work but now holds for any drift (g_t), making the model applicable to assets with strong secular trends.

A cubic correction (\kappa_3 (V-P)^3) is discussed as a way to enforce stronger mean‑reversion for large mispricings, but the main empirical work remains in the linear specification for computational efficiency.

Calibration is performed on monthly spot‑price series from roughly 1800 to the present for four broad asset classes: equity indices, government bonds, commodities, and currencies. Using Bayesian filtering and a “sloppiness” analysis (following Sethna et al.), the authors identify which parameter combinations are tightly constrained by the data and which lie in weakly identified directions. The core parameters (\kappa, \beta, \gamma,) and (\alpha) are well‑estimated, whereas the noise volatilities (\sigma_N) and (\sigma_V) are sloppy.

Empirical findings: (1) Excess volatility is substantial. For equity indices, observed price volatility is about four times the volatility of the calibrated fundamental value; bonds, commodities, and currencies show factors of 2–3. This confirms the classic Shiller excess‑volatility puzzle across a wide historical span. (2) The distribution of mispricings (P_t-V_t) is frequently bimodal, indicating that markets spend considerable time in over‑valued and under‑valued regimes rather than clustering around a single “fair” price. (3) The sloppiness analysis suggests that the market dynamics are dominated by the tug‑of‑war between fundamentalists (pulling prices toward value) and trend‑followers (driving momentum), while stochastic noise plays a secondary, less identifiable role.

Overall, the paper delivers a coherent theoretical framework that reconciles long‑term value drifts with heterogeneous agent behavior, provides a robust calibration methodology for very long historical series, and delivers quantitative evidence that excess volatility and bimodal mispricing are pervasive features inconsistent with the Efficient Market Hypothesis. The authors argue that the medium‑term fate of markets is governed by the balance between trend‑following and value‑based trading, a conclusion that holds across asset classes. Future work is hinted at, including nonlinear demand extensions, multi‑time‑scale trend signals, and more refined estimation of sloppy parameters using higher‑frequency data or machine‑learning‑enhanced Bayesian techniques.