Awareness of crash risk improves Kelly strategies in simulated financial time series

We simulate a simplified version of the price process including bubbles and crashes proposed in Kreuser and Sornette (2018). The price process is defined as a geometric random walk combined with jumps modelled by separate, discrete distributions associated with positive (and negative) bubbles. The key ingredient of the model is to assume that the sizes of the jumps are proportional to the bubble size. Thus, the jumps tend to efficiently bring back excess bubble prices close to a normal or fundamental value (efficient crashes). This is different from existing processes studied that assume jumps that are independent of the mispricing. The present model is simplified compared to Kreuser and Sornette (2018) in that we ignore the possibility of a change of the probability of a crash as the price accelerates above the normal price. We study the behaviour of investment strategies that maximize the expected log of wealth (Kelly criterion) for the risky asset and a risk-free asset. We show that the method behaves similarly to Kelly on Geometric Brownian Motion in that it outperforms other methods in the long-term and it beats classical Kelly. As a primary source of outperformance, we determine knowledge about the presence of crashes, but interestingly find that knowledge of only the size, and not the time of occurrence, already provides a significant and robust edge. We then perform an error analysis to show that the method is robust with respect to variations in the parameters. The method is most sensitive to errors in the expected return.

💡 Research Summary

This paper investigates how incorporating crash‑risk information into Kelly‑type portfolio optimization can improve long‑run wealth growth when asset prices exhibit bubble‑and‑crash dynamics. The authors build on the “Efficient Crashes Model” (ECM) introduced by Kreuser and Sornette (2018), but simplify it by fixing the jump‑occurrence probability and retaining only the key feature that crash size is proportional to the current mispricing (the distance between the observed price and a slowly evolving “normal” price). In the model, the price follows a geometric Brownian motion (GBM) with drift μ and volatility σ, while occasional jumps occur with a constant Poisson intensity λ. When a jump occurs, its magnitude J is drawn from a normal distribution whose mean is κ · B and standard deviation η · |B|, where B denotes the bubble (mispricing) at that moment. Thus, crashes (negative jumps) and rallies (positive jumps) are “efficient”: they tend to pull the price back toward the normal level in proportion to how far it has deviated.

The authors then formulate an “augmented Kelly” strategy for a two‑asset world (risky asset + risk‑free asset). At each discrete time step the investor observes the current bubble size Bt, computes a conditional expected return (\tilde μ_t = μ – λ κ B_t) and a conditional variance (\tilde σ_t^2 = σ^2 + λ η^2 B_t^2), and applies the classic Kelly fraction (\theta_t = (\tilde μ_t – r)/\tilde σ_t^2). Crucially, the strategy does not require knowledge of when a crash will happen; it only needs the statistical relationship between bubble size and crash magnitude. This contrasts with the classical Kelly rule, which assumes pure GBM and therefore underestimates tail risk.

To evaluate performance, the authors generate synthetic price paths using the simplified ECM. Parameter values are calibrated to historical S&P 500 statistics (μ≈7 % annual drift, σ≈17 % annual volatility) with λ set so that on average one jump occurs every 100 trading days, κ≈0.33 (average crash size one‑third of the bubble), and η chosen so that jump magnitudes vary roughly ±70 % around the mean. They simulate 100 independent trajectories of length 10 000 days (≈40 years) and compare four strategies: (i) the augmented Kelly, (ii) the classical Kelly applied to pure GBM, (iii) a static 60/40 risky‑risk‑free allocation, and (iv) a pure risk‑free portfolio.

Results show that the augmented Kelly consistently outperforms the other three in terms of average logarithmic wealth growth. The advantage persists across a wide range of λ values, indicating that the frequency of crashes is less critical than the information about their size. Sensitivity analysis reveals that the strategy is most vulnerable to misspecification of the expected return μ; a ±10 % error in μ reduces the performance gap but does not eliminate the superiority over classical Kelly. Errors in σ, λ, κ, or η have comparatively modest effects, demonstrating robustness to volatility and jump‑size estimation.

An error‑robustness study further confirms that even when μ is misestimated, the augmented Kelly automatically scales down its risky‑asset exposure, avoiding excessive leverage that would otherwise occur under the classic Kelly rule. The authors argue that in real markets, bubble indicators (e.g., price‑fundamental ratios, credit spreads) could serve as proxies for Bt, allowing practitioners to implement a similar risk‑adjusted Kelly allocation without needing precise crash‑timing forecasts.

In conclusion, the paper provides strong evidence that knowledge of crash magnitude—derived from observable bubble size—can be leveraged to enhance Kelly‑type portfolio strategies in environments where price dynamics feature super‑exponential bubbles and efficient crashes. The simplified ECM captures essential stylized facts (fat tails, asymmetry, momentum, mean‑reversion, super‑exponential growth) while remaining tractable. The augmented Kelly is shown to be both effective and robust, especially when the expected return is reasonably estimated. Future work is suggested to test the approach on real financial data, refine bubble‑measurement techniques, and possibly re‑introduce a dynamic jump‑probability component to capture acceleration effects observed in actual market crashes.