The Omniscient, yet Lazy, Investor

Abstract

We formalize the paradox of an omniscient yet lazy investor - a perfectly informed agent who trades infrequently due to execution or computational frictions. Starting from a deterministic geometric construction, we derive a closed-form expected profit function linking trading frequency, execution cost, and path roughness. We prove existence and uniqueness of the optimal trading frequency and show that this optimum can be interpreted through the fractal dimension of the price path. A stochastic extension under fractional Brownian motion provides analytical expressions for the optimal interval and comparative statics with respect to the Hurst exponent. Empirical illustrations on equity data confirm the theoretical scaling behavior.

Figures (4)

• Figure 1: Right-triangle construction per subinterval at resolution $m$. Both triangles share base $\Delta=T/2^m$ and satisfy $\left(\tfrac{T}{2^m}\right)^2=\bar{h}_m^2+W^{2m}c_0^2$. The relation models the scale-dependent roughness of price increments observed empirically in fractal market studies Calvet2002Bouchaud2000.
• Figure 1: Expected profit $R_m$ as a function of trading resolution $m$ in the deterministic framework of Section \ref{['sec:deterministic']}. The curve illustrates the trade--off between the exploitable substructure gain (first term) and proportional plus cognitive costs ($2^m\bar{s} + L(m)$). The interior maximum corresponds to the optimal trading interval $\Delta^\star = T/2^{m^\star}$. Source: Own calculations performed in Julia
• Figure 1: Simulated profit functions $R_m$ under fractional Brownian motion for different Hurst exponents $H\in\{0.40,0.60,0.80\}$. Parameter values: $T=1$, $\kappa=0.5$, $\bar{s}=0.002$, $\lambda=6\times 10^{-4}$, $\alpha=1.4$. Each curve exhibits an interior optimum $m^\star(H)$ that shifts toward finer resolutions as $H$ decreases, in agreement with Theorem \ref{['thm:opt']}. Source: Own calculations performed in Julia.
• Figure 1: Empirical and theoretical profit curves $R_m$ for AAPL daily data (2020--2025). The empirical optimum $m^\star_{\text{emp}}=5$ (red marker) lies near the theoretical prediction $m^\star_{\text{theory}}=6$, illustrating consistency between observed and predicted trading frequencies. Source: Own calculations performed in Julia.

Theorems & Definitions (19)

• Proposition 4.1: Closed-form profit at dyadic level $m$
• Proof 1
• Remark 4.2: Feasibility region and qualitative behavior
• Proposition 5.1: Existence of a maximizer
• Proof 2
• Theorem 5.2: Marginal stopping rule
• Proof 3
• Remark 5.3: Economic interpretation
• Proposition 5.4: Strict unimodality and uniqueness
• Proof 4: Proof sketch