Endogenous Emission–Flow Coupling in Bittensor AMMs

Develop a coupled stochastic model that endogenizes the pool-invariant growth rate k(t) in Bittensor’s Dynamic TAO constant-product automated market makers by linking the growth rate \dot{k}(t) to the net staking flow process through the network’s emission allocation rule (which assigns each subnet a block emission share proportional to max(S_i − L, 0)). Determine how this endogeneity modifies the time-varying CEV volatility parameter δ(t), the integrated variance over an option’s life, and resulting European option prices.

Background

In the emissions extension, the paper shows that deterministic injections of both tokens deepen the pool without changing spot, producing a time-varying CEV volatility δ(t)=2σ_F/√k(t) and an option price that depends on integrated variance. For tractability, the analysis treats k(t) as growing at an exogenous linear rate over the option horizon.

However, under Bittensor’s Dynamic TAO design, each subnet’s TAO pool injection per block depends on an exponentially weighted flow measure S_i via the emission allocation rule. This creates a feedback loop in which higher inflows increase emissions, which deepen liquidity and compress volatility. Capturing this mechanism requires modeling the endogenous coupling of \dot{k}(t) to the flow process, which the authors defer.

References

Bittensor's emission allocation eq:emission introduces a feedback loop: high staking flows raise $S_i$, increasing the subnet's emission share, deepening the pool, and compressing volatility. Modeling this endogenous $\dot{k}$ coupled to the flow process is left for future work.

eq:emission:

eTAO,i=Eblock⋅max⁡(Si−L,0)∑jmax⁡(Sj−L,0),e_{\mathrm{TAO},i} = E_{\text{block}} \cdot \frac{\max(S_i - L, 0)}{\sum_{j} \max(S_j - L, 0)},

— Option Pricing on Automated Market Maker Tokens  (2603.29763 - Maymin, 31 Mar 2026) in Remark (Emissions as a dividend yield), Section 5.6 (Extension: Token Emissions)