Protocol

Width selection

The desk maps volatility and APR into a log-width, or into no position. Costs are impermanent loss and rebalance. If APR is too thin or volatility is too violent, the optimum is to sit out.

State of the pool

Let S_t be the pool mid. TRUSS LP treats log-price as a jump-diffusion, not as a smooth drift. Jumps are the point: a quiet Brownian path understates how often a tight range is blown out.

dS_t / S_t = μ dt + σ dW_t + J dN_tW is Brownian motion. N is a Poisson process with intensity λ. J is the relative jump size, drawn independently (Kou / Merton style). Drift μ is not a trading signal — it cancels in the HODL-relative book.

Effective variance, the quantity that actually stresses a range, is

σ_eff² = σ² + λ E[J²]

Figure 1. The same drift with and without jumps. Width policy is set from effective volatility, not from the quiet path.

Width

A concentrated position is an interval [S_L, S_U] around the current mid. Log-width is

w = ln(S_U / S_L), S_L = S₀ e^−w/2, S_U = S₀ e^w/2

For a fixed notionals V, virtual liquidity on a Uniswap v3-style AMM scales like 1 / w at leading order: a tighter band posts more active liquidity, earns more of the fee stream, and exits more often.

L(w) ∝ V / (√S_U − √S_L) ∼ V / (S₀^1/2 w)

Expected payoff

Let η be the advertised fee-plus-emissions APR on in-range capital, τ(w) the expected fraction of time the mid stays inside the band, n_reb(w) the expected number of exits, and c the all-in cost of one rebalance (gas, slippage, and the protocol swap margin). Over a horizon T the desk maximises

Π(w) = η V T · φ(w) · τ(w, σ_eff) − C_IL(w, σ_eff) − n_reb(w) cφ(w) is the concentration factor. τ falls as w shrinks or σ_eff rises. n_reb grows like barrier hits of a diffusion: roughly (σ_eff / w)² T.

Time-in-range for the diffusive part is the probability that a Brownian motion has not left [−w/2, w/2]. A working closed form, then thinned by jumps, is

τ(w) ≈ e^−λT erf( w / (2 σ √(2T)) ) + (1 − e^−λT) τ_jump

Impermanent loss

A CFMM is short realised volatility. For a full-range constant-product book the second-order loss versus holding the same mix (loss-versus-rebalancing) is

LVR_full ≈ (σ_eff² / 8) V T

Concentration multiplies gamma. In the symmetric-band approximation used by the desk,

C_IL(w) = (σ_eff² / 8) V T · κ(w), κ(w) = (w₀ / w) · (e^w/4 + e^−w/4) / 2

κ(w) → ∞ as w → 0: an infinitely tight range is a pure options short with no time in market. That is why “just use the highest APR pool at min width” is not a strategy.

Rebalance costs

Every time the mid exits, the vault must burn the NFT (or equivalent), swap residual inventory, and mint again — or sit out. Expected hit rate for a diffusion with absorbing barriers at ±w/2 is of order

n_reb(w) ≈ (σ_eff / w)² T

Each event costs gas plus spread plus the protocol’s swap margin. Jumps add an almost-sure exit whenever |J| > w/2, so λ itself is a rebalance intensity. High-vol pools with fat tails are expensive even when the APR table looks green.

The optimum, and sitting out

w* = arg max_w>0 Π(w | σ, λ, η, V, c)

Compare that peak to doing nothing: hold 50/50, earn zero LP fees, pay zero IL and zero rebalance. If max Π ≤ 0, the policy is w = ∞ — no position.

Figure 2. Expected net payoff against log-width. The peak is w*. If the whole curve sits below idle 50/50, the desk holds no LP position.

In (σ_eff, η) space the sit-out region is everything below a convex frontier η*(σ_eff). That frontier is exactly what you would expect: thin APR or violent vol, and LP is a worse inventory than holding.

Figure 3. Sit-out frontier η*(σ_eff). Below the curve, expected fees cannot cover IL and rebalance costs, so capital stays idle.

How the desk uses this

State	Typical σ_eff	Width	Why
Tight	Low, few jumps	Narrow band	Concentration pays; exits are rare; IL is small.
Moderate	Routine two-sided vol	Medium band	Fees still cover gamma and the odd rebalance.
Sit-out	High vol or jump-rich, or η too low	No LP	max Π < idle. Capital stays 50/50 in the vault.

Those three states are what the How-to tape draws. The tape is realised volatility of a financial price — oscillation of returns, not a slope.

The live bot and the simulation share this objective. If a pool’s APR cannot pay for its own σ_eff, it will not stay at the top of the rank table, and the bot will not keep a position there.