Dynamic LP

Dynamic LP Strategy at the Deleverage Point — Break-Even Analysis & Binance Funding Comparison

Overview

This document develops a dynamic LP strategy for Derion pools that keeps the larger side of (A, B) pinned at the deleverage point, effectively maintaining linear leverage. We derive the break-even interest rate for a BTC pool with no premium, then compare it against real-world Binance perpetual funding rates.

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1. Protocol Recap

Derion is a fully on-chain power perpetuals AMM with three pool sides: A (Long), B (Short), and C (LP). The pool state is the tuple ⟨R, α, β⟩ where R is total reserve, α and β are the long and short coefficients.

1.1 Payoff Curves

For leverage parameter k and normalized price x (= spot / MARK):

Long payoff:

$$\Phi(x) = \begin{cases} \alpha x^k & \text{if } x \le \left(\frac{R}{2\alpha}\right)^{1/k} \quad \text{(power branch, leverage = k)} \\ R - \frac{R^2}{4\alpha x^k} & \text{otherwise} \quad \text{(asymptotic branch, leverage → 0)} \end{cases}$$

Short payoff:

$$\Psi(x) = \begin{cases} \beta x^{-k} & \text{if } x \ge \left(\frac{2\beta}{R}\right)^{1/k} \quad \text{(power branch, leverage = −k)} \\ R - \frac{R^2 x^k}{4\beta} & \text{otherwise} \quad \text{(asymptotic branch, leverage → 0)} \end{cases}$$

LP (Counterparty Liquidity):

$$\Omega(x) = R - \Phi(x) - \Psi(x)$$

1.2 Key Quantities

From the pool state, the reserves of each side are: rA = Φ(x), rB = Ψ(x), rC = R − rA − rB. The deleverage prices are the inflection points:

$$\text{dgA} = \text{MARK} \times \left(\frac{R}{2\alpha}\right)^{1/k} \qquad \text{dgB} = \text{MARK} \times \left(\frac{2\beta}{R}\right)^{1/k}$$

1.3 Interest & Premium

Interest decays both α and β by factor $2^{-t/H}$, transferring value from Long+Short → LP:

$$\alpha' = \alpha \times 2^{-t/\text{INTEREST\_HL}}, \qquad \beta' = \beta \times 2^{-t/\text{INTEREST\_HL}}$$

The effective interest rate charged to each side is:

$$\text{side}[A].\text{interest} = \text{interestRate} \times \frac{K}{k_{\text{eff},A}}$$

Premium transfers from the larger side to the smaller side + LP, pro-rata:

$$\text{PR}_{\text{long}} = \text{PR}_{\max} \times \frac{r_A^2 - r_B^2}{R \times r_A}$$

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2. The Deleverage Point & Leverage Regimes

Each side of the curve has two regimes separated by the inflection point (= deleverage point):

Regime	Long (Side A)	Short (Side B)
Power branch (full leverage)	x < (R/2α)^{1/k} → leverage = k	x > (2β/R)^{1/k} → leverage = −k
Asymptotic branch (deleveraged)	x > (R/2α)^{1/k} → leverage → 0	x < (2β/R)^{1/k} → leverage → 0

The effective leverage of side A at price x:

$$k_{\text{eff}}(A) = \frac{x \cdot \Phi'(x)}{\Phi(x)}$$

Branch	k_eff	rA
Power	k (full)	< R/2
At inflection	k (exact)	= R/2
Asymptotic	kR/(4αx^k − R) → 0	> R/2

Key Insight: When a side crosses its deleverage point, its payoff value exceeds R/2 and its leverage compresses toward zero. The larger side of (A, B) is the one closest to or past its deleverage point. At exactly the deleverage point, that side's reserve = R/2 and leverage = k.

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3. The Dynamic LP Strategy

Goal: Always keep the larger of (rA, rB) pinned at approximately the deleverage point — i.e., the larger side's reserve ≈ R/2. This ensures the dominant side stays at exactly linear leverage k.

3.1 Mechanism

1. Monitor the pool state. After each price move, check if max(rA, rB) has moved away from R/2.

2. If price rises (rA grows), rA may exceed R/2 and enter the asymptotic branch. The LP adds liquidity (increases R) so the deleverage point shifts up and rA returns to ≈ R/2.

3. If price falls (rB grows), the LP adjusts symmetrically to keep rB at its deleverage boundary.

4. Net effect: At all times, max(rA, rB) ≈ R/2, both sides maintain effective leverage ≈ k.

Price rises → rA > R/2 → LP adds R (adjusts α) → rA = R/2 restored
Price falls → rB > R/2 → LP adds R (adjusts β) → rB = R/2 restored

3.2 What "Linear Leverage" Means

At the deleverage point, the payoff function transitions from convex (power branch) to concave (asymptotic branch). At this exact point, the value is R/2 and the instantaneous leverage is k. For any further movement in the winning direction, leverage compresses. By always rebalancing to keep the dominant side at this point, the LP ensures:

The dominant side's PnL behaves approximately linearly in x (leverage ≈ k for small moves), not super-linearly (deep in power branch) and not sub-linearly (deep in asymptotic branch).

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4. LP PnL Under This Strategy

4.1 LP Exposure at the Deleverage Point

Under the constraint max(rA, rB) = R/2, assume the Long side is dominant (rA ≈ R/2):

$$r_A = \alpha x^k = R/2, \quad r_B = \beta x^{-k} \text{ (small)}, \quad r_C = R/2 - \beta x^{-k}$$

4.2 LP Gamma

The LP's gamma at the deleverage point (from the power payoff second derivative):

$$\Gamma_{\text{LP}} \approx -\frac{k(k-1) \cdot R}{2x^2}$$

This is negative gamma — the LP loses from price volatility (impermanent loss).

4.3 Impermanent Loss Rate

For a GBM price process with volatility σ:

$$\frac{dIL}{dt} = \frac{1}{2} \times |\Gamma_{\text{LP}}| \times \sigma^2 \times x^2 = \frac{k(k-1) \cdot R \cdot \sigma^2}{4}$$

Per unit of LP capital (≈ R/2):

$$\boxed{\text{IL rate per LP capital} = \frac{k(k-1) \cdot \sigma^2}{2}}$$

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5. Break-Even Interest Rate

With no premium (premium rate = 0), the only income for the LP is the interest rate.

5.1 Interest Income

The interest rate r decays rA and rB, transferring value to LP:

$$\text{Income per unit R} = r \times \frac{r_A + r_B}{R}$$

With the deleverage constraint (rA ≈ R/2, rB small), rA + rB ≈ R/2.

5.2 Break-Even Condition

Setting interest income ≥ IL cost:

$$r \times \frac{1}{2} \ge \frac{k(k-1) \cdot \sigma^2}{4}$$

$$\boxed{r_{\text{break-even}} = \frac{k(k-1) \cdot \sigma^2}{2}}$$

For k = 2 (BTC pool): This simplifies elegantly to:

$$\boxed{r_{\text{break-even}} = \sigma^2}$$

The break-even interest rate equals BTC's variance.

5.3 After Protocol Fee

Derion takes 1/5 of interest income to LP as protocol fee. The gross break-even becomes:

$$r_{\text{gross}} = \frac{k(k-1) \cdot \sigma^2}{2 \times 0.8} = \frac{k(k-1) \cdot \sigma^2}{1.6}$$

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6. Numerical Results for BTC

6.1 Break-Even by Volatility

Using formula $r = k(k-1)\sigma^2 / 2$ (gross, before protocol fee):

k	σ = 40%	σ = 50%	σ = 60%	σ = 70%	σ = 80%
2	16%	25%	36%	49%	64%
4	96%	150%	216%	294%	384%
8	448%	700%	1008%	1372%	1792%

6.2 Converting to Derion's Half-Life

$$\text{INTEREST\_HL} = \frac{\ln 2}{r / \text{seconds\_per\_year}} = \frac{0.693}{r / 31{,}536{,}000}$$

Scenario (k=2)	Annual Rate	Daily Rate	Half-Life
σ = 50% (low vol)	25%	0.068%	~1,012 days
σ = 60% (mid vol)	36%	0.099%	~703 days
σ = 70% (high vol)	49%	0.134%	~516 days
σ = 80% (very high)	64%	0.175%	~395 days

BTC Pool (k=2, σ=60%): Break-even interest ≈ 36% annualized (≈ 0.10% daily), half-life ≈ 703 days. After 20% protocol fee → pool needs ~45% gross, half-life ≈ 562 days.

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7. Comparison with Binance Perp Funding

7.1 Binance BTC Perp Funding — The Benchmark

Binance charges BTC perpetual funding every 8 hours. The baseline (when perp ≈ spot) is 0.01% per 8h, annualizing to ~10.95%. In practice, funding varies dramatically:

Market Regime	Typical 8h Rate	Annualized
Baseline / neutral	0.0100%	~10.95%
Mild bull (normal trending)	0.01–0.03%	~11–33%
Strong bull (2024 BTC run)	0.03–0.06%	~33–66%
Extreme euphoria (peaks)	0.05–0.15%	~55–165%
Bear / capitulation	−0.01 to 0.00%	~−11 to 0%

Key data points:

• BTC aggregate funding was overwhelmingly positive in 2024, only 26 days negative.

• OI-weighted funding has been positive >85% of the time over the past 2 years.

• Many exchanges use a base interest rate that skews default funding to ~10.95% annualized.

• Peak: OI-weighted average reached 109% annualized on Feb 28, 2024.

• Long-run average: approximately 11–22% annualized; in bull years like 2024, closer to 15–30%.

7.2 Head-to-Head Comparison

	Binance BTC Perp	Derion BTC LP (k=2)
Rate type	Market-driven + 10.95% base	Pool-configured (INTEREST_HL)
Long-run average	~11–22% annualized	Needs σ² = 25–64%
Bull market avg	~20–30% annualized	σ=50% → needs 25%
Peak euphoria	55–165% annualized	σ=70% → needs 49%

7.3 Visual Scale

0%        20%        40%        60%        80%       100%+
|----------|----------|----------|----------|----------|
██████░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░  Binance base (11%)
██████████████░░░░░░░░░░░░░░░░░░░░░░░░░░░░  Binance bull avg (25%)
████████████████████░░░░░░░░░░░░░░░░░░░░░░  Derion BE σ=50% (25%)
██████████████████████████░░░░░░░░░░░░░░░░  Derion BE σ=60% (36%)
████████████████████████████████████░░░░░░  Derion BE σ=70% (49%)
██████████████████████████████████████████  Binance peak 2024 (60%+)

7.4 Scenario Analysis

Scenario	BTC σ	Derion Needs	Binance Pays	Verdict
Low vol regime	40%	16%	~11–15%	⚠️ Tight / underwater
Normal vol	50%	25%	~15–25%	⚠️ Marginal
Elevated vol + bull	60%	36%	~25–60%	✅ Often covered
High vol + euphoria	70%+	49%+	~50–100%+	✅ Typically covered

7.5 Structural Differences

Apples-to-apples on 2× leverage exposure:

	Binance (2× levered perp)	Derion (k=2 power perp)
Funding per unit capital	2 × 11% = 22% (baseline)	r (pool interest rate)
Liquidation risk	Yes (at ~50% adverse move)	None (asymptotic curve)
Gamma / convexity	Zero (linear payoff)	Positive (power payoff)

Derion traders get positive convexity and no liquidation — valuable features that justify paying a higher funding rate. The LP bears the cost of that convexity, which is exactly the σ² term.

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8. Key Observations

8.1 The Variance-Funding Relationship

For k=2, the break-even is simply σ². This is not coincidental — it mirrors option theta. A power-2 perpetual's gamma cost is proportional to the variance, just as ATM option theta is ½Γσ²S². The Derion LP is essentially writing a power perpetual, and σ² is its theoretical fair premium.

Binance's funding rate is set by market forces (supply/demand of leverage) plus a fixed base, not directly tied to realized volatility. This creates a structural mismatch that can work in either direction.

8.2 High Vol and High Funding Are Correlated

The crucial nuance: high funding rates and high volatility tend to occur together. Bull markets produce both high σ (increasing LP costs) and high funding (increasing LP income). Empirically during peak euphoria, funding can reach 100%+ annualized — well above σ² ≈ 49–64%. But during quiet periods (σ ≈ 40%), base funding of 11% falls short of the 16% break-even.

8.3 Premium Rate Closes the Gap

Our break-even formula assumes zero premium. In practice, under the dynamic strategy (rA ≈ R/2, rB small), the imbalance is large → premium income is significant. This additional income means the actual break-even interest rate is lower than σ², potentially making the position profitable even at Binance's base rate of ~11%.

8.4 Higher k Pools

The break-even scales as k(k−1), making high-leverage pools require dramatically higher interest:

• k=4: needs 6σ² → ~216% at σ=60%

• k=8: needs 28σ² → ~1008% at σ=60%

Only very short-term positions are economical for traders in high-k pools.

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9. Conclusion

Main Result: For a Derion pool with leverage k, underlying volatility σ, and no premium, the LP break-even interest rate under the dynamic deleverage-point strategy is:

$r_{\text{break-even}} = \frac{k(k-1) \cdot \sigma^2}{2}$

For a BTC pool with k=2 and typical σ ≈ 60%, this gives r ≈ 36% annualized (≈ 0.10% daily).

vs Binance: The Derion LP needs roughly 2–3× Binance's base rate but only about 1–1.5× the bull-market average. Including premium income (material under the dynamic strategy) and the vol-funding correlation, the strategy is viable in trending/active markets but marginal in quiet ones — structurally equivalent to being short volatility, which is exactly what an LP position is.

Practical Considerations

• Rebalancing frequency: Discrete rebalancing introduces tracking error. The break-even is a lower bound; actual required rates may be 10–30% higher.

• Premium as buffer: Under the dynamic strategy, persistent imbalance generates premium that can cover 30–50% of the IL cost.

• Protocol fee: 20% of interest to LP is taken as protocol fee — gross pool rate must be 1.25× the net break-even.

• Gas costs: On-chain rebalancing has transaction costs that reduce net LP returns.

• Comparison to Squeeth: For Opyn's Squeeth (k=2), the funding rate is approximately σ²/year. Derion's LP break-even at k=2 is also σ², confirming the mathematical equivalence of the gamma cost.