Re-Leverage Arbitrage — Toy Example Workflow

Step-by-step mechanics of the VirtualPool re-leverage arbitrage for the Yield Basis LEVAMM with 2× leverage (WBTC market).

CryptoPool1M crvUSD / 10 BTCLP Supply2,000,000Collateral200,000 LPDebt100,000 crvUSDLeverage2×LEV_RATIO4/9Fee0.3%

1Initial State (T0) — Balanced, BTC = 100,000 crvUSD

Curve CryptoPool

stables (crvUSD)1,000,000

BTC10.0000

LP supply2,000,000

price_scale100,000 crvUSD/BTC

LP oracle price1.0000 crvUSD/LP

LEVAMM StateDTV = 50.00%

collateral200,000 LP

debt100,000 crvUSD

x₀ (computed)300,000.00

x₀−debt (x_initial)200,000.00

invariant k = x₀×coll40,000,000,000

LEVAMM price/LP1.0000 crvUSD/LP

Premium vs oracle0.00%

At DTV = 50%: x₀ = 3 × debt = 300,000 | x_initial = 2 × debt = 200,000 | LEVAMM marginal price = x_initial / collateral = 200,000 / 200,000 = 1.00 crvUSD/LP — exactly equal to the oracle LP price. No premium, no discount. The system is at equilibrium.

2After BTC Pump (T1) — BTC = 110,000, price_scale still lagging at 100,000

CryptoPool (after ext. arb)

stables (crvUSD)1,048,808.85

BTC9.534626 BTC

price_scale100,000 (lagging!)

LP price @ market1.048809 crvUSD/LP

LP price @ scale1.001136 crvUSD/LP

LEVAMM StateDTV = 47.67%Premium

collateral200,000 LP (unchanged)

debt100,000 crvUSD (unchanged)

x₀ @ new oracle328,129.17 (grew!)

x_initial = x\u2080\u2212debt228,129.17

Δ x_initial+28,129.17

LEVAMM price/LP (no fee)1.140646 crvUSD/LP

Oracle LP price1.048809 crvUSD/LP

Marginal premium+8.76%

Why does x₀ grow? cv = p₀ × collateral = 1.048809 × 200,000 = 209,761.77. The formula x₀ = (cv + √(cv² − 4·cv·4/9·debt)) / (2·4/9) yields a larger x₀ because cv grew. Since debt is unchanged, x_initial = x₀ − debt grows by 28,129 crvUSD. LEVAMM now bids 8.76% above oracle LP price.

3Workflow Diagram — VirtualPool Re-Leverage Arbitrage (i = 1)

Arbitrageur

EOA / Bot

0.001 BTC

(= 110.00 crvUSD)

VirtualPool.exchange

i = 1 (BTC → crvUSD)

in0.001 BTC

out127.4512 crvUSD

internally:

1Flash Loan

Borrow crvUSD proportional to BTC input

flash = in × S / B
= 0.001 × 1,048,808.85 / 9.534626
= 110.00 crvUSD

2add_liquidity

Symmetric deposit: flash + BTC → LP tokens

in: 110.00 crvUSD + 0.001 BTC
lp = supply × in / B
= 2,000,000 × 0.001 / 9.534626
= 209.7618 LP

3LEVAMM.exchange(1, 0, LP)

Sell LP to LEVAMM, receive crvUSD (hyperbolic curve)

x_initial = 228,129.17
y_after    = 200,209.7618
x_after    = ⌈x_init × coll / y_after⌉ = 227,891
gross      = 238.1657 crvUSD
out (0.3% fee) = 237.4512 crvUSD

4Repay Flash & Collect Profit

LEVAMM paid: +237.45 crvUSD
Repay flash: −110.00 crvUSD
Net out: +127.4512 crvUSD
BTC cost: −110.0000 crvUSD
PROFIT: +17.4512 crvUSD
(+15.865% on BTC cost)

4LEVAMM State Change — Before vs After Re-Leverage Arbitrage

Metric	Before (T1)	After arb	Delta	Interpretation
Collateral (LP)	200,000.00	200,209.76	+209.7618	LEVAMM received LP
Debt (crvUSD)	100,000.00	100,237.45	+237.4512	LEVAMM minted new debt
x₀ (virtual reserve)	328,129.17	328,133.18	+4.0151	Recomputed at same oracle
DTV	47.67%	47.74%	+0.06%	↑ toward 50% target

Both collateral AND debt increase. LEVAMM is re-leveraged: it now holds more LP collateral, funded by more crvUSD debt. DTV moves from 47.67% toward the 50% target (now 47.74%). The “premium” paid to the arbitrageur is the protocol’s cost to restore its 2× leverage target — analogous to option theta cost for maintaining delta exposure.

5DTV Sensitivity — Profit vs Oracle LP Price Deviation

Oracle LP	DTV	LEVAMM/LP	Premium	out_amt	BTC cost	Net Profit	Profit %	Note
0.9500	52.63%	0.8398	-11.60%	65.1211	89.2737	-24.1526	-27.05%
0.9800	51.02%	0.9387	-4.22%	85.8045	95.5665	-9.7620	-10.21%
1.0000	50.00%	1.0000	+0.00%	99.0000	99.7618	-0.7618	-0.76%	T0 balanced
1.0200	49.02%	1.0589	+3.81%	111.5289	103.9570	+7.5719	+7.28%
1.0500	47.62%	1.1440	+8.95%	129.2201	110.2499	+18.9702	+17.21%
1.0488	47.67%	1.1406	+8.76%	128.1657	110.0000	+18.1657	+16.51%	T1 actual
1.0800	46.30%	1.2261	+13.53%	146.3502	116.5427	+29.8075	+25.58%
1.1000	45.45%	1.2796	+16.33%	157.2422	120.7379	+36.5043	+30.23%
1.1500	43.48%	1.4102	+22.63%	184.6167	131.2260	+53.3906	+40.69%

Profit is positive when DTV < 50% (BTC price rose, oracle LP price rose). At small input (0.001 BTC) price impact is negligible, so profit % ≈ marginal premium. At larger inputs the hyperbolic curve’s price impact erodes the premium — this is why real arbitrageurs trade in small increments or use TWAP execution. Break-even around DTV ≈ 50%.

6De-Leverage Direction (T2) — BTC Drops to 90,000

T2 State: DTV > 50%

BTC market price90,000 crvUSD

LP oracle price0.948683 crvUSD

LEVAMM DTV52.70% (> 50%)

x₀ (shrank)267,042.47

x_initial167,042.47

LEVAMM/LP discount-11.96%

Arb directioni = 0 (crvUSD → BTC)

i=0 Step-by-Step (in = 1,000 crvUSD)

r0fee = S*(1-fee)/supply0.474342

K = r0fee * collateral94,868.33

b = x₀−debt + in − K73174.14

D = b\u00b2 + 4*K*in5,733,927,874.41

flash = (\u221aD \u2212 b) / 21274.28 crvUSD

out_BTC = flash * B / S0.01416 BTC

BTC value at 90,0001,274.28 crvUSD

Cost (crvUSD in)1,000 crvUSD

Net Profit+274.28 crvUSD (+27.43%)

De-lever arb: the arbitrageur provides crvUSD, the VirtualPool uses a quadratic formula (not simple proportion) to compute the flash loan needed for a symmetric LP burn. LEVAMM sells LP cheaply (discount), arbitrageur removes LP from Curve pool for more crvUSD than paid → profit. DTV falls back toward 50%.

7Key Formulas

LEVAMM get_x0 — Virtual Reserve Anchor

# cv = oracle value of LP collateral
cv  = p_o × collateral
k   = 4/9  # LEV_RATIO for 2x leverage

x0  = (cv + √(cv² − 4·cv·k·debt)) / (2·k)

# At DTV=50%:  x0 = 3*debt,  x_initial = 2*debt
# At DTV<50%: x0 grows  -> premium (re-lever)
# At DTV>50%: x0 shrinks -> discount (de-lever)

LEVAMM get_dy(1, 0, lp) — LP → crvUSD

x_initial = x0 − debt
y_after   = collateral + lp_amount
x_after   = ⌈x_initial × collateral / y_after⌉

out = (x_initial − x_after) × (1 − fee)

# Hyperbolic invariant:  x_initial × collateral = k
# LEVAMM overpays when x_initial grew (DTV < 50%)

VirtualPool i=1 (Re-Lever, BTC Rises)

# Simple proportion — NO quadratic
flash = in_btc × S / B
#   S = crvUSD in pool,  B = BTC in pool

lp    = supply × in_btc / B
#   symmetric add_liquidity proportional to pool

out   = AMM.get_dy(1, 0, lp) − flash
#   LEVAMM overpays LP when x_initial is inflated

VirtualPool i=0 (De-Lever, BTC Falls)

# Quadratic equation
r0fee = S × (1−fee) / supply
b     = x0−debt + in_crvusd − r0fee×collateral
D     = b² + 4 × r0fee × collateral × in_crvusd

flash = (√D − b) / 2
out   = flash × B / S
#   BTC from symmetric remove_liquidity

8Summary

Re-lever profit (0.001 BTC)

+17.4512 crvUSD

Input: 0.001 BTC = 110.00 crvUSD · +15.865%

LEVAMM marginal premium at T1

+8.76%

DTV dropped to 47.67% · BTC pumped to 110k

DTV restored after arb

47.67% → 47.74%

Moving toward 50% target · LP coll ↑ debt ↑

Flash loan (borrowed & repaid)

110.00 crvUSD

No capital at risk · Same-tx atomicity

LP minted from Curve pool

209.76 LP

Symmetric deposit at pool ratio · 0.105% of coll

De-lever profit (1,000 crvUSD)

+274.28 crvUSD

BTC dropped to 90k · DTV 52.70% · +27.43%

Complete mechanics in one sentence: When BTC rises, the LEVAMM oracle anchor inflates x₀, creating a premium above Curve LP fair value — arbitrageurs exploit this by providing BTC, flash-borrowing crvUSD, minting LP, and selling it to LEVAMM for a profit; the side effect is that LEVAMM gains leverage (DTV rises toward 50%). The symmetric mirror image occurs when BTC falls. Every arbitrage trade nudges the system back toward its 2× leverage target, making arbitrageurs the market’s automatic leverage-rebalancing mechanism.

Valueverse.