♈Skew-Dependent Premium/Discount Function

Perps Pricing Functions

The skew-dependent premium/discount function is central to the operational mechanism of LP-Delta-Neutral Perpetual Futures. It operates on the principle that the market price of a perpetual contract can be adjusted to either a premium or a discount relative to the spot price based on the current delta skew of the LPs' positions:

$$P_{\text{perp}} = (1 + \text{Premium/Discount}) \cdot P_{\text{index}}$$

From LP’s POV

\begin{eqnarray} f(x) &=& \text{Premium/Discount} = \frac{\text{Perp Price} - \text{Index Price}}{\text{Index Price}} = -c \cdot x \cdot |x| \\ x &=& \frac{LP_{\text{long}} - LP_{\text{short}}}{LP_{\text{AUM}}} = \frac{\text{Trader}_{\text{short}} - \text{Trader}_{\text{long}}}{LP_{\text{AUM}}} \end{eqnarray}

From Trader’s POV

\begin{eqnarray} f(x) &=& \text{Premium/Discount} = \frac{\text{Perp Price} - \text{Index Price}}{\text{Index Price}} = c \cdot x \cdot |x| \\ x &=& \frac{LP_{\text{short}} - LP_{\text{long}}}{LP_{\text{AUM}}} = \frac{\text{Trader}_{\text{long}} - \text{Trader}_{\text{short}}}{LP_{\text{AUM}}} \end{eqnarray}

Where:

• c is a constant factor, the genesis setting for vUSDC perp is 11.11

• x represents the LP's delta position, calculated as the difference between LP’s short position sizes and LP’s long position sizes, normalized by the total assets under management (AUM)

• Index Price: Spot TWAP price that combines both Oracle Prices and Index Prices

• Perp Price: Current Price of Perpetual Future (same for long/short)

• LP AUM: Total amount of liquidity for this trading pair, calculated by dollar

• Premium/Discount: The deviation of Perp Price from Index Price

The function outputs a value for f(x) that dictates the necessary adjustment to the perp price to incentivise trades towards delta neutrality. For instance, if the traders' short dominates, the LP's position is overexposed to long positions (delta imbalance), the function will adjust the perp price downwards, encouraging traders to take long positions and rebalance LPs' position to neutrality. Conversely, if the trader long dominates, then LPs' positions are overexposed to short positions (delta imbalance), and the perp price is adjusted upwards to incentivise short positions.

---

Numerical Example

Time	LPs' Position	Traders' Position	Total AUM	Perp Price
T1	0	0	1,000,000	70,000
T2	30,000 Long	30,000 Short	1,000,000	69,300 (Adjusted by -1%)

Let's apply the formula with numerical values to illustrate its impact:

Assuming:

• LP AUM (Total Assets Under Management of the LP pool) = $1,000,000

• After Trade Net Traders’ Short Position Size = -$30,000 (meaning traders are net short)

• After Trade Net LP’s Long Position Size = +$30,000 (equivalent net long to offset traders' net short)

• Before Trade BTC Index Price = $70,000

We start by calculating the LP's delta position:

$$x = \frac{\text{Net LP Long Position Size}}{\text{LP AUM}} = \frac{30,000}{1,000,000} = 0.03$$

Applying the premium/discount function

$$f(x) = -c \cdot x \cdot |x| = -1.11 \cdot 0.03 \cdot |0.03| = -1\%$$

The premium/discount is a 0.1% discount on the perp price due to the net short position of traders (and net long position of LPs). Next, we need to apply this to the index price to find the perp price:

$$P_{\text{perp}} = (1 + \text{Premium/Discount}) \cdot P_{\text{index}} = (1 - 0.009999) \cdot 70,000 = \$69,300$$

After applying the discount for delta neutrality, the perp price is approximately $69,300. This adjusted price aims to incentivise traders to take long positions to balance the overall delta of the LP pool, moving towards delta neutrality.

Through this dynamic and responsive pricing mechanism, vDEX's LP-Delta-Neutral Perpetual Futures offer a robust framework for managing liquidity in a decentralised volatile environment. By maintaining LPs' delta-neutrality, the platform protects them from undue market risks while promoting a healthy, liquid, and efficient market for trading derivatives in DeFi.