Inverse vs Linear Contracts

Understanding Inverse vs. Linear Contracts

A linear contract follows a straightforward pricing model and is commonly used in many derivative markets. The price of a linear contract is expressed as the value of the underlying asset relative to the base currency. For example, a BTCUSDT perpetual contract is a linear contract, where BTC is quoted in USDT, and all margin and P&L calculations are denominated in USDT.

An inverse contract, on the other hand, is denominated in a fixed amount of the quote currency. For instance, in a BTCUSD perpetual contract, each contract represents $1 worth of Bitcoin at any price level. BTCUSD is considered an inverse contract because it is quoted in BTC/USD, but the underlying value is derived from USD/BTC, effectively following the formula: 1 / (BTC/USD). This inverse structure allows traders to hedge USD values efficiently while aligning with the spot market convention of quoting asset prices in USD.

Convexity Implications

Convexity (also referred to as Gamma) measures the second derivative of a contract's value relative to price changes. In inverse perpetual contracts, this relationship differs from the straightforward linear movement seen in linear contracts.

For a linear contract, the payoff is calculated as:

Contract Multiplier × (Entry Price - Exit Price)

For an inverse contract, the payoff follows a different formula:

Contract Multiplier × (1 / Entry Price - 1 / Exit Price)

This introduces convexity into the pricing model.

Example:

A trader enters a long position with 50,000 BTCUSD contracts at an entry price of $10,000 per BTC.

If the price rises to $11,000:

• Profit = 50,000 × 1 × (1/10,000 - 1/11,000) = 0.4545 BTC

If the price instead drops to $9,000:

• Loss = 50,000 × 1 × (1/10,000 - 1/9,000) = -0.5556 BTC

The greater loss in the downward move is due to the inverse, non-linear nature of the contract. Conversely, if the trader were short, the profit from a price drop would be larger than the loss from a price increase due to the same convexity effect.