Pricing model and leverage
teeter is separated into two tranches, each with a different formula for determining NAV.
Senior tranche (bToken)
The net asset value (NAVb) of the senior tranche, or bond side, of teeter can be determined by the following equation:
n: days since the asset was injected
N: days since initial investment (time t0)
f: all fees, i.e., trading and time fees
Qu: quantity of underlying assets
P0: price of the underlying asset at time t0
Subordinated tranche (xToken)
The net asset value (NAVx) of the subordinated tranche, or perpetual contracts / leveraged token side, of teeter can be determined by the following equation:
Qu: quantity of underlying assets
Qfs: quantity of funds sold
Qfa: quantity of funds available
Pt: price of the underlying asset at time t
P(t-1): price of the underlying asset at time t-1
f: all fees, i.e., trading and time fees
Leverage
Our smart contract will lock in the underlying asset at several times the market value of leveraged tokens. Due to natural price fluctuations in the market, the income of these assets will be distributed to leveraged token holders. For example, if a user makes a $100 purchase in the 3x long pool of ETH/USDC, the smart contract will lock up $300 worth of ETH in the contract, and the user will enjoy $300 worth of ETH floating return. When the price goes up by 10%, the value of the ETH locked in becomes $330, and the floating return of $30 goes to the user. The value of the principal goes from $100 to $130, which is a 3x (leverage multiple) return.
We use the following formula for calculating the leverage at any time t, or Lt:
Example: subordinated NAV(xToken price)
At time t0, the net value of the subordinated fund is 1, as denoted by the column in pink. The leverage multiple is 2 and the value of underlying asset is $20 000. The value of the underlying position is $2 000 000
At t1, the price is up by 10% and hence the value of the underlying assets has become $22 000, corresponding to a NAVx of 1.1996. Because the leverage multiple at t0 = 2 and expense deductions are taken into account, the net worth is now close to 1.2. Note that while the price has increased by 10%, the net worth of the user is now up 20%, resulting in a new leverage multiple of 1.83. However, the number of underlying assets has remained constant at 100.
At time t3, however, the price has gone to 10% lower than the initial price. Given a price at t3 of $18 000, the NAVx of the user is now 0.7989. Because the initial leverage multiple was 2 and the expense deduction is taken into account, the net value is approximately 0.8. This results in a higher leverage multiple of 2.25 to compensate for the price decline. The price is down 10% and the NAVx of the user is down 20%.
The relationship between leverage multiples and the value of a user's subordinated fund are briefly illustrated in the graph below for a hypothetical asset price trajectory:
In this graph, we're assuming that the user is inputting 300 tokens of an asset that was originally valued at $20 000 / token. The blue bars represent the aforementioned hypothetical asset price trajectory while the pink line (left vertical axis) represents the resulting leverage multiple. The right vertical axis in black represents the total value of the underlying assets, determined by 300 times the current price.
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