## Uniswap V2 TWAP In this lesson, we'll discuss the concept of Time Weighted Average Price (TWAP) within the context of Uniswap V2. ### What is TWAP? The Time Weighted Average Price (TWAP) is the average price of a token (X) in terms of another token (Y). This is calculated over a specific time period (t_i ≤ t < t_{i + 1}) where Δt is the duration of the price. ### Calculating TWAP To calculate the TWAP, we'll use the concept of cumulative price and a specific formula. The cumulative price (c_j) up to time t_j is calculated as the sum of the price for each time period: ```javascript c<sub>j</sub> = cumulative price up to t<sub>j</sub> = ∑<sub>i = 0</sub><sup>j-1</sup> Δt<sub>i</sub>.p<sub>i</sub> ``` We can then expand the formula for the cumulative price from time t_k to t_n as: ```javascript ∑<sub>i = k</sub><sup>n-1</sup> Δt<sub>i</sub>.p<sub>i</sub> = Δt<sub>k</sub>.p<sub>k</sub> + ... + Δt<sub>n-1</sub>.p<sub>n-1</sub> ``` This can be further simplified as: ```javascript ∑<sub>i = k</sub><sup>n-1</sup> Δt<sub>i</sub>.p<sub>i</sub> = (Δt<sub>0</sub>.p<sub>0</sub> + ... + Δt<sub>k-1</sub>.p<sub>k-1</sub> + Δt<sub>k</sub>.p<sub>k</sub> + ... + Δt<sub>n-1</sub>.p<sub>n-1</sub>) - (Δt<sub>0</sub>.p<sub>0</sub> + ... + Δt<sub>k-1</sub>.p<sub>k-1</sub>) ``` This then becomes: ```javascript ∑<sub>i = k</sub><sup>n-1</sup> Δt<sub>i</sub>.p<sub>i</sub> = c<sub>n</sub> - c<sub>k</sub> ``` Therefore, the TWAP from time t_k to t_n can be calculated as: ```javascript TWAP from t<sub>k</sub> to t<sub>n</sub> = (c<sub>n</sub> - c<sub>k</sub>) / (t<sub>n</sub> - t<sub>k</sub>) ``` ### TWAP to Current Time We cannot directly calculate the TWAP from a specific time (t_k) to the current time (t), as we do not know how long the current price will remain at its current value. To solve this, we can use a technique which sets the current time (t) to t_{n + 1}, and the current price (p) to p_n, which results in: ```javascript ∑<sub>i = k</sub><sup>n</sup> Δt<sub>i</sub>.p<sub>i</sub> = ∑<sub>i = k</sub><sup>n-1</sup> Δt<sub>i</sub>.p<sub>i</sub> + Δt<sub>n</sub>.p<sub>n</sub> = c<sub>n</sub> + (t - t<sub>n</sub>).p ``` Using this, we can then calculate the TWAP from time t_k to t_{n + 1} using the formula: ```javascript TWAP from t<sub>k</sub> to t<sub>n + 1</sub> = (c<sub>n</sub> + (t - t<sub>n</sub>).p - c<sub>k</sub>) / (t<sub>n + 1</sub> - t<sub>k</sub>) ``` This formula will allow us to estimate the TWAP up to the current time. ### Example **Diagram:** [Diagram of a simple example with time and price data points] We have the following price and timestamp data: | t_i | p_i | Δt_i.p_i | c_i | |---|---|---|---| | 1 | 1000 | - | - | | 3 | 100 | (3 - 1)1000 = 2000 | 2000 | | 4 | 1300 | (4 - 3)100 = 100 | 3100 | | 7 | 1200 | (7 - 4)1300 = 3900 | 7000 | | 11 | 1500 | (11 - 7)1200 = 4800 | 11800 | We want to calculate the TWAP from time 4 to 11: ```javascript TWAP from 4 to 11 = (c<sub>11</sub> - c<sub>4</sub>) / (t<sub>11</sub> - t<sub>4</sub>) = (11800 - 3100) / (11 - 4) ``` This gives us: ```javascript TWAP from 4 to 11 = 1271.43 ``` This demonstrates how we can calculate the TWAP up to the current time.