## Time-Weighted Average Price (TWAP) We're going to talk about the TWAP calculation, which can be useful when we want to have a more accurate representation of the average price of a token over a specified period. The equation we'll be using is: TWAP from $T_k$ to $T_{n+1} = \frac{C_{n+1}-C_k}{T_{n+1}-T_k}$ This is the TWAP from time $t_k$ to $t_n$. ## TWAP Example Let's say we have a token that starts at 1000 at time t = 1, and the price changes as follows: | Time | Price | |---|---| | 1 | 1000 | | 3 | 1100 | | 4 | 1300 | | 7 | 1200 | | 11 | 1500 | Let's say that we want to find the TWAP from time t = 4 to time t = 11. To calculate the TWAP, we can create a table to keep track of our calculations. We'll need four columns in the table: | Time ($t_i$) | Price ($p_i$) | $Δt_i * p_i$ | Cumulative Price ($c_i$) | |---|---|---|---| | 1 | 1000 | | | | 3 | 1100 | | | | 4 | 1300 | | | | 7 | 1200 | | | | 11 | 1500 | | | We can fill out the first two columns with the data from our price history. Next, we can calculate the delta of t * p_i for each time period. This is calculated by multiplying the price by the duration that the price remained at that level. For example, for time t = 1, the price was at 1000 for 2 seconds (from time t = 1 to time t = 3). So, the delta of t * p_i for this period is: (3 - 1) * 1000 = 2000 We can fill out the table with the rest of our calculations: | Time ($t_i$) | Price ($p_i$) | $Δt_i * p_i$ | Cumulative Price ($c_i$) | |---|---|---|---| | 1 | 1000 | 2000 | 2000 | | 3 | 1100 | 1100 | 3100 | | 4 | 1300 | 3900 | 7000 | | 7 | 1200 | 4800 | 11800 | | 11 | 1500 | | | The cumulative price for each row is calculated by adding the $Δt_i * p_i$ for that row to the cumulative price of the previous row. The final step is to use the equation to calculate the TWAP. The latest cumulative price, $c_n$, is 11,800. The cumulative price at time t = 4, $c_k$, is 3100. We are taking the TWAP from time t = 4 to time t = 11, so the values of $t_n$ and $t_k$ are 11 and 4, respectively. This gives us the following calculation for TWAP: (11,800 - 3,100) / (11 - 4) = 1242 The TWAP from time t = 4 to time t = 11 is approximately 1242. We can see that this makes sense, as the price was at 1300 for 3 seconds and at 1200 for 4 seconds, making the TWAP a little less than 1250. This is just a basic example of how to calculate TWAP.