UniV3 Strategy - Part5, UMAMI-WETH
Forecast UMAMIETH Price in the next 1 and 2 hours with Autoregressive Moving Average Model
Previously, I demonstrated how to use AutoRegression (AR) to predict the next hour BTCETH price. Today, I’ll show you how to predict the next hour UMAMIETH (UMAMI/ETH) price with the AutoRegressive Moving Average (ARMA) model. Because ARMA is built on top of AR, I recommend you to read my previous article first to connect the dots.
Why are we interested in predicting the short term movement of token A price / token B price? Because if we can successfully do that, we can better set the lower and upper price limits and hence play the in-and-out liquidity provision game better. This is a game sophisticated UniV3 users have been playing. For example, they’d identify a small pool with $1M ~ $10M TVL, provide liquidity for as short as 1 ~ 3 hours to rack up trading fees while minimizing impermanent loss. On the other hand, short term price movement is predictable because price momentum tends to carry forward in a short time window.
The UMAMI-WETH 1bp pool on arbitrum has a TVL of $657K at the moment, so it seems a good choice for demonstration. But the recipe and python code I provide here can be easily adopted to other pairs. So let’s get started.
First, we need to download hourly UMAMI and WETH price data from Arbitrum. Thanks to DeFiLlama’s data API and the defillama2 package, we can easily do that in python. For this demo, I set the download period between 2022-11-01 and 2022-12-09. It took ~11 minutes to download the data. I then divided UMAMI price by WETH price to get UMAMIETH price.
There are no missing values in the prices so we don’t need to worry about imputation or filling. I then ran some summary statistics and plotted UMAMIETH to gain a quick understanding of the data.
The second step is to prepare data for model building and forecasting. Using an arbitrary cutoff time, 2022-12-03 23:00:00, I split UMAMIETH into training and test sets. The training set has 792 prices and the test set 121. I then scaled the data so that all values are in the range of 0 and 1. I used the min and max of UMAMIETH in the training set to scale the test series to avoid data leaking. The following table shows scaled values from the training and test series.
The third step is to determine parameter values that give the best model fit to the training data. AutoRegression (AR) model only uses historical prices to predict future prices, whereas AutoRegressive Moving Average (ARMA) model uses both historical prices and historical prediction errors to predict the future. ARMA model has two parameters, p and q , where p is the number of lagged series and q is the number of past prediction error series. For example, ARMA(1, 1) is short for autoregressive moving average of order (1, 1), and its model equation is
Y(t) = a + b1*Y(t-1) + c1*error(t-1) + error(t)and ARMA(2, 1) has a model equation of
Y(t) = a + b1*Y(t-1) + b2*Y(t-2) + c1*error(t-1) + error(t)We want to find p, q such that ARMA(p, q) fits the training data best. We can use the ar_select_order() function from the statsmodels package to automatically determine the best p. I ran it on my training data and got p = 2, telling me to use prices at last hour and two hours ago as predictors.
To determine the best q value, I fit both ARMA(2, 1) and ARMA(2, 2) and examined their residuals by looking at the model diagnostic plot.
Both correlograms of the residuals show almost all lollipop sticks squashed down inside the blue 95% confidence band except at lag 15. This says both ARMA(2, 1) and ARMA(2, 2) are able to explain away the autocorrelation in the price series and hence are good model choices. The tiny peak at lag 15 is not a concern because it's expected that 5% out of the 20 lags will be non-zero due to sampling variation.
So shall we go with ARMA(2, 1) or ARMA(2, 2)? To make that choice, I looked at the coefficient estimates of ARMA(2, 1) and found `ar.L2` is not statistically significant and hence can be dropped. This contradicts to what the auto-selection algo suggested that the best AR order is 2.
I then looked at the coefficient estimates of ARMA(2, 2) and found that although `ar.L2` becomes significant after adding `ma.L2`, `ar.L1` is not anymore. But obviously we can't drop `ar.L1`, so let’s go with the parsimonious ARMA(1,1).
I fit an ARMA(1, 1) on the training set and obtained the following diagnostic plots. The correlogram (bottom right) looks good and we are good to move to the next step.
Iterative Training and Forecasting
Now that we know ARMA(1, 1) fits our data best. Let's train an ARMA(1, 1) model and use it to make forecasts iteratively on a rolling basis. That is, as we iterate over the timestamps in the test series, we will
train an ARMA(1, 1) on the training set,
make a 2-step forecast, i.e., predict next-hour and next-2-hour prices simultaneously,
grow the training set by adding the test set observation at that timestamp to the end of the training series. Because we have a small dataset and a simple model, we don’t need to keep the training size fixed to speed up the computation.
Afterwards, we’ll reverse the scaling operation so that we have the forecast values and the actuals in the original scale. We can then plot them. Notice the next hour forecasts are better than the next-2-hour forecasts. This is expected as it’s more difficult to predict the distant future than the near future.
Finally, we should calculate a single metric to summarize the error. Mean Absolute Percent Error (MAPE) is a good choice.
Code & Data
Python notebook. You can easily adopt the code to any UniV3 pairs.
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