Applying a Window function to calculate differences in pySpark

Lag function can help you resolve your use case.

from pyspark.sql.window import Window
import pyspark.sql.functions as func

### Defining the window 
Windowspec=Window.orderBy("day")

### Calculating lag of price at each day level
prev_day_price= df.withColumn('prev_day_price',
                        func.lag(dfu['price'])
                                .over(Windowspec))

### Calculating the average                                  
result = prev_day_price.withColumn('daily_return', 
          (prev_day_price['price'] - prev_day_price['prev_day_price']) / 
prev_day_price['price'] )

You can bring the previous day column by using lag function, and add additional column that does actual day-to-day return from the two columns, but you may have to tell spark how to partition your data and/or order it to do lag, something like this:

from pyspark.sql.window import Window
import pyspark.sql.functions as func
from pyspark.sql.functions import lit

dfu = df.withColumn('user', lit('tmoore'))

df_lag = dfu.withColumn('prev_day_price',
                        func.lag(dfu['price'])
                                 .over(Window.partitionBy("user")))

result = df_lag.withColumn('daily_return', 
          (df_lag['price'] - df_lag['prev_day_price']) / df_lag['price'] )

>>> result.show()
+---+-----+-------+--------------+--------------------+
|day|price|   user|prev_day_price|        daily_return|
+---+-----+-------+--------------+--------------------+
|  1| 33.3| tmoore|          null|                null|
|  2| 31.1| tmoore|          33.3|-0.07073954983922816|
|  3| 51.2| tmoore|          31.1|         0.392578125|
|  4| 21.3| tmoore|          51.2|  -1.403755868544601|
+---+-----+-------+--------------+--------------------+

Here is longer introduction into Window functions in Spark.