What happens when you buy an in the money put?

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When you buy an “in the money” put option, it means the strike price of the put is above the current market price of the underlying asset. In your example, if you buy a SPY 550 put when SPY is trading at 522, the put is already in the money by 28 points (550 - 522).

If SPY rises to 525, the intrinsic value of your 550 put would decrease to 25 points (550 - 525), but it would still be in the money. The 520 put, on the other hand, would be closer to being at the money, which means it would have less intrinsic value than the 550 put but might have more extrinsic value due to the time value and implied volatility.
So, if SPY continues to rise, both the 550 and 520 puts will decrease in value, but the 550 put will always have a higher intrinsic value as long as SPY remains below 550. However, the rate at which their prices change can differ due to the factors

The percentage loss of the two put options would not necessarily be the same if SPY rose to 525. This is because the sensitivity of an option’s price to changes in the underlying asset’s price, known as the option’s delta, varies for different strike prices.

The 550 put, being further in the money compared to the 520 put when SPY is at 522, would have a higher absolute delta value (closer to -1). This means it would lose more in dollar terms for a small increase in the SPY price compared to the 520 put. However, because the 550 put also has a higher initial value due to its intrinsic value, the percentage loss might be less compared to the 520 put, which has a lower absolute delta and lower initial value.

Here’s a simplified example to illustrate this:

550 Put: Let’s say it has a delta of -0.7 and an initial value of $30. If SPY increases by $3 (from 522 to 525), the put’s price would decrease by approximately $2.10 (0.7 * $3). The new value of the put would be $27.90, resulting in a percentage loss of approximately 7%.
520 Put: If this put has a delta of -0.5 and an initial value of $5, the same $3 increase in SPY would decrease the put’s price by $1.50 (0.5 * $3). The new value of the put would be $3.50, which is a percentage loss of 30%.
As you can see, even though the 550 put loses more in dollar terms, the percentage loss is greater for the 520 put because of its lower initial value and the fact that its price is more sensitive to changes in the underlying asset’s price relative to its lower initial value.