What is the intuition behind a positive theta for European long puts?

I've googled extensively for an answer to this question. Very similar (if not identical) questions have popped up in this same website (example) but I never find the answers to be clear and/or precise.

I've played around with the equation for theta (for puts), and indeed it becomes positive for S << K [ie. deep in the money (itm)]. But intuitively why is this so? One explanation I've read is "Well, if you're deep itm, wouldn't you want to be closer to maturity?" The answer is yes, but I'd also want the same thing for deep itm calls and yet theta is negative for calls. So why is it that theta only becomes positive for deep itm puts and not deep itm calls?

Also, as an aside and this may just be a trivial thing, but why is theta in general called "the time decay" when theta can be both positive and negative?

• The put payoff is bounded by $K$. If $S\ll K$, then it can't really get any better for you. What do you want to wait for? For a call option, even if $S\gg K$, the asset price is unbounded and can always rise even further. Sep 30 at 18:59

It’s just the effect of interest. If you are long a deep ITM European put, it is worth the PV of K minus the stock price. But one day later the PV of K has grown a bit. That’s it. It’s the opposite for calls because you have to pay the K, so bringing the date closer costs you money. This is all assuming interest rates are positive.

First, let's go back to basics to answer why theta can be both positive and negative, and why it's referred to as time decay? At it's core, an option's value is composed of two components:

1. intrinsic value, and
2. time value.

As time passes, the proportion of the 'time value' gradually decreases until the option is worth exactly its intrinsic value at its expiration. Theta is simply the rate at which the option losses its value as time passes (all other market conditions remaining unchanged). Hence theta is offen referred to as time decay.

As you have mentioned, although theta can be positive (where time value is negative), almost all options lose value as time passes. This is why the convention has been to express theta as a negative number.

Instances of negative time value and hence postive theta are relatively rare and assume European option contracts deep in the money (ITM) with stock-type settlement. This positive theta or negative time value is the effect of interest rates.

In the case of positive interest rates, for deep ITM puts the present value of the strike (K) less the underlying price (S) can increase day-to-day and hence have a positive theta. In this case, the present value of the strike (K) has increased in value. If this were an American option, everyone would exercise the option today to earn interest on the intrinsic value.

To consider the same circumstances with European calls, one has to imagine a world with negative interest rates. In this case, a deep ITM call option's present value of the underlying price (S) less the strike (K) can increase day-to-day. In this case you (the holder of the call option) have to pay the strike (K), and in a world with negative interest rates would be receiving interest on this amount with each passing day.