Why do ATM options intuitively have higher Time Value (Extrinsic Value) than Out- and In-The-Money options?

Question

I'm trying to get some intuition concerning the Black Sholes Formula and in doing so I've come across these graphs:

Trying to understand the intrinsic value relationship with Options Value was trivial but as soon as you touch the extrinsic value (specifically the Time Value) it all goes south:

How is Time Value affected by an increase in Stock Price, two seemingly unrelated variables? Admittedly I can get my mind around the relationship between OTM and ATM options: Since the lower the stock price is, the lower the likelihood for the options to make a comeback and enter the In-The-Money zone, assuming a random market.

But that thought can't apply to ITM options. Since ITM options are already profitable, but you're at plus minus 0 since you paid for the intrinsic value, I don't understand how this scenario differs from ATM options.

In other words, regardless of what your strike price is relative to the stock price, your profits are fully dependent on how your stock performs from the moment you buy the stock. Even if the option is far into the money from the beginning since you put a low strike price, it doesn't affect its potential to generate further profits or losses. So why would your time value diminish? In that case, wouldn't you just always buy ITM options since they essentially offer the same deal but without you having to pay for that high Time Value?

I've tried to search as to why this might be, but all sources are protected by payment walls (link) or else their answers don't really tackle the relationship but somehow stall or avoid answering the question by rephrasing it. For instance, I've tried Reddit, this site, these link link S.E. ones and this Quora one.

The Reddit one is a good example: He answers the questions by essentially saying that as the Stock Price increases you get less Time until expiration, which means that your potential for further profits decreases. This would make sense, but the question still stands then, how does an increase in Stock Price affect your Time until expiration, it just doesn’t connect.

Luckily I found another quant S.E. question that appears to ask the very same thing here. But the person who answers it (Robert James) gives an answer using quite complex terms.

So if somebody out there could help me give some intuition as to why this phenomenon exists and not just that it exists, or else try and translate Mr James's answer into plain English, it'd be great. I'm sorry if this is a nooby question but I can't be asked to prompt more questions, only to get more of Chat GPT's unhelpful answers. It's driving me insane.

Answer 1

For an OTM option, time value represents the likelihood that the option ends up in-the-money, increasing its value over its current "intrinsic" value of zero.

For an ITM option, time value represents the protection that an option gives if the stock were to move below the strike - buyers pay a premium over its intrinsic value for that downside protection. The more ITM you are, the lower the probability that you need that protection (and the less you're willing to pay for it).

As you approach ATM from either direction, these two facets of time value both increase (probability of expiring ITM and probability of needing protection), giving you the "tent" shape that you observe.

Answer 2

Another way to look at it:

The PnL gain from delta hedging is proportional to gamma and always positive. Gamma is highest at the ATM so you have higher PnL gain from the dynamic hedge - and this represents extrinsic value (value due to volatility)

This ignores theta (movement due to intrinsic value/irrespective of vol).