The BS assumes you delta hedge with the underlying to achieve put call parity and in that case they're the same option with the theta in the options and the cost to carry the hedge.

Given that theta is an appox. or mean of expectation it's impossible to say. Sometimes there is a clear cut gradual decay intraday. But on my platform it is expressed as a drop in IV. So which is it? Sometimes it's difficult to say whether you are seeing time decay or IV contraction. You may be able to tell that day or the next, or not at all. If the underlying is moving and the option prices are moving then you need experience and a keen eye to surmise decay. If IV rises there may be no decay that day or negative time decay. It's easiest to see time decay intraday on expiry Friday. But there are times where the ATM option may hold considerable time value until the last hour or two.

Let's go back and remember that an option is the sum of two things - time value and intrinsic value. Spin, the call at a given strike can be more expensive or the put can be more expensive depending on where the underlying is. But that's the result of changing intrinsic value, not time value. Intrinsic value is irrelevant to our discussion, so let's concentrate on time value. The put and the call at the same strike have the same time value. If anyone is confused on that point, then please think it through and get unconfused. It is a fundamental, bedrock truth about options and if you have that wrong, you have the whole thing wrong. If it weren't true, then there would be no synthetic puts and calls, no conversions or reversals, no put-call parity. Gamma, vega and theta are all aspects of time value. So if the put and call at a given strike have the same time value - which they do - they have the same gamma, vega and theta. HOWEVER, it is true that time value is discounted by the cost of carry of the option (do not confuse with cost of carry of the underlying). So if IBM is at 100, then the 180 put is much more expensive than the 180 call, and the time value of the 180 put is discounted by the interest you will be paying on those 80 points of intrinsic value. Since that cost of carry calculation modifies the time value, it also slightly modifies the gamma, vega and theta.

The way to decay option time intra day is to use seconds to expiration, and then take out each second. The actual details of the way the software does it is more optimized than this, but this is the general idea.

Intraday time decay is pretty much irrelevant other than for a theoretical exercise on a forum. The markets going to price the options they way it perceives them to be around fair value and theyâre not going to be clicking off seconds in a day. This even goes for the last day of trading in a contract. In a thinly traded option on expiration day pricing may be more subjective since there will be so few trades but in a deep option the price on a day like expiration is going to be set by supply and demand rather than how an option decays per second. The point of the thread in pretty moot in the context of the real world.

Thanks for the explanation. Let me be more specific about what I meant about calls being higher priced than puts. Obviously (as you noted), the call at a given strike can be more expensive than the put or the put can be more expensive than the call due to where the underlying is (one ITM, one OTM). That's due to intrinsic value. However, assuming no dividend, if you adjust for the intrinsic value amount, the call will be higher priced than the put due to the carry cost (extrinsic value). Of course, if you're right on top of expiration, the carry cost will be almost nothing and they'll be equal. I understand your explanation of the cost of carry of the 180 put and I've seen that in a pricing model (the discounting) but I'm confused as to where it's coming from theoretically. Practically, I understand that if you buy a put with 80 pts of intrinsic that there's a carry cost. That makes sense. But I'm lost on where it's coming from in the put-call parity equation (or other equation). The gist of what I've taken from the various contributors is that theta is the same for puts and calls with the same terms if you make an exception for carry cost.

Spin, I think what you're saying is that if IBM is at 100, then the 100 call will be more expensive than the 100 put due to cost of carry of the underlying. That's true, but it's due to "where the underlying is." That's because the pricing model uses the forward price of IBM as the underlying price, not the current price of IBM. In other words, imagine there's a year remaining until expiration and interest rates are 10%. So the cost of carry of IBM for 1 year will be $10 (the calculation is slightly different but I'm simplifying here). The "forward price" of IBM then is 100+10=$110. As far as your pricing model is concerned, you're pricing a 100 put and a 100 call with IBM at 110. So of course the call is more expensive than the put. Now let's turn to part II - the cost of carry of the option itself. In the example I gave with the 180 call and put with IBM at 100, I was talking about discounting the 180 call by the cost of carry OF THE OPTION. This is completely separate and different from calculating the forward price of IBM based on the cost of carry of IBM itself. In other words, imagine IBM is at 100, and the 180 calls are worth 1.00. If interest rates were zero, the 180 put would have to be worth 180+1=81 (intrinsic value plus time value). But let's say interest rates are 1% and there's a year remaining in your option. If you buy that 180 put for 81, hold it for a year and IBM doesn't move, how much have you lost? Of course you lost a point in time value. But you also lost the money you DIDN'T make by keeping that $81 in T-bills all year, earning 1% - about .81. Altogether you'd have lost 1+.81=$1.81. But your pricing model has a heart, and takes pity on you. It figures it would be unfair for you to lose the dollar in time value AND the .81 in lost T-bill interest. It would be doubly unfair because the person who sold you that option would EARN an extra .81 by taking the proceeds of selling that option and buying T-bills. So right at the start, in Solomon-like fashion it discounts the option by the cost of carry - the T-bill interest that you would lose and the seller would earn. It will subtract that .81, and spit out a fair value of 81 - .81 = 80.19. Make sense?

Good explanation DMO. the carrying cost of the option is the opportunity cost of the funds committed elsewhere.