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Hello, I have a question about Floating Strike Lookback Options (aka "Standard Lookback Options"), and I think it best to provide an example.

To give an example of a Standard Lookback Option , let's say a one-year Standard Lookback CALL option (full period, by the way; not partial period) is created on July 1, 2019, but a person (say, John Smith) doesn't buy it until January 1, 2020.

Let's say the stock price as of New Year's Day 2020 ("time 0") is $200/share, and the lowest it fell during the six months following its creation date but before John bought it was $140/share.

My question is, per the Standard Lookback Call Option pricing formula which I already understand, when calculating the price

Or would the lowest value per share be the current $200 stock price, and likewise, whatever happened in between the option's creation date and purchase date would be deemed irrelevant, as far as John Smith is concerned?

I know that if we value the option at time "t" to see how much it's worth then (0<t<T, with "T" being the option's expiration date), we would obviously take the lowest it has gotten since he bought the option at time 0 and use that as our minimum so far (that's the pricing formula I do understand, as noted in the second paragraph of this email,) but I'm just wondering what our minimum value would be when pricing it as of time 0, the date on which John Smith actually bought the Lookback Call Option in question.

Like I said, would that minimum value be $140? Or would it be equal to the $200 current stock price as if to say the $140 low it reached between its creation date six months ago and the actual purchase date didn't matter?

To put it another way, when you buy a Standard Lookback Call Option on any given date, does the minimum value on that date ("time 0") always equal the current stock price?

Or does the period between the option's creation date and purchase date have some relevance after all?

To give an example of a Standard Lookback Option , let's say a one-year Standard Lookback CALL option (full period, by the way; not partial period) is created on July 1, 2019, but a person (say, John Smith) doesn't buy it until January 1, 2020.

Let's say the stock price as of New Year's Day 2020 ("time 0") is $200/share, and the lowest it fell during the six months following its creation date but before John bought it was $140/share.

My question is, per the Standard Lookback Call Option pricing formula which I already understand, when calculating the price

*ON THE DAY WHICH JOHN PURCHASED IT*, or time 0, would $140 be used as the lowest value the stock has fallen in the six months between its creation date and its purchase date?Or would the lowest value per share be the current $200 stock price, and likewise, whatever happened in between the option's creation date and purchase date would be deemed irrelevant, as far as John Smith is concerned?

I know that if we value the option at time "t" to see how much it's worth then (0<t<T, with "T" being the option's expiration date), we would obviously take the lowest it has gotten since he bought the option at time 0 and use that as our minimum so far (that's the pricing formula I do understand, as noted in the second paragraph of this email,) but I'm just wondering what our minimum value would be when pricing it as of time 0, the date on which John Smith actually bought the Lookback Call Option in question.

Like I said, would that minimum value be $140? Or would it be equal to the $200 current stock price as if to say the $140 low it reached between its creation date six months ago and the actual purchase date didn't matter?

To put it another way, when you buy a Standard Lookback Call Option on any given date, does the minimum value on that date ("time 0") always equal the current stock price?

Or does the period between the option's creation date and purchase date have some relevance after all?

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