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In addition to Wikipedia, YouTube and other internet sources, I'm reading Timothy Crack's "Basic Black-Scholes: Option pricing and trading". Most of these sources suggest that it is fairly obvious that an option has time-value. But I am not yet convinced -- could someone please clarify the following reasoning.

To me it seems that any time-value an (American) option might have is due pretty much exclusively to the volatility of the underlying asset (and partly on the riskless rate, but I'd like to ignore this for now), and so an option on a hypothetical asset with no volatility should have no time-value. My reasoning is as follows: suppose first that we have a hypothetical asset with no volatility (suppose also that the riskless rate is zero). This means the asset's price is deterministic. In this case, the value of a call option must be [the present value of] $\text{max}(S_m - K,0)$ where $S_m$ denotes the maximum that the spot price ever reaches (possibly infinite) and $K$ denotes the strike price. This remains the value of the call until the (first) time $t=t_1$ that the underlying asset reaches price $S_m$. After that time, the value of the option becomes $\text{max}(S_{m,2}-K,0)$ where $S_{m,2}$ is the highest price the underlying asset reaches after $t_1$, and so on. Indeed this reasoning does demonstrate that the option value changes in time but not because of "time-value". In particular one could exercise the option at time $t=0$ thereby obtaining the stock at the strike price $K$ and then simply wait until time $t=t_1$ (assuming it exists) to sell it, thus realizing the maximum value of the option without holding it beyond time $t=0$. This strategy would be the same whether time $t_1$ is tomorrow or next year, and so the value is the same in both cases, thus the value is independent of time to any maturity date of the option. So, since there seems to be no time-value when the volatility is zero, any time-value must be entirely due to non-zero volatility.

Where am I going wrong here? Thanks.

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I think its since you're taking the hypothetical situation of the stock deterministically having the same price for each time.

Yes, in that case it would not have time value.

But usually we think of it as the future being ucertain, and the assumtion for time value is that the longer the duration till expiery, the more value there is for speculation.

If some option expires in an hour, then the price will be much more determinant on the intrinsic value (the difference beteen strike and asset price), as there is less time and less chance for price to move a lot (variance). But the price of the option also has the extrinsic part which in turn is affected by time value.

In reverse, if there is a long time to expiery, the price difference between the price of the option and the intrinsic value could be much bigger, due to time value.

https://www.investopedia.com/terms/t/timevalue.asp

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  • $\begingroup$ Hi! Thanks very much for your answer. I indeed accept that we usually think of the future as uncertain, but it seems to me that uncertainty about the future price of the underlying asset is literally the same thing as "volatility". Is there a meaningful way to distinguish those two? I think this is another way of expressing my original question really. Also, maybe I picked you up wrong, but when I say that the hypothetical no-volatility asset is deterministic, I don't necessarily regard that as implying that the price is "the same for each time" -- what did you mean by that? Thanks again $\endgroup$
    – salanfaer
    Commented Jan 21 at 18:34
  • $\begingroup$ hi! when you say 'suppose first that we have a hypothetical asset with no volatility', I understand that as the price being constant. Also, the difference between two terms, maybe think of how volatility describes the past, as measure for predicting the future with a degree of certainty. If we see volatility some form of variance, this is usually calculated with some past window. (or any other measure if you want like another indicator or market). $\endgroup$
    – Iloos
    Commented Jan 22 at 20:06
  • $\begingroup$ Back to time-value, if we have no volatility, it becomes certain there is no change in price. with some volatility, the future is uncertain, leaving room for speculation. This is then part of the extrinsic value, affected by the amount of time remaining. So with more time, there is more potential in change of variance. If you set the change of variance to constant (being none in youur case) then there is no time value. But if the variance is unpredictable then that creates value for speculators, based on other factors like politics or buisness $\endgroup$
    – Iloos
    Commented Jan 22 at 20:10
  • $\begingroup$ if i understand you correctly, you say that: if i eliminate variance, the price can still change, so time value (based on variance) is not valid? i'm a bit confused with what you mean with a) asset with no volatility, b) price Sm can change for a change in t. If the price changes, then that means there is volatility right? the room for speculation, the change of Sm, is the time value. S should be constant, based on the intinsic value of the stock and the price gap of the option and the market. However, S can vary more than that, which i see as influenced by the time value. $\endgroup$
    – Iloos
    Commented Jan 22 at 20:16
  • $\begingroup$ Hi Iloos, thanks very much for sticking with me on this. I'll try and clarify my thinking a bit and if you're inclined to let me know your thoughts after that, I'll be very grateful. First, to be more precise, although I accept that as you say volatility is usually measured from past fluctuations (this is a kind of "implied volatility" if I understand terms correctly), I want to think of it for the moment in the following more idealistic way: the (log of the) stock price is a stochastic process made up of a deterministic part plus a non-deterministic part $\endgroup$
    – salanfaer
    Commented Jan 23 at 20:20

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