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option price = intrinsic value + time value where intrinsic value (in other words payoff at N) is defined generally as difference between the underlying asset price and strike price (order depending on the type of the option of course).

In the quantitative pricing models only the difference between the underlying price and strike seem to be modeled. For example the price of a call assuming no arbitrage possible is presented as

$V_{(0)}=\bar{V_n}=E^*[\bar{V_n}]=E^*[\frac{(S_n^j-K)}{S_n^0}]$

how the time value is defined/reflected here?

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    $\begingroup$ I think you are missing an $e^{-rT}$ in front of the expectations operator. $\endgroup$
    – Alex C
    Jan 20, 2016 at 1:15
  • $\begingroup$ $S_n^0$ in the denominator represents the discount $\endgroup$
    – Michal
    Jan 20, 2016 at 1:18
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    $\begingroup$ Time value is reflected in expectation operator. $\endgroup$
    – Neeraj
    Jan 20, 2016 at 6:26
  • $\begingroup$ Time Value is simply "the value that comes from future stock paths that go up", it is captured by taking the expectation in the future, i.e. at maturity. The expectation includes the positive payoff value of those paths (and zero for those paths that go down). $\endgroup$
    – nbbo2
    Jan 20, 2016 at 16:47
  • $\begingroup$ so the time value is contained in the expectation together with the intrinsic value but they cannot be directly split, right? it is confusing in relation to such approaches where the premium is divided in time value and intrinsic (Intrinsic value + Time value + Volatility value = Price of Option) nasdaq.com/investing/options-guide/pricing-options.aspx $\endgroup$
    – Michal
    Jan 21, 2016 at 0:10

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time value "appears" from two sources a) convexity of payoff function max(S-K,0) b) settlement of stock in future (at option expiry). if u recall put-call parity: C-P=Forward and consider statement max(S-K,0) [this is call]-max(K-S,0)[this is put]=S-K. you see that call has time value (convexity), put has time value (convexity), but C-P does not have time value (no convexity), payoff of forward is S-K. the only "time value" of forward is "cost of carry" (interest rate and dividends). but nobody names it as time value, it is cost of carry. so, in sum, time value of option comes from payoff function which is convex.

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