Why Convex Option Payoffs Have Time Value

Summary

The document asks how option time value appears in quantitative pricing when a pricing expression focuses on the underlying price and strike. The answer links time value to the convex payoff of an option and to the fact that settlement occurs at a future expiry. Because a call or put payoff is convex, uncertainty about the future underlying price can give the option value beyond its immediate exercise value.

Put-call parity helps clarify the explanation: the call-minus-put payoff corresponds to a forward payoff, which is linear rather than convex. The response therefore attributes option time value to payoff convexity, while describing the forward’s financing and dividend effects as cost of carry rather than option time value. This is a conceptual explanation rather than a full derivation: it does not specify a volatility model, discounting conventions, or how to calculate time value for a particular option. The distinction is useful when separating intrinsic value, convexity-related option value, and forward carry.

Key ideas

• Option time value is associated with the convexity of the call or put payoff.
• Future settlement means an option’s value reflects uncertainty about the underlying price at expiry.
• Put-call parity relates the difference between call and put values to a forward payoff.
• A forward payoff is linear, while interest and dividends affect its cost of carry.

Tags

# option time value in the pricing models


# option time value in the pricing models












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?

## Answer by mxzzzzz (score 1)

https://quant.stackexchange.com/a/22869

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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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.