# Rigorous definition of the two values of a European call

Assume a BS model. For a European call option with strike $K$ and expiry $T$, its intrinsical value at time $t$ is defined to be $(S_t-K)_+$ i.e. the payoff we could get if we immediately exercised the option. It can be shown that it is less than its real BS value $c_t$. So we denote its time value $W_t$ as the difference between the two: $$W_t = c_t - (S_t-K)_+>0.$$ Now we further decompose $W_t$ into two parts $W_t^{de}$ and $W_t^{dc}$, with $W_t^{de}$ meaning "the value of being able to defer exercise" and $W_t^{dc}$ "the value of being able to defer cash flows arising from sale or purchase of asset".

Definitions given in finance class:

Deferral-of-exercise value (insurance value) is always positive. It is most valuable for at-the-money options, not worth much if option is deep in or out of the money since the decision is already fairly clear then.

Deferral-of-cash-flows value (cost of carry) is more complicated and differs between calls and puts.
- A call holder may in principle pay over $K$ and receive asset. The deferral value is in not choosing to pay over $K$ and meanwhile receive interest on it. The value of this strategy is positive.
- But for a put deferring receipt of the strike price $K$ means loss of its interest, so it is negative.

But I am never quite clear on the following questions:

1). is such a decomposition conventional?

2). if so, how to define $W_t^{de}$ and $W_t^{dc}$ mathematically (at least in the BS model)? I really have some trouble understanding the vague definitions given in my class.