Decomposing European Call Time Value into Exercise and Cash Flow Deferral
Summary
The document introduces a Black–Scholes view of a European call’s time value: the option price minus intrinsic value. It asks whether this amount can be divided into value from postponing the exercise decision and value from postponing cash flows tied to acquiring or selling the underlying asset. A finance-class explanation describes exercise deferral as insurance value, especially relevant near the money, and cash flow deferral as a carry effect, with interest on an unpaid call strike described as beneficial.
The text is mainly a conceptual question, not a derivation or a worked example. It gives no mathematical definitions for the proposed components and does not establish whether the decomposition is conventional. Its discussion mentions puts to illustrate that delayed receipt of strike proceeds can have the opposite carry effect. Readers should treat the component labels and intuition as the subject of inquiry rather than as a fully specified model or result.
Key ideas
- European call time value is defined as the option value less intrinsic value.
- The document asks whether time value can be split into exercise deferral and cash flow deferral.
- It presents exercise deferral as insurance value that may matter most near the money.
- It describes interest on an unpaid call strike as a positive cash flow deferral effect.
- The document leaves the decomposition’s convention and mathematical definitions unresolved.
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Full text
# Rigorous definition of the two values of a European call
# 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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.