Why a Call with Zero Strike Has the Underlying’s Payoff

Summary

The document asks how to interpret a claim about option-model coherence as the strike approaches zero. Its answer starts from the call’s expiry payoff: the greater of zero and the underlying price minus the strike. At a zero strike, and assuming the underlying price cannot be negative, that payoff equals the underlying price itself. The option therefore has the same terminal payoff as holding one unit of the underlying asset.

This explains why the limiting option value is linked to the value of the underlying, or to its forward value when described at a future date. The response is an intuitive payoff argument, not a full derivation of option prices before expiry. It does not discuss discounting, dividends, model assumptions, or how skew affects prices for strikes close to—but not equal to—zero, so those details require additional analysis.

Key ideas

• A call option pays the positive part of the underlying price minus its strike at expiry.
• With a zero strike and a nonnegative underlying, the call payoff equals the underlying price.
• The zero-strike payoff matches holding the underlying at expiry, connecting its value to the forward value.
• The explanation does not derive pre-expiry pricing or analyze nonzero strikes near the limit.

Tags

# Detailing a proposition about option pricing model coherence


# Detailing a proposition about option pricing model coherence












On page 4 of this paper, the author states the following:

> "Looking at the limit case, when the strike tends towards 0, we should have the price of a forward contract and it should not depend on the equity skew around 0."

I cannot interpret mathematically what the author stated. Could you please provide a mathematical explanation of what the author said in words?

Thank you!

## Answer by will (score 3, accepted)

https://quant.stackexchange.com/a/34463

The value of a call option at expiry is $V=\mathrm{max}(0, S_t-K)$.

If you set $K=0$, then you have $V=\mathrm{max}(0, S_t)$, and since $S\geqslant0$, $\mathrm{max}(0, S_t) = S_t$ - i.e. ie's equivalent to holding the stock, which at expiry you'll expect to be worth the whatever the forward is.

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.