Replicating a Cash-or-Nothing Call with a Tight Vanilla Call Spread

Summary

The question examines how to interpret the units in a finite-difference approximation for a domestic cash-or-nothing call replicated with a narrow vertical spread of vanilla calls around the strike. It compares the difference in call prices at strikes on either side of the target strike with the spread width, and asks why the resulting quantity appears dimensionless when the binary option payoff is cash-denominated.

The document states the setup and the apparent dimensional mismatch, but contains no answer or resolution. It therefore serves as a focused prompt about option units and spread replication rather than a complete pricing explanation. Any application would need to specify the currency and payoff normalization, as well as the exact strike and price conventions used.

Key ideas

• A cash-or-nothing call can be approximated using vanilla call prices around its strike.
• The proposed approximation divides a call-price difference by the width between strikes.
• The question is whether the resulting units match those of the binary option payoff.
• The document provides no answer, derivation, or empirical evidence to settle the issue.

Tags

# Units when replicating a digital call with vanilla spread


# Units when replicating a digital call with vanilla spread












This is an embarassingly basic question, but in the course of studying windmill adjustment in binary FX options, I unexpectedly got stuck figuring out the correct units. Say we have a domestic binary call ("cash-or-nothing") and we replicate with a tight call spread around the strike. Theoretically, we have

$$ C_b \approx \frac{C_v(K - \epsilon)-C_v(K + \epsilon)}{2\epsilon} $$

for $h \to 0$. The denominator and numerator here share the same units, right? So the result is unitless, but that doesn't make any sense.

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