# How to replicate a digital call option

Call Option S=100 K=100 Payoff=1 (option is not available) How can i replicate this (payoff) with calls and puts with strike prices with multiples of 5\$

Thanks for help

• I'm actually not convinced that you can replicate a binary option with vanilla options, even with arbitrary strike prices. Reasoning: a binary option's payout graph has an infinite slope at the strike price, whereas all vanilla options (and underlyings) have finite-slope graphs. I don't think you can add finite-slope combinations to get infinite slope, unless you use an infinite number of them.
– user59
Commented Oct 15, 2011 at 19:37
• The payout is discontinuous, it's the delta that has extreme slope around the strike. Commented Sep 24, 2013 at 21:44

A digital call option (cash-or-nothing) can be replicated with two call options with different Strike. When we make the delta infinitely small and assume we have arbitrary strike prices. We get:

• Quite a concise answer, just to allow a few more words for the interested reader: fieldrecordings.wordpress.com/2011/01/07/… Commented Feb 21, 2018 at 1:48
• @SmallChess what did you use to draw that diagram?
– Mike
Commented Dec 26, 2019 at 22:05
• There is a mistake in the sign in the last equality (missing negative sign). Commented Dec 13, 2023 at 14:55

use a vertical spread and delta hedge it.