# 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
Oct 15 '11 at 19:37
• The payout is discontinuous, it's the delta that has extreme slope around the strike. Sep 24 '13 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/… Feb 21 '18 at 1:48
• @SmallChess what did you use to draw that diagram?
– Mike
Dec 26 '19 at 22:05

use a vertical spread and delta hedge it.