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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

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  • $\begingroup$ 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. $\endgroup$ – user59 Oct 15 '11 at 19:37
  • $\begingroup$ The payout is discontinuous, it's the delta that has extreme slope around the strike. $\endgroup$ – Joshua Sep 24 '13 at 21:44
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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:

enter image description here

enter image description here

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    $\begingroup$ Quite a concise answer, just to allow a few more words for the interested reader: fieldrecordings.wordpress.com/2011/01/07/… $\endgroup$ – h.alex Feb 21 '18 at 1:48
  • $\begingroup$ @SmallChess what did you use to draw that diagram? $\endgroup$ – Mike Dec 26 '19 at 22:05
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use a vertical spread and delta hedge it.

http://www.wilmott.com/messageview.cfm?catid=3&threadid=65988

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    $\begingroup$ Just want to tell you that the link is dead. $\endgroup$ – Henrik Sep 5 '16 at 12:07
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Going back to the original question, there is no static replication. This is clear from the first answer above which states that a call spread with an infitesimal difference between their strikes is needed. To arrive at an approximate replication, we need the probability of the underlying fixing in the USD 5 interval between the calls to be small, hence:

  1. Far away from the binary's strike, at large volatilities and long expiries, the call-spread replication could be good enough.

Going towards the strike at limited volatlity and time to expiry, the delta of the binary rises above that of the replica. Adjusting the positions in the options of the replica would probably not liquidity-wise be feasible - hence the second answer:

  1. Add a dynamic delta hedge to cover the delta mismatch near the strike of the binary.
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