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I recently made a post that was closed right away because it wasn't focused and asked too many questions. In that post, I asked five questions that were related but different. It looks like stack exchange won't allow that to happen, so I will ask only one question: what is your exposure when you, as a market maker, write/sell a binary call option? Many posts here have talked about how to approximately replicate the payoff binary option using two calls, but none of them seem to clarify what the exposure is.

In other words, when you sell a binary call and now you want to hedge this exposure, what exposure are you looking at? Are you looking at the exposure due to you shorting this binary call? Are you using the portfolio of two calls to approximate the payoff of this binary option? And because it is approximate, there is still some mismatch (hence exposure), and therefore you need to hedge that residual exposure?

Thank you very much for your time.

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Let’s say you short a 1 month binary call struck at usd100 with a payoff of usd1. Let’s say the current stock price is usd80. Then your exposure is simply that if the stock gaps upwards to above 100 at expiration, you will lose a dollar. Unlike a regular call option, it doesn’t matter how far above 100 the stock goes, you just lose a dollar.

If you want to hedge this exposure, the best you can do might be to approximately hedge by a replication strategy using regular call options. For example , you buy a 1 month 99-100 call spread. This pays one dollar if the stock ends up above 100, just like the binary option. If also pays zero below 99, just like the binary. The only difference in payoff between the call spread and the binary is that if the stock ends up between 99 and 100, you will make a windfall because the call spread will pay a positive amount while the binary pays zero. This difference typically you would not try to hedge, you would just leave it.

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  • $\begingroup$ I see! Thank you very much! I now understand! So, when people (quant and traders) say hedge a binary call with a portfolio of calls with spread, they really mean use this portfolio to approximately replicate the payoff of a binary call. Because this replication is not perfect, there is still some mismatch, as you point out, if the stock price ends up between 99 and 100. But apparently, you don't try to 'hedge' that. In other words, in practice, when you use a portfolio of calls with a relatively small spread, you consider yourself 'hedged' already. $\endgroup$
    – Chen Lizi
    Jul 5, 2023 at 5:07
  • $\begingroup$ Yes that is correct. $\endgroup$
    – dm63
    Jul 5, 2023 at 9:18

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