Pricing Binary Options with Call-Spread Replication and Hedging Margins
Summary
The document explains a practical way to quote a binary call by approximating its payoff with a call spread. A seller of the binary can buy a spread around the strike; the spread’s cost provides a replication-based estimate of the option’s price. The theoretical value is described as obtainable from Black–Scholes or as the limiting value of progressively narrower call spreads.
Because a finite-width spread does not perfectly match the binary payoff, the hedge leaves residual exposure near the strike. The replies frame a premium on either side of the theoretical value as compensation for replication and hedging costs, and give an example where the seller’s hedge costs 0.61 while the buyer-side hedge is worth 0.59. This suggests a bid–offer range rather than one exact traded price. The figures are illustrative, and the discussion does not quantify hedge risk or specify how the spread width should be selected.
Key ideas
- A call spread can approximate a binary call payoff and provide a replication-based price estimate.
- A finite-width spread leaves residual exposure, especially around the strike.
- A seller may quote above theoretical value to account for replication and hedging costs.
- Different hedge directions can imply a bid–offer range around the theoretical value.
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# How to 'price' a binary option as a writer or seller of it? # How to 'price' a binary option as a writer or seller of it? So, suppose you use calls spread to approximately replicate the payoff and therefore 'hedge' the position of shorting a binary call. Suppose the binary call's time to maturity is 1 month. The underlier current price is 100 dollars. You buy a call with strike 99.5 dollars and sell a call with strike 100.5 dollars and they both have time to maturity which is 1 month. So, my question is how would you price/charge the binary option you sell then? Suppose the theoretical price of a binary option is 0.6 dollars, which can be obtained using Black-Scholes or take the limit of calls spread to zero...which is just math. But is this the real price or the real way of pricing a binary option? I imagine there are two parts to the pricing of a binary option in practice: the replication cost and the hedging cost. The replication cost is easy to understand and should be close to 0.6 dollars. But because it is not perfect hedging, there is some residual exposure. For instance, if the stock price ends up being 100.4 dollars, then you yourself need to cough up and pay a little. So as the seller of the binary option, you would definitely charge a little for this risk you are taking, correct? I heard trader/quant/financial engineer use the word 'bump'. I imagine this must be it. Again, thank you very much for your time. ## Answer by QuantNero (score 1) https://quant.stackexchange.com/a/76034 You would add a small premium on both sides to account for hedging risk and replication cost. That's essentially the bump.. ## Answer by dm63 (score 0) https://quant.stackexchange.com/a/76028 As I mentioned in your other recent question, if you have sold the binary option, you would buy a 99-100 call spread (which costs say 0.61 dollars) thereby giving you a slight ‘overhedge’. If you have bought the binary call, you would sell a 100-101 call spread, which is worth say 0.59 dollars , also giving you a slight overhedge. Your bid- offer on the binary call is therefore 0.59- 0.61 dollars.
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