# I can’t understand why the premium of two butterflies with same strike but different broadness are approximately the same

Consider the following premiums of calls option with different strikes. C90 = 57.35 C95 = 52.55 C100 = 47.3 C105 = 42.9 C110 = 38.25 In this case, the butterfly 90-100-110 cost 1 and the 95-100-105 cost 0.85. I expected them to be very different. By drawing the payoff curve, I expected the payoff to be at least 4 times higher.. Thanks !

• You do not say how you got these prices. These problems are often caused by using prices of transactions that did not occur at the same time (non-simultaneity problem). It would be better to use the bid-ask midpoint of the different options rather than possibly stale transaction prices. Oct 7, 2022 at 14:02
• Yes of course, my bad. These prices are yesterday prices for Apple on Bloomberg. Prices for C90 C95 C105 and C110 are ask price and C100 is bid price Oct 7, 2022 at 14:31
• What is the payoff of the butterfly if the price of Apple at expiry is the current market price? Oct 7, 2022 at 22:22
• You mean if the current market price is the middle of the butterfly right ? Then it’s the difference between the two closest calls. It’s 5 in my example Oct 8, 2022 at 21:10
• No, I meant your butterflies. Do you mind showing your calculations? At least 4 times is generally not possible. If one costs 1, the other 0.85, you have this guaranteed loss outside of your range. Oct 8, 2022 at 21:34

The following argument shows that the price of the big fly should be approximately 4* the price of the small fly: Consider the portfolio of large flies B(0,10,20)+B(10,20,30)+……B(80,90,100)+….+all the way to infinity. This portfolio pays exactly 10 at maturity. Hence each fly represents 1/10 of the probability of being in the range of the fly, in a sense. This approximation works if we approximate each fly by a digital payoff , for example B(90,100,110) represents 1/10 the probability that the stock finishes in the range (95,105). A similar argument for the small flies shows that B(95,100,105) represents 1/20 of the probability of being in the range (97.5,102.5). Then it’s easy to see that these should be in the approximate ratio 4:1. This was intuitively obvious by looking at the payoff diagrams. As others point out , any large deviation from this in the marketplace is almost certainly due to asynchronous data or transaction costs.

In most realistic scenarios, the price of Apple will end up outside the range of your proposed butterflies and you simply pay or receive the costs. Insofar, your argument with going long the 90-100-110 and shorting the 95-100-110 twice sounds plausible given your numbers. However, bear in mind what @nbbo2 wrote in the first comment:

These problems are often caused by using prices of transactions that did not occur at the same time (non-simultaneity problem).

If you look up the option prices on Bloomberg (OMON), you see the following picture on October 6th.

As you can see, the volume column (Volm) is either not showing any volume, or very low volumes for almost all strikes you consider (you did not specify the expiry date but it should be similar for all expiries). If you now look at GIT, you can see when these prices were observed.

There is a considerable mismatch in terms of timing, which means you are not using prices that reflect the actual costs of your butterflies if you were to buy them simultaneously.

If you price it with OVME, you can use the OTC mode to use a VOL surface (called BVOL here), that you could also look up on OVDV. Doing this gives the following butterfly prices.

56.8076371  | 52.0521078
-94.6852675 | -94.6852675
38.1983043  | 42.7308402
------------------------
=0.3206740  | =0.0976805


In this case, your argument no longer holds.

Regarding your last comment, you do not have 99.5 and 100.5 but 95 and 105. In any case, the prices you looked at were almost certainly not from the same time period. How to (best) assess risk neutral probabilities using option prices is a separate question really.