ASM 2.7

[CASACT]

Hi, I just started studying for MFE and I am a little confused by the examples given in this book. In question 2.7 They give three strike prices and three option prices and ask you how many of each you will buy or sell to exploit the mispricing.

Now, I understand how the arbitrage works, however, I dont understand how, if given only three examples, you can infer which of the three is mispriced with convexity. I see how you could do it with four, but whos to say which two options were correctly priced and which two were not? In the book, EVERY example has the middle one as the mispriced option which is frustrating. I dont know how it would work out if the last one were mispriced. I also dont understand the logic in determining how many of each to buy or sell. I saw how they worked out the formulas and given a situation where the middle option is priced incorrectly I could tell you how many of each to buy, but I dont know why or what logic goes into the formulas that decide how much of each to buy. If anybody could help it would be greatly appreciated so I can put a close to the first few chapters. Thanks!!

[Abraham Weishaus]

It doesn't matter whether the middle one is overpriced or the top or bottom are underpriced, or whether they're all mispriced. All that matters is that the relative prices are inconsistent. You always buy the one that's relatively underpriced and sell the one that's relatively overpriced in proportions that have a net cost of zero.

[CASACT]

It doesn't matter whether the middle one is overpriced or the top or bottom are underpriced, or whether they're all mispriced. All that matters is that the relative prices are inconsistent. You always buy the one that's relatively underpriced and sell the one that's relatively overpriced in proportions that have a net cost of zero.

Thank you so much Abraham, that really helped, I got all caught up on thinking that there were 2 "correctly" priced options. I didnt even think about the fact that you could still take advantage of the mispricing even if all three were priced wrong. I am still a bit confused about selling the one thats relatively overpriced in proportions that have a net cost of zero. I apologize for my lack of knowledge again but this has been holding me up a bit. I have already made it through chapter 3 and some of 4 and this is the only problem before this that I don't understand completely.

[Abraham Weishaus]

Actually, for this particular situation (non-convex option prices), you buy options so that the net payoff is at least zero. If the middle one is underpriced, you accomplish this by buying that one and selling the two others as if you're trying to recreate the middle one with linear interpolation. For example, if the strikes are 40, 45, 65, you'd sell 4/5 of the 40 and 1/5 of the 60. You always get money at the start (because of the non-convexity), and you never have a net payoff later, so you're guaranteed a profit.

It is also possible to arbitrage in a way to make the initial cost zero. Just increase the amount of the ending options you're buying proportionally until your net cost is zero. You're guaranteed a positive payoff in that case. However, the textbook does it the first way.