QuoteOriginally posted by: Zedr0nIt's all about which arbitrage we are seeking for... I'm coming from the side of mathematical definition of arbitrage. There is an arbitrage opportunity which occurs if the forward price isn't consistent with the theoretical one - this arbitrage opportunity holds true in any case. Now, that I call general arbitrage opportunity.Here we have a model: the probability that S_T is in (0,90) is positive - this is an assumption, yes, it is a weak one, but still. As a matter of fact we can assume black-scholes framework and proceed as Alli first told - get the implied vol from option A, compare it with the implied vol from option B - if there is a difference then we have an arbitrage. I believe that the answer to the original question is the one lesliejinyu proposed, but it's worth pointing out that strictly speaking it's not always an arbitrage opportunity. Or there is a possibility that this was a "trick" question...I say: "Here we sell 9 puts and buy 8 puts, we have here the arbitrage opportunity", Interviewer: "Ok, what if we now that S is binomial with positive probability that S_T=0 and S_T > 90", I: "Umm, well, you know, it's not reasonable assumption". Do you really think it's going to look good?I wouldn't bet my career on a whim, I'd rather show both sides, i.e. there is no arbitrage opportunity regardless of model(give the example of binomial model), but if we make the following assumptions then there is this arbitrage opportunity(I'd also provide the opportunity of just selling the put if S_T>90 a.s.)...Yes, Zedr0n, you are right. from a mathematical point of view it is an arbitrage opportunity only if one assumes probability that S_T \in ]0, 90[ is positive (btw, it always holds when we are in a BS framework, i.e. if one uses implied vol. as you suggested, =>implicitly one assumes BS=>continuous value process=> probability that S_T \in ]0, 90[ is positive ). Dont get me wrong. I am not trying to rule out the case that the state of the world is driven by a particular binomial model, under which there might not be arbitrage opportunity for given prices. However, I am not quite sure whether it is a good idea to give complicated (but correct) answer when a simple one (but still correct, loosely speaking) is available. if i am not mistaken, the goal of the question is to assess people's responses to numbers (prices) and the ability of finding out arbitrage opportunities as soon as possible. if one wants to signal his/her mathematics knowledge, one can do it probably in many other ways, but not providing such an counter-example in this case because the example is TRIVIAL.

QuoteYes, Zedr0n, you are right. from a mathematical point of view it is an arbitrage opportunity only if one assumes probability that S_T \in ]0, 90[ is positive (btw, it always holds when we are in a BS framework, i.e. if one uses implied vol. as you suggested, =>implicitly one assumes BS=>continuous value process=> probability that S_T \in ]0, 90[ is positive ). I actually meant that if we assume BS then if the implied vols from the two options are different then we have an arbitrage opportunity as we can hedge them(at least one of the prices is wrong then), it's not about the probability of (0,90). The point was that we can often find such a model when there exists an arbitrage opportunity if we search long enough QuoteDont get me wrong. I am not trying to rule out the case that the state of the world is driven by a particular binomial model, under which there might not be arbitrage opportunity for given prices. However, I am not quite sure whether it is a good idea to give complicated (but correct) answer when a simple one (but still correct, loosely speaking) is available. if i am not mistaken, the goal of the question is to assess people's responses to numbers (prices) and the ability of finding out arbitrage opportunities as soon as possible. if one wants to signal his/her mathematics knowledge, one can do it probably in many other ways, but not providing such an counter-example in this case because the example is TRIVIAL.I'm not sure about the goal of this question myself - and assume the worst Maybe it's just because I'm a maths major, the trivial questions on exams always were the worst

QuoteOriginally posted by: Zedr0nQuoteYes, Zedr0n, you are right. from a mathematical point of view it is an arbitrage opportunity only if one assumes probability that S_T \in ]0, 90[ is positive (btw, it always holds when we are in a BS framework, i.e. if one uses implied vol. as you suggested, =>implicitly one assumes BS=>continuous value process=> probability that S_T \in ]0, 90[ is positive ). I actually meant that if we assume BS then if the implied vols from the two options are different then we have an arbitrage opportunity as we can delta-hedge them, it's not about the probability of (0,90).What I want to say is if we use implied vol. and delta hedge, we implicitly assume BS. Consequently, there is a strict probability of ]0, 90[ and the short 9 puts and long 8 puts portfolio works. Backing to the origin, one might just use short 9 and long 8 (then buy-and-hold and use the rest of time for beers/coffees) instead of delta hedging two positions from 0 to T.

QuoteWhat I want to say is if we use implied vol. and delta hedge, we implicitly assume BS. Consequently, there is a strict probability of ]0, 90[ and the short 9 puts and long 8 puts portfolio works. Backing to the origin, one might just use short 9 and long 8 (then buy-and-hold and use the rest of time for beers/coffees) instead of delta hedging two positions from 0 to T.Yes, that's right =) I was thinking ahead about different prices for A and B when there is no trivial arbitrage And what is the differences between these arbitrage opportunities then - just the strength of assumptions basically.Oh, well, I'm pretty sure jerrylee25 wanted your answer, not this discussion

yes, i put my fullstop here.

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Sorry for digging up an old post.The answer by lesliejinyu is very nice. Though I am wondering what is the best/usual way to identify if there is an arbitrage opportunity with two products? Say how can we tell that the $8 put with strike 80 is overpriced compare to the $9 put with strike 90? If the put with strike 80 is at $8, what would be the arbitrage free price for put with strike 90?