Pricing Derivatives When Market Data Implies Arbitrage

# Answer by Dimitri Vulis (score 3, accepted)


# Is it fair in an introductory stochastic calculus/derivatives pricing class to ask for the price when absence of arbitrage is violated?












Re close votes: I believe this is a fair kind of opinion-based question because it's like those ethics questions in academia se or workplace se or because it's pedagogical.

Context: I'm actually asking this in the context of this maths education se question I have.

I have a pedagogical question to ask.

Pricing when arbitrage is possible through Negative Probabilities or something else --> in a stochastic calculus exam I took for my master's in applied maths 7 years ago, we were expected to give a price to something when all our formulas for pricing assumes absence of arbitrage but yet the absence of arbitrage assumption is violated! (And we were not told anything like price is automatically zero if this assumption is violated, if it's even true.)

I remember

- I asked during exam (this part was in English, though the next part wasn't)

> Me: Sir/Mme, are you sure we can do this (determine the price), if you are allowed to say? Instructor: What do you mean? Me: Well, sir/mme, I mean...what if we cannot determine it? Instructor: If it cannot be determined, then it cannot be determined. If it can be determined, then determine it. Me: (smiles to myself thinking that I've figured out the 'trick')

- when returning our exams, our stochastic calculus professor said (assuming I translated good enough from Tagalog/the Philippine language to English) 'Apparently, there is arbitrage. I was lenient in the checking here. But I have a feeling there is a way to price it.' (And there could be?) Oh so it wasn't a trick. It was a mistake. Interesting.

Ok anyway in retrospect, I think it would've been pretty unfair if it were given really as a trick and people were penalised if they didn't realise there was arbitrage. I distinctly remember correction ink over 'not needed' for something that supposedly (dis/)proves there was arbitrage in the market. I recall it was the existence of a strictly positive state price vector or something. I don't really remember finance anymore. But I do remember that the proof that there isn't arbitrage in a given market is not needed in computing the price...assuming there isn't arbitrage.

Question 1: Would it have been fair to really give this in class as a trick with the answer supposedly as 'There exists arbitrage(, so the formula doesn't apply).' ?

Question 2: Does your answer to Question 1 depend on whether or not there were no such exercises or homeworks with questions like 'Determine which of the following markets don't/have arbitrage. For the ones that don't, determine the prices in each market.' (well in my case, I don't recall there were any...based on what happened above)

## Answer by Dimitri Vulis (score 3, accepted)

https://quant.stackexchange.com/a/68039

This is an opinion-based question.

In practice, sometimes one needs to calculate mark to market and also various risk measures under risk scenarios that stress/perturb market data so much that arbitrage becomes possible. (See (*) below for more information.)

Hence, 1 provided there was some class discussion of heuristics for dealing with market data that admits arbitrage it's a perfectly reasonable question, 2 yes, it's an important omission if the question of how to deal with market data that admits arbitrage never arose in class.

I taught undergraduates, and I discussed this in class. However, I wouldn't ask it on an exam.

(*) More information:

Suppose for concreteness that you're trying to price a credit default swap that has 4 years left to maturity. Suppose that you're given market data - a collection of credit default swap quotes for various maturities up to 10, or even 30 years, and recovery assumption(s) and risk-free interest rates. Suppose further that as you try to calculate the survival probabilities from the market data, assuming, as usual, constant hazard rate between quotes, you see that the quotes admit arbitrage (e.g. have decreasing probability of default / negative hazard rate, or even negative probabiliy of survival or of default). In my opinion, it would not be wrong to always stop at this point, throw an exception, and say that the swap cannot be priced because the market data admits arbitrage. However you could also check that the negative hazard rate is after the maturity of your swap. In this case you could still price your swap. However it would be a good practice to tag your outputs with some metadata warning about the questionable market data. (I would not be comfortable silently assuming that the problem in the market data must be after the maturity of the swap.)

However once you get into repricing your swap under risk scenarios, you may want to be more lenient. Sometimes people subject credit curves to large stresses, e.g. every credit spread widens 100 bps in parallel, or every credit spread widens 30%. (Because of convexity, this is not the same as 100 times the impact of a 1bp move.) If your credit curve has many quotes, it's quite possible that the stressed curve implies negative hazard rate. If you just throw an exception and refuse to re-price the swap under the risk scenario, then you will have a lot of risk measures that fail to calculate. You can decide that your pricing model still "kinda works" if the hazard rate is negative but still greater than some slightly negative threshold - I woulnd't feel comfortable allowing this for mark to market, but it may be the acceptable for risks. Of course you should try to tag the outputs with metadata indicating that perturbed market data allowed arbitrage and whether the negative hazard rate arose after the swap's maturity.