# Do different prices under different models admit arbitrage?

There are many models for interest rate. If two people use two different models to price the same interest rate derivative, and come to two different prices, doesn't that admit an arbitrage? How should I think about this problem? What is done in practice?

• Buy low, sell high... May 25 '20 at 20:02
• To specifically answer the question, it is arbitrage if and only if you can buy the derivative from one dealer and sell it to another. Thus, the offered side of dealer A must be lower than the bid side of dealer B. This condition may not be true even if the mid markets of the two dealers are different.
– dm63
May 25 '20 at 20:34

This phenomenon is not limited to interest rate derivatives. Any time a product is priced to model - be it equity derivatives, commodity derivatives, simple cash products whose price is not observable but is interpolated from other simple cash products - there is some uncertainty about the model price from

• choice of different models or methodologies

• choice of inputs (market data, assumptions) even when using the same model

when two firms trade an OTC derivative, it is normal for their mark-to-markets on the same trade to differ slightly. If the derivative is anything more complicated than a vanilla interest rate swap, it is not very unusual for their MTMs to differ by a lot. It is not unusual to have disputes over marks, especially when one side wants to unwind the trade. The traders are incentivized to make their unrealized P&L look at optimistic as possible, rather than to predict accurately what the P&L might be if the trades are unwound.

It is a problem. Here are some ways in which firms defend themselves from it:

• when there are several possible models available, price the trade with several of them. You typically choose one as your official model, but you are aware of the difference between your chosen model and the others. In some countries, regulators actually require you to keep track of the difference between your official mark to market and the most pessimistic one from alternative models.

• a group independent from the trading desk should try to verify the rates used as inputs to your model and keep track of the uncertainty caused by the uncertainty about the rates. For example, if the trading desk marks some rate to "95", and a third-party data vendor says the rate is "somewhere between 93 and 96", you figure out which rate in this range is the most pessimistic, and keep track of the difference. If the rate cannot be observed at all, but, for example, can theoretically be somewhere between 0 and 100, you again study the variance that can arise from this rate.

An arbitrage opportunity might arise in theory if two (or more) dealers differ so much in their valuation of the same exotic product, that you could buy it from one and immediately sell the other at a profit, even after all the bid-ask and tranaction costs. Or maybe some news just hit the wires and one of the dealers hasn't priced it in yet. I've actually seen this happen a few times when the seller screws up when providing a quote to a client. I.e. the scenario went like:

Buy side to dealer 1: how much do you ask for X?

Dealer 1: I want $$\N$$.

Buy side to dealer 2: how much do you bid for X?

Dealer 2: I'd pay $$c\N$$, where $$c>1$$.

Buy side to dealer 1: I'll take a yard!

Buy side to dealer 2: Here's a yard!

Typically, very soon (usually the same day) either dealer 1 realizes that he asked too low or dealer 2 realizes that he bid too high. (The controls I mentioned are there to help the dealer realize this and not let the trader hide his screw-up.)

• Thanks very much! May 25 '20 at 21:48