Arbitrage, Negative Rates, and Market Incompleteness

Summary

The document considers whether an inverted term structure or negative zero-coupon rates necessarily imply arbitrage, and whether arbitrage can exist in an incomplete market. The answers distinguish curve shape from arbitrage: one response says a negative rate above negative one hundred percent means holding cash has a cost, while another states that a zero-rate curve by itself can take any shape without creating arbitrage.

On market completeness, the discussion separates missing hedgeable instruments from arbitrage opportunities. An unhedgeable risk cannot itself be turned into a risk-free arbitrage, but arbitrage may exist in other parts of the same market. The replies are brief and do not establish assumptions about compounding, instruments, or pricing conventions; the initial question about an inverse term structure is also left unclear. The conclusions therefore need to be read in the context of the models and definitions used.

Key ideas

• An inverted rate curve alone does not establish an arbitrage opportunity.
• A negative zero-coupon rate can represent a cost of holding cash rather than an arbitrage.
• Market incompleteness means some risks cannot be hedged with available traded instruments.
• A market may be incomplete in one area and still contain arbitrage elsewhere.
• The discussion is concise and leaves conventions and assumptions unspecified.

Tags

# Questions on arbitrage


# Questions on arbitrage












I have the following questions about arbitrage that I am unsure of.

- Will an inverse term structure rate imply arbitrage possibilities?

- Will negative zero coupon rates imply arbitrage possibilities?

- Is it possible to have arbitrage possibilities in an incomplete market?

## Answer by M. Jeunesse (score 4)

https://quant.stackexchange.com/a/26423

1) not sure to see what you mean by inverse term structure rate

2) no if $r>-100\%$, $r<0$ means having cash on you will cost you something

3) If there is an unhedgeable risk in your market, it is not complete. So you cannot build an arbitrage based on this risk (since it is unhedgeable, it is pure bet). However, you could have arbitrage elsewhere in your market.

## Answer by user29970 (score 2)

https://quant.stackexchange.com/a/27474

- If we are talking about zero rates, then any curve is arbitrage free.

- No.

- Yes. Although in general arbitrage and incompleteness are in different directions (not enough tradeable instruments --> incomplete, too many traceable instruments --> potential arbitrage), it's of course possible for a model to be incomplete in one place and arbitrageable in another.

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