The no-arbitrage principle is one of the strongest in capital markets theory. Without it, we open the door to (not slight but) gross irrational behavior.

## What is arbitrage?

An arbitrage opportunity exists when an investor can construct a portfolio that will earn a **riskless profit**. Strictly speaking, it should also be a zero investment portfolio (i.e., short some assets, long other assets), although borrowing can also be considered the short position.

The simplest example is one in which the law of one price is violated: asset X sells at different prices in two markets (assume the same or no transaction costs). In this case, you short asset X with the higher price and buy asset X with the lower price. Net proceeds are positive and there’s no risk because the two positions offset each other.

### FX market example

A less straightforward example could look like this:

- Currency A / Currency B exchange rate (today): 1:1
- Currency A / Currency B exchange rate (1-year forward): 1:1
- Currency A 1-year risk-free rate: 10%
- Currency B 1-year risk-free rate: 1%

With these conditions, you can earn a riskless profit by doing the following:

- Borrow 100 units of B for 1 year at 1%
- Exchange 100 units of B for 100 units of A
- Invest 100 units of A for 1 year at 10%
- With the forward rate at parity (1:1), enter into a forward contract to exchange 110 units of A for 110 units of B

In one year:

- Collect 110 units of A (your initial investment of 100 plus interest of 10)
- Close your forward contract: deliver 110 units of A, and receive 110 units of B
- Repay your debt: 101 units of B (100 of principal plus 1 of interest)
- Take home your riskless profits: 9 units of B

## No-arbitrage principle

The principle states that market prices will move to eliminate arbitrage opportunities.

What makes the principle so strong is that, if an arbitrage opportunity arises, **any single investor should want to take an infinite position** in it. This is regardless of the investor’s portfolio size or risk tolerance. This implies that we don’t need the entire universe (or a large set) of investors to correct the mispriced assets. We only need one investor to identify the arbitrage window, and that investor alone, by taking an infinite position, should bring enough pressure to restore equilibrium.

*Side note: arbitrage is different than risk-return mispricings, in which we assume that all investors, depending on their individual portfolio size and risk tolerance, make portfolio tilts until, in aggregate, price equilibrium is restored.*

## Derivatives

In the FX example above, the forward contract is a derivative. The market value of derivatives is completely determined by the prices of other securities. This characteristic makes it possible to achieve exact pricing for derivatives through the no-arbitrage principle.

*Side note: conversely, in the world of primeval instruments such as stocks or bonds, things get a little blurry because the prices of primeval instruments do not depend solely and strictly on the observable price of other securities. For example: to arbitrage an equity portfolio with another (short one portfolio, long the other), you need to make assumptions about correlations, volatility, and returns to believe that one portfolio dominates the other. More concerning, you need to assume that portfolio characteristics will remain constant over your investment horizon.*

## Is the no-arbitrage principle simply an expression of the risk-return relationship?

It appears so, since a higher return should be accompanied by higher risk, and earning profit without taking risk should not be possible.

I’m undecided, but leaning towards concluding that **the no-arbitrage principle is the extreme value of the risk-return relationship function**.