The textbook "The Mathematics of Financial Derivatives: A Student Introduction" by Paul Wilmott, Sam Howison, and Jeff Dewynne (Amazon affiliate link) is a standard introductory text, and describes arbitrage in the following fashion.

This [arbitrage] can be loosely stated as "there is no such thing as a free lunch." More formally, in financial terms, there are never any opportunities to make an instantaneous risk-free profit. (More correctly, such opportunities cannot exist for a significant length of time before prices move to eliminate them.)

## Simple Example of Arbitrage

Let's assume that we have annual coupon credit risk free bonds that trade along side zero-coupon bonds (or strips). We are also in a magical world where bid/offer spreads are zero. We now look at the pricing of a 2-year 5% bond alongside a 1-year and 2-year zero coupon. A $100 face value of the bond pays $5 in one year, and $105 in two years. Meanwhile, a portfolio of $5 face value of the 1-year zero coupon bond, and $105 of the zero coupon bond pays generates exactly the same cash flows.

We then see that if the portfolio of zero coupon bonds (as structured above) does not have the exact same price as the bond, we buy the cheap investment, and sell the expensive. We generate an immediate cash flow that is the difference in prices, and have a portfolio that generates zero net cashflows in the future (so it allegedly has no risk, at least if we can hold it to maturity).

(Note that the usual way of describing this is to look at a portfolio with zero net cost that generates guaranteed cash flows in the future. We would construct this version by buying more of the cheaper investment so that net cost is $0. Since we have a larger long position, it generates guaranteed positive future cash flows. The Technical Appendix gives a formal definition.)

Generating an immediate cash flow with (allegedly) no risk and no investment implies an infinite rate of return, which is very attractive. Since one of the most worrying thing in financial markets is seeing somebody else make money, what would happen is that pricing should shift to eliminate this opportunity.

## Bid/Offer (Normally) Eliminates this Possibility

In practice, we cannot transact with no costs. The bid (price someone else is willing to pay you for the instrument) is less than the offer (or ask; the price that they are willing to sell you the instrument). If we are in a dealer market, the dealer will quote prices so that the bid/offer precludes arbitrage. Theoretically, one might be able to construct an arbitrage by using more than one dealer, or trading on an exchange where related instruments have distinct order queues. Although high-frequency traders might achieve this feat, it is not something you would normally bank on.

The apparent exception might be sufficiently exotic derivatives, where some dealers might not be able to price them properly. Although trying to structure such arbitrage positions was an activity historically, my understanding is that the risks in warehousing positions was not properly taken into account.

One can interpret the profits generated by market makers as a form of arbitrage. We can view them as having negative transaction costs, and if they are managing to lay off their risk at the end of the day, they have locked in arbitrage profit versus their counter-parties.

## More Complicated As Risks Are Added

Arbitrage is more complex when we start adding risks to cash flows. The simplest example would be buying a 6% corporate bond at par ($100) and selling a 5% Treasury at par ($100). The net investment is zero, yet the position is supposed to pay $1 per year in cash flows.

The reason why this is not an arbitrage is that the corporate bond as default risk. (We assume the Treasury has none for this example. Debt/GDP ratio bugs might jump up and down about that, but it is standard in financial mathematics.) We are not guaranteed the cash flows from the corporation. The spread between the two bond yields is meant to compensate for the default risk.

The more interesting case is when we start to look at options. The option pay-off depends upon the future price of the underlying. We need to start looking at probability distributions of the instruments, and see whether we can construct a portfolio with guaranteed profits with no initial investment. This is where the exotic derivatives come in -- if they are sufficiently exotic, people might not be able to catch such a possibility.

## Statistical "Arbitrage"

From a historical perspective, statistical arbitrage is an example of financial market participants calling something an "arbitrage" when it was not so. Heading into 1998, investors had latched onto the theory of using historical statistics to generate an implied probability distribution for future prices and spreads. The idea was to generate relative value positions with zero cost, but allegedly were always going to be profitable if held to maturity.

Since the statistical techniques leaked out, apparently everyone was looking at the same trade structures. The nastiness of the 1998 LTCM crisis was partly the result of "everyone" having to unwind the same trades at the same time. (The above comment was based on market gossip; I was a brand-new junior analyst at the time, and I so I was out of the information loop.)

It seems unlikely that this fad will come back, but it could pop up once again if enough people retire.

## Risk Warehousing and Financing

There are number of hidden risks that do not show up in basic option pricing models. The first issue is just the problem of warehousing the risk -- although you may have guaranteed profits if the instruments are held to maturity, the instruments can still move in price in the meantime. You might face a margin call ahead of maturity. More importantly, the positions need to be financed, and use up risk limits.

For rates derivatives, the financing cost is correlated with the instruments that you are trading. That makes analysis much messier than for derivatives that are allegedly unrelated to interest rates (e.g., currency derivatives). My rule of thumb for fixed income was that most "arbitrage" opportunities was just mis-pricing of financing risks.

## Final Remarks

- The no arbitrage condition is a constraint added to pricing algorithms that ensure that everything is internally consistent.
- Unless one is a market maker or high-frequency trader, you normally expect to be facing prices that are arbitrage-free. As such, scanning the market for such opportunities is not going to be a major use of your time. This is a case where mathematical models do help shape expected outcomes.

## Technical Appendix

Although I would recommend the Wilmott-Howison-Dewynne text as an introduction to derivatives, the cleanest technical definition in a text I own is the one in Section 4.5 of Riccardo Rebonato's

*Interest-Rate Option Models*. (The text is somewhat out-of-date, but it is one that I have handy).Without using mathematical notation, the definition is as follows. (Since I dropped the notation, the following is a paraphrase, and not a direct quotation.)

*Arbitrage is a self-financing trading strategy of zero initial value and of non-zero final value. That is, no-arbitrage implies that if in all the states of the world at time T the portfolio value is greater than or equal to zero. and also in at least one state of the world it has strictly positive value, then its set up cost today must be strictly positive.*

*Alternatively, something that gives a strictly positive amount in some states of the world, and a strictly negative amount in no state of the world, must cost something today.*

This ends up looking slightly different than my discussion of the bond payouts. We might not have guaranteed profits with the portfolio, but it costs nothing, but offers a positive payout. One can think of the arbitrage portfolio as a free option. It might not pay off, but since it is free, you can buy as much as you want.

(c) Brian Romanchuk 2020

I have read this post and re-read. I remain confused.

ReplyDeleteIt seems to me that the case of instant trading is intermingled with cases of

ownership by the trader for measurable lengths of time.If arbitrage is defined as the act of taking advantage of the difference in value as judged separately in two different markets, then we can focus more upon market conditions and less on the motives of transient owners.

Perhaps you can better explain how a mathematical model can price-in transient ownership conditions.