Arbitrage in Financial TheoryThe usual definition used in mathematical finance would look like follows.
An arbitrage trade is a set of positions in financial instruments that have a net cost of $0, but have a guaranteed positive return on a fixed investment horizon.Financial theory blithely assumes that everyone can finance any position at the risk-free rate, and so if an arbitrage existed, everyone would put that trade into place. The buying pressure on long positions, and the selling (borrowing) pressures on short/funding legs of the trade would drive relative prices in a way so that the expected return would go to zero.
During the financial crisis, it was easy to find the funds that were heavily influenced by finance academics -- they blew sky high because of leverage. The hidden assumptions about the availability of financing was their undoing.
A more realistic definition for arbitrage from a fixed income perspective is: a set of trades that guarantee a return on the position equity above the risk-free rate. This definition includes the previous standard definition of arbitrage; you could take the "$0 down" arbitrage trade, and layer it on top of a Treasury bill. Although this definition sounds restrictive, in an ideal world with fixed income managers sitting on funds with positive assets under management, we still would expect the profits to be squeezed out of arbitrage trades.
This definition underlies the formal definition used in arbitrage-free curves (discussed later). However, in fixed income trading, one often encounters the word "arbitrage" used to describe other activities.
The reason why this mathematical definition is not too useful in practice is that such arbitrage trades don't normally exist. If you are on the buy-side (an investor that is not a market-maker), no dealer is going to quote you prices that have internal arbitrage possibilities normally. Furthermore, you face a bid-offer spread which moves you further from being able to lock in an arbitrage position. The only way you lock in an arbitrage is to find a dealer that is quoting prices that are off market. If you are on the sell side (market maker), you may be able to match up flows that create an arbitrage. However, what you are really doing is locking in your bid-offer transaction income. Of course, you need to make sure you do not quote prices that are too off market, as otherwise your counterparties will rip your face off.
If the products being traded are sufficiently complex, you may not be able to quote internally consistent (arbitrage-free) prices. This is what happened in the early days of fixed income quantitative analysis; however, such opportunities have largely been crushed out of the market, courtesy of arbitrage-free yield curve models.
Statistical ArbitrageStatistical arbitrage was the product of quantitative genius of the 1990s, but it fell into disfavour after the trading style blew sky high during the LTCM crisis in 1998. (I started working in finance a few months before that crisis erupted.)
The idea of statistical arbitrage was that you loaded up on a bunch of mean-reverting trades, and you used sophisticated stochastic calculus to prove that you have portfolio with a hypothetical $0 investment, and a positive expected return. As is often the case, the sophisticated stochastic mathematics failed in the real world, and these trade structures blew sky high, because everyone used the same statistical screens to find trades, and they all ended up on the same side of the trade.
This episode largely eliminated the loose use of the word arbitrage in finance. However, the term did creep back into respectability, and so I believe that some people would describe some trade structures as being an arbitrage.
An example is the following structure:
- Buy an index-linked gilt.
- Sell short a nominal gilt, putting yourself into a breakeven trade (link to primer). If held to maturity, you make money based on the difference between realised inflation and the breakeven inflation rate (modulo technical factors like financing differentials).
- You can then hedge out your inflation exposure using inflation swaps.
This may have been described as an arbitrage trade to risk managers, as the net profits when held to maturity appeared to be guaranteed, and were not just a prediction of a statistical model. (This description would depend upon the people involved, and how strong memories of LTCM were.)
Anyone familiar with the nooks and crannies of the Financial Crisis would know immediately that this trade structure was one of the trades that also blew sky high. Index-linked gilts were trading at LIBOR+150 basis points (on an asset swap basis, if that means anything to you), which most quants pre-2007 would have said was practically impossible,
The weak link in the trade, like almost all arbitrage trades, was that there was a funding leg to the trade. Since funding is a fixed income instrument, people should have realised that such trades were not risk-free. This is why a good fixed income analyst should believe that there are no arbitrage trades (outside of instruments that are netted, like futures); every trade exposes you to some risk, the only question is whether that risk is priced properly. If a non-trivial trade looks like it has no risk, it just means that you are measuring risk wrong.
This is why really should use internally consistent pricing rather than arbitrage free when discussing mathematical concepts like yield curves (discussed next). Unfortunately, finance academics are unlikely to adapt their theories to real world experience, so we are stuck calling them arbitrage-free curves by convention.
Option-Free Arbitrage-Free Yield CurvesThe first step in discussing internally consistent (arbitrage free) fixed income pricing is to cover instruments without embedded options: vanilla bonds (like most government bonds) and swaps.
There are many ways of specifying such a curve, but the key is that any method specifies a set of discount factors for cash flow dates. (This primer discusses discount, zero, and par curves.) If the traded instruments are priced in an internally consistent (arbitrage free) manner using the discount curve, the prices of all instruments match the discounted value of their cash flows.
If instruments do not lie on the curve, we can structure trades that do the following:
- buy instruments (receive fixed) for instruments that are cheap versus the predicted value;
- sell (pay fixed) the instruments that are expensive.
In some cases, it may be able to lock in a trade that meets the strict definition of a pure arbitrage; the more likely situation is that you have locked in an extremely attractive relative value trade.
Since market makers are not in the business of handing their customers extremely attractive relative value trades, it is safe to assume that any price run you receive is going to have internally consistent pricing. This is certainly the case for swaps, where there are no instrument inventory effects. Bond pricing is going to vary from ideal pricing in most cases, but these reflect various technical factors (financing differentials, inventory squeezes, tax effects). These technical factors mean that arbitrage trades among the various bonds are best interpreted as exposure to various idiosyncratic risk factors.
Unfortunately, the specification of a discount curve is not unique. The first problem is that zero coupon bonds are generally quite illiquid, we quite often have more cash flow dates than observed prices.
For example, imagine that we want to specify the discount rates for the cash flow dates associated with a 2-year semi-annual coupon bond. The bond has 4 cash flow dates (3 intermediate coupons, and then the final coupon and principal), but there is only one observed price. There is an infinite number of possible discount factor combinations that allow us to recover the observed price. That is, there is a constraint on the weighted average of the discount factors, but this constraint is not enough to provide uniqueness.
The second problem is that we often want to specify the discount function at dates other than cash flow dates. This is either intermediate dates, or at points beyond the longest quoted maturity. (If you use continuous time, there is an infinite number of intermediate points. In practice, you price based on a settlement date, and so there are about 250 settlement days to worry about in a year. In either case, the number of intermediate dates is greater than the number of quoted instrument prices.)
The solution to these problems is to impose a restriction on the form of the discount function. It can be done by directly setting the form of the discount function, or it may be done by setting the functional form of the zero rate curve, or even the forward rate curve.
The manner in which we choose the functional form for the discount curve is more of an art than a science. We need to decide what features we want, and choose the form accordingly. For example, one could try to make the curve at any given day look as smooth and well-behaved as possible. Such a curve is useful for structuring relative value trades on a given day. However, such smoothing often introduces noise into the time series of yield curve values. As a result, if you want to create curves that will primarily be used for time series analysis, you may want to avoid too many smoothing manipulations in the curve fitting.*
Instruments with OptionsThe addition of optionality (including callable bonds) makes the curve creation exercise more difficult. The option-free yield curves can be fit each day, and there is no need for there to be a relationship between the curve on one day and the next. The addition of options makes the determination of internal consistency harder.
This is because we can use dynamic hedges to lock in the value of realised volatility. If we can buy an option priced with an implied volatility less than the future realised volatility, we would make a profit based on the hedges associated with that option. These profits are realised even if the option expires out of the money. (This is typically referred to as delta-hedging; it is not too difficult to understand if you look at an example of such a trading strategy, and not attempt to go straight from the mathematics. I will not attempt to provide such an example here.)
I have seen various attempts to downplay the importance of the Black-Scholes option-pricing results, on the basis of various theories about probability theory or distributions. The reality is that the key insight of the paper is that the fair value of an option is derived from a hedging strategy, and not the pricing formula itself.
As a result, in order to have internally consistent prices, we need to take into account implied volatilities, and the realised volatility of interest rates has to be consistent with the implied volatilities. For an option-free instruments, the form of an internally-consistent yield curve could morph every day, and it would not matter. Once we worry about realised volatility, we need to ensure that forward rates act in a sensible fashion from day-to-day.
The fact that the discount rate shows up in the fair value of an option price makes fixed income options a mathematical nightmare. The discount rate itself is a fixed income instrument, and thus is correlated with the underlying instrument of the option. As a result, the fixed income option world is much less exotic than other areas of finance, such as currencies.
This inherent complexity shows up in the mathematics of various modern arbitrage-free curve models that are used in academia. Under most circumstances, it is possible to explain results in applied mathematics to a non-technical audience. Although it might be difficult to explain any particular piece of mathematics, you can first describe why we are interested in a topic, and then fit the particular piece of mathematics within the bigger picture.**
In the case of arbitrage-free curves, you are largely stuck with saying: if we do not do all this mathematical argle-bargle, the modelled yield curves are not consistent with observed prices. If pressed on why this is the case, "because" is the best answer I can come up with.
Concluding RemarksPure arbitrage trades are an important concept in mathematical finance. In real world fixed income trading, such trades are not really encountered; we are always exposing ourselves to some risk, and it is always a good idea to understand what risks you are running before you put on the trade.
Arbitrage-free yield curves, so long as they take volatility into account, are inherently complicated. This complexity has the result that a lot of modern academic and central bank research into the properties of yield curves is unfortunately obscure.
* As an example, we could imagine that we could drop bonds that are too far off the fitted curve, under the argument that they are outliers. (If such bonds are used within the fitting, they drag the curve towards themselves, creating a hump in the otherwise smooth curve.) However, relative valuations of bonds are typically stable (auto-correlated), and so a bond's distance from a fitted curve does not change much. If the bond is near the cutoff for being considered an "outlier," it will jump back and forth from being considered as part of the fitting. This will cause an irregular oscillation in the time series of the curve.
** I would give as an example the following bit of mathematics.
|From "A Polytope Based Algorithm to Compute Regions of Attractions: Planar Case," B.G. Romanchuk.|
(c) Brian Romanchuk 2017