Relative Value and the Basis Trade

Bagaria, Sarthak

Chapter 22 Relative Value and the Basis Trade

We look at what the other side of the industry does with the same machinery. A relative value desk is not trying to price a derivative correctly; it is trying to find two things that should be worth the same and are not. We derive the decomposition of a fixed income return into carry, roll-down and yield change, construct the curve trades that isolate one factor from another, and work through the cash-futures basis in enough detail to see why the trade exists and why it periodically destroys the people doing it. Who is on the other side, and why they stay there, is the question the rest of the chapter returns to: the same ideas cover selling volatility and what that position is short, the retail versions of these trades, and how much a signal about a future release is worth, which is a mutual information measured in nats and sized by a log-optimal rule.

22.1 What This Chapter Is

Everything so far has been derivation. A model was posited, a measure found, a price computed, and the result was true given the assumptions. This chapter describes trades that people put on to make money.

The subject here is not whether these trades work. It is what they are: what a position consists of, how its profit and loss decomposes, and — the part that is genuinely mathematical and genuinely useful — where the risk actually sits, which is frequently not where it appears to sit. A trade whose P&L looks like a small steady income and whose risk is a rare enormous loss is a recognisable object, and recognising it is worth more than any view about whether it is currently cheap.

The reader who wants to know whether to put a trade on will not find it here. The reader who wants to know what they are holding will.

Structure (What makes a spread a trade).

A model tells you a spread is wide. That is not a signal, and treating it as one is the characteristic error of the subject. Two further things are required before a wide spread is a trade, and neither is a modelling output.

A structural reason. Somebody has to be on the other side for a reason that is not an opinion — a regulatory constraint that forces a pension fund to hold a particular instrument, an index rule that requires a bond to be sold on a date, a balance sheet cost that makes a dealer unwilling to warehouse a position, a settlement convention that nobody can arbitrage away. When such a reason exists, the spread is a price being paid for a service and can be collected. When it does not, the wide spread is more likely to be the model’s error than the market’s, and §22.9’s negative number is the case where telling the two apart is the whole problem.

A horizon. Convergence is not a date. A spread that is two standard deviations wide and genuinely mean reverting still takes longer to close than the intuition suggests, and goes further against the position first. That is a question about first passage times rather than about forecasting, it is answerable, and the next section answers it — because the usual way of getting it wrong is not misjudging direction but misjudging how long being right takes.

The model’s job in all of this is narrower than it looks. It supplies the spread and the decomposition of the P&L; it has nothing to say about either of the two conditions above. Which is why the discipline in relative value work lives in chapter 20’s estimation rather than in the pricing.

22.2 Who Is On the Other Side

The structural reason the structure above demands is not an abstraction. It belongs to a specific institution, with a specific mandate and a specific regulator, and knowing who that is — and what they are required, rather than persuaded, to do — is most of what separates a genuine relative value trade from a model’s own noise.

Repo, which is how balance sheet actually gets priced

Almost everything below is financed the same way, so the mechanism comes first, before the players.

Definition 22.1 (Repurchase agreement).

A repo is a sale of a security today, combined with an agreement to repurchase the same security at a fixed later date for a fixed, slightly higher price. Economically it is a collateralised loan — the seller borrows cash and posts the security as collateral — structured as two trades rather than one so that, in a bankruptcy, the lender already owns the collateral rather than standing in a queue for it. The difference between the two prices, annualised, is the repo rate: the interest paid on the cash. The lender advances less than the collateral’s market value, the haircut, to stay protected if the collateral’s price falls before it can be sold.

Most repo is general collateral: the cash lender does not care which bond is posted, any similar one will do, and the rate sits close to a short-term benchmark. A bond becomes special when enough people need that specific security — to deliver into a short, to meet a settlement obligation, because it is the benchmark issue hedges are executed in — that they will accept a lower return on their cash purely to secure it. The repo rate on a special bond falls below the general collateral rate, and in a genuine squeeze it can fall to zero or below: the cash lender is then paying to lend, because what they are actually buying is the bond, not the yield. That is the mechanism behind §22.11’s on-the-run richness and what happened to Salomon’s target note in 1991: a repo rate is not a property of the borrower’s credit, it is a price for a specific piece of collateral, and it moves with demand for that collateral exactly like any other price.

The players

Pension funds owe payments decades into the future, frequently linked to inflation or wages, and are marked against those liabilities’ own discounted value rather than against a market index. That creates a structural demand for very long duration and for inflation exposure, in quantities a fund’s own assets rarely supply directly — which is why liability-driven investing uses swaps and long gilts or Treasuries, often levered through repo, to close the gap cheaply rather than buying enough of the underlying bond outright. The lever is exactly what turns a slow, structural trade into a fast one under stress: in September 2022 a rise in UK gilt yields fell on levered liability-driven investing positions as a fall in collateral value, triggering margin calls that could only be met by selling gilts, which pushed yields higher and triggered the next round of calls — the same feedback loop the mortgage convexity hedging entry of §22.11 describes, with a margin call standing in for a rebalancing trigger and the Bank of England’s emergency purchases standing in for the intervention that finally broke it.

Insurers face a related but differently shaped constraint. Solvency II in Europe and risk-based capital rules in the US charge capital against a mismatch between the timing of assets and the timing of liabilities, so an insurer’s own capital cost falls when it buys assets that match liability cashflows closely — long-dated credit, structured or illiquid assets that qualify for favourable treatment under the rules, specifically because they match rather than because they are attractively priced. The constraint creates demand for an asset shape, and a security that fits that shape can trade rich to an otherwise identical one that does not, for the same reason the on-the-run bond does.

Banks and dealers are constrained by capital and leverage-ratio rules, most acutely at quarter ends when those ratios are reported, and by the balance sheet a repo book requires to warehouse anything. Chapter 23 works out the consequence for quoting; here the consequence is that a dealer is structurally reluctant to hold inventory, structurally short the optionality clients want to buy, and the natural supplier of exactly the balance sheet covered interest parity’s cross-currency basis prices.

Hedge funds and relative value desks are the balance sheet that steps into the gap dealers leave — unconstrained by the same regulatory capital rules, but constrained instead by financing that can be pulled and by investors who can redeem, both on a timescale shorter than the trade’s own horizon. That asymmetry is the horizon condition named above, in institutional form: the trade is right, and the capital available to hold it is not guaranteed to survive as long as being right takes.

Corporates issue the debt the rest of this list trades around, and hedge the interest rate and currency exposure that issuing it creates — paying floating and receiving fixed on newly issued debt, or the reverse, and buying the caps that bound a floating liability. That one-directional demand — borrowers wanting caps, issuers wanting the option to call — is exactly the flow “Selling Volatility, and What That Is Short” identifies as what leaves the dealer community structurally short optionality and paid to be.

Asset managers run mandates benchmarked to an index rather than to a liability, and are marked to market continuously rather than against a discounted cashflow, which pushes them towards the more capital-efficient instrument for a given exposure — futures over cash bonds, a swap over a bond where the mandate allows it — exactly the preference behind the cash-futures basis below.

None of this is a claim that any of these institutions is behaving irrationally. Each is optimising something — a funding ratio, a capital ratio, a tracking error — that is not the price of the instrument it trades, and §22.11 is a catalogue of what is left over when several of them optimise different things against the same market.

22.3 How Long Being Right Takes

Take the simplest possible model of a converging spread: an Ornstein-Uhlenbeck process, as in chapter 20,

dX_{t}=-\kappa X_{t}\,dt+\sigma\,dW_{t},

(22.1)

entered when $X$ sits two stationary standard deviations from zero, held until it returns. The mean reversion is certain — this is not a case where the trade might be wrong about direction. Everything below is what happens when it is right.

Calculation 22.2 (First passage is not the half-life).

The half-life $\ln 2/\kappa$ is the number everyone quotes and it answers a different question. It says how fast an expectation decays: $\mathbb{E}[X_{t}]=X_{0}e^{-\kappa t}$ halves in that time. It does not say how long a path takes to reach zero, and the two are not close.

Simulating (22.1) from two standard deviations, with a one-year half-life:¹¹1estimation::ConvergenceTrade.

half-life $\ln 2/\kappa$	$1.00$ years
mean time to first reach the mean	$2.10$ years
mean worst level reached first, in deviations	$2.48$

So the trade takes more than twice the half-life, and before converging it goes about half a deviation further against the position than the level it was entered at. Neither number is available from the half-life, and both are what size the trade.

Remark (Where a stop turns a winning trade into a loss).

The mean worst excursion is an average, so a stop placed at it is hit about half the time. Placing it closer is worse:

Stop, in deviations	Fraction stopped out before converging
$2.5$	$39\%$
$3.0$	$12\%$
$4.0$	under $1\%$

Entered at two deviations, a stop half a deviation away loses a trade that was right about direction two times in five. At twice the entry width the stop stops binding altogether, and the trade’s risk becomes the holding period rather than the loss.

That is the regime to be in, and reaching it is a sizing decision rather than a stop-placement decision: the position has to be small enough that sitting through twice the entry width is tolerable. A desk that sizes to its stop rather than to its horizon has built a trade that loses when it is right.

Structure (The same first passage problem as the doubling strategy).

This is chapter 4’s doubling strategy in respectable clothing.

There, a strategy on a martingale was certain to reach its target and the drawdown before it had a tail so heavy that its mean was infinite — which is why admissibility has to bound the loss rather than its expectation. Here the process is mean reverting rather than a martingale, so the tail is far lighter and the mean excursion is finite. But the structure is the same: a bet that is certain to win, whose risk lives entirely in what happens before it does, and whose sizing is therefore governed by a first passage distribution rather than by an expected return.

The difference is what makes relative value a business and doubling a fallacy. Mean reversion makes the excursion’s distribution thin enough to survive with finite capital, and (22.1)’s $\kappa$ is exactly what controls how thin. Which puts an uncomfortable amount of weight on knowing $\kappa$ .

Remark (And $\kappa$ is the parameter that is estimated badly).

Chapter 20 measures that mean reversion estimated from a finite sample comes out too fast, and chapter 14 explains why from the spectrum: a deviation decays more quickly than the slowest mode at first, so fitting one exponential to a whole sample returns a rate above the true gap. The bias is one-signed.

Follow that through to the trade. An overstated $\kappa$ is an understated half-life, which is an understated horizon. With the true half-life at one year and the estimate thirty per cent fast — comfortably inside what a decade of data delivers — the planned holding period is $1.63$ years against a realised $2.10$ .²²2measured. The desk expects to be out in twenty months and is still in the position at twenty-five.

Every part of that error points the same way. The horizon is longer than planned, so the funding cost is larger than budgeted; the position is held through more of the excursion distribution, so the drawdown is deeper than modelled; and the capital is committed longer, so the return on it is lower than advertised. There is no compensating error in the other direction.

22.4 Whether the Spread Converges at All

The section above began by granting that the mean reversion is certain, and everything in it followed from that. What happens when it is not granted?

Chapter 20 has already ruled out the obvious approach. Fitting (22.1) to a history returns a positive $\hat{\kappa}$ whether or not there is any mean reversion to find — on a random walk the estimate is positive with probability approaching one — so “the regression found convergence” establishes nothing. What is needed is not a better estimate but a test, with the random walk as the null hypothesis rather than as an alternative nobody considered.

Chapter 20’s Dickey-Fuller test is the natural tool here rather than an imported one, and its critical value of about $-2.86$ rather than $-1.65$ is exactly the correction that a non-stationary null requires.³³3estimation::unit_root_test.

Calculation 22.3 (What the test can actually see).

The test is correctly sized — a genuine random walk passes for mean reverting about five per cent of the time, as asked. The question is the other error: how often does the test find mean reversion that is really there?

Half-life	1 year	3 years	5 years	10 years
$6$ months	$6\%$	$11\%$	$17\%$	$52\%$
$1$ year	$6\%$	$7\%$	$8\%$	$18\%$
$2$ years	$6\%$	$5\%$	$6\%$	$8\%$

Read the middle row. A spread that genuinely reverts with a one-year half-life, watched for five years, is identified as reverting eight times in a hundred.⁴⁴4measured. The test is not broken; it is being asked to separate two hypotheses that a sample of that length barely distinguishes.

And the pattern across the table is the same shape chapter 20 found for the estimation bias. Power depends on the number of half-lives the sample spans and not on its length or its resolution: a one-year half-life over five years and a two-year half-life over ten give the same answer, though one history is twice the other.⁵⁵5measured. Sampling the spread hourly would add observations and no power at all.

Structure (Which is why the structural condition does the work).

Put Calculation 22.3 beside the two conditions this chapter opened with and the two halves of its argument fit together.

A test that finds real convergence eight times in a hundred cannot be what a desk relies on. Worse, it is not merely uninformative but dangerous under search: given $n$ instruments there are many combinations to weight, and looking through enough of them will produce one that passes at five per cent whether or not anything converges. A screen that tests a thousand spreads finds fifty by construction. The statistic that was weak as evidence becomes actively misleading as a filter.

So the structural reason demanded earlier is not a piece of good practice sitting alongside the statistics. It is what the statistics cannot supply. A butterfly is a curvature by construction, a cash-futures basis is tied to financing by an arbitrage that must close at delivery, an on-the-run spread is a liquidity premium with a named mechanism — and each of those is a reason to expect convergence that does not come from the sample, so it is not consumed by having searched the sample. The test is then a check on a prior rather than a way of forming one, which is the only role its power supports.

That is also the honest difference between this and the equity pairs trading the same mathematics is usually taught with. There the combination is typically discovered by search over a universe, and the discovery is exactly the procedure Calculation 22.3 says will manufacture false positives. Here the combination is usually written down first, for a reason, and the data is asked only whether it disagrees.

22.5 Carry, Roll-Down, and the Rest

Start with the simplest possible question. I buy a bond, fund it, and hold it for three months. Where does the money come from?

Write $y(T)$ for the yield of a $T$ -maturity bond, $D$ for its duration and $h$ for the horizon. Over the horizon two things happen: the bond gets older, and the curve moves. Expanding the price change,

\underbrace{\frac{\Delta P}{P}}_{\text{total return}}\approx\underbrace{y(T)\,% h}_{\text{coupon}}\;\underbrace{-\;r\,h}_{\text{funding}}\;\underbrace{-\;D% \big{(}y(T-h)-y(T)\big{)}}_{\text{roll-down}}\;\underbrace{-\;D\,\Delta y}_{% \text{yield change}}\;+\;\tfrac{1}{2}C(\Delta y)^{2}.

(22.2)

The first two terms are the carry: what the bond pays less what the funding costs. The third is the roll-down: even if the curve does not move at all, a five year bond becomes a four-and-three-quarter year bond, and on an upward sloping curve that means its yield falls and its price rises. The fourth is the only term involving an actual change in the market, and the fifth is the convexity of chapter 7.

Calculation 22.4 (How the known part compares with the unknown).

Equation (22.2) separates what is known from what is not and says nothing about their sizes, which is the comparison that decides whether a carry trade is a harvest or a bet. Both are available: the known part from today’s curve, and the risk from the realised volatility of the same maturity’s yield over the same horizon.

Taking the Treasury curve and a quarter’s holding period, with the one-month bill as funding: the known part is exact discount-factor arithmetic against today’s curve, and the risk is realised yield volatility over four decades of history where a tenor has been quoted that long, and one year where it has not.⁶⁶6risk::what_the_curve_currently_pays_to_hold_by_maturity, against the panel in public/marketdata.

Maturity	yield	carry $+$ roll	risk over the quarter	ratio
$1$ y	$4.43\%$	$96$ bp	$37$ bp	$2.57$
$2$ y	$4.71\%$	$28$ bp	$89$ bp	$0.32$
$5$ y	$4.83\%$	$-13$ bp	$230$ bp	$-0.06$
$10$ y	$4.96\%$	$-4$ bp	$400$ bp	$-0.01$
$30$ y	$5.29\%$	$-269$ bp	$444$ bp	$-0.61$

Remark (One year is an exception, and the long end has inverted).

From two years out the pattern is the one the two conditions above lead you to expect: the known part is a small fraction of the risk, a directional bet with a small tilt in its favour rather than an income stream with noise attached.

One year does not fit it. A ratio of $2.57$ is not a small tilt, and the reason is visible in the curve itself: the one month bill sits close to half a point below the one year bill, a genuinely steep front end that even four decades of yield volatility does not dominate. Whether that persists is a question about tomorrow’s curve, not a property asserted of curves in general — the same calculation answers it again whenever it is asked.

And the long end has inverted rather than merely underperformed. The thirty year known return is comfortably negative, not the largest number on the curve: the twenty year point sits above both its neighbours, so a bond ageing from thirty towards twenty years ages into a higher yield and a lower price, roll-down running backwards. The worst risk-adjusted place to hold duration on this curve is not “the longest maturity” as a rule; it is wherever the curve’s own local slope happens to be working against you, and on this curve that is squarely the long end.

The important structural feature of (22.2) is that the first three terms are known today. They are properties of today’s curve and the passage of time, not forecasts. Only the fourth is uncertain.

That is what makes carry-and-roll trades attractive and what makes them dangerous. A position with positive carry and roll makes money on every day the market does not move. It loses money when the market moves against it, and the losses are proportional to duration, which is to say much larger than the daily income. A carry trade is therefore short a large, rare loss and long a small, steady gain — which is the payoff diagram of a sold option, assembled without buying or selling one.

Carry and roll-down
Carry alone
Roll-down alone

Figure 22.1: Carry and roll-down along the US Treasury curve, annualised, over the cost of funding for three months. Carry generally rises with maturity — a longer bond typically yields more over the funding rate — but it is not immune to the curve’s own shape: it peaks in the twenties and eases back by thirty, tracking the same local inversion roll-down responds to more sharply. Roll-down depends on the slope of the curve where the bond sits, so it is largest where the curve is steepest and turns negative where the curve inverts, which on this curve is between twenty and thirty years. The total is the sum, and it is not monotone in maturity: the point of the curve that pays best to hold is not the longest one.

Data: US Department of the Treasury, daily par yield curve rates, as of 2026-09-22 (par yield, semiannual coupon, actual/actual). Retrieved from https://home.treasury.gov/interest-rates-data-csv-archive.

Show the model behind this figure (2 functions)

bootstrap_parquant/src/curve.rs

/// Bootstrap a curve from par yields.
///
/// `tenors` are maturities in years, strictly increasing; `par_rates` the quoted
/// par yields as decimals; `freq` the coupon frequency of the quoted instrument.
/// Tenors shorter than one coupon period are treated as a single payment at
/// maturity, which is what a deposit or a bill is.
///
/// The scheme matters here as well as afterwards: the coupon dates of a ten year
/// instrument fall between the nodes, so the value of a node depends on how the
/// curve is read between the earlier ones. This is why the three schemes give
/// three different curves rather than three readings of one curve.
pub fn bootstrap_par(
    tenors: &[f64],
    par_rates: &[f64],
    freq: f64,
    interp: Interp,
) -> Curve {
    let mut curve = single_pass(tenors, par_rates, freq, interp);

    if interp == Interp::MonotoneConvex {
        // A single sweep is not enough here, and the reason is the scheme's
        // defining property rather than a shortcoming of the sweep. The forward
        // at a node is built from the buckets on both sides of it, so while node
        // k was being solved the bucket beyond it did not yet exist, and the
        // shape assumed for it was wrong. Re-solving every node against the
        // finished curve and repeating converges quickly, because the
        // dependence on the far side is weak.
        //
        // This is the non-locality the chapter warns about, arriving as a
        // concrete cost: the other three schemes are done in one pass.
        for _ in 0..100 {
            let before = curve.yields.clone();
            for k in 0..curve.times.len() {
                resolve_node(&mut curve, k, tenors[k], par_rates[k], freq);
            }
            let moved = curve
                .yields
                .iter()
                .zip(&before)
                .fold(0.0f64, |worst, (a, b)| worst.max((a - b).abs()));
            if moved < 1e-15 {
                break;
            }
        }
    }

    curve
}

Curve::dfquant/src/curve.rs

/// The discount factor `P(0,t)`.
pub fn df(&self, t: f64) -> f64 {
    (-self.integrated(t)).exp()
}

Remark.

Note how much of this survives without any model at all. The decomposition is arithmetic and the figure is the curve of chapter 7 with a subtraction applied. No volatility, no measure, no calibration. That is characteristic of relative value work: most of the analysis is careful bookkeeping, and the modelling enters only when an option is involved.

22.6 Curve Trades

A directional position in one bond is a bet on the level of rates, which is a macroeconomic view rather than a relative value one. The relative value version isolates the shape.

Chapter 12 noted that a principal component analysis of curve changes finds three factors: a level factor moving all rates together, a slope factor steepening or flattening, and a curvature factor moving the middle against the ends. Between them they explain almost all of the variance.

This immediately suggests what to trade. Pick three points on the curve, and choose notionals $w_{1},w_{2},w_{3}$ so that the position has no exposure to the first two factors:

\sum_{i}w_{i}D_{i}\,\ell_{i}=0,\qquad\sum_{i}w_{i}D_{i}\,s_{i}=0,

where $\ell$ and $s$ are the loadings of the level and slope factors on each point and $D_{i}$ the durations. Two equations in three unknowns leave a one-dimensional family, and fixing the overall size picks one. The result is a butterfly: long the middle against the wings, or the reverse.

What is left is exposure to curvature and to nothing else — so if the middle of the curve is dear relative to the two ends, this position monetises exactly that, whatever the level and slope do.

That is the claim, and on a real curve it is only approximately true.

Calculation 22.5 (How neutral the neutral position is).

Take the equal-weighted $1:-2:1$ fly on the two, five and ten year points — the version that is used because it needs no estimate at all, being exact for any move that is linear in maturity — and measure how much of its daily variance the first two principal components still explain, on the committed Treasury history.⁷⁷7risk::factor_neutrality, on the same daily curves as the factor decomposition above.

	explained by level and slope
Outright five year	$98.1\%$
$1:-2:1$ butterfly	$26.4\%$

So the weighting does most of what it claims and not all of it. An outright position is the level factor and essentially nothing else; the fly removes three quarters of the factor exposure, and a quarter of what it does is still level and slope. Its own volatility is around two basis points a day, so the residual is not a rounding error — it is a directional position of real size sitting inside a trade that is described as non-directional.

The gap is the difference between the weighting that is exactly right for a straight-line move and the weighting that is right for the moves this curve actually makes. Fitting the weights to the estimated loadings closes it, at the cost of the next remark.

Remark (The weights are a modelling choice).

The loadings come from a covariance matrix estimated over some window, and the estimate depends on the window. A butterfly hedged with loadings from a calm period is not hedged in a volatile one, because the factors themselves rotate.

This is chapter 7’s lesson about interpolation, and chapter 24’s about model risk, arriving in a third place: the hedge is a property of an estimate, and the estimate is a choice. A desk running these trades is exposed to the estimation window in a way that no risk report shows, because the risk report uses the same window.

Calculation 22.6 (Removing the third factor too).

A fourth point turns the fly’s two constraints into three: choose weights on four points so the position is null to level, slope and curvature at once, leaving one degree of freedom for size. Add the thirty year point to the existing two, five and ten year fly and solve for the weights that null all three fitted loadings at once — a condor.⁸⁸8risk::condor_neutrality, the null vector of the same three-equation system the butterfly’s own weights above solve with one fewer point, here on fitted loadings rather than asserted as $1:-2:1$ .

	2y	5y	10y	30y
Weight	$1.00$	$-0.03$	$-0.63$	$0.27$

Where the naive fly left a quarter of its variance in level and slope, this position’s exposure to all three factors together is indistinguishable from zero: nothing measurable is left of level, slope or curvature, only whatever the curve’s own three-factor decomposition does not span.

Remark (What is left cannot be told from noise).

Chapter 6 called a position like this one a statistical arbitrage: a factor-neutral portfolio with a genuine expected return, compensation for whatever the factors do not span rather than for bearing level, slope or curvature risk. Whether this residual has one is a different question, and chapter 6’s own point that volatility can be estimated from one realisation and drift cannot answers it before the position is even priced. Its mean daily move over the committed year is about a tenth of a basis point against a volatility of eight basis points a day — a $t$ -statistic of about $0.2$ , indistinguishable from the zero a position with no edge at all would produce.⁹⁹9risk::the_condor_is_genuinely_neutral_and_its_own_return_is_unmeasurable. A year of daily data was never going to settle this. The condor’s neutrality is exact and measured; its edge, if it has one, is not — which is why a structural reason, not this regression, is what would actually make a residual like this one collectible.

Calculation 22.7 (Not choosing a window).

There is no window that is right, and that can be shown rather than argued. Take a hedge ratio that genuinely wanders and estimate it two ways, over a month and over a year. When the ratio moves slowly the long window wins comfortably; when it moves five times faster the short window wins instead.¹⁰¹⁰10measured. The better choice depends on a rate of rotation that is not observed, so choosing a window is guessing at it while pretending to be doing something else.

The alternative is to stop choosing. Write the ratio as a state that drifts and the regression as a noisy view of it,

\beta_{t}=\beta_{t-1}+\eta_{t},\qquad y_{t}=\beta_{t}\,x_{t}+\varepsilon_{t},

and this is chapter 21’s filter with a different state in it — no new machinery, and the same recursion. Its gain settles at a level fixed by the ratio of the two variances, so the estimate is exponentially weighted with a memory implied by how fast the ratio is thought to move rather than declared in advance. Told the truth about that rate, it beats both windows at every rate.¹¹¹¹11measured.

Remark (What the filter has actually changed).

Not as much as the last paragraph suggests.

The filter needs the rate of rotation, which is the window question in different clothes. Give it a rate within about a factor of two of the truth and its advantage survives; tell it the ratio moves five times more slowly than it does and it is worse than having simply taken the long window.¹²¹²12measured. Being told too fast is the more forgiving of the two errors, which is the usual asymmetry: a filter that distrusts its own history recovers, one that will not update does not.

So the gain is not that a parameter has been removed. It is that the parameter has been replaced by a better posed one. “How many days of history should the weights use” is a question about the estimator with nothing outside the data to answer it; “how fast do the factors rotate” is a question about the curve, and chapter 12’s factor analysis is the sort of thing that could answer it. That is a real improvement and a modest one, and it is the honest description of what filtering buys here.

22.7 The Cash-Futures Basis

Now the trade that made the news, and the one where the mathematics is richest.

A Treasury futures contract does not reference a single bond. The short may deliver any bond from a defined basket, and to make bonds of different coupons and maturities comparable the exchange applies a conversion factor $\text{CF}_{i}$ to each. Delivering bond $i$ against a futures priced at $F$ earns the short the invoice amount

\text{Invoice}_{i}=\text{CF}_{i}\times F+\text{accrued interest}_{i}.

Definition 22.8 (The basis).

For a deliverable bond with clean price $P_{i}$ ,

\text{Basis}_{i}=P_{i}-\text{CF}_{i}\times F.

The short delivers whichever bond costs least to buy and deliver, so the futures price is set by the cheapest to deliver — the bond minimising the basis. That is an option held by the short, and like every option in these notes it has value, is worth more when volatility is higher, and changes hands as the market moves: a large enough rally changes which bond is cheapest, so the futures contract’s effective underlying switches.

The standard way to see whether the basis is rich or cheap is to ask what financing rate makes the trade break even.

Definition 22.9 (Implied repo rate).

The implied repo rate is the return earned by buying the bond today, funding it to delivery, and delivering it into the futures:

\text{IRR}=\frac{\text{Invoice at delivery}-\text{Purchase price today}}{\text% {Purchase price today}}\times\frac{1}{\tau},

with $\tau$ the time to delivery.

Compare it with the actual repo rate at which the bond can be financed. If the implied repo rate exceeds the actual, the package — buy the bond, sell the future, fund in repo — earns more than it costs, and that difference is the basis trade.

Calculation 22.10 (What the basis trade actually is).

Set it out as a portfolio, which is the only way to see the risk.

-

Buy $N$ face of the cheapest to deliver bond, at price $P$ .
-

Finance it in the repo market: post the bond as collateral, receive $NP(1-h)$ in cash, where $h$ is the haircut. Put up $N P h$ of your own money.
-

Sell $N\times\text{CF}$ of futures — the amount that delivers exactly the bonds held, since the invoice pays CF per unit of futures price. Selling futures requires initial margin rather than the notional, so this leg costs little cash.

The futures position is pinned by the bonds held and by nothing else: it is set so that the delivery obligation is exactly what is owned.

With the legs matched, the position is close to hedged: a rise in yields loses money on the bond and makes it on the future. What remains is the basis itself, which converges to zero at delivery — so the profit is the difference between the implied and actual repo rates, earned with near certainty if the position is held to delivery.

So the leverage does not come from the futures at all. It comes entirely from $h$ , the haircut on the bond leg, which is what lets a large holding of bonds be carried against a small amount of capital; the futures margin is a further call on capital rather than a multiplier of the position. With a haircut of $1\%$ , $\$100$ of bonds requires $\$1$ of equity to hold, so a basis of two basis points annualised becomes a return on that equity of two hundred. The trade is not attractive because the edge is large. It is attractive because the edge is small and the leverage is enormous.

Why does the gap exist at all? Because the two sides of it are wanted by different people. Asset managers who want duration without balance sheet buy futures, pushing them rich. Someone must take the other side, hold the cash bond, and finance it, and that is a use of balance sheet that banks have been progressively less willing to supply since capital rules began charging for it. Hedge funds stepped into that gap. The basis is, in effect, the price of balance sheet.

22.8 Why It Breaks

The trade is hedged against the thing it looks exposed to, and exposed to two things it does not look exposed to. Both are about funding rather than about rates.

The margin on the futures leg is settled daily; the gain on the cash leg is not. Chapter 8 derived this asymmetry to compute the future-forward basis, where it was worth a basis point or two. Here it is a cash flow: if yields fall, the short futures position loses and must post variation margin today, while the matching gain on the bond is unrealised. A hedged position with no economic exposure can therefore demand cash without limit in the interim.

The haircut is not constant. It is set by the lender and rises when markets are volatile. A haircut moving from $1\%$ to $3\%$ triples the capital the position requires, and a fund that does not have it must sell. Selling the cash leg widens the basis, which marks every other holder of the trade to a loss and raises their margin calls too.

That is the whole mechanism. At $1\%$ haircut the leverage is a hundred to one. A move in the basis of ten basis points — unremarkable in a stressed week — is ten percent of capital. A move of a hundred is the fund.

Remark (This is chapter 4’s admissibility condition, in the wild).

There is a genuine and not merely rhetorical connection to the theory.

Chapter 4 met a strategy that was an arbitrage under the definition then in use: the doubling strategy, which reaches a certain profit almost surely by being willing to sustain unbounded losses along the way. The repair was to require admissibility — that the wealth process stay above a fixed level — and we observed that without it no market is arbitrage-free, because the mathematics permits a trade that no counterparty would fund.

The basis trade is that condition being tested by reality. Held to delivery with no financing constraint it converges, and the profit is nearly certain. Held with a hundred times leverage and daily margin, the interim losses are exactly what admissibility rules out, and the strategy fails not because the arbitrage was not there but because surviving to collect it was not financed.

The lesson chapter 4 drew from a piece of measure theory is the same one: a profit you cannot fund your way to is not a profit.

Exercise (Sizing the spiral).

A fund holds $\$100$ of the basis trade at a $2\%$ haircut, so $\$2$ of capital. The haircut rises to $4\%$ . Show that, holding the position, the fund needs $\$2$ more capital; and that if instead it deleverages to its existing capital it must sell half the position. Then argue that if enough funds do this at once the basis widens, and show that a widening of $20$ basis points on the remaining position wipes out a further tenth of the capital. This is the loop, and its speed is set by the haircut, not by rates.

22.9 Swap Spreads, and a Negative Number That Should Not Exist

One more spread, because it makes a point about what these trades measure.

The swap spread is the swap rate of chapter 7 less the yield of the matched-maturity government bond. Naively it should be positive: a swap has bank credit behind it and a government bond does not, so the swap should pay more.

At long maturities it has been persistently negative, which appears to say that lending to a bank is safer than lending to the government. It does not. The swap is collateralised, and chapter 7 derived what that means: a fully collateralised trade carries essentially no credit exposure, and is discounted at the collateral rate. The bond is not collateralised — it must be bought, funded and held on a balance sheet that is charged for it.

So the swap spread is not a credit spread at all. It is the price of balance sheet again, the same quantity the basis measures, seen from another direction. Once the capital rules made holding bonds expensive, the spread that compensated for that could and did go negative, and stayed there.

Remark.

The general lesson: when a spread that theory says should have one sign persistently has the other, the usual explanation is not that the market is wrong. It is that the theory omitted a constraint that binds — here, that balance sheet is finite and charged for. Chapter 7’s collateral derivation and chapter 24’s capital reserves are two faces of that constraint, and this spread is its price.

22.10 Selling Volatility, and What That Is Short

Everything so far has been a trade in a rate or a spread between rates. There is a second family, at least as large, in which the instrument is an option and the quantity being traded is volatility itself.

The position is a swaption or a cap sold and then delta hedged, so that the direction of rates is removed and what is left is the difference between the volatility charged and the volatility that arrives. The structural reason is on the other side and is the one chapter 23 describes: the flow is one-directional. Borrowers want caps and issuers want the option to call, so the dealer community is structurally short optionality and is paid to be. Chapter 10 locates the same premium inside a model, where it sits in the drift parameters $\kappa$ and $\theta$ and not in the vol-of-vol $\eta$ , which is measure-invariant.

Calculation 22.11 (What a hedged short option actually earns).

Sell a call at an implied volatility $\sigma_{i}$ , hedge it with the delta computed at $\sigma_{i}$ , and let the world realise $\sigma_{r}$ . Over one rebalancing interval the position is delta neutral, so by chapter 5’s expansion the profit and loss over that interval is the theta collected less the gamma paid,

-\Theta\,dt-\tfrac{1}{2}\Gamma\,(dS)^{2},

and the theta-gamma identity of that chapter, $\Theta=-\tfrac{1}{2}\sigma_{i}^{2}S^{2}\Gamma$ at zero rates, substitutes for the first term. Since $(dS)^{2}=\sigma_{r}^{2}S^{2}dt$ along the realised path, the two collapse into

\text{P\&L}\;=\;\int_{0}^{T}\tfrac{1}{2}\,\Gamma_{t}\,S_{t}^{2}\,\big{(}\sigma% _{i}^{2}-\sigma_{r}^{2}\big{)}\,dt.

(22.3)

Simulating the hedge and accumulating (22.3) along the same paths are different computations — the first forms the payoff and uses the delta, the second never forms a payoff and uses the gamma — and they agree.¹³¹³13measured.

Read (22.3) carefully, because it does not say what the trade is usually described as saying. It is not a bet on realised variance. It is a bet on realised variance weighted by gamma, and gamma is large only near the strike and near expiry. A move of a given size contributes according to where the underlying was when it happened, so two paths delivering identical realised variance pay different amounts.

How different is worth measuring. Selling a one-year at-the-money option at $22\%$ into a world realising $18\%$ earns $1.59$ on average, which is exactly the difference between the two Black-Scholes premiums and is the sense in which the edge is real.¹⁴¹⁴14measured. But restricting to the paths that realised $18\%$ to four decimal places, the outcomes still run from $0.49$ to $2.94$ .¹⁵¹⁵15measured. The dispersion remaining after the forecast has been made exactly right is wider than the entire average edge.

Structure (Short the path, not the variance).

A desk selling volatility describes itself as short variance and forecasts variance accordingly, but (22.3) says the exposure is to a gamma weighted average, and the weighting is a fact about the path rather than about its total. Being right about volatility and wrong about where the moves land is a losing position.

So the trade is short the shape of the realised path, and the chapter’s usual question has the usual answer: it is short a constraint, namely that the hedge is discrete and the gamma is concentrated. Chapter 5 has already drawn the same object: two markets with identical total variance and different jump content produced identical premiums and entirely different distributions of outcome. This is that figure’s trading counterpart.

It also explains the instrument. A variance swap pays realised variance with no gamma weighting at all, which is exactly the exposure the hedged option fails to give, and that — not any view on volatility — is what it is for. Chapter 15 has in fact already built the object, though not under that name: the second moment of the swap rate comes out as an integral of swaption prices with equal weight at every strike, which is a variance swap on the rate in all but the contract. The uniform weighting is not an accident — expanding a payoff in options gives each strike the weight $g^{\prime\prime}(K)$ , and the second moment has $g^{\prime\prime}=2$ everywhere.

The familiar $1/K^{2}$ weighting is the same construction asking for a different variance. Buying the variance of the logarithm means replicating $-2\ln(F_{T}/F_{0})$ , whose second derivative is $2/K^{2}$ , and that is the equity convention because equity variance is quoted in proportional terms. A rates desk buying basis points squared weights uniformly. The two are not competing formulas for one instrument; they are the same formula for two.

Three further trades in this family:

The volatility term structure. Selling short-dated and buying long-dated volatility is a trade on how implied volatility is interpolated between expiries, and what it is short is the event calendar. Chapter 9 shows why: a smooth interpolation through a payroll or a meeting date reads the lump as a term structure and marks every expiry between the quotes wrong. The trade collects the marking error and is short the possibility that the event is bigger than the lump priced for it.

Skew. Selling a receiver and buying a payer against it is a position in the smile’s slope, and chapter 10 says the slope is a statement about the correlation between the rate and its volatility. What that is short is not a rate move but a change in that correlation — and the mortgage hedging flow of the next section is precisely a mechanism that changes it, which is why the two trades are less independent than a risk report shows.

Conditional and curve volatility. A swaption on a spread is worth an amount that depends on the correlation between the two rates, and chapter 13 measures how much: marking correlation moves a price by percentage points where the choice between two well-specified models moves it by hundredths. A trade in spread volatility against outright volatility is therefore short a correlation marking, which is an estimate from chapter 21 rather than a quote, and the least observable input in the book.

22.11 Six More, and What Each One Is Short

The two trades worked through above are not a representative sample, and a reader who saw only them would take away that relative value is about bonds against futures. It is not. What follows is a catalogue rather than a derivation — each entry is a trade that has been put on at scale, with the structural reason that made it collectible and the exposure that made it dangerous.

On-the-run against off-the-run. The most recently auctioned Treasury of a given maturity trades rich to an almost identical bond issued a few months earlier. The reason is not credit and not cash flows, which are nearly the same; it is that the new bond is the benchmark, is what hedges are executed in, and finances more cheaply in repo because everyone wants to borrow it. The spread is therefore the price of liquidity and of financing, and it can be collected by buying the old bond and selling the new one. What the position is short is liquidity itself — and liquidity premia widen precisely when a position needs to be unwound. The trade was a signature of Long Term Capital Management, and in 1998 the spread went from rich to richer while the leverage that made it worth doing forced the unwind. Chapter 4’s admissibility condition is the abstract version: a trade financed by an unbounded drawdown is not a trade, and the drawdown here was a mark-to-market on a position whose thesis was correct.

The squeeze. A short position in a specific deliverable bond is not short a price. It is short the ability to borrow that bond, and if someone corners the supply, the cost of maintaining the short is set by them. Salomon Brothers demonstrated this at the two year Treasury auction of May 1991 by acquiring a dominant share of the issue, after which the repo rate on the note collapsed and the shorts paid. The general point is structural and worth carrying: for a deliverable instrument the funding leg has a different risk profile from the price leg.

Inflation-linked against nominal. A Treasury inflation-protected bond plus an inflation swap that converts its cash flows to fixed replicates a nominal Treasury exactly — a static replication of the kind chapter 15 builds, with no model in it at all. The replication has been persistently cheaper than the nominal bond it replicates, at times by a wide margin, and the mispricing survived for years rather than minutes. It is as close to a textbook arbitrage as the market offers, and it persisted for the reason everything else in this section persists: capturing it requires holding a levered, balance-sheet-intensive position to maturity, and the institutions with the balance sheet had better uses for it.

Covered interest parity. Borrowing dollars directly and borrowing yen and swapping them into dollars should cost the same, since the forward rate is a ratio of numeraires and chapter 17 derives it with no freedom in it. Since 2008 they have not cost the same, the gap widens reliably at quarter ends when balance sheets are reported, and it is a cross-currency basis rather than a mistake. The trade is to supply the scarce thing — dollars, on a balance sheet — and what it is short is the regulatory constraint that made them scarce, which is to say a rule that can change.

Mortgage convexity hedging. A homeowner holds an option to prepay — refinancing the loan when rates have fallen enough that a new, cheaper one is worth the cost of arranging — so a mortgage portfolio is short that option. Its duration shortens when rates fall, because more of the pool is then expected to prepay and return principal early, and lengthens when rates rise, because refinancing slows and the expected life extends: the opposite of an ordinary bond, whose duration moves the other way. Whoever holds that exposure must hedge it, and the hedge is mechanical: rates fall, duration has shortened below target, so the hedger receives fixed to buy duration back; rates rise, duration has lengthened, so the hedger pays fixed to sell it off. Buying duration into a rally and selling it into a sell-off is trading with the move rather than against it, so this flow — proportional to the change in the aggregate duration of a very large portfolio, and forced rather than discretionary — amplifies whatever rate move triggered it rather than damping it. And because the amplification is large, mechanical and recurring, it supports a standing bid for the instrument that lets a hedger own the same exposure outright instead of chasing it with rebalancing — a swaption — which is why this flow supports swaption implied volatility. Two trades follow: buying that optionality ahead of a move expected to trigger the flow, a direct bet that realised volatility will run ahead of what is priced; or trying to anticipate the flow itself, the rate level at which a wave of refinancing starts and the rebalancing it forces, rather than volatility in general. Both are short the same thing: the stability of prepayment behaviour, which is a model output and one of the least reliable ones in the industry.

The auction cycle. Dealers must absorb each new issue, and their capacity to warehouse it is finite. Yields have tended to rise into an auction and fall after it, a pattern with a named mechanism rather than a curve-fit, and it is the closest thing in this chapter to a pure statistical arbitrage with a structural reason attached. What it is short is dealer capacity: the pattern is strongest when balance sheet is scarce, which is also when the position cannot be financed.

Structure (The same short, six times).

Set the six side by side and one thing is common to all of them. None is short a price. Each is short a constraint — financing, borrowing, balance sheet, regulation, a behavioural model, warehousing capacity — and each pays a premium that exists because the constraint binds on somebody.

That is not a coincidence and it is the reason these trades exist at all. A genuine mispricing of a price, in a liquid market with many participants, is arbitraged away in the time it takes to notice. What is not arbitraged away is a premium paid for absorbing a constraint, because absorbing it requires the very thing that is scarce. So the surviving trades are exactly the ones where the return is compensation for a service rather than a reward for being right.

Why is a mispriced price removed so fast?

Not because it is hard to see. A cash-futures basis, a triangular inconsistency in currencies, a bond trading through its own strip — each is arithmetic on quoted numbers, computed identically by everyone with the same feed. Detection is not the scarce input and confers no advantage, which is why no desk earns anything from noticing.

It is removed fast because it is a race for a finite quantity. A mispricing is not a pool of free money but a bounded amount available at a stale price, and it goes to whoever reaches it first. That makes the return to speed winner-take-most rather than proportional, and a payoff of that shape is competed to whatever the physical limit happens to be — which is why the participants who do this have spent their capital on the distance between two machines rather than on models.

And that is exactly what separates the two classes. A mispricing removable by a fast trade needs no balance sheet: the position is opened and closed, nothing is carried, and the only scarce input is time, which is bought once and then owned. A premium for absorbing a constraint cannot be competed away by speed at all, because the scarce input is the ability to hold — collateral, capital, warehousing, the willingness to sit through Calculation 22.2’s excursion — and no amount of speed supplies it. Speed removes the arbitrages that can be closed in a moment and is powerless against the ones that must be carried for months, which is why the surviving trades all look alike.

The fast trader’s profit is the slow quoter’s adverse selection as seen in chapter 23: a quote that can be picked off before it is pulled is exactly the toxic flow that chapter measures and prices into the spread.

It follows that the risk of the whole class is one risk. All six lose money in the same circumstance — when the constraint tightens rather than relaxes, which is when financing is withdrawn, balance sheets contract and everyone holding the premium needs to stop. That is why relative value books that appear diversified across trades are not, and why chapter 24’s stress work has to be run on the constraint rather than on the instrument.

22.12 What Retail Actually Trades

Everything above is desk-scale: bonds financed in size, spreads held to maturity, balance sheet an individual account does not have. A retail account trades a narrower set of instruments, built mostly around two things — a rate ETF with a liquid, exchange-listed options market, and a callable or structured note — and the question this chapter has asked of every trade above applies to these too: what is the position short, and is there a reason to expect it is paid for.

Selling optionality on a rate ETF

A long-dated Treasury ETF such as TLT has a liquid, listed options market, which an individual bond or a swap does not — and because it trades in the same equity-market structure, the size check chapter 21 runs against an equity index, implied over realised by one to three volatility points, applies to it directly rather than needing translation the way a swaption would. Selling a covered call against a holding of it — or, equivalently, a cash-secured put to acquire it below today’s price — is the retail-accessible version of “Selling Volatility, and What That Is Short” above: short gamma, financed by the premium, and earning it for the same underlying reason a dealer’s short-swaption book does — variance is a priced risk, and a hedge against it costs a premium to hold. The edge, where there is one, is the same edge; the position is smaller and the counterparty is a market maker rather than a client hedging a liability, but the source of the premium has not changed.

What has changed is everything working against collecting it. A retail account pays the ETF’s own expense ratio and the option’s own bid-offer, and cannot warehouse the position the way a dealer’s balance sheet can if the premium on offer is not enough for the risk taken — the same one to three points, collected in smaller size against larger frictional costs.

Buying optionality: leverage, not edge

Buying a call or a put on the same ETF, or on a Treasury future for more leverage and less liquidity, is the mirror position, and it inherits the mirror problem. The premium a covered call collects is paid by whoever is on the other side of it: a bought option is, on average, betting against the same variance risk premium the seller of one collects. That does not make the trade wrong. The premium is compensation for a real, one-signed risk, and a trader with a genuine view about direction can be right about it while still paying a structural headwind on the instrument used to express that view. It means the honest accounting for a leveraged directional options position is the view’s own expected payoff, net of a premium priced against the buyer before the position is even opened — a structural reason is what would make a position collectible, and leverage on its own is not one.

Trading the event calendar

The event-lump mispricing “The volatility term structure” trades institutionally, as a calendar spread across two expiries, is executable directly on a single name or a single ETF: buy a straddle or a strangle expiring just after a scheduled event — an FOMC meeting, a payrolls print, an earnings date — and the position is a bet that the actual move exceeds what the option’s own implied volatility already prices for it. Chapter 9’s point about smoothing through the event applies here exactly as it does institutionally: a market maker who has not modelled the event separately from the surrounding calendar marks every expiry near it off an interpolated volatility that is neither the event’s own size nor the quiet days around it, and the mispricing — too large or too small — is what a straddle bought or sold around the date is short. Selling the same structure is the mirror position, financed by the same over-pricing an event lump can carry when the interpolation is too generous, and it is short exactly what the trade is built to collect from: a move bigger than the market left room for, on the one day size is least forgivable.

Why the retail version is usually defined-risk

A dealer selling volatility warehouses the tail because chapter 23 shows a partial hedge is the arithmetic optimum once hedging costs money, and a dealer’s balance sheet is deep enough to survive being wrong before it is right again. A retail account, or a small fund, does not have that — §22.2 already made the point that the capital available to hold a position is not guaranteed to survive as long as being right takes, and for an individual account “not guaranteed” is closer to “essentially never.” So the retail version of every short-volatility position above is usually built with the loss capped from the outset — an iron condor rather than a naked short strangle, a credit spread rather than a naked short option — paying away part of the premium collected to buy back the tail a dealer would simply hold. That is not a different trade; it is the same premium, minus the price of the one thing a small account cannot self-insure.

Structured notes, and who actually prices the option

The third common vehicle is not a strategy an investor builds but one sold pre-built: an autocallable, a reverse convertible, a note marketed as “enhanced income,” each of which is a bond plus a short option position packaged as a single security with a headline coupon. The economics are the covered call above, with one difference that matters: the issuer prices the embedded option once, keeps a spread for structuring and distribution, and the buyer never sees a premium to compare against a quoted market price the way the ETF option above has one sitting next to it. A structural reason is present here too, just on the other side of the trade from where it usually sits in this chapter — it is the issuer, not the buyer, who is the natural seller of a service, structuring and distribution and balance sheet, and is paid for it. That the product is marketed as an investment rather than sold as an option is exactly what makes the issuer’s side of the premium collectible without ever being quoted against anything the buyer could check.

22.13 Where an Edge Lives: Information as a Filtration

Everything so far in this book has fixed a filtration $\mathcal{F}_{t}$ at the outset and asked what prices must be, given it. That construction has been so uniform that it is easy to miss what it assumes: that everyone sees the same thing. An edge is what happens when they do not, and the machinery to say so precisely is already in place.

Let $\mathcal{G}_{t}$ be the market’s filtration, the one under which the discounted price is a martingale by chapter 4, and let $\mathcal{F}_{t}\supseteq\mathcal{G}_{t}$ be a trader’s, containing something extra. A martingale in the smaller filtration need not be a martingale in the larger one. Under a technical condition it remains a semimartingale, and acquires a drift:

\underbrace{dS_{t}=\sigma_{t}\,dW_{t}}_{\text{in }\mathcal{G}}\qquad% \longrightarrow\qquad\underbrace{dS_{t}=\sigma_{t}\alpha_{t}\,dt+\sigma_{t}\,d% \tilde{W}_{t}}_{\text{in }\mathcal{F}}.

(22.4)

That drift $\alpha$ is called the information drift. The alpha a trader is looking for is not a metaphor borrowed from regression. It is a drift term, it exists only relative to a filtration, and its magnitude is the amount by which one information set disagrees with another about where the price is going. The literature that works this out is the literature on enlargement of filtrations, developed to study insider trading; the mathematics is indifferent to how the extra information was obtained.

Example 22.1 (A signal, and what it is worth).

Take the tractable case, where the extra information is a single number seen at the outset. Let $W$ drive the price over $[0,1]$ and let the trader observe

L=\rho W_{1}+\sqrt{1-\rho^{2}}\,Z,

with $Z$ independent noise — a view on the terminal value, correlated $\rho$ with it.

Fix $t$ and work conditionally on $\mathcal{F}_{t}$ , which is to say on $W_{t}$ . Split the signal at $t$ ,

L\;=\;\underbrace{\rho W_{t}}_{\text{known}}\;+\;\underbrace{\rho\,(W_{1}-W_{t% })+\sqrt{1-\rho^{2}}\,Z}_{\text{not}},

so the unknown part has variance $\rho^{2}(1-t)+(1-\rho^{2})=1-\rho^{2}t$ . Write $V=L-\rho W_{t}$ for it: the piece of the signal the market’s own filtration cannot yet see, though the trader — who has observed $L$ itself — already knows its value exactly. Its variance is what the market would still assign it, and that shrinks as $t$ runs on: the price path itself increasingly confirms what $L$ already implied.

Now take the next increment $\Delta W$ over $[t,t+h]$ . Given $\mathcal{F}_{t}$ , the pair $(\Delta W,V)$ is jointly Gaussian with

\operatorname{Var}(\Delta W)=h,\qquad\operatorname{Var}(V)=1-\rho^{2}t,\qquad% \operatorname{Cov}(\Delta W,V)=\rho h,

the covariance because $L$ loads on every increment of $W$ with weight $\rho$ . Gaussian conditioning is then one line,

\mathbb{E}\big{[}\Delta W\,\big{|}\,\mathcal{F}_{t},L\big{]}\;=\;\frac{% \operatorname{Cov}(\Delta W,V)}{\operatorname{Var}(V)}\,V\;=\;\frac{\rho\,h}{1% -\rho^{2}t}\,\big{(}L-\rho W_{t}\big{)},

and dividing by $h$ gives the drift the enlarged filtration sees,

\alpha_{t}\;=\;\frac{\rho\,(L-\rho W_{t})}{1-\rho^{2}t}.

(22.5)

So the information drift is a regression coefficient — the slope from projecting the next increment on the part of the signal not yet realised. That is the same object chapter 4 found when it wrote the hedge ratio as $d\langle V,S\rangle/d\langle S\rangle$ and read it as a regression slope. Alpha and a hedge ratio are the same kind of thing.

The second moment follows without further work. Unconditionally $V\sim N(0,1-\rho^{2}t)$ , so squaring (22.5) and taking expectations,

\mathbb{E}[\alpha_{t}^{2}]\;=\;\frac{\rho^{2}\,(1-\rho^{2}t)}{(1-\rho^{2}t)^{2% }}\;=\;\frac{\rho^{2}}{1-\rho^{2}t},

which rises as $t\to 1$ : the signal is worth most at the end, when little of it is left unconfirmed and the remaining uncertainty has collapsed onto it. What that drift is worth needs a criterion for sizing a position, and the criterion is not the obvious one.

Remark (Why maximise ${\mathbb{E}[\ln X_{T}]}$ and not ${\mathbb{E}[X_{T}]}$ ).

Put a fraction $\pi$ of wealth in an asset with $dS/S=\mu\,dt+\sigma\,dW$ and the rest in cash at zero, so $dX/X=\pi\mu\,dt+\pi\sigma\,dW$ and $X_{T}=X_{0}\exp\big{(}(\pi\mu-\tfrac{1}{2}\pi^{2}\sigma^{2})T+\pi\sigma W_{T}% \big{)}$ . Taking the expectation directly,

\mathbb{E}[X_{T}]=X_{0}\exp\!\big{(}(\pi\mu-\tfrac{1}{2}\pi^{2}\sigma^{2})T% \big{)}\,\mathbb{E}\big{[}e^{\pi\sigma W_{T}}\big{]}=X_{0}\exp\!\big{(}(\pi\mu% -\tfrac{1}{2}\pi^{2}\sigma^{2})T\big{)}\,e^{\frac{1}{2}\pi^{2}\sigma^{2}T}=X_{% 0}\,e^{\pi\mu T},

the Itô correction and the lognormal mean adjustment cancelling exactly. That is linear and unbounded in $\pi$ : maximising $\mathbb{E}[X_{T}]$ says take infinite leverage, an answer produced entirely by a vanishing set of paths with an astronomical outcome, while the leverage needed to reach them puts almost every other path into ruin. $\ln X_{T}$ is what is additive across time — a sum of log-returns — so its average over many periods converges to its expectation, and maximising $\mathbb{E}[\ln X_{T}]$ maximises the almost-sure long-run growth rate of compounded wealth rather than a mean dominated by outcomes that are never actually reached.

The point sharpens because $\ln X_{T}$ is Gaussian, and a Gaussian is symmetric about its mean: ${\mbox{Pr}\left[[\right]}\ln X_{T}<\mathbb{E}[\ln X_{T}]]=\tfrac{1}{2}$ exactly, so

\mathrm{median}(X_{T})\;=\;e^{\mathbb{E}[\ln X_{T}]}.

Maximising $\mathbb{E}[\ln X_{T}]$ is therefore exactly maximising the wealth level half of all paths beat — a typical outcome, not a hypothetical average one. The gap between the two objectives is now a number rather than a slogan: for a lognormal, mean equals median times $e^{\mathrm{Var}(\ln X_{T})/2}$ , so

\frac{\mathbb{E}[X_{T}]}{\mathrm{median}(X_{T})}\;=\;e^{\frac{1}{2}\pi^{2}% \sigma^{2}T},

growing without bound in $\pi$ . That ratio measures precisely how far $\mathbb{E}[X_{T}]$ has drifted from what almost every path actually delivers.

By Itô,

d\ln X_{t}=\Big{(}\pi\mu-\tfrac{1}{2}\pi^{2}\sigma^{2}\Big{)}dt+\pi\sigma\,dW_% {t},

so the long-run growth rate is the bracket, and maximising it over $\pi$ gives $\pi^{\ast}=\mu/\sigma^{2}$ and

g^{\ast}\;=\;\tfrac{1}{2}\Big{(}\frac{\mu}{\sigma}\Big{)}^{\!2}.

(22.6)

The achievable growth rate is half the squared Sharpe ratio, and nothing about the horizon or the wealth enters.

Structure (A hard boundary on leverage, invisible to ${\mathbb{E}[X_{T}]}$ ).

The growth rate $g(\pi)=\pi\mu-\tfrac{1}{2}\pi^{2}\sigma^{2}$ is a downward parabola, zero at $\pi=0$ and again at $\pi=2\mu/\sigma^{2}=2\pi^{\ast}$ , positive strictly between them and negative outside. So wealth grows without bound almost surely for $0<\pi<2\pi^{\ast}$ , and collapses to zero almost surely for $\pi>2\pi^{\ast}$ — a hard ruin boundary at exactly twice the growth-optimal leverage, not a soft or probabilistic one. $\mathbb{E}[X_{T}]=X_{0}e^{\pi\mu T}$ never sees it: it keeps rising for every $\pi>0$ , describing paths that, past $2\pi^{\ast}$ , almost surely do not occur. Maximising $\mathbb{E}[\ln X_{T}]$ instead sits at $\pi^{\ast}$ , in the interior of the region that actually grows.

Remark (Why this objective, and not some other utility).

The choice of $\ln$ here is not a risk-aversion parameter fitted to taste. For repeated bets at a constant leverage, Kelly (1956) and Breiman (1961) proved something stronger: fix $\pi^{\ast}=\mu/\sigma^{2}$ and compare against any other constant leverage $\pi^{\prime}\neq\pi^{\ast}$ . Then

\frac{X_{T}(\pi^{\ast})}{X_{T}(\pi^{\prime})}\;\to\;\infty\qquad\text{almost % surely as }T\to\infty.

Not on average — on almost every path, the log-optimal strategy eventually overtakes any fixed competitor, by a factor that grows without bound. Among constant-leverage strategies, $\mathbb{E}[\ln X_{T}]$ is therefore not one reasonable objective among several: it is the one whose maximiser almost surely dominates every other fixed one in the long run. The adapted, time-varying case — which is what $\pi_{t}^{\ast}$ below actually is — needs a separate argument, given after it.

Remark (The optimum is myopic, which is why $\pi_{t}$ can track $\alpha_{t}$ ).

$\pi^{\ast}=\mu/\sigma^{2}$ was derived for constant $\mu,\sigma$ , and (22.5)’s $\alpha_{t}$ is not constant — it moves with $t$ and $W_{t}$ . The substitution below is licensed by a specific property of log utility. Since

\ln X_{T}=\ln X_{0}+\int_{0}^{T}\Big{(}\pi_{t}\mu_{t}-\tfrac{1}{2}\pi_{t}^{2}% \sigma_{t}^{2}\Big{)}\,dt+\int_{0}^{T}\pi_{t}\sigma_{t}\,dW_{t},

and the second integral has expectation zero, maximising $\mathbb{E}[\ln X_{T}]$ over adapted $\pi_{t}$ reduces to maximising the first integral’s integrand pointwise at every $t$ , since nothing couples the choice at one time to the choice at another. The instant-by-instant optimum, using whatever $\mu_{t},\sigma_{t}$ are known at $t$ , is therefore exactly globally optimal. That is what licenses $\pi_{t}^{\ast}=\mu_{t}/\sigma_{t}^{2}$ with a time-varying drift below, continuously rebalanced as $\alpha_{t}$ itself evolves — a property specific to log utility, not a general fact about portfolio choice.

Remark (The almost-sure dominance carries over to the adapted case).

Algoet and Cover (1988) extend Kelly and Breiman’s result exactly this far, with no restriction on how $\mu_{t},\sigma_{t}$ move: the myopic strategy $\pi_{t}^{\ast}$ almost surely dominates every other adapted strategy $\pi_{t}^{\prime}$ in the same sense, $X_{T}(\pi^{\ast})/X_{T}(\pi^{\prime})\to\infty$ almost surely. The mechanism is the pointwise inequality already used above, $g(\pi_{t}^{\ast})\geq g(\pi_{t}^{\prime})$ at every $t$ — so the drift of $\ln\big{[}X_{T}(\pi^{\ast})/X_{T}(\pi^{\prime})\big{]}$ accumulates without bound wherever $\pi_{t}^{\prime}$ differs from $\pi_{t}^{\ast}$ , while what is left is a martingale whose fluctuation grows only like $\sqrt{T}$ , too slowly to catch a drift growing like $T$ . So tracking $\alpha_{t}$ with $\pi_{t}^{\ast}=(\theta+\alpha_{t})/\sigma$ carries the same almost-sure dominance as the constant-leverage case, not merely the weaker property of maximising an expectation.

Now read (22.4) through (22.6). In the enlarged filtration the price acquires an extra drift of $\sigma\alpha_{t}$ , so its Sharpe ratio rises from $\theta$ to $\theta+\alpha_{t}$ and the optimal position becomes $\pi_{t}^{\ast}=(\theta+\alpha_{t})/\sigma$ . The informed investor’s growth is $\tfrac{1}{2}(\theta+\alpha_{t})^{2}$ against the uninformed investor’s $\tfrac{1}{2}\theta^{2}$ , and the difference is $\theta\alpha_{t}+\tfrac{1}{2}\alpha_{t}^{2}$ . The signal is unbiased — $\mathbb{E}[\alpha_{t}]=0$ , since $L$ and $W_{t}$ both have mean zero — so the cross term vanishes in expectation, and the informed investor’s advantage over the uninformed one’s $\tfrac{1}{2}\theta^{2}$ is $\tfrac{1}{2}\mathbb{E}[\alpha_{t}^{2}]$ exactly, whatever $\theta$ itself happens to be.

The value of the signal is second order in it: it comes from the variability of the perceived drift and not from its level, which is why what emerges below is an information quantity rather than a return. Integrating over the horizon,

\frac{1}{2}\int_{0}^{1}\mathbb{E}[\alpha_{t}^{2}]\,dt=\frac{1}{2}\int_{0}^{1}% \frac{\rho^{2}\,dt}{1-\rho^{2}t}=-\tfrac{1}{2}\ln\big{(}1-\rho^{2}\big{)},

(22.7)

which is exactly the mutual information of the jointly Gaussian pair $(W_{1},L)$ , checked here rather than taken on faith. Both are standard Gaussians with correlation $\rho$ , so $I(W_{1};L)=h(W_{1})-h(W_{1}\mid L)$ . The unconditional entropy is $h(W_{1})=\tfrac{1}{2}\ln(2\pi e)$ ; conditioning on $L$ leaves $W_{1}$ Gaussian with variance $1-\rho^{2}$ , the same conditioning fact used throughout this example, so $h(W_{1}\mid L)=\tfrac{1}{2}\ln\big{(}2\pi e(1-\rho^{2})\big{)}$ . Subtracting,

I(W_{1};L)\;=\;\tfrac{1}{2}\ln(2\pi e)-\tfrac{1}{2}\ln\big{(}2\pi e(1-\rho^{2}% )\big{)}\;=\;-\tfrac{1}{2}\ln(1-\rho^{2}),

matching (22.7) exactly.

Kelly proved the general statement in 1956, for a gambler receiving tips over a noisy wire: the growth rate a log-optimal bettor can achieve from a side channel equals that channel’s mutual information, whatever the channel is. The derivation above is the continuous-time instance — half a squared Sharpe ratio, integrated — and it arrives at an information quantity because that is what the theorem says it must. Which also fixes the units in which a signal should be valued. Not in basis points of edge, which depend on how the position is sized, but in nats, which do not.¹⁶¹⁶16estimation::information_value simulates the drift and confirms both (22.7) and the pointwise second moment, which is where an algebra error would hide.

Example 22.2 (What the datasets actually are, in rates).

In fixed income the enlargement usually has a specific and mundane shape: the market is waiting for a scheduled government statistic, and something observable earlier is correlated with it.

Payrolls and unemployment are released monthly and are among the largest scheduled moves in the front end of the curve. Job postings, payroll-processor records and staffing volumes are observable continuously and are correlated with the release. Consumer spending arrives in retail sales and in the consumption component of output; card transaction panels and point-of-sale feeds see much of it first. Inflation is released monthly and scraped prices, freight rates and commodity fixings move before it. Wages appear in the employment cost series and in earnings, and are visible earlier in posted salaries. Activity shows up in satellite imagery of shipping, in electricity load, in traffic. Geopolitical and policy events are read from filings, transcripts and news well before they are quantified anywhere official.

Each of these is an enlargement $\mathcal{F}_{t}\supseteq\mathcal{G}_{t}$ of exactly the kind (22.4) describes, and (22.7) says what each is worth. That reframing does real work, because it replaces the question people usually ask — is this dataset novel, is it large, is it hard to obtain — with the only question that determines the value: how much does it move the conditional distribution of the release, given everything already public. A card panel covering a small and unrepresentative slice of spending may be genuinely novel and worth nothing. A widely available series may still carry information if nobody has bothered to condition on it correctly.

Two features specific to this setting are worth naming. The information has a known expiry: the release date is when $\mathcal{G}$ catches up completely and the drift goes to zero by construction, so the horizon in (22.7) is not a modelling choice but a published calendar. And the release itself is a measurement of the same underlying quantity with its own error, so a signal can be right about the economy and wrong about the print — the tradeable object is the statistic, not the world, and the conditional mutual information that matters is with respect to the number that gets published.

Structure (The value of a dataset is a mutual information).

Equation (22.7) is a special case of a theorem: the additional expected logarithmic utility available to a trader whose filtration is enlarged by a random variable equals the mutual information between that variable and the market.¹⁷¹⁷17Amendinger, Imkeller & Schweizer (1998) identify the value with mutual information in general, under a condition on the enlargement that guarantees the price remains a semimartingale in the larger filtration; Pikovsky & Karatzas (1996) solve the enlarged-filtration portfolio problem the value function comes from. Kelly’s horse race and this chapter’s Gaussian signal are both special cases of the same theorem, not two different results. The question “what is this dataset worth” therefore has an answer, in nats, and the answer does not depend on what the dataset is made of.

Three consequences.

Novelty is not the criterion; conditional information is. What (22.7) measures is the information in $L$ given what is already known. A dataset that is a function of the observable price history has zero conditional mutual information and is worth exactly nothing, however unusual its provenance — and this is a testable condition rather than a matter of judgement. Most datasets sold as alternative fail it.

The value is quadratic in the correlation at the low end. Expanding (22.7), a signal with $\rho=0.1$ is worth about a quarter of one with $\rho=0.2$ , not a half. Weak signals are worth very much less than their correlation suggests, and the last increment of correlation is worth the most, which is the mathematical form of the observation that being slightly better informed than the market is nearly useless.

Alpha decays by ceasing to exist, not by being crowded. The drift in (22.4) is defined relative to $\mathcal{G}$ . As others acquire the same data, $\mathcal{G}$ grows to contain it, and $\alpha$ does not shrink — it becomes zero, because the information is no longer information. That is a sharper statement than the usual one about crowding, and it predicts the right shape: the decay is driven by who else has the data rather than by how much capital is deployed against it.

Remark (The chapter’s other alpha, and what it shares with this one).

The projection above is not the only place this chapter’s regression appears. Chapter 6 showed that in an arbitrage-free curve, a portfolio spanned by the level, slope and curvature factors earns exactly the factor premia and no more — its own market price of risk $\theta$ , read from the portfolio side. Regress the excess returns of the bonds on a curve against their loadings on those three factors and the residual is a portfolio with no factor exposure at all; the butterfly earlier in this chapter, which neutralises level and slope and bets on curvature, is the same projection stopped one factor short.

Where that residual carries a drift — and deciding whether it does is the hard part — it is also called an alpha, and the construction is (22.5)’s again: there, the next return increment was projected onto the unconfirmed part of a signal, run through time; here, each bond’s return is projected onto its factor loadings, run across bonds at one moment. Same operation, cross-sectional instead of temporal.

The economics are not the same, and the mutual-information valuation above does not transfer. The residual is not information about anything — it still holds what the factors do not span, a bond’s liquidity, its specialness in repo, the flow behind it, so it is a statistical arbitrage rather than an actual one. It is compensation for absorbing a constraint, in the sense the six trades earlier in this chapter used the word, not a signal about a future realisation. What does carry over is the sizing mathematics alone: once the residual has an estimated mean and variance, $\pi^{\ast}=\mu/\sigma^{2}$ prices and sizes a position in it exactly as it does a position in $S$ , applied to a different asset rather than derived differently. The pricing theory behind the factor premia is Ross’s Arbitrage Pricing Theory; relative value inverts it, taking the residual to be not quite zero and measuring by how much.

Two honest limits on all of this. The mutual information is an upper bound, attained by an investor with no transaction costs, no position limits and no financing constraint, and the two conditions above’s structural conditions are exactly the reasons that investor does not exist; the earlier sections of this chapter are a catalogue of what the bound loses to reality. And the binding difficulty in practice is not acquiring information but estimating $\alpha$ from it, which is chapter 20’s identification problem in yet another guise. Equation (22.7) tells you the size of the prize. It does not tell you that you can find it, and chapter 19 argues that this — rather than compute — is what the constraint has always been.

References

-

Kelly, J. L. (1956). A new interpretation of information rate. Bell System Technical Journal, 35(4), 917–926.
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Breiman, L. (1961). Optimal gambling systems for favorable games. Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability, 1, 65–78.
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Algoet, P. H., & Cover, T. M. (1988). Asymptotic optimality and asymptotic equipartition properties of log-optimum investment. Annals of Probability, 16(2), 876–898.
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Burghardt, G., & Belton, T. (2005). The Treasury Bond Basis. McGraw-Hill.
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Barth, D., & Kahn, J. (2021). Hedge funds and the Treasury cash-futures disconnect. OFR Working Paper 21-01.
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Schrimpf, A., Shin, H. S., & Sushko, V. (2020). Leverage and margin spirals in fixed income markets during the Covid-19 crisis. BIS Bulletin 2.
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Ross, S. A. (1976). The arbitrage theory of capital asset pricing. Journal of Economic Theory, 13(3), 341–360.
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Litterman, R., & Scheinkman, J. (1991). Common factors affecting bond returns. Journal of Fixed Income, 1(1), 54–61.
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Klingler, S., & Sundaresan, S. (2019). An explanation of negative swap spreads. Journal of Finance, 74(2), 675–710.
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Krishnamurthy, A. (2002). The bond/old-bond spread. Journal of Financial Economics, 66(2–3), 463–506.
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Jegadeesh, N. (1993). Treasury auction bids and the Salomon squeeze. Journal of Finance, 48(4), 1403–1419.
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Fleckenstein, M., Longstaff, F. A., & Lustig, H. (2014). The TIPS-Treasury bond puzzle. Journal of Finance, 69(5), 2151–2197.
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Du, W., Tepper, A., & Verdelhan, A. (2018). Deviations from covered interest rate parity. Journal of Finance, 73(3), 915–957.
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Perli, R., & Sack, B. (2003). Does mortgage hedging amplify movements in long-term interest rates? Journal of Fixed Income, 13(3), 7–17.
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Lou, D., Yan, H., & Zhang, J. (2013). Anticipated and repeated shocks in liquid markets. Review of Financial Studies, 26(8), 1891–1912.
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Amendinger, J., Imkeller, P., & Schweizer, M. (1998). Additional logarithmic utility of an insider. Stochastic Processes and their Applications, 75(2), 263–286.
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Pikovsky, I., & Karatzas, I. (1996). Anticipative portfolio optimization. Advances in Applied Probability, 28(4), 1095–1122.

Chapter 22 Relative Value and the Basis Trade

22.1 What This Chapter Is

Structure (What makes a spread a trade).

22.2 Who Is On the Other Side

Repo, which is how balance sheet actually gets priced

Definition 22.1 (Repurchase agreement).

The players

22.3 How Long Being Right Takes

Calculation 22.2 (First passage is not the half-life).

Remark (Where a stop turns a winning trade into a loss).

Structure (The same first passage problem as the doubling strategy).

Remark (And κ is the parameter that is estimated badly).

22.4 Whether the Spread Converges at All

Calculation 22.3 (What the test can actually see).

Structure (Which is why the structural condition does the work).

22.5 Carry, Roll-Down, and the Rest

Calculation 22.4 (How the known part compares with the unknown).

Remark (One year is an exception, and the long end has inverted).

Remark.

22.6 Curve Trades

Calculation 22.5 (How neutral the neutral position is).

Remark (The weights are a modelling choice).

Calculation 22.6 (Removing the third factor too).

Remark (What is left cannot be told from noise).

Calculation 22.7 (Not choosing a window).

Remark (What the filter has actually changed).

22.7 The Cash-Futures Basis

Definition 22.8 (The basis).

Definition 22.9 (Implied repo rate).

Calculation 22.10 (What the basis trade actually is).

22.8 Why It Breaks

Remark (This is chapter 4’s admissibility condition, in the wild).

Exercise (Sizing the spiral).

22.9 Swap Spreads, and a Negative Number That Should Not Exist

Remark.

22.10 Selling Volatility, and What That Is Short

Calculation 22.11 (What a hedged short option actually earns).

Structure (Short the path, not the variance).

22.11 Six More, and What Each One Is Short

Structure (The same short, six times).

22.12 What Retail Actually Trades

Selling optionality on a rate ETF

Buying optionality: leverage, not edge

Trading the event calendar

Why the retail version is usually defined-risk

Structured notes, and who actually prices the option

22.13 Where an Edge Lives: Information as a Filtration

Example 22.1 (A signal, and what it is worth).

Remark (Why maximise 𝔼⁢[ln⁡XT] and not 𝔼⁢[XT]).

Structure (A hard boundary on leverage, invisible to 𝔼⁢[XT]).

Remark (Why this objective, and not some other utility).

Remark (The optimum is myopic, which is why πt can track αt).

Remark (The almost-sure dominance carries over to the adapted case).

Example 22.2 (What the datasets actually are, in rates).

Structure (The value of a dataset is a mutual information).

Remark (The chapter’s other alpha, and what it shares with this one).

References

Remark (And $\kappa$ is the parameter that is estimated badly).

Remark (Why maximise ${\mathbb{E}[\ln X_{T}]}$ and not ${\mathbb{E}[X_{T}]}$ ).

Structure (A hard boundary on leverage, invisible to ${\mathbb{E}[X_{T}]}$ ).

Remark (The optimum is myopic, which is why $\pi_{t}$ can track $\alpha_{t}$ ).