The Basis Trade

Two prices for the same asset must agree by delivery day. When they don't, the gap between them, know as the basis, is either free money or a very patient way to lose it.

Defining the basis

For any asset with a futures contract, such as a Treasury note, a barrel of oil, or an equity index, two prices exist at once: the spot price you would pay to own it right now, and the futures price you can lock in today for delivery later. The basis is the difference.

At the instant the contract expires it converges to spot, because delivery forces it to. So \(B_T = 0\): the basis is a number engineered to hit exactly zero on a known date. That is what makes basis trades feel like arbitrage.

Why the basis isn't zero today

If you could buy the asset now and hold it for free, spot and futures would already be equal. They are not, because holding costs something (financing) and sometimes pays something back (coupons, dividends, convenience yield). This is the cost-of-carry.

Rearranged, the “fair” basis is just the net cost of carrying the asset to delivery.

When the market's basis drifts away from this fair value, a mechanical trade appears: buy the cheap side, sell the expensive side, hold to delivery, collect the difference.

The Treasury Basis Trade

The canonical basis trade is in Treasury futures. A futures contract there is not written on a single bond but on a notional bond, and the short may deliver any of several eligible issues, each scaled by a conversion factor \(\mathrm{CF}\) that approximately equalises them.

Term	Definition	Meaning
Gross basis	\(P_0 - F_0\cdot \mathrm{CF}\)	Raw price gap, before financing.
Carry	coupon accrued − repo cost	What you earn (or pay) while holding to delivery.
Net basis (BNOC)	gross basis − carry	The real economics; also the price of the delivery option.
Implied repo rate	break-even financing rate	Compare with actual repo: if IRR > repo, cash-and-carry pays.

The bond with the highest implied repo rate is referred to as the cheapest to deliver (CTD), and the short's right to switch between issues is a genuine option with value. In the simplified model below there is a single underlying and no delivery option, so net basis and gross basis coincide up to carry (this is a deliberate simplification, and one of the reasons the model's payoff comes out cleaner than in reality).

The trade, leg by leg

The classic version is cash-and-carry: futures look rich relative to fair value, so you buy the underlying and sell the future against it, financing the purchase until delivery.

To deliver exactly one unit at expiry, buy \(n = e^{-yT}\) units at inception and reinvest the income; holdings then grow to \(n\,e^{yT}=1\). Netting the legs, the profit is fixed at inception regardless of where the market goes.

Because \(\Pi\) is tiny relative to \(S_0\) the trade only pays if it is run with leverage \(L\), posting equity \(E = S_0/L\). The return on that equity is

Financing asymmetries

Everything risky about this trade follows from one fact: the two legs are financed in completely different ways.

This gives two independent ways to be forced out:

• Margin channel. The basis widens, variation margin exceeds equity, and one is stopped out at a loss.
• Funding channel. The repo lender raises the haircut or refuses to roll the loan at all. As a consequence, the trade must be unwound even if the basis has not moved.

March 2020 was largely the second channel. Repo funding for Treasuries became scarce and expensive very suddenly, and leveraged basis positions were unwound on a funding shock rather than a pricing one.

A small stochastic model

Model the spot as geometric Brownian motion and the basis mispricing \(\varepsilon_t\) as a mean-reverting Ornstein–Uhlenbeck process, damped so that it must vanish at expiry:

\begin{align} dS_t &= \mu S_t\, dt + \sigma S_t\, dW_t, \\ d\varepsilon_t &= -\kappa\, \varepsilon_t\, dt + \sigma_b\, dZ_t, \\ F_t &= S_t\, e^{(r-y)(T-t)}\; e^{\varepsilon_t \frac{T-t}{T}} . \end{align} Notice that the factor \((T−t)/T\) forces \(F_T = S_T\) exactly, regardless of the noise process.

Discretised with an Euler–Maruyama scheme on a daily grid:

\begin{equation} \varepsilon_{i+1} = (1-\kappa\,\Delta t)\,\varepsilon_i + \sigma_b\sqrt{\Delta t}\; z_i , \qquad z_i \sim \mathcal N(0,1). \end{equation}

Then, the mark-to-market value of the position is

\begin{equation} V_t = \underbrace{n\left(S_t\,e^{y t} - S_0\, e^{r t}\right)}_{\text{cash leg net of financing}} \;+\; \underbrace{F_0 - F_t}_{\text{short futures leg}}, \qquad n = e^{-yT}, \end{equation} so that \(V(T) = F_0 − S_0 e^{(r−y)T} = \Pi\) exactly, with no residual spot exposure

The funding shock is modelled as Poisson shock with hazard rate \(h\).

\begin{equation} \mathbb{P}\big(\text{funding shock in } [t,\,t+dt]\big) = 1 - e^{-h\,dt} \approx h\,dt . \end{equation}

Here, \(h\) is the expected shocks per year; a shock forces an immediate unwind at the prevailing \(V\), independent of the margin channel. The position exits at whichever comes first: \(V_t < -S_0/L\) (margin call), a funding shock, or delivery. Note that the margin barrier is by construction the total loss of your equity, so a margin call is a \(-100\%\) return on equity.

The distribution of outcomes

Running 2000 paths on the same shock buffer gives the classic basis-trade profile: a spike at the small convergence profit, and a left tail of forced liquidations whose weight grows with \(\sigma_b\), \(h\) and \(L\) — “picking up nickels in front of a steamroller”.

How much leverage?

Leverage does not change the path of \(V_t\) at all: it only moves the barrier. That makes the whole leverage sweep computable from a single set of paths: for each path the times at which \(V_t\) set a new running minimum is recorded, and that short list determines the first-passage time for every barrier at once.

The shape is the whole argument. Expected return rises linearly in \(L\) while the trade survives, then turns over as the barrier climbs into the body of the distribution: optimal leverage is finite, and well short of the maximum the haircut would permit. Push past the peak and you are taking more risk for less expected return.

When the arbitrage breaks

Cash-and-carry only looks riskless if you can fund the position all the way to delivery. Two episodes show what happens when that assumption fails.

• 1998

  Long-Term Capital Management ran highly levered relative-value and basis trades across bond markets. When Russia defaulted and spreads widened instead of converging, margin calls forced unwinds at the worst possible moment, turning positions that were “right” on paper into realised losses. The fund was recapitalised in a Fed-organised private rescue.

• 2020

  Hedge funds running the Treasury cash–futures basis at high leverage were caught by the March COVID shock. The basis widened sharply and repo funding tightened at the same time — both channels firing at once — and the resulting forced deleveraging amplified dislocation in what is normally the world's most liquid market.

The lesson is not that the mathematics is wrong. Convergence at delivery really is close to certain. It is that leverage converts a small, temporary, adverse move into a permanent exit, and exits happen before the sure thing gets to pay.

References

[1] CME Group — Understanding the Treasury futures basis
[2] BIS Quarterly Review — Leverage and margin spirals in fixed income markets (March 2020)
[3] Office of Financial Research — notes on hedge-fund Treasury basis exposure
[4] Federal Reserve FEDS Notes — Treasury market functioning, March 2020
[5] Financial Stability Board — leverage in non-bank financial intermediation

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