Rate and index spreads: 2s10s, NOB, NQ/ES and Europe versus the U.S..

Betting on the slope, not on the level

The 2s10s slope is the difference between the 10-year yield and the 2-year yield (lesson 1.2.7). A view on it (that the curve will steepen, or flatten) is not a view on rates: the curve can steepen with rising rates (the 10-year rises more) or with falling rates (the 2-year falls more). To trade only the slope with futures you have to buy one and sell another in such a way that a parallel move of the whole curve neither gains nor loses.

The tool is the DV01 of each future: how many dollars a contract gains or loses if the yield of its reference bond falls one basis point. The data (tipos3_dv01.csv) give it for the contract held of each rate future. They are synthetic data: TPA, TPB, TPC and TPD imitate the U.S. Treasury 2, 5 and 10-year notes and bond (ZT, ZF, ZN, ZB), and TPE the Bund; each contract is the forward price of a standard bond with a 6% coupon (the model's “CTD”, with a conversion factor of 1).

import numpy as np
import pandas as pd

dv = pd.read_csv("tipos3_dv01.csv", parse_dates=["fecha"])
cols = ["mercado", "contrato", "liquidacion", "rendimiento_ctd",
        "vencimiento_ctd_anos", "duracion_modificada", "dv01"]
dv.loc[dv["fecha"] == "2025-12-31", cols].set_index("mercado")

        contrato  liquidacion  rendimiento_ctd  vencimiento_ctd_anos  duracion_modificada    dv01
mercado
TPA       TPAH26   104.273438           3.6651                 2.147                2.040   42.55
TPB       TPBH26   108.750000           3.7311                 4.447                3.973   43.21
TPC       TPCH26   112.546875           3.8286                 6.816                5.729   64.47
TPD       TPDH26   121.625000           4.1319                16.216               11.056  134.46
TPE       TPEH26   126.910000           2.5060                 8.878                7.240   91.88

A TPC contract gains or loses 64.47 USD per basis point; a TPA one, 42.55, even though its notional is twice as large (200,000 versus 100,000): the 2-year bond is much less sensitive to rates. To match the DV01 of one TPC contract you need $h = 64.47 / 42.55 = 1.52$ TPA contracts. That is the DV01-neutral hedge:

where “short” and “long” refer to the maturity, not to the sign: the steepener is long $h$ contracts of the short maturity and short one of the long maturity.

flowchart TB
  M[Curve moves] --> P[Parallel]
  M --> E[Steepens]
  M --> A[Flattens]
  P --> P0[Steepener: 0]
  E --> E1[Steepener: gains]
  A --> A1[Steepener: loses]

The NOB (notes over bonds) is the same spread between the 10-year note and the bond. The ratio is not fixed: the DV01 of each future changes with the level of rates and with the bond that is delivered.

d01 = dv.pivot(index="fecha", columns="mercado", values="dv01")
ratios = pd.DataFrame({"2s10s (TPA per TPC)": d01["TPC"] / d01["TPA"],
                       "NOB (TPC per TPD)": d01["TPD"] / d01["TPC"]})
print(ratios.describe().loc[["min", "50%", "max"]].round(2))
ratios.resample("YE").mean().round(2).T.iloc[:, ::3]

     2s10s (TPA per TPC)  NOB (TPC per TPD)
min                 1.41               1.94
50%                 1.59               2.23
max                 1.87               2.93
fecha                2010-12-31  2013-12-31  2016-12-31  2019-12-31  2022-12-31  2025-12-31
2s10s (TPA per TPC)        1.52        1.55        1.62        1.68        1.56        1.53
NOB (TPC per TPD)          2.03        2.11        2.30        2.53        2.22        2.15

In the data, the 2s10s needs between 1.41 and 1.87 TPA contracts for each TPC one, and the NOB between 1.94 and 2.93 TPC contracts for each TPD. CME lists these spreads as instruments with fixed ratios: 2:1 in the 2-year versus the 10-year (TUT) and 2:1 in the 10-year versus the bond (NOB), among others, which according to CME itself are not expected to change except for substantial market changes. A fixed ratio is convenient (a single order, a single price), but it is only DV01-neutral when it matches the ratio of the DV01s. Hedges by notional, by DV01 and by volatility, and how to choose among them, are the subject of lesson 3.2.4.

The DV01-neutral steepener

Before building the spread it is worth checking that the DV01 says what it promises. The daily P&L of a long in TPC (with the same contracts from one day to the next, as in lesson 3.2.1), divided by the change in the yield of its reference bond in basis points:

cur = pd.read_parquet("futuros3_curvas.parquet")
ctr = pd.read_csv("futuros3_contratos.csv",
                  parse_dates=["vencimiento", "fecha_roll"])
esp = pd.read_csv("futuros3_especificaciones.csv",
                  index_col="mercado")

def cambio(mercado, log=False):
    """Daily change of the contract held at the previous close
    (the first that has not reached its fecha_roll; lesson 3.2.1)."""
    c = cur[cur["mercado"] == mercado].merge(
        ctr[["contrato", "vencimiento", "fecha_roll"]], on="contrato")
    px = c.pivot(index="fecha", columns="contrato",
                 values="liquidacion")
    d = (np.log(px) if log else px).diff().stack()
    c = c[c["fecha"] < c["fecha_roll"]].sort_values(
        ["fecha", "vencimiento"])
    ley = c.groupby("fecha")["contrato"].first().shift(1).dropna()
    idx = pd.MultiIndex.from_arrays([ley.index, ley.to_numpy()])
    return pd.Series(d.reindex(idx).to_numpy(), index=ley.index)

def pyg(mercado):
    """Daily P&L per contract, in the contract's currency."""
    return cambio(mercado) * esp.loc[mercado, "multiplicador"]

usd_pb = pyg("TPC") / (dv[dv["mercado"] == "TPC"].set_index("fecha")
                       ["rendimiento_ctd"].diff() * 100)
print(f"P&L of a long in TPC per bp rise in the CTD: "
      f"{usd_pb.median():.1f} USD (mean DV01 "
      f"{d01['TPC'].mean():.1f})")

P&L of a long in TPC per bp rise in the CTD: -70.1 USD (mean DV01 70.4)

A long in TPC loses a median 70.1 USD for every basis point the yield rises, and the mean DV01 of the series is 70.4. With that, the steepener:

yld = dv.pivot(index="fecha", columns="mercado",
               values="rendimiento_ctd")

def empinador(corto, largo):
    """Long h contracts of the short maturity and short 1 of the long,
    h = long DV01 / short DV01 of the previous session."""
    h = (d01[largo] / d01[corto]).shift(1)
    p = (h * pyg(corto) - pyg(largo)).dropna()
    nivel = yld[largo].diff().reindex(p.index) * 100       # bp
    pendiente = (yld[largo] - yld[corto]).diff().reindex(
        p.index) * 100
    return p, nivel, pendiente

p, nivel, pend = empinador("TPA", "TPC")
dp = (yld["TPC"] - yld["TPA"]).iloc[[0, -1]].diff().iloc[-1]
print(f"Cumulative P&L 2010-2025: {p.sum():,.0f} USD per TPC "
      "contract")
print(f"Change in the CTD slope over the period: {100 * dp:+.0f} bp")
print(f"Correlation of the daily P&L with the change in slope: "
      f"{p.corr(pend):.2f}; with the change in level: {p.corr(nivel):.2f}")

Cumulative P&L 2010-2025: -29,452 USD per TPC contract
Change in the CTD slope over the period: -193 bp
Correlation of the daily P&L with the change in slope: 0.99; with the change in level: 0.49

The figure at the top sums it up: between 2010 and 2025 the synthetic slope (yield of the 10-year reference bond minus that of the 2-year) went from about 200 bp to negative values, with a total change of −193 bp, and the steepener lost 29,452 USD per TPC contract. Its daily P&L has a correlation of 0.99 with the change in slope. The cumulative P&L is not exactly the slope times the DV01, because over 16 years the DV01s, the reference bond of each contract and the financing of the two legs change; but day to day it is.

The correlation with the level (0.49) is not zero. Is the position neutral to the level or not? To separate the two things, a regression of the daily P&L on the change in the 10-year yield (“level”), alone and together with the change in slope:

def betas(p, nivel, pend):
    filas = {}
    for per, (a, b) in {"2010-2015": ("2010", "2015"),
                        "2016-2025": ("2016", "2025")}.items():
        y, l, s = p.loc[a:b], nivel.loc[a:b], pend.loc[a:b]
        sola = np.polyfit(l, y, 1)[0]
        X = np.column_stack([np.ones(len(y)), l, s])
        _, b_niv, b_pend = np.linalg.lstsq(X, y, rcond=None)[0]
        filas[per] = {"nivel_sola": sola, "nivel_con_pend": b_niv,
                      "pendiente": b_pend}
    return pd.DataFrame(filas).T.round(1)

pd.concat({"2s10s": betas(p, nivel, pend),
           "NOB": betas(*empinador("TPC", "TPD"))})

                 nivel_sola  nivel_con_pend  pendiente
2s10s 2010-2015        34.0            -0.2       71.7
      2016-2025        15.1             1.1       69.5
NOB   2010-2015        13.0             0.1      156.6
      2016-2025        14.2             1.0      158.2

With the slope in the regression, the beta to the level is almost zero (−0.2 and 1.1 USD per bp in the 2s10s; 0.1 and 1.0 in the NOB) and the beta to the slope is almost the DV01 of the long-maturity leg: 69.5-71.7 USD per bp in the 2s10s (DV01 of the 10-year, 70-71) and 156.6-158.2 in the NOB (DV01 of the bond, 155-163). It is the known answer from the data README, and the regression recovers it. The position is neutral to a parallel shift. But the beta to the level alone, without the slope, is 34 USD per bp in the 2s10s of 2010-2015 and 13-15 in the other cases. The explanation lies in how the two maturities moved:

dy = yld.diff() * 100
vol = dy.groupby(dy.index.year >= 2016)[["TPA", "TPC", "TPD"]].std()
vol.index = ["2010-2015", "2016-2025"]
vol.round(2)

mercado     TPA   TPC   TPD
2010-2015  4.00  6.22  6.33
2016-2025  6.95  7.13  7.30

In 2010-2015, with short-term rates near zero, the 2-year yield moved with a daily standard deviation of 4.00 bp versus 6.22 for the 10-year. A rise in the 10-year was almost always a steepening, so, in practice, the steepener behaved largely like a short in the 10-year note. DV01-neutral means neutral to a parallel move, not neutral to what the curve actually does. From 2016, with the 2-year as volatile as the 10-year (6.95 versus 7.13 bp), the beta to the level alone falls to 15.

NQ/ES: one index against another

The most traded index spread in the U.S. is the Nasdaq-100 versus the S&P 500 (NQ/ES): a bet on tech stocks against the broad market. In the data, RVB imitates the E-mini Nasdaq-100 and RVA the E-mini S&P 500. In the model, the log return of each future is its own drift (7.0% a year in RVA, 9.5% in RVB), plus a beta to a common market factor (1.00 and 1.15) and its own noise. With the same contracts from one day to the next:

r = pd.DataFrame({m: cambio(m, log=True)
                  for m in ["RVA", "RVB", "RVE"]}).dropna()
beta = r.cov().loc["RVB", "RVA"] / r["RVA"].var()
print(f"Beta of RVB to RVA: {beta:.3f}; correlation: "
      f"{r['RVB'].corr(r['RVA']):.2f}")
for nombre, s in {"notional (1:1)": r["RVB"] - r["RVA"],
                  "beta": r["RVB"] - beta * r["RVA"]}.items():
    media, vol = s.mean() * 252, s.std() * 252 ** 0.5
    t = media / vol * (len(s) / 252) ** 0.5
    print(f"Neutral by {nombre}: drift {media:+.1%}/year, "
          f"vol {vol:.1%}, t = {t:.2f}, "
          f"correlation with RVA {s.corr(r['RVA']):+.2f}")

Beta of RVB to RVA: 1.151; correlation: 0.91
Neutral by notional (1:1): drift +3.2%/year, vol 10.4%, t = 1.21, correlation with RVA +0.28
Neutral by beta: drift +1.4%/year, vol 10.0%, t = 0.56, correlation with RVA +0.00

The measured beta (1.151) recovers the planted one (1.15). There are two usual ways to build the spread:

• Notional-neutral (one dollar long for one dollar short): it is the one CME describes with its spread ratio, the ratio of the notionals of the two contracts computed with the spot index. It keeps the difference in betas: 1.15 − 1 = 0.15 of the market. In the data, its correlation with RVA is 0.28 and its drift, +3.2% a year.
• Beta-neutral (1.15 dollars of RVA per dollar of RVB): it removes the market factor. Its correlation with RVA is 0.00 and its drift is the pure relative one, which in the model is $9.5 - 1.15 \cdot 7.0 = 1.45$% a year. The measured one is +1.4%.

The estimate of the beta-neutral drift is almost exact, and even so it means nothing: with a volatility of 10% a year and 16 years, the standard error of the mean is $10/\sqrt{16} = 2.5$ points, and $t = 0.56$. The notional-neutral spread looks better (t = 1.21, +50% cumulative in the figure), but its advantage comes almost entirely from the 0.15 of market it carries inside, which gains while the stock market rises and loses when it falls (the 2016 stretch of the figure is the most visible). It is the same point as in lesson 3.2.1 with the crude calendar: what is called a “spread” can be, in part, a directional position.

Converting to contracts:

idx = pd.read_csv("futuros3_indices_contado.csv",
                  parse_dates=["fecha"])
s = idx[idx["fecha"] == "2025-12-31"].set_index("indice")["contado"]
nominal = s[["RVA", "RVB"]] * esp.loc[["RVA", "RVB"], "multiplicador"]
print(nominal.round(0).to_string())
n_b = 1e6 / nominal["RVB"]
print(f"Long RVB: {n_b:.1f} contracts; short RVA notional-neutral: "
      f"{n_b * nominal['RVB'] / nominal['RVA']:.1f}; "
      f"beta-neutral: {beta * n_b * nominal['RVB'] / nominal['RVA']:.1f}")

indice
RVA    335966.0
RVB    435294.0
Long RVB: 2.3 contracts; short RVA notional-neutral: 3.0; beta-neutral: 3.4

One million USD in RVB is 2.3 contracts (435,294 USD of notional each as of 2025-12-31); the short side, 3.0 RVA contracts notional-neutral or 3.4 beta-neutral. With large contracts, a one-million spread cannot be sized precisely: the micros (RVAM, RVBM, a tenth of the multiplier, lesson 3.1.1) exist, among other things, for that. And the beta is not fixed: the second exercise measures it year by year.

Europe versus the U.S.: the currency

The Europe versus U.S. spread adds a difficulty: the legs are quoted in different currencies. In the data, RVE imitates the Euro Stoxx 50 and is quoted in QEV (the Vesperian evro, which imitates the euro); RVA, in USD. If an investor buys the European stocks with dollars, they have the European index and also the currency: their return in USD is that of the index in QEV plus that of the QEV against the USD. With a future there is no need to exchange the notional: a margin is posted and the P&L is settled every day, in QEV. The currency risk is limited to that accumulated P&L and to the margin.

flowchart LR
  C[European cash] --> N[Notional in QEV]
  N --> X1[Fully exposed]
  F[RVE future] --> P[Result in QEV]
  P --> X2[Only the result]

fx = pd.read_parquet("fx3_diario.parquet")
x = fx[fx["divisa"] == "QEV"].set_index("fecha")["usd_por_unidad"]
r_fx = np.log(x).diff().reindex(r.index)
contado_usd = r["RVE"] + r_fx          # European index converted to USD
futuro_usd = np.expm1(r["RVE"]) * x.reindex(r.index) / x.shift(
    1).reindex(r.index)                # P&L of the future, in USD
for nombre, s in {"RVE in QEV": r["RVE"],
                  "RVE in USD (cash)": contado_usd,
                  "RVE future, P&L in USD": futuro_usd}.items():
    print(f"{nombre:24s} vol {s.std() * 252 ** 0.5:.1%}  "
          f"correlation with RVA {s.corr(r['RVA']):.2f}")

RVE in QEV               vol 20.9%  correlation with RVA 0.86
RVE in USD (cash)        vol 22.9%  correlation with RVA 0.78
RVE future, P&L in USD   vol 20.9%  correlation with RVA 0.86

In QEV, the European index has a volatility of 20.9% and a correlation of 0.86 with RVA. Converted to USD as cash, the volatility rises to 22.9% and the correlation falls to 0.78: the currency adds noise that has nothing to do with the stock market. The P&L of the future converted to USD each day stays like the index in QEV (20.9% and 0.86). A Europe versus U.S. spread with futures is, therefore, almost a spread in local currency; with cash, it is also a position in the currency. How to manage that risk in a multi-currency portfolio is the subject of lesson 3.3.3.

What this changes in a system

• The ratio decides the bet. DV01-neutral, the position bets on the slope; with another ratio, also on the level. Notional-neutral, an index spread also bets on the market; beta-neutral, it does not.
• Neutral to one move is not neutral to all. Near zero rates, a DV01-neutral steepener has a beta to the level of 34 USD per bp. The residual exposure is measured on data from the regime, with a regression.
• Ratios move. The DV01 and the beta change over time; listed spreads with a fixed ratio are convenient, but not always neutral. Redoing the ratio every day has an execution cost (lesson 3.2.5).
• A relative drift is not validated with 16 years. With 10% volatility, you need a drift of more than 5 points a year for the t to exceed 2.
• The currency, only in the P&L. With futures, the currency risk is that of the P&L and the margin; with cash, that of the notional.

Exercises

1. Build the 2s10s with the fixed 2:1 ratio (two TPA contracts for one TPC) and repeat the beta regression. What exposure to the level remains with the slope in the regression and why does it have that sign?

p_fijo = (2 * pyg("TPA") - pyg("TPC")).dropna()
betas(p_fijo, nivel.reindex(p_fijo.index),
      pend.reindex(p_fijo.index))

           nivel_sola  nivel_con_pend  pendiente
2010-2015        23.8           -19.6       91.1
2016-2025         1.0           -16.6       87.7

2. Estimate the beta of RVB to RVA in each year from 2010 to 2025. How much does it vary and what consequence does it have for a beta-neutral spread calibrated with one year of data?

por_ano = r.groupby(r.index.year).apply(
    lambda g: g["RVB"].cov(g["RVA"]) / g["RVA"].var())
print(por_ano.round(2).to_frame("beta").T.to_string())
print(f"From {por_ano.min():.2f} to {por_ano.max():.2f}")

fecha  2010  2011  2012  2013  2014  2015  2016  2017  2018  2019  2020  2021  2022  2023  2024  2025
beta   1.15  1.21  1.15  1.12  1.13  1.15  1.14  1.15  1.17  1.06  1.15   1.1  1.15  1.19  1.16  1.17
From 1.06 to 1.21

3. A manager with a portfolio of European stocks in USD wants to bet that Europe will outperform the U.S. without adding currency risk. Compare doing it with futures (long the European future, short the U.S. one) and with cash (buying European stocks and selling a U.S. ETF).