Calendar spreads, butterflies and condors.

A spread buys one month and sells another

A crude oil future is not one contract: it is eighteen, one per delivery month, all alive at the same time (lesson 1.2.3). Buying one and selling another of the same market is a calendar spread (also intra-commodity spread). The two contracts respond almost identically to whatever moves the price of crude, so the direction largely cancels out. What remains is the difference between them: the slope of the curve between those two months, which depends on the cost of carrying the underlying, on storage and on the scarcity of the moment (lesson 3.1.2).

The curves in this lesson are those of level 3 (futuros3_curvas.parquet): 23 fictitious markets that imitate real contracts, with the settlement of all listed contracts in every session from 2010 to 2025. They are synthetic data: the levels resemble history, but they are not market data.

import numpy as np
import pandas as pd

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")
dia = cur[(cur["mercado"] == "PEC")
          & (cur["fecha"] == "2025-06-04")].merge(
    ctr[["contrato", "vencimiento"]]).sort_values("vencimiento")
print(len(dia), "PEC contracts listed on 2025-06-04")
dia[["contrato", "vencimiento", "liquidacion", "volumen"]].head(4)

18 PEC contracts listed on 2025-06-04
   contrato vencimiento  liquidacion  volumen
6    PECN25  2025-06-20        53.35   256676
8    PECQ25  2025-07-22        53.23   288298
10   PECU25  2025-08-20        53.23    93695
12   PECV25  2025-09-22        53.32    34244

PEC imitates NYMEX WTI (1,000 barrels per contract). The July contract (PECN25) expires on June 20, but it already has less volume than the August one: liquidity has moved to the next contract before expiration, which is what the fecha_roll column of futuros3_contratos.csv marks. To build structures it is convenient to number the legs starting from the contract being held, the first one that has not yet reached its roll date:

def patas(mercado, n=4):
    """1st, 2nd… contracts of each session. The 1st is the one
    held: the first that has not yet reached its fecha_roll."""
    c = cur[cur["mercado"] == mercado].merge(
        ctr[["contrato", "vencimiento", "fecha_roll"]],
        on="contrato")
    c = c[c["fecha"] < c["fecha_roll"]]
    c = c.sort_values(["fecha", "vencimiento"])
    c["k"] = c.groupby("fecha").cumcount() + 1
    return c[c["k"] <= n]

pec = patas("PEC")
pec.loc[pec["fecha"] == "2025-06-04",
        ["k", "contrato", "liquidacion"]]

       k contrato  liquidacion
68564  1   PECN25        53.35
68566  2   PECQ25        53.23
68568  3   PECU25        53.23
68570  4   PECV25        53.32

With the legs numbered, each structure is a vector of weights on them. The calendar is the difference of two legs; the butterfly, the difference of two consecutive calendars that share the center leg; the condor, the difference of two calendars that do not share it:

flowchart LR
  S[Outright +1] --> C[Calendar +1 −1]
  C --> B[Butterfly +1 −2 +1]
  C --> K[Condor +1 −1 −1 +1]

Buying the 1st − 2nd calendar (long the near, short the next) gains if the near contract rises more than the next one: if the curve steepens in backwardation or flattens in contango. Buying the butterfly gains if the center month falls relative to the average of its neighbors.

The level series and the P&L with the same contracts

The easiest series to build is the spread level: in each session, the settlement of the 1st minus that of the 2nd.

niv = pec.pivot(index="fecha", columns="k", values="liquidacion")
cal = niv[1] - niv[2]                    # USD per barrel
print(cal.describe().round(2).to_string())
print(f"Sessions in backwardation (1st > 2nd): {(cal > 0).mean():.1%}")

count    4024.00
mean        0.15
std         1.64
min        -3.54
25%        -1.04
50%         0.14
75%         1.15
max         4.94
Sessions in backwardation (1st > 2nd): 53.5%

The spread of the synthetic crude ranges from −3.54 USD per barrel (deep contango) to +4.94 (backwardation) and spends slightly more than half of the sessions in backwardation. That series is useful to describe the curve, but it is nobody's P&L. On the roll date, “1st” and “2nd” become other contracts and the series jumps without any position having gained or lost that jump, the same problem as the continuous series of lesson 1.4.6:

rolls = ctr.loc[ctr["mercado"] == "PEC", "fecha_roll"]
salto = cal.diff()
en_roll = salto.index.isin(rolls)
print(f"Mean absolute change on roll days: "
      f"{salto[en_roll].abs().mean():.3f} USD/barrel")
print(f"Other sessions:                    "
      f"{salto[~en_roll].abs().mean():.3f} USD/barrel")

Mean absolute change on roll days: 0.270 USD/barrel
Other sessions:                    0.156 USD/barrel

On roll days the series changes on average 70% more than on other days, and that change is an artifact. The correct P&L is computed with the contracts the position held at the previous close: the weights are fixed with the legs of session $t-1$ and applied to the changes of those same contracts between $t-1$ and $t$. With $M$ the multiplier:

def cambios(mercado, n, log=False):
    """Daily change of each leg k = 1..n with the contracts the
    structure held at the previous close (no roll jumps)."""
    p = patas(mercado, n)
    px = cur[cur["mercado"] == mercado].pivot(
        index="fecha", columns="contrato", values="liquidacion")
    d = (np.log(px) if log else px).diff().stack()
    ley = p.pivot(index="fecha", columns="k",
                  values="contrato").shift(1).dropna()
    out = {k: d.reindex(pd.MultiIndex.from_arrays(
        [ley.index, ley[k]])).to_numpy() for k in ley.columns}
    return pd.DataFrame(out, index=ley.index)

def pyg(mercado, pesos):
    """Daily P&L in the contract's currency, per structure."""
    c = cambios(mercado, len(pesos))
    mult = esp.loc[mercado, "multiplicador"]
    return (c @ np.array(pesos) * mult).dropna()

ESTRUCTURAS = {"simple": [1], "calendario": [1, -1],
               "butterfly": [1, -2, 1], "condor": [1, -1, -1, 1]}
vol = pd.DataFrame({m: {n: pyg(m, w).loc["2025"].std()
                        for n, w in ESTRUCTURAS.items()}
                    for m in ["RVA", "MTO", "PEC", "PEG",
                              "GRM", "GNA"]}).T
vol.round(0)

     simple  calendario  butterfly  condor
RVA  3957.0        41.0       25.0    24.0
MTO  1622.0        15.0       10.0     8.0
PEC  1258.0       149.0       29.0    46.0
PEG  1816.0       321.0      100.0   159.0
GRM   302.0        53.0       28.0    35.0
GNA  1695.0       637.0      639.0   596.0

The table is the figure at the top of the lesson: standard deviation of the daily P&L per contract in 2025, in USD. In the RVA index (it imitates the E-mini S&P 500), the calendar moves one hundredth of what the future moves; in gold, even less. In crude, the calendar keeps 12% of the risk and the butterfly 2.3%. In corn and gasoline, the calendar keeps about 18%. Natural gas is the exception this lesson explains further below: its butterfly (639 USD) moves as much as its calendar (637).

What remains of the direction

A calendar is not direction neutral by construction. If the two contracts moved by the same amount, their difference would have no risk; they do not. In commodities, the near contract reacts more to shocks that are expected to revert (the Samuelson effect, measured in lesson 3.1.2), so long the 1st and short the 2nd keeps a small long:

for m in ["RVA", "PEC", "GRM", "GNA"]:
    d = pd.concat([pyg(m, [1]), pyg(m, [1, -1])], axis=1,
                  keys=["simple", "cal"]).dropna()
    beta = d.cov().loc["cal", "simple"] / d["simple"].var()
    print(f"{m}: beta {beta:+.3f}  correlation "
          f"{d.corr().loc['cal', 'simple']:+.2f}")

RVA: beta -0.003  correlation -0.20
PEC: beta +0.106  correlation +0.92
GRM: beta +0.238  correlation +0.89
GNA: beta +0.228  correlation +0.74

In crude, for every 1,000 USD the near leg gains, the calendar gains 106. It is small in size, but the correlation of 0.92 says that the crude calendar is, above all, a scaled-down version of the outright position. In corn and natural gas the beta is larger (0.24 and 0.23). Whoever trades a calendar as if it were “neutral” and combines it with an outright position in the same market is adding exposure, not hedging it. The hedges that correct this beta (by notional, by DV01, by volatility) are the subject of lesson 3.2.4.

In the index the beta is almost zero, for a different reason: an index calendar does not depend on shocks that revert, but on financing. An index future is worth the spot price compounded at the interest rate minus dividends (lesson 3.1.2), so the difference between two expirations is, almost exactly, the net financing of the section that separates them:

idx = pd.read_csv("futuros3_indices_contado.csv",
                  parse_dates=["fecha"])
s = idx[idx["indice"] == "RVA"].set_index("fecha")
rva = patas("RVA", 2)
f = rva.pivot(index="fecha", columns="k", values="liquidacion")
v = rva.pivot(index="fecha", columns="k", values="vencimiento")
dias = v.sub(v.index.to_series(), axis=0).apply(
    lambda c: c.dt.days)
carry = (s["tipo_1m"] - s["rentabilidad_dividendo"]) / 100
teo = s["contado"].to_numpy()[:, None] * np.exp(
    carry.to_numpy()[:, None] * dias / 365)
eq = pd.DataFrame({"real": f[2] - f[1],          # 2nd − 1st
                   "teorico": teo[2] - teo[1]}).dropna()
print(f"Correlation: {eq.corr().iloc[0, 1]:.4f}; median "
      f"difference {(eq['real'] - eq['teorico']).abs().median():.2f}"
      " points")
eq.resample("YE").mean().round(1).loc["2019":]

Correlation: 0.9997; median difference 0.14 points
            real  teorico
fecha
2019-12-31   2.9      3.0
2020-12-31  -7.9     -7.9
2021-12-31 -11.7    -11.7
2022-12-31   2.0      1.8
2023-12-31  30.1     30.1
2024-12-31  41.4     41.6
2025-12-31  39.1     38.9

The financing formula explains the index calendar with a correlation of 0.9997 and a median error of 0.14 points, less than its tick (0.25). With rates below the dividend yield (2020 and 2021), the deferred contract trades below the near one; with rates of 4-5% (2023-2025), about 40 points above. For whoever holds the future and rolls it every quarter, that spread is the price of the roll (lesson 3.1.3). For whoever trades it, it is a bet on short-term rates and dividends, not on the stock market.

flowchart TB
  C[Calendar] --> I["Index: r − q"]
  C --> B["Bond: coupon and repo"]
  C --> A[Storable: inventory]
  C --> E[Gas: season]

What moves a calendar depends on the market: net financing in indices and currencies; the coupon, the financing and the cheapest-to-deliver bond in bonds; storage and scarcity in crude, metals and grains; and, in natural gas and electricity, the jump between seasons.

Reduced margins

A clearing house does not require margin per leg, but for the risk of the portfolio (the SPAN of lesson 3.1.3). If a calendar moves 12% of what the outright position moves, its margin can be, approximately, 12% of the outright position's. In the data, margen_spread_calendario was set this way: the same number of daily standard deviations that the outright margin covers, applied to the standard deviation of the spread. The calculation with the 23 markets:

base = esp[esp["mercado_base"].isna()].index
filas = {}
for m in base:
    s1 = pyg(m, [1]).loc["2025"].std()
    s2 = pyg(m, [1, -1]).loc["2025"].std()
    filas[m] = {"riesgo": s2 / s1,
                "margen": esp.loc[m, "margen_spread_calendario"]
                / esp.loc[m, "margen_inicial"]}
rel = pd.DataFrame(filas).T.sort_values("riesgo")
print(rel.round(3).T.to_string())

          RVH    MTO    RVC   RVB   RVA    RVE    MTP    DQE    TPD    TPE    TPC    TPB    PEC    TPA    PEH    GRM    PEG    GRT    GRS    GRA    GRH    GNA    ELB
riesgo  0.004  0.009  0.009  0.01  0.01  0.012  0.013  0.022  0.033  0.045  0.053  0.073  0.118  0.136  0.146  0.175  0.177  0.181  0.185  0.192  0.199  0.376  0.389
margen  0.030  0.030  0.030  0.03  0.03  0.029  0.030  0.029  0.033  0.043  0.052  0.072  0.120  0.140  0.147  0.179  0.174  0.186  0.178  0.188  0.195  0.388  0.438

From bonds to electricity, the two rows almost coincide: the calendar margin is proportional to its risk, from 3.3% of the outright margin in the 30-year bond (TPD) to 44% in electricity (ELB). The indices, gold, silver and the currency stay at 3%, even though their relative risk is between 0.4 and 2.2%: the data generator applies a floor of 3% of the outright margin. Real clearing houses also apply minimums and surcharges to spreads (for example, between expirations that cross a delivery period) and publish them with their margin parameters.

A reduced margin is, above all, leverage:

m = esp.loc["PEC"]
precio = niv[1].loc["2025"].median()
print(f"Notional of one leg: {10 * precio * m['multiplicador']:,.0f}"
      " USD")
print(f"Margin for 10 separate longs and 10 shorts: "
      f"{20 * m['margen_inicial']:,.0f} USD")
print(f"Margin for 10 calendar spreads: "
      f"{10 * m['margen_spread_calendario']:,.0f} USD")

Notional of one leg: 618,500 USD
Margin for 10 separate longs and 10 shorts: 112,000 USD
Margin for 10 calendar spreads: 6,700 USD

With 6,700 USD of margin you control 618,500 USD of notional in each leg, more than 1.2 million across the two. The notional figure says nothing about the risk, which is that of the spread; but the margin does not say how much can be lost either.

Butterflies and condors: curvature

If the calendar leaves the slope, the butterfly tries to remove it as well. To see what remains you need to know how the whole curve moves, and that is what a principal component analysis (PCA) is for: the eigenvectors of the covariance matrix of the daily changes of the first twelve contracts. In rate and commodity curves, the first three are usually read as level (the whole curve at once), slope (the short end against the long end) and curvature (the center against the ends). In natural gas:

r = cambios("GNA", 12, log=True).dropna()
val, vec = np.linalg.eigh(np.cov(r.T))
val, vec = val[::-1], vec[:, ::-1]
vec = vec * np.sign(vec[0])           # positive loading on the 1st
print("Explained variance:", (val[:4] / val.sum()).round(3))
cargas = pd.DataFrame(vec[:, :3], index=r.columns,
                      columns=["nivel", "pendiente", "curvatura"])
cargas.T.round(2)

Explained variance: [0.841 0.118 0.022 0.008]
             1     2     3     4     5     6     7     8     9     10    11    12
nivel      0.50  0.42  0.36  0.31  0.27  0.24  0.22  0.20  0.19  0.18  0.17  0.17
pendiente  0.55  0.31  0.11 -0.04 -0.16 -0.24 -0.29 -0.31 -0.31 -0.29 -0.27 -0.25
curvatura  0.38  0.02 -0.28 -0.41 -0.40 -0.30 -0.09  0.10  0.25  0.33  0.32  0.27

The level explains 84.1% of the variance and loads more on the near contracts (the Samuelson effect again: from 0.50 on the first to 0.17 on the twelfth). The slope (11.8%) separates the first four from the rest. The curvature (2.2%) is positive at the ends and negative in the center. The exposure of each structure to each factor is the product of its weights by the loadings, and the split of its variance among the three factors, that exposure squared times the variance of each one:

pesos = pd.DataFrame({n: (w + [0] * 12)[:12]
                      for n, w in ESTRUCTURAS.items()},
                     index=cargas.index)
expo = pesos.T @ cargas
print(expo.round(2))
reparto = expo ** 2 * val[:3]
(100 * reparto.div(reparto.sum(axis=1), axis=0)).round(0)

            nivel  pendiente  curvatura
simple       0.50       0.55       0.38
calendario   0.08       0.24       0.35
butterfly    0.02       0.04       0.05
condor       0.03       0.09       0.23
            nivel  pendiente  curvatura
simple       84.0       14.0        1.0
calendario   37.0       45.0       19.0
butterfly    58.0       33.0        9.0
condor       31.0       29.0       40.0

The calendar reduces the exposure to the level from 0.50 to 0.08, and its variance within these three factors is split between level (37%), slope (45%) and curvature (19%): it is above all a slope bet with some level, the beta of the previous section. The butterfly is barely exposed to any of the three (0.02, 0.04 and 0.05): its +1 −2 +1 weights are almost orthogonal to the three loadings, and how its small exposure is split matters little. The condor is exposed mainly to curvature (0.23). According to this PCA, the gas butterfly should have very little risk; the table at the beginning says that in 2025 it moved 639 USD a day, as much as the calendar. What is missing is not in these three factors.

But the PCA by order of expiration averages contracts of different months: the 3rd is October in July and March in December. In a market with seasons, what matters is which delivery months the structure crosses. Natural gas has a winter factor (in the data model, it loads +0.5 from November to March and −0.5 from April to October), and a butterfly whose wings fall on the two sides of that jump does not neutralize it:

g = cur[cur["mercado"] == "GNA"]
mes = g["contrato"].str[3].map("FGHJKMNQUVXZ".index)
n = (g["contrato"].str[4:].astype(int) + 2000) * 12 + mes
px = g.assign(n=n).pivot(index="fecha", columns="n",
                         values="liquidacion")
d = px.diff() * esp.loc["GNA", "multiplicador"]
hoy = px.index.year * 12 + px.index.month - 1
por_mes = {}
for c in px.columns[1:-1]:
    fly = d[c - 1] - 2 * d[c] + d[c + 1]
    ok = (c - hoy >= 3) & (c - hoy <= 9) & (px.index.year >= 2016)
    por_mes.setdefault("FGHJKMNQUVXZ"[c % 12], []).append(fly[ok])
riesgo = pd.Series({k: pd.concat(v).std()
                    for k, v in por_mes.items()})
riesgo.reindex(list("FGHJKMNQUVXZ")).round(0).to_frame("usd_dia").T

            F     G      H      J     K     M     N     Q     U      V      X     Z
usd_dia  54.0  55.0  391.0  489.0  45.0  31.0  43.0  44.0  71.0  422.0  511.0  35.0

Each column is a butterfly of three consecutive delivery months, identified by its center month (futures month code: F January, G February… Z December), when it has 3 to 9 months left. Eight of the twelve move between 31 and 71 USD a day. The four that cross a change of season—centered in March and April (February-March-April and March-April-May) and in October and November—move between 391 and 511: seven or eight times more. They are the structures that turn a gas butterfly into a bet on the winter, and the ones that most surprise whoever treats it as “almost riskless”.

Listed spreads and sign conventions

Calendars, butterflies and condors are not only built leg by leg: exchanges list them as instruments with their own order book. An order on the spread is filled in full or not at all, without the risk of being left with a single leg. In the data, spreads3_cotizados.parquet has the closing book of the 1st − 2nd calendar of six markets:

cot = pd.read_parquet("spreads3_cotizados.parquet")
fila = cot[(cot["mercado"] == "PEC")
           & (cot["fecha"] == "2025-06-04")].iloc[0]
print(fila[["spread", "liquidacion", "bid", "ask"]].to_string())
leg = niv.loc["2025-06-04"]
print(f"1st − 2nd with the settlements: {leg[1] - leg[2]:.2f}")

spread         PECN25-PECQ25
liquidacion             0.12
bid                     0.12
ask                     0.13
1st − 2nd with the settlements: 0.12

The PECN25-PECQ25 spread settles at 0.12 USD per barrel, the difference of the legs' settlements, and its book quotes one tick wide (0.12 to 0.13). Comparing that book with the “synthetic” spread obtained by crossing the bid-ask spreads of the two legs, and what it costs to execute one leg and wait for the other (legging risk), is the subject of lesson 3.2.5.

Before sending an order on a listed spread you have to read its sign convention, because it is not the same for every product. The CME Globex documentation defines, among others:

What this changes in a system

• The P&L, with the same contracts. A 1st − 2nd level series is useful to describe the curve; for a backtest you have to fix each day's contracts and roll the whole structure on a date decided in advance.
• Neutral is not riskless. The crude calendar keeps a beta of 0.11 and a correlation of 0.92 with the outright position; the gas butterfly that crosses a season moves like a calendar.
• The small margin is leverage. Position sizing in spreads is decided by their measured risk (and by their worst day), not by what the margin lets you open.
• Each market has its own engine. In indices, the calendar is financing; in commodities, storage and seasons. A rule that works in one need not work in another.
• The sign, first of all. Near minus deferred or deferred minus near: it depends on the exchange, the product and the data vendor.

Exercises

1. Check that a natural gas condor is exactly the difference of two calendars (1st − 2nd and 3rd − 4th) and measure in 2025 the risk of each one and their correlation. Why does the condor (596 USD) not have less risk than the 1st − 2nd calendar (637)?

c = cambios("GNA", 4) * esp.loc["GNA", "multiplicador"]
condor = c[1] - c[2] - c[3] + c[4]
dos_cal = (c[1] - c[2]) - (c[3] - c[4])
print("Identical:", np.allclose(condor.dropna(), dos_cal.dropna()))
d = pd.DataFrame({"cal_1_2": c[1] - c[2], "cal_3_4": c[3] - c[4]})
print(d.loc["2025"].std().round(0).to_string())
print(f"Correlation: {d.loc['2025'].corr().iloc[0, 1]:.2f}")

Identical: True
cal_1_2    637.0
cal_3_4    332.0
Correlation: 0.38

2. With the margins from the PEC specification sheet, compare ten longs in the contract being held with ten 1st − 2nd calendars in 2016–2025: the worst day in USD, how many times the initial margin and in how many sessions the day's loss exceeds the entire margin.

for nombre, w, marg in [
        ("10 longs", [1], 10 * m["margen_inicial"]),
        ("10 calendars", [1, -1],
         10 * m["margen_spread_calendario"])]:
    p = 10 * pyg("PEC", w).loc["2016":]
    peor = p.min()
    print(f"{nombre}: worst day {peor:,.0f} USD = "
          f"{-peor / marg:.1f} times the margin; "
          f"days exceeding the margin: {(p < -marg).sum()}")

10 longs: worst day -92,800 USD = 1.7 times the margin; days exceeding the margin: 9
10 calendars: worst day -13,000 USD = 1.9 times the margin; days exceeding the margin: 10

3. A system buys 10 ESU9-ESZ9 spreads on CME (type EQ) thinking it is long September and short December. What position does it actually have and what happens to it if short-term rates rise and the dividend yield does not change?