Intercommodity spreads: crack, spark and crush.

Processing margins and relative value

A calendar spread (lesson 3.2.1) pits two months of the same product against each other. An intercommodity spread pits different products against each other. The most traded ones are the margins of an industry: what whoever turns one product into another earns. A refinery buys crude and sells gasoline and diesel (the crack); a combined cycle power plant buys gas and sells electricity (the spark); a crusher buys soybeans and sells meal and oil (the crush).

flowchart LR
  C[Crude] --> R[Refinery]
  R --> G[Gasoline]
  R --> D[Diesel]
  S[Soybeans] --> M[Crusher]
  M --> H[Meal]
  M --> A[Oil]
  N[Gas] --> E[Power plant]
  E --> L[Electricity]

The physical link is what makes these spreads interesting. If the refining margin rises a lot, refineries process more crude and sell more gasoline, and the margin falls; if it drops below the cost of operating, they shut down units and the margin recovers. The spread cannot drift without limit away from what it costs to transform the product, at least over horizons of months. Other intercommodity spreads do not have that anchor: the gold/silver ratio or wheat versus corn are relative value bets between products that partly substitute for each other or that have moved together in the past.

flowchart TB
  P[Two markets] --> V{Physical link?}
  V -- yes --> T[Processing margin]
  V -- no --> R[Relative value]
  T --> RV[Reversion possible]
  R --> SA[No anchor]

The level 3 data have the three processing margins and two relative value spreads, with the known answer in spreads3_verdad.csv: which component of the model moves each spread, how fast it reverts and which ones are controls. They are synthetic data: the markets imitate WTI, RBOB gasoline, New York Harbor diesel, Henry Hub gas, PJM peak electricity, the soybean complex, wheat, corn, gold and silver, but the prices are not market prices.

Units and contract ratios

Before subtracting two prices you have to put them in the same unit. Each contract is quoted in its own:

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")
mercados = ["PEC", "PEG", "PEH", "GNA", "ELB", "GRS", "GRH",
            "GRA", "GRT", "GRM", "MTO", "MTP"]
esp.loc[mercados, ["imita", "multiplicador", "unidad_cotizacion",
                   "meses"]]

                                imita  multiplicador   unidad_cotizacion         meses
mercado
PEC                WTI Crude Oil (CL)         1000.0          USD/barril  FGHJKMNQUVXZ
PEG                RBOB Gasoline (RB)        42000.0           USD/galón  FGHJKMNQUVXZ
PEH               NY Harbor ULSD (HO)        42000.0           USD/galón  FGHJKMNQUVXZ
GNA        Henry Hub Natural Gas (NG)        10000.0           USD/MMBtu  FGHJKMNQUVXZ
ELB      PJM Western Hub Peak mensual          800.0             USD/MWh  FGHJKMNQUVXZ
GRS                     Soybeans (ZS)           50.0     centavos/bushel       FHKNQUX
GRH                 Soybean Meal (ZM)          100.0  USD/tonelada corta      FHKNQUVZ
GRA                  Soybean Oil (ZL)          600.0      centavos/libra      FHKNQUVZ
GRT            Chicago SRW Wheat (ZW)           50.0     centavos/bushel         HKNUZ
GRM                         Corn (ZC)           50.0     centavos/bushel         HKNUZ
MTO                         Gold (GC)          100.0            USD/onza        GJMQVZ
MTP                       Silver (SI)         5000.0            USD/onza         HKNUZ

The conversions that define each spread:

In the crack, the gasoline or diesel contract has 42,000 gallons, which are 1,000 barrels: the same volume as the crude one. One product contract against one crude contract is the 1:1 crack, and the exchange treats it that way in its examples. The 3:2:1 (three of crude, two of gasoline, one of diesel) imitates the yield of a typical U.S. refinery, which produces roughly twice as much gasoline as distillates.

In the spark, the heat rate is the efficiency of the power plant: the MMBtu of gas it burns to produce one MWh. The EIA publishes the spark spread with a reference heat rate of 7,000 Btu/kWh (7 MMBtu/MWh), that of a modern, efficient combined cycle plant; and the spark does not count the plant's other costs (gas transport, maintenance, emissions), so a positive spark is not a profit.

In the crush, the CME guide starts from the physical yields: a 60-pound bushel gives about 44 pounds of meal and 11 of oil (plus hulls and shrink). With 44/2,000 = 0.022 short tons of meal per bushel and the contract sizes, the proportion in contracts is:

bushels = 10 * 5000                    # 10 soybean contracts
harina_t = 44 / 2000 * bushels         # short tons
aceite_lb = 11 * bushels               # pounds
print(f"Meal: {harina_t:,.0f} t = {harina_t / 100:.2f} contracts")
print(f"Oil: {aceite_lb:,.0f} lb = "
      f"{aceite_lb / 60000:.2f} contracts")

Meal: 1,100 t = 11.00 contracts
Oil: 550,000 lb = 9.17 contracts

The figure does the calculation with the July 2025 contracts on June 4: the meal from one bushel is worth 7.48 USD, the oil 5.52 and the soybeans cost 10.89, so the gross crush margin is 2.12 USD per bushel.

Ten soybean contracts give exactly eleven of meal and 9.17 of oil. CME facilitates the board crush as 30 contracts: 10 of soybeans, 11 of meal and 9 of oil. The oil ends up 2% short: with small whole numbers, the ratio is never exact. The second exercise does the same search for the spark spread.

The same delivery month in every leg

A July crack is July gasoline against July crude. Mixing months adds to the position a calendar spread nobody asked for. The legs also have different calendars: soybeans list seven months a year, meal and oil eight, and only six are common (January, March, May, July, August and September). For each session, the code looks for the first delivery month listed in every leg whose roll date has not arrived in any of them, and computes the daily change with the previous day's contracts, as in lesson 3.2.1:

cm = cur.merge(ctr[["contrato", "mes_entrega"]], on="contrato")
px = cm.pivot_table(index="fecha", columns=["mercado", "mes_entrega"],
                    values="liquidacion")

def mes_comun(mercados, meses=None):
    """Delivery month held at the same time in every leg: the
    first one listed in all of them whose first roll date has not
    arrived."""
    c = ctr[ctr["mercado"].isin(mercados)]
    r = c.groupby("mes_entrega").agg(
        n=("mercado", "nunique"), roll=("fecha_roll", "min"))
    r = r[r["n"] == len(mercados)]
    if meses:
        r = r[[int(m[5:]) in meses for m in r.index]]
    roll = r["roll"][::-1].cummin()[::-1]
    pos = np.searchsorted(roll.to_numpy(), px.index.to_numpy(),
                          side="right")
    pos = np.minimum(pos, len(r) - 1)
    return pd.Series(r.index.to_numpy()[pos], index=px.index)

def patas_mes(mercados, meses=None):
    """Today's price of each leg in today's common month and today's
    change of each leg in yesterday's common month (no jumps)."""
    mes = mes_comun(mercados, meses)
    dif = px.diff()
    def toma(tabla, m, meses_):
        idx = pd.MultiIndex.from_arrays([mes.index, meses_.to_numpy()])
        return tabla[m].stack().reindex(idx).to_numpy()
    nivel = pd.DataFrame({m: toma(px, m, mes) for m in mercados},
                         index=mes.index)
    cambio = pd.DataFrame({m: toma(dif, m, mes.shift(1))
                           for m in mercados}, index=mes.index)
    return nivel, cambio, mes

niv, dif, mes = patas_mes(["PEG", "PEC"])
mes.loc["2025-06-02":"2025-06-06"]

fecha
2025-06-02    2025-07
2025-06-03    2025-07
2025-06-04    2025-07
2025-06-05    2025-07
2025-06-06    2025-07
dtype: str

On June 4, 2025, the crack being held is the July one in both legs. With that function the six spreads are built in their units:

SPREADS = {
    "crack_gasolina": (["PEG", "PEC"], None,
                       lambda p: 42 * p.PEG - p.PEC),
    "crack_gasoleo": (["PEH", "PEC"], None,
                      lambda p: 42 * p.PEH - p.PEC),
    "spark": (["ELB", "GNA"], None, lambda p: p.ELB - 7 * p.GNA),
    "crush": (["GRS", "GRH", "GRA"], [1, 3, 5, 7, 8, 9],
              lambda p: 0.022 * p.GRH + 0.11 * p.GRA - p.GRS / 100),
    "trigo_maiz": (["GRT", "GRM"], [3, 5, 7, 9, 12],
                   lambda p: p.GRT / p.GRM),
    "aceite_harina": (["GRA", "GRH"], [1, 3, 5, 7, 8, 9, 10, 12],
                      lambda p: p.GRA / p.GRH),
}
UNIDAD = {"crack_gasolina": "USD/barrel", "crack_gasoleo": "USD/barrel",
          "spark": "USD/MWh", "crush": "USD/bushel",
          "trigo_maiz": "ratio", "aceite_harina": "ratio"}
niveles = {}
for nombre, (ms, meses, f) in SPREADS.items():
    niveles[nombre] = f(patas_mes(ms, meses)[0])
tabla = pd.DataFrame(niveles).describe().T
tabla["unidad"] = pd.Series(UNIDAD)
tabla[["unidad", "mean", "min", "max"]].round(3)

                    unidad    mean    min     max
crack_gasolina  USD/barrel  21.318 -8.886  65.059
crack_gasoleo   USD/barrel  31.515  9.177  78.124
spark              USD/MWh  15.711 -2.250  74.091
crush           USD/bushel   1.034 -0.289   3.611
trigo_maiz           ratio   1.355  1.019   1.779
aceite_harina        ratio   0.132  0.087   0.251

The average crack of the synthetic gasoline is 21.3 USD per barrel and it even turns negative (−8.89); the diesel one, higher and more stable, never falls below 9.18. The spark also turns negative (−2.25 USD/MWh): there are days when burning gas to produce peak electricity does not even cover the fuel. The average crush is 1.03 USD per bushel. The figure at the top shows the two cracks: the gasoline one oscillates much more, with a very marked annual cycle. Before interpreting it, it is worth looking at the average level by delivery month:

mes_gas = patas_mes(["PEG", "PEC"])[2]
nivel_mes = niveles["crack_gasolina"].groupby(mes_gas.str[5:]).mean()
nivel_mes.round(1).to_frame("usd_barril").T

              01    02    03    04    05    06    07    08    09    10   11    12
usd_barril  13.6  14.9  23.1  31.1  33.8  32.8  29.5  26.7  19.2  11.0  9.3  11.1

The gasoline crack for May delivery is worth on average 33.8 USD; the November one, 9.3. Summer gasoline is a different product (more expensive to produce) and less winter gasoline is sold, so the gasoline curve has a fixed seasonal shape. The cycle at the top is, above all, the switch from one delivery month to the next.

Risk: it depends on the correlation of the legs

A two-leg spread with volatilities $\sigma_a$ and $\sigma_b$ and correlation $\rho$, with equal weights in logarithms, has a volatility of

The more correlated the legs are, the less risk remains. Measured with the same contracts from one day to the next:

LOG = {
    "crack_gasolina": lambda p: np.log(42 * p.PEG / p.PEC),
    "crack_gasoleo": lambda p: np.log(42 * p.PEH / p.PEC),
    "spark": lambda p: np.log(p.ELB / (7 * p.GNA)),
    "crush": lambda p: np.log((0.022 * p.GRH + 0.11 * p.GRA)
                              / (p.GRS / 100)),
    "trigo_maiz": lambda p: np.log(p.GRT / p.GRM),
    "aceite_harina": lambda p: np.log(p.GRA / p.GRH),
}
filas = {}
for nombre, (ms, meses, _) in SPREADS.items():
    nivel, cambio, mes = patas_mes(ms, meses)
    antes = nivel.shift(1)            # yesterday, yesterday's contract
    r = np.log((antes + cambio) / antes).dropna()     # same contract
    s = LOG[nombre](antes + cambio) - LOG[nombre](antes)
    a, b = ms[0], ms[-1]
    filas[nombre] = {"pata_a": a, "vol_a": r[a].std() * 252 ** 0.5,
                     "pata_b": b, "vol_b": r[b].std() * 252 ** 0.5,
                     "correlacion": r[a].corr(r[b]),
                     "vol_spread": s.std() * 252 ** 0.5}
riesgo = pd.DataFrame(filas).T
num = ["vol_a", "vol_b", "correlacion", "vol_spread"]
riesgo[num] = riesgo[num].astype(float).round(2)
riesgo

               pata_a  vol_a pata_b  vol_b  correlacion  vol_spread
crack_gasolina    PEG   0.39    PEC   0.36         0.91        0.16
crack_gasoleo     PEH   0.38    PEC   0.36         0.93        0.14
spark             ELB   0.48    GNA   0.42         0.81        0.29
crush             GRS   0.19    GRA   0.27         0.71        0.10
trigo_maiz        GRT   0.29    GRM   0.24         0.83        0.17
aceite_harina     GRA   0.28    GRH   0.25         0.49        0.27

The legs of the crack move with a correlation of 0.91 (gasoline) and 0.93 (diesel): the spread keeps 14-16% annual volatility versus 36-39% for each leg. The spark, at 0.81, keeps 29%; oil versus meal, at 0.49, 27%: almost as much as each leg. In the crush the table shows soybeans and oil (0.71), but the spread includes all three legs, and its volatility is the lowest of all, 10%. An intercommodity spread has more risk than a calendar spread of the same market (lesson 3.2.1), because two different products are less alike than two months of the same one.

That lower risk is also reflected in the margin, although less than in calendars. CME explains that, thanks to the offset between the legs, a crack can require up to 30% less capital than an outright position in gasoline or diesel, and in its example with micros the crack requires 480 USD versus a diesel leg that, on its own, requires 599: 19% less. A crude calendar in the data requires 12% of the outright margin, not 70-80%.

Reversion: planted versus measured

The physical link suggests that processing margins revert. In the data they do by construction: each spread's own component is an Ornstein-Uhlenbeck process with speed $\kappa_P$, and its half-life in sessions is $\ln 2 / \kappa_P \cdot 252$. The useful question is whether a standard method recovers it. The method is the one from lesson 2.1.5: an AR(1) of the level of the log-spread, from which its mean by delivery month is first removed (the seasonal shape of the curve from the previous section), and an ADF unit root test:

from statsmodels.tsa.stattools import adfuller

ver = pd.read_csv("spreads3_verdad.csv", index_col="spread")

def adf_p(x):
    """ADF p-value; result_object=False keeps the usual tuple
    and avoids the statsmodels 0.15 FutureWarning."""
    return adfuller(x, result_object=False)[1]

def vida_media(nivel, clave):
    """AR(1) of the level minus its mean by key (delivery month)."""
    x = (nivel - nivel.groupby(clave).transform("mean")).dropna()
    b = np.polyfit(x.shift(1).dropna(), x.diff().dropna(), 1)[0]
    return np.log(2) / -b if b < 0 else np.inf

res = {}
for nombre, (ms, meses, _) in SPREADS.items():
    nivel, _, mes = patas_mes(ms, meses)
    s = LOG[nombre](nivel)
    clave = mes.str[5:]
    x = (s - s.groupby(clave).transform("mean")).dropna()
    res[nombre] = {"plantada": ver.loc[nombre, "vida_media_sesiones"],
                   "medida": vida_media(s, clave),
                   "p_adf": adf_p(x),
                   "control": ver.loc[nombre, "control"]}
pd.DataFrame(res).T.astype({"plantada": float, "medida": float,
                            "p_adf": float}).round(
    {"plantada": 1, "medida": 1, "p_adf": 4})

                plantada  medida   p_adf control
crack_gasolina      43.7    36.6  0.0000   False
crack_gasoleo       58.2    40.5  0.0000    True
spark               29.1    17.5  0.0000   False
crush               69.9    54.6  0.0000   False
trigo_maiz         174.7    71.4  0.0003   False
aceite_harina      218.3   157.6  0.0522    True

The ADF rejects the unit root in five of the six spreads and the AR(1) recovers the order of magnitude of the reversion, but with a systematic bias: every measured half-life is shorter than the planted one. In the spark, 17.5 sessions versus 29.1; in the gasoline crack, 36.6 versus 43.7; in wheat versus corn, 71.4 versus 174.7, less than half. It is the bias of the autoregressive estimator in finite samples that lesson 2.1.5 measured with simulations, to which is added that each spread mixes its own component with leftovers of the legs' common factors.

Oil versus meal is the hard case: it reverts (planted half-life of 218 sessions, almost a year), but so slowly that the ADF stays at p = 0.052 with 16 years of data. A test at 5% would deem it non-stationary.

A ratio without an anchor: gold versus silver

The gold/silver ratio is the classic example of relative value: how many ounces of silver one ounce of gold buys. In the data it is, by construction, a random walk with episodes of drift: there is nothing that brings it back to a mean.

def simple(mercado):
    c = cur[cur["mercado"] == mercado].merge(ctr, on=["mercado",
                                                     "contrato"])
    c = c[c["fecha"] < c["fecha_roll"]].sort_values(
        ["fecha", "vencimiento"])
    return c.groupby("fecha")["liquidacion"].first()

ratio = simple("MTO") / simple("MTP")
lr = np.log(ratio)
uno = pd.Series("", index=lr.index)
print(f"Gold/silver ratio: from {ratio.min():.1f} to {ratio.max():.1f}")
print(f"Whole sample: half-life {vida_media(lr, uno):.0f} "
      f"sessions, ADF p {adf_p(lr):.2f}")
for ini in range(2010, 2023, 3):
    t = lr.loc[str(ini):str(ini + 2)]
    print(f"{ini}-{ini + 2}: half-life "
          f"{vida_media(t, uno.loc[t.index]):4.0f}, "
          f"ADF p {adf_p(t):.2f}")

Gold/silver ratio: from 39.5 to 108.2
Whole sample: half-life 698 sessions, ADF p 0.58
2010-2012: half-life  183, ADF p 0.37
2013-2015: half-life   61, ADF p 0.11
2016-2018: half-life  153, ADF p 0.84
2019-2021: half-life  136, ADF p 0.66
2022-2024: half-life  111, ADF p 0.51

The ratio ranges from 39.5 to 108.2 in 16 years. The AR(1) of the whole sample gives a half-life of 698 sessions (almost three years), which sounds like “slow reversion”; in three-year windows it gives between 61 and 183 sessions, which sounds like tradable reversion. But the ADF does not reject the unit root either in the whole sample (p = 0.58) or in any window (from 0.11 to 0.84). The regression of the change on the level almost always gives a negative coefficient in a finite sample, even if the process does not revert: the half-life comes out finite because the estimator is biased, not because there is reversion. It is the highlighted point in the previous figure.

What this changes in a system

• Units, first. Each leg in its quotation unit, converted before subtracting; a unit error gives a different spread.
• The same month in every leg. Otherwise, the spread carries a calendar inside it. With different delivery calendars, only the common months count.
• Whole ratios. The formula gives fractions; contracts do not. 10:11:9 in the crush, 25:14 in the spark: with few contracts, rounding leaves exposure to one leg. Hedges by notional, DV01 or volatility are the subject of lesson 3.2.4.
• Tested reversion. The measured half-life is shorter than the real one, and a ratio without an anchor also has a measured half-life. Reversion is tested and explained.
• Seasonal shape is not drift. The level by delivery month is priced in; the drift, if it exists, is studied separately (lesson 3.2.6). The leg-by-leg backtest, with its costs and the risk of executing one leg without the other, is the one in lesson 3.2.5.

Exercises

1. Build the daily P&L in USD of the 3:2:1 crack (two gasoline contracts and one diesel contract against three of crude) and check that it is the sum of two gasoline cracks and one diesel crack. Compare its risk in 2025 with the sum of the risks of the three cracks separately.

nivel, cambio, mes = patas_mes(["PEG", "PEH", "PEC"])
usd = 1000 * (2 * 42 * cambio.PEG + 42 * cambio.PEH
              - 3 * cambio.PEC)                 # 2 PEG + 1 PEH − 3 PEC
gas = 1000 * (42 * cambio.PEG - cambio.PEC)     # 1:1 crack
die = 1000 * (42 * cambio.PEH - cambio.PEC)
print("Equal to 2 gasoline cracks + 1 diesel crack:",
      np.allclose(usd.dropna(), (2 * gas + die).dropna()))
r25 = pd.DataFrame({"usd": usd, "gas": gas, "die": die}).loc["2025"]
print(f"Daily risk of the 3:2:1: {r25['usd'].std():,.0f} USD")
print(f"Sum of the separate risks: "
      f"{2 * r25['gas'].std() + r25['die'].std():,.0f} USD")
print(f"Correlation of the two cracks: {gas.corr(die):.2f}")

Equal to 2 gasoline cracks + 1 diesel crack: True
Daily risk of the 3:2:1: 1,982 USD
Sum of the separate risks: 2,529 USD
Correlation of the two cracks: 0.18

2. With the contract sizes for electricity (800 MWh) and gas (10,000 MMBtu) from the specification sheet and a heat rate of 7, find the whole contract ratio that replicates the spark spread with the least error, with up to 25 electricity contracts.

mwh = esp.loc["ELB", "multiplicador"]          # MWh per contract
mmbtu = esp.loc["GNA", "multiplicador"]        # MMBtu per contract
exacto = 7 * mwh / mmbtu                       # GNA per ELB
print(f"Gas contracts per electricity contract: {exacto:.2f}")
mejor = []
for n_elb in range(1, 26):
    n_gna = round(n_elb * exacto)
    if n_gna:
        error = n_gna / n_elb / exacto - 1
        mejor.append((abs(error), n_elb, n_gna, error))
for _, a, b, e in sorted(mejor)[:4]:
    print(f"{a:2d} ELB : {b:2d} GNA  error {e:+.1%}")

Gas contracts per electricity contract: 0.56
25 ELB : 14 GNA  error +0.0%
16 ELB :  9 GNA  error +0.4%
 9 ELB :  5 GNA  error -0.8%
18 ELB : 10 GNA  error -0.8%

3. A crusher hedges its margin by selling the board crush (sells meal and oil, buys soybeans). Explain which risk it hedges, which risk it does not hedge and why the result depends on the delivery month it chooses.