--- title: "3.2.2 Intercommodity spreads: crack, spark and crush" description: "Spreads between markets: crack, spark, crush, gold/silver and wheat/corn. Units, contract ratios, risk and measured versus planted reversion." url: "https://pythonparatrading.com/en/python-trading-course/level-3/intercommodity-spreads-crack-spark-crush/" lang: "en" image: "https://pythonparatrading.com/og/en/python-trading-course/level-3/intercommodity-spreads-crack-spark-crush.png" --- # 3.2.2 · Intercommodity spreads: crack, spark and crush > A refinery buys crude and sells gasoline; a power plant buys gas and sells electricity; a crusher buys soybeans and sells meal and oil. Their margins are spreads between two or three different futures, with different units and a physical link that ties them together. This lesson builds them with the same delivery month in every leg, measures their risk and checks whether their reversion is the one planted in the data. - Course: [Level 3: Derivatives: futures, spreads, forex and options](https://pythonparatrading.com/en/python-trading-course/level-3/) - Module: 3.2 · Futures spreads - Duration: 65 min - Prerequisites: [3.2.1 · Calendar spreads, butterflies and condors](https://pythonparatrading.com/en/python-trading-course/level-3/calendar-spreads-butterflies-condors/), [2.1.5 · Cointegration: Engle-Granger, Johansen and the half-life of mean reversion](https://pythonparatrading.com/en/python-trading-course/level-2/cointegration-engle-granger-johansen-half-life/) ## By the end of this lesson - Build the processing spreads (1:1 and 3:2:1 crack, spark, crush) and the relative value ones (gold/silver, wheat/corn) converting units and with the same delivery month in every leg. - Go from the formula of a spread to the whole number of contracts of each leg (10:11:9 in the crush, 1:1 in the crack) and measure the rounding error. - Measure the risk of a spread versus that of its legs and relate it to the correlation between them. - Estimate the half-life of the reversion, compare it with the known answer and recognize the false positive of a ratio without a physical link. ## 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**). ```mermaid 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. ```mermaid 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: ```python 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"]] ``` ```output 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: | Spread | Formula | Unit | Conversion | |---|---|---|---| | 1:1 gasoline crack | $42 \cdot \text{PEG} - \text{PEC}$ | USD/barrel | 42 gallons per barrel | | 1:1 diesel crack | $42 \cdot \text{PEH} - \text{PEC}$ | USD/barrel | 42 gallons per barrel | | 3:2:1 crack | $(2 \cdot 42 \cdot \text{PEG} + 42 \cdot \text{PEH} - 3 \cdot \text{PEC})/3$ | USD/barrel | two of gasoline for one of diesel | | Spark | $\text{ELB} - 7 \cdot \text{GNA}$ | USD/MWh | heat rate of 7 MMBtu per MWh | | Crush | $0.022 \cdot \text{GRH} + 0.11 \cdot \text{GRA} - \text{GRS}/100$ | USD/bushel | 0.022 t of meal and 11 lb of oil per bushel | | Gold/silver | $\text{MTO} / \text{MTP}$ | ratio | ounces of silver per ounce of gold | | Wheat/corn | $\text{GRT} / \text{GRM}$ | ratio | same bushels | 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: ```python 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") ``` ```output Meal: 1,100 t = 11.00 contracts Oil: 550,000 lb = 9.17 contracts ``` *Fig. — Anatomy of the soybean crush: the value of what comes out of a bushel against what it costs (July 2025 contracts, 4 June session).* Horizontal bars in USD per bushel with the July 2025 contracts on 4 June: the meal from one bushel (0.022 short tons at 340.2 USD) is worth 7.48; the oil (11 pounds at 50.17 cents), 5.52; the soybeans cost 10.89. The crush margin, highlighted, is 7.48 + 5.52 − 10.89 = 2.12 USD per bushel. 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: ```python 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"] ``` ```output 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: ```python 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) ``` ```output 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: ```python 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 ``` ```output 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 $$ \sigma_{\text{spread}} = \sqrt{\sigma_a^2 + \sigma_b^2 - 2 \rho \, \sigma_a \sigma_b}. $$ The more correlated the legs are, the less risk remains. Measured with the same contracts from one day to the next: ```python 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 ``` ```output 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: $$ \Delta x_t = a + b \, x_{t-1} + \varepsilon_t, \qquad \text{half-life} = \frac{\ln 2}{-b}. $$ ```python 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}) ``` ```output 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 ``` *Fig. — Mean-reversion half-life planted in the data against the one measured with an AR(1), six spreads and the gold/silver control.* Log scatter. Every spread sits below the diagonal: the measured half-life is shorter than the planted one (spark 17.5 against 29.1 sessions; gasoline crack 36.6 against 43.7; crush 54.6 against 69.9; wheat/corn 71.4 against 174.7; oil/meal 157.6 against 218.3). Highlighted, the gold/silver ratio, a random walk with no reversion (infinite half-life), gives a measured half-life of 698 sessions. 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. ```python 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}") ``` ```output 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. **Solution.** ```python 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}") ``` ```output 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 ``` The 3:2:1 is, exactly, two gasoline cracks plus one diesel crack. Its daily risk in 2025 (1,982 USD) is lower than the sum of the separate risks (2,529 USD) because the two cracks are weakly correlated with each other (0.18): their own components are independent in the data model, and they only share crude, which cancels out in both. Combining margins of different products diversifies. 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. **Solution.** ```python 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%}") ``` ```output 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% ``` An 800 MWh electricity contract burns 5,600 MMBtu of gas with a heat rate of 7: 0.56 gas contracts. The exact ratio with small numbers is 25 electricity contracts for 14 of gas; 16:9 leaves an error of 0.4% and 9:5, of 0.8%. With fewer contracts the error grows: with 2:1, the gas ends up 11% short. In a small position, the “exact” spark spread does not exist, and what is left over is an outright position in gas. 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. **Solution.** Selling the crush locks in today the margin of the months it chooses: if meal and oil later fall relative to soybeans, what it loses in the physical business it gains in the futures. It hedges the price of the three products in Chicago, not its own yield (if its plants get less meal per bushel than 0.022 tons, the real margin is different), nor the local basis (the difference between the price in its area and that of the contract), nor the operating costs, which are not in the formula. The month matters because only six months are common to the three legs and because each delivery month has its own crush level: the shape of each product's curve (old crop versus new crop) is priced into each contract. The hedge for the July month does not lock in the October margin. ## Self-assessment 1. Why is a 1:1 gasoline crack done with one gasoline contract for each crude contract? - A. Because both are quoted in USD - B. Because the refinery produces the same amount of gasoline as crude - C. Because the exchange requires trading the legs in the same quantity - D. Because the gasoline contract (42,000 gallons) is equivalent to 1,000 barrels, the same volume as the crude one *Answer D: Because the gasoline contract (42,000 gallons) is equivalent to 1,000 barrels, the same volume as the crude one.* A barrel has 42 gallons: 42,000 gallons are 1,000 barrels, the size of the crude contract. That is why the crack in USD per barrel is 42 × product price − crude price, and one contract on each side replicates it without rounding. 2. The CME board crush is traded as 10 soybean contracts, 11 of meal and 9 of oil. Where does that proportion come from? - A. From the physical yields: 50,000 bushels give 1,100 short tons of meal (11 contracts of 100) and 550,000 pounds of oil (9.17 contracts of 60,000, rounded to 9) - B. From the liquidity of each contract - C. From matching the notional of the three legs - D. It is a convention unrelated to the process *Answer A: From the physical yields: 50,000 bushels give 1,100 short tons of meal (11 contracts of 100) and 550,000 pounds of oil (9.17 contracts of 60,000, rounded to 9).* A 60-pound bushel gives about 44 pounds of meal and 11 of oil (CME crush guide). With contracts of 5,000 bushels, 100 short tons and 60,000 pounds, the closest whole proportion with few contracts is 10:11:9; the oil ends up 2% short. 3. In the data, the measured half-life of every spread with reversion is shorter than the planted one (for example, 17.5 versus 29.1 sessions in the spark). Why? - A. Because the data have errors - B. Because of the bias of the AR(1) estimator in finite samples: the coefficient comes out smaller than the true one and the reversion looks faster (lesson 2.1.5) - C. Because the reversion speeds up over time - D. Because the planted half-lives are in calendar days *Answer B: Because of the bias of the AR(1) estimator in finite samples: the coefficient comes out smaller than the true one and the reversion looks faster (lesson 2.1.5).* The OLS estimator of the autoregressive coefficient is biased downward, more so the slower the reversion is relative to the sample. In addition, the measured half-life mixes the reverting component with other factors of the spread. What matters is the sign of the error: a backtest calibrated with the measured half-life expects a faster reversion than the real one. 4. The gold/silver ratio of the data gives a half-life of 698 sessions over the whole sample and of 61 to 183 in three-year windows. What do you conclude? - A. That it reverts slowly and can be traded over long horizons - B. That it reverts quickly in short windows - C. Nothing in favor of reversion: it is a random walk by construction, the ADF does not reject (p = 0.58) and an estimated AR(1) always gives a finite half-life - D. That shorter windows should be used *Answer C: Nothing in favor of reversion: it is a random walk by construction, the ADF does not reject (p = 0.58) and an estimated AR(1) always gives a finite half-life.* 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 of how the estimator is built. The test (ADF) is what says whether there is reversion; here, it does not reject at 5% in any window. 5. The gasoline crack of the data is worth on average 33.8 USD per barrel in the May contracts and 9.3 in the November ones. Is it a seasonal opportunity? - A. Yes: buy the crack in November and sell it in May - B. Yes, because the difference is more than 20 USD - C. No, because the gasoline crack has no seasonality - D. Not necessarily: it is the seasonal shape of the curve (summer gasoline is a different product and trades higher), and the market already prices it into each contract *Answer D: Not necessarily: it is the seasonal shape of the curve (summer gasoline is a different product and trades higher), and the market already prices it into each contract.* The average level by delivery month is a property of each contract, known from the day it is listed. Profiting from it would require the May contract to rise while it is held, and that is a drift, something else. Whether it exists and whether it is tradable is studied in lesson 3.2.6. ## Key points - An intercommodity spread pits different markets against each other: processing spreads (crack, spark, crush) have a physical link; relative value ones (gold/silver, wheat/corn), only one of substitution or of history. - Before subtracting you have to convert units (42 gallons per barrel, heat rate of 7 MMBtu per MWh, 0.022 tons of meal and 11 pounds of oil per bushel) and use the same delivery month in every leg. - The contract ratio comes from the sizes: 1:1 in the crack, 10:11:9 in the crush, 25:14 in the spark of the data; small ratios round. - The risk of the spread depends on the correlation of its legs: 16% a year in the gasoline crack (legs with 0.91 correlation) versus 27% in oil/meal (0.49). - The measured half-life is shorter than the planted one in every spread that reverts; the gold/silver control, a random walk, gives a finite half-life and an ADF that does not reject. ## Resources ### Documentation - [CME Group: Trading Crack Spreads (1:1 and 3:2:1 crack, conversion to barrels, margin offsets)](https://www.cmegroup.com/articles/2024/trading-crack-spreads.html) · cmegroup.com - [CME Group: An Introduction to Crack Spreads (1:1, 3:2:1, 5:3:2 and 2:1:1 ratios)](https://www.cmegroup.com/articles/whitepapers/an-introduction-to-crack-spreads.html) · cmegroup.com - [CME Group: Soybean Crush Reference Guide (yields per bushel and board crush)](https://www.cmegroup.com/education/files/soybean-crush-reference-guide.pdf) · cmegroup.com - [CME Group: Soybean Crush Spreads (10 soybeans, 11 meal, 9 oil)](https://www.cmegroup.com/trading/agricultural/grain-and-oilseed/soybean-crush-spreads.html) · cmegroup.com - [U.S. EIA: An introduction to spark spreads (2013)](https://www.eia.gov/todayinenergy/detail.php?id=9911) · eia.gov - [U.S. EIA: Today in Energy, Daily Prices (spark spread with a reference heat rate of 7,000 Btu/kWh)](https://www.eia.gov/todayinenergy/prices.php) · eia.gov ### Data - [Level 3 data: the truth of the spreads (planted parameters and controls)](https://pythonparatrading.com/curso/datos/nivel-3/spreads3_verdad.csv) · Download · spreads3_verdad.csv Previous lesson: [3.2.1 · Calendar spreads, butterflies and condors](https://pythonparatrading.com/en/python-trading-course/level-3/calendar-spreads-butterflies-condors/) Next lesson: [3.2.3 · Rate and index spreads: 2s10s, NOB, NQ/ES and Europe versus the U.S.](https://pythonparatrading.com/en/python-trading-course/level-3/interest-rate-and-index-spreads/)