A pricing formula is read by its level and behaves through its parts. Each input is a different exposure, a slope or lag ties the price to another market and earlier months, a cap or floor makes the exposure change size, and two correct calculations of one formula can still disagree.
A pricing formula is usually read by its level: an index, a premium or discount, perhaps a currency. That reading is accurate and it is not enough, because the formula behaves through its parts. Each input is a different kind of exposure, and only some of them can be hedged. A weight or a lag ties the price to another market and to months that may already have passed. A cap, a floor or a change of slope makes the exposure grow or shrink as the market moves. And the arithmetic itself, the order in which averages, conversions and roundings are applied, is often not fully written down, so two correct calculations of the same formula can produce two different prices.
This article takes a formula apart into those parts: what it reads from, which of its inputs it leans on and when, where that lean changes, who does the arithmetic and with which conventions, and what a record of the formula has to hold before its exposure can be hedged and its invoice can be checked. It is written for the people who live with a formula after it has been agreed: the middle office who turn it into a position, the risk team who hedge it, the settlements team who compute it every period, and the traders who are asked why the number on the invoice is not the number they expected.
If you want the mechanism from the beginning, start with A formula is read by its level and behaves through its parts. If you already price on formulas and want the part that produces small, recurring disagreements, start with Two correct calculations of the same formula can produce different prices.
A formula is read by its level and behaves through its parts
The level of a formula tells you roughly what the price will be. Its parts tell you what the price will do: which markets move it, by how much, in which periods, and under which conventions it is finally computed.
Most formulas can be summarised in one line. An average of a published index over a period, plus or minus an agreed amount, converted into the invoicing currency. That line is how the trade is described on the desk, how it is booked, and how it is remembered. It answers the question the trade was negotiated around, which is whether the price is good.
It does not answer the questions that arrive afterwards. How much of this contract is exposed to the index, and to which months of it? Does that amount stay the same if the index doubles? Is any of the price already decided before delivery? Which party's arithmetic produces the invoice, and would the other party's arithmetic produce the same number? Every one of those answers is in the formula, but none of them is in the one-line summary of it.
The gap matters because the summary is what usually travels. It is what goes into the trade record, what the position report reads, and what a new member of the team learns. The parts that decide the behaviour tend to stay where they were first written: in the contract, in a schedule attached to it, or in the workbook of whoever computes the invoice.
Each input to a formula is a different kind of exposure, and only some of them can be hedged
A formula combines inputs that look alike on paper and are nothing alike in practice. A published market price, an exchange rate, a measured quality, a freight assessment and a fixed number each move for different reasons, are supplied by different parties, and are reachable by different hedges, or by none.
The market index is the part everyone sees, and it is usually the part a desk is set up to hedge. It is published by a third party, it moves with the market, and there is often an instrument that tracks it closely enough to offset it.
An exchange rate enters whenever the index is quoted in one currency and the contract is invoiced in another. It is a second market exposure inside the same formula, with its own source, its own fixing time and its own hedge, and it is frequently managed by a different team from the one that manages the commodity price.
A quality adjustment is a different kind of input altogether. Where a formula escalates or discounts the price according to a measured property of the delivered goods, part of the price is set by a laboratory result rather than by a market. No market instrument hedges it. It is known only after the goods have been sampled and analysed, and its value is argued from the analysis, not from a published page.
A freight or logistics component, where one is built in, follows a market of its own with its own assessments, and is often the component least visible to whoever is hedging the commodity.
The fixed premium or discount looks like the simplest input because it does not move. It is also the one most often treated as though it were the whole of the agreed price, which is how a desk can be confident it has a good price and still be wrong about what that price is exposed to.
Read this way, a single formula is a small portfolio. Adding its exposure into a position report as one quantity against one index records the largest part of it and silently drops the rest.
A weight, a slope or a lag moves the exposure to another market and into other months
Many formulas do not follow one index one-for-one. They follow a weighted mix of indices, a fraction of an index from a different commodity, or an average of months that ended before the delivery began. Each of these changes how much exposure the contract carries, to which market, and when that exposure stops moving.
A weighted formula, which takes part of its price from one index and part from another, is exposed to both, in the proportions of the weights. That is obvious once stated and easy to lose once booked, because a trade record that holds a single index field has nowhere to put the second one.
A slope applies a fraction of an index to produce the price. In some long-term gas and LNG supply contracts, for example, the price is set as a slope times an average of an oil index, plus a constant. The contract is for gas and its price exposure is largely to oil, in a quantity set by the slope rather than by the volume delivered. A hedge sized to the delivered volume, in the delivered commodity, is a hedge of the wrong quantity in the wrong market.
A lag places the averaging period before the delivery period. When the price for this month's deliveries is the average of an index over several earlier months, part of the price is already known on the first day of delivery, and part of the exposure for future deliveries is fixed day by day in months that have nothing to do with when the goods move. Two consequences follow. The exposure ends earlier than the delivery schedule suggests, so a hedge that runs to the delivery dates runs past the risk. And when the lag window overlaps several delivery periods, one day's index print feeds the price of more than one delivery, which a report that assigns each print to one delivery will count once.
None of this is unusual. It is how a buyer and a seller in one market agree a price by reference to another market they both trust more. What makes it costly is that the formula's lean, which market and which months, is exactly the information needed to hedge it, and exactly the information a one-line booking leaves out.
A cap, a floor or a change of slope makes the exposure change size as the market moves
A formula with a cap, a floor or a kink is not a straight line. Its sensitivity to the index is one amount in one range of prices and a different amount, sometimes zero, in another. The exposure therefore changes as the market moves, without anyone trading, and a hedge sized once is sized for only one part of the range.
A cap fixes a maximum price. Below the cap the price follows the index; above it, the price stops moving. The buyer under a capped formula holds protection against high prices, and the seller has given it. That is an option, and an option has value whether or not anyone priced it when the cap was agreed.
A floor is the same thing in the other direction, protecting the seller and costing the buyer. A formula with both, a collar, holds two options at once, one granted by each party.
Some formulas change slope rather than stopping altogether: the price follows the index steeply in a middle band and more gently outside it. The effect is the same in kind. The amount of exposure per unit of index depends on where the index is, so the quantity that has to be hedged is not a fixed property of the contract. It is a function of the market.
Two practical problems follow, and both are quiet. The first is that the exposure on a capped or kinked formula changes as prices approach the kink, so a position report that carries a constant quantity is correct in one range and wrong in another, and nothing on the report says which range the market is in. The second is that the option inside the formula is a position in its own right. Whoever holds the short side of a cap is carrying an exposure that grows precisely when the market moves against them, and it is rarely recorded anywhere except in the clause that created it.
A cap is not the only choice that changes hands inside a deal. A quantity tolerance, a port range and a quality band are options too, and the minute that agrees the price rarely prices them. Trade Execution: Why the Price Is the Part You Will Argue About Least
Two correct calculations of the same formula can produce different prices
A formula as written usually states its inputs and its operations, but not every choice needed to compute it. The order in which averaging, conversion and rounding are applied, and the factors used to convert units, can each change the result. When both parties compute the price, each with reasonable choices, the two results can differ without either being wrong.
Take a formula that averages a dollar index over a month and invoices in another currency. One way to compute it is to average the index first and convert the average at one exchange rate. Another is to convert each day's quotation at that day's rate and then average the converted figures. Unless the rate was constant through the month, those two calculations produce different numbers. Both are defensible readings of a formula that says "the monthly average, converted".
Rounding behaves the same way. Rounding each daily quotation before averaging gives a different result from averaging unrounded quotations and rounding only the final price. The difference per unit is tiny. Multiplied across a cargo, and repeated every period for the life of a contract, it becomes a recurring difference that someone has to explain, and it reappears each time because nothing about the cause has changed.
Unit conversion adds a third source. A price quoted per barrel and invoiced per tonne needs a conversion factor, and so does a price quoted in one energy unit and settled in another. Whether that factor is a fixed number written into the contract or is derived from the measured properties of the actual cargo is itself a term, and two parties who each assumed the other answer will compute two different prices from identical index data.
What makes these differences persistent is that recomputing does not resolve them. Each party checks its own arithmetic, finds it correct, and is right. The disagreement is not in the calculation. It is in the conventions the calculation rests on, and those are settled by agreeing the conventions, preferably in writing and preferably before the first invoice.
Here the inputs agree and the arithmetic does not. When the inputs themselves differ, recalculating reproduces the disagreement and only the supporting documents settle it. Commodity Trade Invoicing: Why the Invoice Is Where a Trade First Has to Agree With Itself
The booked formula is a summary, and the parts that change the number are held by whoever computes it
What a trade record typically holds is the index, the premium and the period. The weights, the slope, the lag, the cap, the conversion factor and the rounding convention are often held in the contract text and in the workbook used to compute the invoice. The exposure is reported from the summary; the invoice is computed from the workbook.
This is not carelessness. Formulas vary from contract to contract in ways that a fixed set of fields struggles to hold, and the person who computes the invoice needs the full detail long before anyone else does. So the detail accumulates where the computation happens.
The consequence is that two numbers derived from the same contract are produced from two different descriptions of it. The position report reads the summary and shows a quantity against one index. The invoice is computed from the full formula and arrives at a price that reflects the weights, the lag, the cap and the conventions. When they disagree, the report is the more likely of the two to be wrong, because it was built from less of the formula, and it is often the last place anyone looks.
There is a second consequence, slower and more expensive. When a formula is amended, a cap added at renewal, a weight changed, a conversion basis agreed after a dispute, the change reaches the workbook because the next invoice depends on it. It reaches the trade record only if someone remembers to put it there, and the position report will continue to describe the formula as it used to be.
One formula, five kinds of input
| Market index | Exchange rate | Quality adjustment | Weight, slope or lag | Cap, floor or kink | |
|---|---|---|---|---|---|
| What it does to the price | Sets the level and most of the movement | Converts the price into the invoicing currency | Raises or lowers the price by a measured property of the goods | Sets which market, how much of it, and which months | Changes how much the price moves as the index moves |
| Who supplies the number | The publisher of the index | The rate source named, or assumed, by the contract | The laboratory or surveyor | Nobody. It is a term of the contract | Nobody. It is a term of the contract |
| Can a market instrument offset it | Often, against the right index and months | Often, with a separate currency hedge | No | Only if the hedge is resized to the slope and dated to the lag | Not with a fixed quantity. It needs an option, or a hedge resized as the market moves |
| How it goes wrong | Hedged against the delivery months instead of the pricing months | Managed by another team and missing from the commodity exposure | Treated as a dispute rather than as part of the price | Exposure booked in the delivered commodity at the delivered volume | A constant quantity reported for an exposure that is not constant |
| Where it first exists | The index publication | The rate publication | The analysis certificate | The contract, then the invoicing workbook | The contract, then the invoicing workbook |
The two right-hand columns are the ones worth noticing. They supply no number of their own, so no feed ever delivers them, and they are the parts that decide how much of every other number the contract is exposed to.
A formula has to be recorded as eight things before its exposure and its invoice can be checked
| The record states | Which question it answers | What happens when it is missing |
|---|---|---|
| Every index in the formula, with its weight | Which markets the price follows, and how much of each | The second index is absent from the exposure report |
| The slope and any constant, as separate fields | How much exposure each unit delivered carries | The hedge is sized to the delivered volume |
| The averaging period for each index, as dates | When each part of the price is fixed | The hedge runs to the delivery dates and past the risk |
| Any cap, floor or change of slope, with its levels | Where the exposure changes size | A constant quantity is reported for a changing exposure |
| The currency, the rate source and when the rate is taken | Which currency exposure sits inside the price | The currency exposure belongs to nobody |
| Any quality adjustment, and what it is measured from | Which part of the price waits for a laboratory | A routine adjustment is handled as a dispute |
| The conversion factors, and whether they are fixed or measured | How a quoted unit becomes an invoiced unit | Both parties compute different prices from the same quotes |
| The order of operations and the rounding convention | How the price is actually computed | A small difference recurs on every invoice and is never closed |
None of these fields is difficult to state. The difficulty is that the first and fourth tend to be negotiated by the desk, the fifth and eighth tend to be settled by whoever computes the first invoice, and the sixth is known only to whoever reads the analysis. A formula is complete in the contract and incomplete everywhere else.
Questions people ask about this
What is a pricing formula in a commodity contract?
It is a rule written into the contract for turning future published prices into the price actually paid. A typical formula names one or more indices, an averaging period for each, any premium, discount or slope applied to them, and the currency and units of the result. The price does not exist when the contract is signed; it is produced by applying the rule once the inputs have been published. Formulas let parties agree terms before the market level is known, which is why they are common in physical trading.
What is the difference between a fixed price and a formula price?
A fixed price is a number agreed at the time of the trade. A formula price is a method agreed at the time of the trade that produces a number later. A fixed price is exposed to the market from the moment it is agreed, because the market can move away from it while the goods have not yet changed hands. A formula price carries exposure to the inputs of the formula until each of them is fixed, in the proportions and periods the formula sets. Many contracts mix the two, fixing some components and leaving others to a formula.
How do you hedge a contract priced on a formula?
You hedge each input the formula actually reads, in the quantity and periods the formula actually applies. That means splitting a weighted formula into one exposure per index, sizing an index exposure by the slope rather than by the delivered volume, dating it to the averaging period rather than the delivery period, and treating any currency conversion as a separate exposure. A cap or floor changes the hedge quantity as the market moves, so a formula with one cannot be fully hedged with a fixed quantity. A quality adjustment cannot be hedged in the market at all.
What does a slope and a lag mean in an oil-indexed gas price?
In contracts priced this way, the gas price is set as a fraction of an oil index, the slope, usually plus a constant, and the oil index is averaged over a period that ends before the gas is delivered, the lag. The slope decides how much oil price exposure each unit of gas carries. The lag decides when that exposure is fixed. Together they mean that a gas contract can be largely an oil price position, fixed in months before the gas moves. The specific slope, constant, averaging period and lag differ from contract to contract and are set by the agreement itself.
Why does our invoice price differ from the counterparty's by a small amount every month?
When the difference is small and recurring, and both sides are using the same published quotations, the cause is usually a convention rather than an input. The common ones are the order of operations, averaging before or after currency conversion; the rounding step, rounding each quotation or only the final price; and the conversion factor between units, fixed in the contract or derived from the cargo. Each side's calculation can be correct on its own convention. The fix is to agree the convention explicitly and record it with the formula, so that the next invoice is computed the same way by both parties.
What does a cap or a floor in a pricing formula actually cost?
A cap or a floor is an option. The party it protects holds the option, and the other party has sold it, whether or not anyone discussed its value when it was agreed. Its cost to the party who granted it depends on how likely the market is to reach the level and how far it may go beyond it, so it varies with the market rather than staying fixed. The practical step is less about valuing it precisely and more about recording it, so that the party short the option knows it is, against whom, and at which level.
Should a pricing formula be stored as text or as structured fields?
Both, for different purposes. The contract text remains the authority on what was agreed. Structured fields are what allow a formula to be read by a position report, a hedge calculation and an invoice in the same way. The usual failure is storing a summary in fields and the detail in text, so that reports read the summary and invoices read the detail. If every part listed in the table above has a field, the text and the fields can be compared, and an amendment to one can be checked against the other.
Where should we start if our formula-priced contracts keep producing invoice differences?
Separate the differences that recur from those that do not. A difference that appears every period on the same contract, by a similar small amount, points to a convention: compute one past period both ways, with the counterparty's convention and with yours, and see whether the difference disappears. A difference that appears once and is larger points to an input, a quotation, a rate, an analysis or a date, and is settled from the documents behind that input. Treating the first kind as though it were the second produces the same dispute every month.
Where this lands in a trading system
Nothing above is an argument for buying software. It is an argument about which parts a formula is made of and where each part is kept. Some of those parts map onto system capabilities, and it is worth being specific about which ones, and about where the mapping stops.
The first conclusion is that a formula is several exposures at once, an index, a currency, sometimes a second index, and that each has to be read against the hedges and the invoice of the same contract. That is the job of a CTRM/ETRM system providing complete trading lifecycle management for commodity traders, integrating physical trade, financial hedging, risk control, and settlement in one platform; Time Dynamics' Fusion is one, and the parts of it that bear on this article are complete physical trade lifecycle management and execution, advanced trading management for futures and derivatives markets, comprehensive risk assessment and monitoring capabilities and automated financial settlement and payment processing. Holding them in one place is the condition under which the exposure a formula creates and the invoice it produces are read from the same description of it.
The second conclusion is that the parts which change the number, the weights, the lag, the cap, the conversion factor and the rounding convention, are often held in contract text and in the workbook used to compute the invoice. That is what X-Ray addresses: a non-invasive data processing and analysis platform, whose collection toolkit XDK performs non-invasive automated data collection from databases, Excel files, and web interfaces, and which is built to collect data without disrupting existing systems or workflows. The last property is the relevant one, because the invoicing workbook is where the formula is understood in full, and replacing it is not the first step.
The third conclusion is that a capped or kinked formula has an exposure that changes size as the market approaches a level. Watching a value against a threshold is the shape of real-time risk alerts and monitoring, which is what X-Eagle does; its published description points that capability at trade finance field warnings, and whether it is pointed instead at an index nearing a cap is a configuration question rather than something a platform decides for you. A level that changes the size of a position is an ordinary shape, not an exotic one.
Re-cutting the same contracts by index, by averaging month and by currency is the fourth, and one-click generation of personalized reports with Excel-like interface, X-Sheet, is that kind of work.
One conclusion of this article is deliberately not mapped. Valuing the option inside a cap or a floor, and deciding how much of it to hedge, does not correspond to anything on the published capability list, and the honest thing to do is leave it unmapped rather than stretch a nearby capability across it.
And here is where the rest stops. None of this writes a formula or negotiates one. It does not decide which index, weight, slope or lag is right for a contract, it does not decide whether a cap is worth what it costs, and it does not choose between two reasonable arithmetic conventions or settle a dispute about which one the contract meant. Those are commercial and contractual decisions with money on both sides of them. What a system can settle is narrower and duller: whether every part of a formula is recorded where reports and invoices can both read it, whether an amendment reaches both, and whether a difference between the position you report and the price you invoice can be traced to a part of the formula rather than argued about. That is a large share of the problem. It is not all of it.
A note on the evidence in this article
There are no statistics in this article, and that is a decision rather than an oversight. The observations behind it, about where the parts of a formula tend to be kept and how often a booked summary drifts from the formula that is actually computed, come from material we are not in a position to publish as a citable figure. A number nobody could check would make the argument look stronger and make it worth less.
So the argument is made from the mechanism, and each part of it can be tested against your own contracts. Take three formula-priced contracts that invoiced recently. For each one, write down every index and its weight, the slope and constant, the averaging period, any cap or floor, the currency and rate source, any quality adjustment, the conversion factors and the rounding convention, and note where each item is recorded. Then compute one past period's price twice, averaging before conversion and after it, and compare both with the invoice. The items you could not find in a system, and the size of the difference between the two calculations, are the finding. If your contracts behave differently from what is described here, your contracts are the better evidence.
Practice also varies by market and by contract. Which indices are used, whether slopes, lags, caps or quality adjustments appear at all, and which arithmetic conventions apply are set by the agreements you actually signed and the conventions of the markets you trade in. Where this article describes a mechanism, it describes the shape that mechanism takes, not the terms of any particular market or contract.
The formula is agreed once. It is computed every period, by both sides.