How to Take an Option Structure Apart, Step by Step

Taking an option structure apart means writing every leg on one row as a signed primitive, gathering the rows that share a strike and an end date, and then applying the single substitution that is arithmetic rather than judgement: a bought call sitting beside a written put at one level is an agreement to buy at that level. Whatever survives the reduction is what the position actually obliges.

Underneath that answer is a fact about option positions that makes the whole routine possible. Every option leg is one of four things: a bought call, a written call, a bought put, or a written put, each carrying a strike and an end date. A holding of the reference asset itself is a fifth row of the same shape. A structure is nothing more than a sum of those rows. Taking one apart is therefore arithmetic performed on a list rather than an act of interpretation. The routine below is built to keep it that way, and every step exists to stop judgement leaking back in.

What does taking a structure apart actually produce?

Taking a structure apart produces two things and refuses a third. The first is a statement of what the position obliges at each price, written in the fewest rows that carry the same obligation. The second is a figure for what the assembly cost, or an honestly empty cell where that figure cannot be worked out from the record in front of the analyst. The refusal is a view on which price will turn up. No step in the routine ever asks that question.

Consider a restaurant bill for a moment. A thali is ordered and what comes back on the bill is a name. To anybody checking the bill, the name is no use at all: the bowls, the bread and the sweet are counted, each one is priced, and the total is added up. A structure's name works exactly like the name of the thali, and the first instruction in this routine is to put the name down and count the bowls. A reader who keeps the name in front of them spends the whole routine decomposing a label instead of an obligation.

The routine has eight steps and the order is not decorative. Two of the steps do nothing except prevent something being done too early, and those two are where readers who already know the arithmetic go wrong. A substitution applied to two rows that do not belong together produces a tidy answer that is simply false, so the grouping step sits before any combining is attempted.

Step one: what gets written down before anything else?

One row for each leg, and five columns on every row. Bought or written. Call or put. The level it is struck at. The date it runs to. The number of contracts. Those five columns are the whole of step one, and the rule that governs them fits in a line: a name is not a leg, so the name of the structure goes nowhere on the list.

The word to hold on to for the second column is that every option leg is one of exactly four things. Bought call, written call, bought put, written put. There is no fifth kind of option leg, and the absence of a fifth kind is why the routine can be a routine at all rather than a set of cases. A holding of the reference asset itself does turn up on these lists, and it gets a row of the same five columns with the strike column left blank. A holding has no level written into it and no date on which it stops.

The sign column is doing more work than it looks. Bought or written decides the direction of every figure on the row, so a row whose sign is missing cannot be added to anything. Mark it first, before the kind, before the level. A row that says call, Rs 2,000.00/-, one year and nothing else is not a leg anybody can work with; it is a description of a contract with the most important thing about the position left off.

Step two: why does the grouping happen before any combining?

Because two legs can be read as one thing only when they share both a strike and an end date, and the cheapest way to prevent rows being combined that cannot combine is to sort them before the temptation arises. Grouping runs by strike first, then by end date inside each strike group. A group of one row is still a group, and it travels through the rest of the routine untouched.

Here is the reason, and it is worth having the reason rather than the rule. The substitution in step three works only because two payoff lines bend at the same level on the same day, so the two bends sit on top of each other and cancel. With one leg moved to a different level, the two bends land in different places. The result is a shape with two bends in it, and no substitution in this routine removes a second bend. The rows stay as rows.

Sorting of this kind is already familiar without the name. A month of bills put in a pile does not get added straight away. The bills are sorted by shop, then by date. Two bills from the same shop on the same day might be one purchase split across two slips, and two bills from different shops never are. Legs behave the same way, and the sorting is what shows which pairs are even worth looking at.

Step three: which two rows add up to a single obligation?

Inside one group, a bought call standing next to a written put becomes a commitment to buy the reference asset at that level on that date. The substitution is the only one in the routine that is arithmetic rather than judgement. A reader who has seen it fall out of the two payoff formulas will never again wonder whether it holds at the edges, so it is worth writing out rather than quoting.

Take the two legs at the end date. The bought call pays the amount by which the price exceeds the strike, and nothing at all where the price sits below the strike. The written put costs the amount by which the strike exceeds the price, and nothing at all where the price sits above the strike. Now look at the two conditions side by side. At any price that can be named, at least one of those two amounts is nothing, so the pair never has two live terms at once, and what is left over is the price less the strike.

The mirror sits in the same block and takes one line: a written call together with a bought put at one strike is an agreement to sell at that strike, by the same working with the signs turned round. Both versions produce a synthetic row on the list, and a synthetic row is treated from then on exactly like any other row.

Step four: what happens to the list once a pair has gone?

Cross the substituted pair out, write the agreement in its place, and then read the whole list again from the top. The re-reading is not politeness. A list that had four rows and now has three very often contains a second pair that was invisible while the first pair sat between them, and the routine says to run steps two, three and four again until a whole pass produces no substitution at all.

The reduced list is the same obligation as the original written in fewer rows, and nothing has been thrown away in the reduction. The word reduction invites a reader to think something has been approximated or simplified. Nothing has. If the two lists ever disagreed at any price, the reduction would be wrong. Step five goes looking for exactly that disagreement.

The routine run on the two rows above goes as follows. Step one gives a bought call struck at Rs 2,000.00/- running one year and a written put struck at Rs 2,000.00/- running one year, one contract of each, on a reference asset whose price today is Rs 2,000.00/-. Nothing is paid out to whoever holds that reference asset between today and the end date. A payment during the year would move every figure below. Both rows share the level and the date, so step two puts them in one group. Step three substitutes. Step four leaves a single row: an agreement to buy at Rs 2,000.00/- on that date, and a second pass over a one-row list finds nothing further to do.

Two numbers here are the same and neither was copied from the other. The reference asset costs Rs 2,000.00/- today, and the pair is struck at Rs 2,000.00/-. The agreement between them is not a coincidence. A pair struck level with today's price of the reference asset is what people are describing when they say at the money, and this pair is struck that way so the arithmetic below stays readable. The Rs 2,000.00/- the legs reference is also exposure, not an amount either side has paid.

Step five: why check the reduction at three prices and not one?

Because one price cannot tell a straight line from a bent one. Two shapes that are nothing like each other can pass through the same point, so a reduction tested at one price has been tested nowhere at all. The three prices are chosen rather than convenient: one below every strike on the list, one sitting on a strike, and one above every strike. That set catches a shape that has grown a bend the reduction does not carry, and a shape that has lost one.

At each of the three, the total of what every original row pays is set beside the total of what every reduced row pays. The rule of the step is short: if the totals disagree at any one of the three, the reduction is wrong, and the mistake is in the list rather than in the arithmetic. The remedy is a return to step one, looking for a row signed the wrong way or a leg left off.

The working pair carries four prices rather than three. The fourth is where financing takes today's price of the reference asset by the end date, and it earns its place further down. At Rs 1,600.00/- the bought call pays Rs 0.00/- and the written put costs Rs 400.00/-, so the pair together is minus Rs 400.00/-. At Rs 2,000.00/- both legs are Rs 0.00/- and so is the pair. At Rs 2,130.00/- the pair is plus Rs 130.00/-. At Rs 2,400.00/- it is plus Rs 400.00/-. Every one of those four is the price less Rs 2,000.00/-. The single reduced row pays exactly the same at those four prices.

Step six: what did the assembly cost, and is that figure a payoff or a profit?

Three instructions live in this step and the order matters. First, add the premiums with their signs: premiums paid on bought legs, premiums received on written legs, and the total is a net premiumThe result of setting the amount paid against the amount received on day one, so that only the difference is treated as having moved. that moved on day one. Second, the payoff is read at the end while the premium left at the start, so carry that net premium forward to the end date at the financing rate and subtract it from the payoff line to get the profit line. Third, check the net premium against the identity that fixes one premium once the other is known.

On the working pair: Rs 180.00/- paid for the call, Rs 57.93/- received for the written put, so Rs 122.07/- left the assembler on day one. Financing takes that to Rs 130.00/- by the end date. So the payoff line is the price less Rs 2,000.00/- and the profit line is the price less Rs 2,130.00/-. The profit line reads minus Rs 530.00/- at the lowest of the four checked prices, minus Rs 130.00/- at the strike, nothing at all at Rs 2,130.00/-, and plus Rs 270.00/- at the top. The payoff line and the profit line are two different lines, and the distance between them is the financed net premium and nothing else.

Now the check, and this is where the step slows down. At one strike, the net premium on a call bought and a put written has to equal the reference asset price less the strike in day-one money. Work the second route: Rs 2,000.00/- divided by one plus 6.50 per cent is Rs 1,877.9343/-, and taking that away from Rs 2,000.00/- leaves Rs 122.0657/-. Compare it with what actually left the bank on day one, Rs 122.07/-. The two land 43 hundredths of a paisa apart, and that is a pass rather than a failure.

Write the toleranceThe size of difference a check agrees to ignore, decided and written down before the check is run rather than after it has disappointed somebody. into the check before the check is run, at the paisa. A difference inside that is agreement. A difference outside it is a wrong figure, and the test can now actually find one. RoundingCutting a figure back to a fixed number of places so that a person can write it and say it. The part that is cut off does not come back later. is why the tolerance has to exist at all: the put premium as printed is Rs 57.93/-, and the unrounded figure behind it is Rs 57.9343/-.

One more thing about that gap is worth carrying. Carrying Rs 122.07/- forward at 6.50 per cent gives Rs 130.0046/-. Rs 2,000.00/- financed for the year is Rs 2,130.00/- and the strike is Rs 2,000.00/-, so carrying the exact Rs 122.0657/- forward gives exactly Rs 130.00/-. Both figures print as Rs 130.00/- to the paisa, and the identical print is the whole point: rounding at the start does not blow up into something visible at the end, it stays where it was put.

Step seven: what can the routine not reach on the figures available?

Two things, and they are different from each other. The first is a limit of this working example. A structure spanning two strikes can still be reduced in shape by this routine. The payoff arithmetic needs no premium at all: the bends and the arms come out of the strikes and the signs. A two-strike structure cannot be costed unless a premium exists at each of those strikes, and only one strike carries premiums on the working list. So a two-strike structure gets its shape from the routine and an empty cost cell.

The second is a limit of the record itself, and no amount of care with the routine gets round it. A premium cannot be produced from scratch at any strike. Producing one needs a measure of how far the reference asset might wander before the end date arrives, and how much weight belongs on each landing place. No such measure sits on the working list, so neither premium was modelled, and the two in hand are simply given figures.

Write the name of the missing input into the empty cell, in words, rather than leaving the cell blank or filling it with something plausible. The reason is behavioural rather than mathematical. A plausible figure that nobody actually produced will get used by the next person to pick the card up, and it will get used with no idea that it was invented. A named absence gets looked up. The cell should read something like no premium at this strike, and none can be produced from what is here. Those words are longer than a number and do the reader far more good.

Step eight: what does a finished decomposition establish?

A finished decomposition establishes what the position obliges at each price, and what the part of it that could be costed actually cost. The two together are a great deal, and they are not everything. No step in the routine ever asked which price will turn up on the end date, so a finished decomposition establishes nothing about that. Whether the position is worth holding is established nowhere on the card either.

The question whether the position should be held arrives with three blanks in it. The first blank wants a figure for how widely the reference asset might range by the end date, together with the weight to place on each part of that range, and no such figure was ever set down here. The second wants the reader's own position and the reader's own purposes. Neither can be known from a decomposition. The third wants what the arrangement costs to carry and what it costs to get out of again. No decomposition answers that question, and the three blanks are the reason rather than an excuse. A routine that says what an obligation is has done its job; it has not become a reason to take one on.

A reader who has just watched two lines behave themselves is in the mood to conclude something, so one honest reading of the finished card is worth stating plainly. The card describes an obligation. The card says what each side owes at each price. The description is complete and it is silent about which price arrives, and the two facts are not in tension: together they are what a description of an obligation is.

What does the whole card look like, run end to end?

Here is the output of the routine on the two rows, gathered in one place. Two rows in, one row out, checked at four prices, costed at Rs 122.07/- of net premium against a computed Rs 122.0657/-, with one cost cell empty and named. Every figure in the payoff row is a payoff and every figure in the profit row is a profit, and the two rows are labelled that way so nobody has to remember which is which.

The mirror takes one line and is worth writing on the same card. A written call and a bought put at Rs 2,000.00/- reduce to an agreement to sell at Rs 2,000.00/-, by the same working with the signs turned round, and the net premium of Rs 122.07/- is received on day one rather than paid. Nothing else in the routine changes: the same grouping, the same check at three prices, the same tolerance on the same arithmetic.

Who actually reads each cell of the finished card?

Take the card cell by cell and ask who needs that cell. The answer is a different person for each one, and the difference explains why the routine bothers to produce all four.

The reduced row is read by whoever has to state, in one line, what a book is obliged to do. Four rows of contract names is not a statement anybody can act on; an agreement to buy at Rs 2,000.00/- on that date is. The reduced row turns a screen full of legs into a sentence, and a sentence is what gets carried into a meeting.

The check totals are read by whoever has to put their name to the reduction. The person putting their name to it is not checking the arithmetic; they are checking that the list reduced is the list that actually exists. Three prices, two totals each, and a disagreement anywhere sends the whole thing back. The check is the cheapest audit trailThe kept record of each step of a calculation, in the order it was done, so that somebody else can walk it again without having to ask a single question. available on a position, and cheapness is why it survives in places where far more elaborate checks have been quietly dropped.

The net premium cell is read by whoever reconciles what actually left or arrived in a bank account on day one. The net premium is the only cell in the whole card that corresponds to money that has moved, and it is the cell where nettingSetting amounts owed each way against each other so that only the difference travels between the two sides, rather than each amount making its own trip. matters: Rs 180.00/- went out and Rs 57.93/- came in, and what the statement shows depends on whether the two were settled separately or as one difference of Rs 122.07/-.

The empty cell is read by whoever has to go and get the missing figure. The empty cell is the one most likely to be quietly filled in by somebody who does not know it was left blank on purpose, and that risk is exactly why the words go in it rather than a dash. A household budget works the same way. Leave the electricity line blank on this month's sheet and somebody will pencil in last month's figure; write bill not arrived and somebody goes and finds it.

The reference asset priced at Rs 2,000.00/-, together with the call and the put written against it, are invented.
Educational material. Not advice on any investment, tax, budget or market position.