The Payoff Function: What the Contract Pays and When

A payoff function says what a contract delivers at the horizon as a function of the level then, and nothing else. The payoff is a function of one thing. A price is a function of five. The payoff is the terminal condition the pricing machinery is solved subject to, so it fixes where the answer must end up and leaves everything before it open.

Why any particular contract delivers what it delivers is settled in the instruments subject area. The delivery rule arrives here as a mathematical object already handed over. What kind of object it is, what it fixes and what it refuses to say are the three questions worth asking of it.

The refusal is the interesting part. A payoff functionWhat a contract delivers at the horizon, written as a function of the level at that moment and of nothing else. is a complete specification of one moment and a total silence about every moment before it. The rule pins the answer at exactly one instant and leaves the whole interior of the problem open, and everything the pricing machinery does is work out what that one pinned instant implies earlier. Hold on to the asymmetry: total information at the end, no information at all before it.

What is a payoff function, as an object?

Strip away the word contract for a moment and look at what is actually being handed over. A payoff is a rule that accepts one number and returns one number. Feed it the level of the standard process at the horizon and it returns an amount in rupees. Feed it a different level and it returns a different amount. The input and the output are the whole object. There is no second input, no memory of the route taken to get there, and no reference to when the question is being asked.

The everyday version is worth carrying for everything that follows. A left luggage counter charges by the weight shown on its scale at the moment the bag is collected. The charge rule is a card on the wall: a reading of so many kilograms costs so many rupees. The card is a function of one thing, the reading. The card says nothing about how heavy the bag was when it was handed in, nothing about what happened to it in the meantime, and nothing about what anybody should pay today to secure the right to collect it later. The card is a complete rule for one instant and a complete silence about every other instant, and a payoff function is exactly that card.

The formal version of the card is one line. The line is written for the locked contract on the standard process, the at-the-money contract at a strike of Rs 100/-. The contract arrives already known from elsewhere, and the mathematics here is about the shape of its payoff rule.

Two things in that line deserve a second look. The strike sits inside the rule as a fixed number, not beside the level as a second input. Changing the strike does not give the same function a different argumentA quantity a function accepts as input, as distinct from a constant written into the function itself.; it gives a different function. And the level is the only argument. Once the horizon arrives and the level is read, there is nothing left to decide, nothing left to estimate, and no averaging left to do. The answer is a lookup.

A payoff can therefore be published as a short table without losing anything. Four rows are enough to show every amount the rule returns, and the machinery downstream is told nothing beyond what a table of this kind carries.

Why is it the terminal condition rather than the price?

A price and a payoff answer two different questions, and the words sit close enough together that the difference gets lost. The payoff answers what will be delivered, once. The price answers how much the delivery is worth now, at a moment when the level at the horizon is not yet known. The two are not two readings of one quantity. One is a fact about one instant, and the other is a valuation of the uncertainty that stands in front of it.

The pricing machinery treats the payoff as a terminal conditionThe value a solution is required to take at the end of the time interval it is solved over.. The term terminal condition has a precise job. The Black, Scholes and Merton equation, set out under the Black-Scholes partial differential equation, holds strictly inside the interval, for every time before the horizon, and by itself it has an enormous number of solutions. The equation constrains how a price may change; it does not say which price. The payoff removes that freedom in one stroke by declaring what the answer must equal at the far end.

Notice the shape of that statement. The terminal condition is a boundary specificationA statement of what an answer has to be somewhere, rather than a statement of what the answer is., not a valuation. The condition says where the answer must be, at one edge of the region being solved, and then stands back. A condition of that kind carries no arithmetic and no opinion; it carries a requirement, and the requirement is what turns an equation with many solutions into an equation with one.

The measurement version helps here. Consider the task of reconstructing the temperature in a room over the whole of yesterday. The law that governs how heat spreads is given, and it fixes how the temperature may change from one moment to the next. One complete reading is given too: the temperature at every point in the room at midnight. The law alone would allow countless histories. The law together with that one complete reading pins the history down. The reading is not the history and nobody would confuse the two. The payoff is that midnight reading, and the price is the history.

How does the shape at the horizon differ from the shape before it?

The payoff drawn against the level is two straight segments meeting at a corner. Flat at nil for every level up to Rs 100/-, then rising at exactly one rupee per rupee above it. The price a year earlier, drawn against the same levels, is a single curve that never touches nil, never has a corner, and sits above the payoff at every level without exception. The two pictures are of the same contract and they look nothing alike.

The numbers make the separation concrete. At a level of Rs 80/- the payoff is nil and the price a year earlier is Rs 1.859420/-. At Rs 100/- the payoff is nil and the price is Rs 10.450584/-. At Rs 110/- the payoff is Rs 10/- and the price is Rs 17.662954/-. At Rs 120/- the payoff is Rs 20/- and the price is Rs 26.169044/-. Every one of those four prices is strictly larger than the payoff at the same level, and the gap is widest exactly at the strike, where the payoff has nothing to say at all.

So why does the corner disappear? Because the price at any earlier time is an average of the payoff taken over a spread of possible finishing levels, and averaging a cornered function over a spread rounds the corner off. Standing at Rs 100/- with a year to run, the process may finish anywhere; the levels that pay nothing and the levels that pay a great deal are both inside the spread, and the average of the two sides of the corner lands above the corner itself.

The measurement version, again, and it is the one to remember. A footpath runs dead level and then, at one exact spot, starts to climb. Walked, it gives up the precise stride where the slope changes. Photographed from far away through frosted glass, the corner is gone; there is a gentle bend instead. Nothing about the path changed. The blur did it. Time remaining is that blur, and it acts on the payoff exactly the way frosted glass acts on a corner: the more of it there is, the rounder the shape and the higher the shape sits above the corner it came from.

A shape that is angularHaving a corner, so that the slope jumps from one value to another with no value in between. at the horizon is smoothHaving no corner, so that the slope moves continuously through every value between its extremes. everywhere before it, and the difference shows in the slopes rather than in the eye. The payoff has a slope of nil below the strike and a slope of one above it. At the strike itself the two sides disagree, so it has no slope at all. The price has a slope that moves continuously through every value between nil and one, and at the strike a year out that slope is 0.636831.

What does the payoff fix, and what does it leave entirely open?

Take the two halves in order. The second half is where readers lose their footing. The payoff fixes the whole of the horizon. Every level, every amount, no exceptions and no approximations. Hand somebody the rule and they can answer the question at the horizon for a level nobody has thought of yet, without asking anything further. In that one instant the payoff is not merely informative; it is complete.

The payoff leaves open every other instant. The rule is silent on the worth of the contract today. The rule is silent on how the worth moves as the level moves. The rule is silent on how fast the worth decays as the horizon approaches, and it carries no opinion at all on how much the level tends to move about. Movement is the one input the answer turns out to be most sensitive to. The payoff is complete about one moment and empty about all the rest, and the pricing machinery exists precisely to fill the emptiness it leaves.

The gap between the two shapes is the visible measure of what has been left open. The gap has two named parts worth having: the intrinsicWhat the contract would deliver if the horizon were now, read straight off the payoff rule. amount is what the payoff rule returns at the current level, and the time valueThe gap between the price and the payoff at the same level, which is what the possibility of movement is worth. is everything the price carries beyond it. On the locked parameters the second of those is not a rounding detail. At the strike it is the whole price.

Why does a payoff take one argument while a price takes five?

The sharpest way to hold the difference is a count rather than a feeling. The payoff reads one number. The price reads five. Everything else about the comparison follows from that count, so nothing else needs remembering.

The payoff reads the level at the horizon. The list ends there. The price reads the level now, the strike written into the rule, the rate, the volatility, and the time remaining. Change any one of the five and the price changes. The payoff never consulted the four that are not the level, so changing any of those does not move it a paisa.

The volatility entry is the one to dwell on. The volatility does not appear in the payoff at all, and yet it is the input that carries most of the value at the strike. Set it aside and the price at Rs 100/- with a year to run collapses toward the small amount the rate alone would justify; raise it and the price rises with it. The single most important input to the answer is invisible in the object that specifies the contract. Nothing demonstrates more cleanly that the payoff is not a compressed price.

Where does the payoff enter the machinery?

At one point. Not at several, not continuously, and not as an assumption threaded through the working. The payoff is supplied once, as the statement of what the answer must equal at the horizon, and every other number the machinery produces is worked backward from that one statement.

There are two standard routes to a price, and the payoff enters both at the same single place. On the equation route it is the terminal condition attached to a partial differential equation that holds strictly inside the interval. On the expectation route it sits inside a single expectation taken under the risk-neutral measure Q and then discounted. Different notation, same entry point.

The practical consequence deserves its own line. The payoff is the only place a contract enters the calculation, so changing the contract means changing one function and nothing else at all. The measure stays. The discounting stays. The process stays. The whole apparatus is indifferent to which rule is standing in that slot, and that indifference is why the same machinery prices contracts it was never separately designed for.

What does the collapse look like when it is run?

Everything above is a claim about a motion, and a pair of still pictures cannot carry a motion. The control below moves one thing only, the time remaining, from a full year down to nothing at all. The price curve is recomputed at every position from the same locked parameters and redrawn, and the shaded region between it and the payoff is the value the payoff never mentions.

Two things to watch as the control comes down. The shaded region shrinks from every side at once rather than sliding along, and the price slope at the strike climbs toward the jump the payoff has there without ever reaching it until the time remaining is gone. The corner is not approached gradually by the shape; it arrives only at the instant the time remaining hits nothing, and one moment before that the shape is still smooth.

What goes wrong when the payoff shape is read as the price shape?

The payoff picture is the one everybody has seen, so the error has a specific form and is easy to make. Somebody wants the worth of the contract six months before the horizon. The angular picture sits in front of them. They locate the current level on the horizontal scale, read the height of the angular line at that point, and write it down. Every step of that procedure is executed correctly. The procedure is answering a different question.

Who has to hold this distinction, and where does it bite?

Anybody who builds a pricing routine meets the payoff as code rather than as a picture, and that is where the one argument rule shows its practical worth. A lattice routine, a solver on a grid and a stratified expectation are three different kinds of machinery, but each of them calls the payoff at exactly one place: at the last layer of the lattice, at the last row of the grid, and inside the average. Everything else in each routine is about the process and the discounting, not about the contract.

So the working test for whether a payoff has been coded correctly is a question about arguments rather than about answers. If the function being called needs the time, it is not a payoff. If it needs the volatility, it is not a payoff. If it needs to know the route the level took, then it is a rule for a different kind of contract and it will not sit in a slot built for one argument. A payoff that asks for anything except the level at the horizon has been written wrong, and the wrongness is visible in the arguments before a single number is computed.

The distinction bites in a second place, in how results are read back. A routine that reports a value close to the payoff at the same level is reporting that very little time remains, or that the volatility supplied was very small, and the two look identical in the output. For this contract the price sits above the payoff everywhere until the horizon, so a routine that reports a value below the payoff at some level has a defect. Both checks follow from the shape rather than from the code, so both are available to somebody who has never seen the routine.

The everyday version of the same discipline runs as follows. A weighing scale that gives a different reading depending on how long someone stands on it is broken, and that is evident without opening it. A payoff rule that gives a different amount depending on when it is asked is broken in exactly the same way, and for exactly the same reason: the object was specified to read one thing, and it is reading two.