The Fundamental Theorems of Asset Pricing, Stated Plainly

The first theorem links no arbitrage to the existence of a pricing measure: one holds exactly when the other does. The second links completeness to that measure being the only one. Together they say that the assumption buys a price, and buys exactly one price only when every payoff can be built. Neither says how to find anything.

Two results carry the whole of this subject, and both of them are usually met in the wrong order. The notation comes first, a line of symbols with a double arrow in the middle, and the words arrive afterwards as a translation of something already skipped. Meeting the symbols before the sentences is why the two results get misquoted more than anything else in derivative pricing. Both statements read perfectly well in ordinary words, with no symbols at all, and both are set out that way below, before a single letter of notation appears. The notation comes second, as a compression of sentences already agreed to, and by then it has nothing left to hide.

Every worked number runs on one invented quantity, called the standard process and written S with a time subscript. The standard process starts at Rs 100/-. Its volatility is 20 per cent a year. The risk-free rate is 5 per cent a year, continuously compounded, so cash grows by a factor of 1.051271 over the one year horizon and the discount factor back is 0.951229. A single change to the worked instance, adding one outcome and nothing else, moves it from the world where both theorems bite to the world where only the first one does, and both worlds can be computed to six decimal places.

What do the two results say, before any notation at all?

Here are both results, in words. Both repay a second reading. Everything that follows is these two sentences worked out.

The two sentences just given are the first fundamental theoremThe result linking the absence of arbitrage to the existence of a set of pricing weights, in both directions. and the second fundamental theoremThe result linking the ability to build every payoff to those pricing weights being the only ones, in both directions. of asset pricing. The set of weights has a name, the pricing measure, and its construction is covered separately rather than introduced here. The ability to build every payoff out of what is already traded has a name too, completenessThe property of a market in which every payoff can be reproduced exactly by trading what is already priced.. Completeness itself is defined separately.

Consider what neither sentence contains. Neither says how to find the weights. Neither says what the weights are. Neither says whether a particular market has the building property or not. Each one says that two things go together, and stops. The entire content of both results is that two conditions are the same condition wearing different clothes. Saying so is worth a great deal, and it is not the same as being handed a number.

The abstraction is doing a lot of lifting. Here is an everyday version. Suppose a set of scales is handed over, without any sight of the weights inside, together with an assurance that whatever weights are inside the machine, they are consistent: the same object placed on it twice reads the same, and two objects together read the sum of their separate readings. The assurance is a statement of exactly the kind these theorems make. The assurance rules out an enormous amount of nonsense, and it does not say what any object weighs.

What does the first result link, and which way does the link run?

The first theorem links two conditions. On one side, no arbitrage: there is no trade that can be put on today, costing nothing, that can never lose and might win. On the other side, the existence of a pricing measure: a set of weights on the future outcomes, positive on every outcome the real world thinks is possible, under which the discounted value of the traded process is on average today's value.

The result says those two conditions hold together and fail together. Not that the first causes the second, not that the second is evidence for the first. No arbitrage and the existence of a pricing measure are the same condition. The names attached to it are Harrison and Kreps, 1979, and Harrison and Pliska, 1981, who established it in the finite and the discrete-time settings, and Delbaen and Schachermayer, 1994, whose version handles the continuous-time case, where the plain phrase no arbitrage has to be sharpened before the statement is true at all. The sharpening is real work and is covered separately, and the shape of the result survives it exactly.

The equivalent word in that term table is the one people drop, and dropping it breaks the result. A set of weights that put zero on an outcome the real world thinks can happen would price away a genuine possibility, and the theorem would be false. The pricing measure is allowed to disagree with the real world about how likely anything is, and is not allowed to disagree about what is possible.

What does the second result link, and which way does that link run?

The second theorem also links two conditions, and it takes the first one as already settled. Assume there is no arbitrage, so at least one pricing measure exists. Then: the market is complete exactly when that pricing measure is the only one.

UniquenessThere being exactly one object with the stated property, rather than at least one. is the whole of the second result, and it is a much stronger claim than existence. Existence says the set of pricing measures is not empty. Uniqueness says it has exactly one member. Between those two sits everything interesting: the set could be empty, could have one member, or could have infinitely many, and which of the three holds decides whether a payoff has a price, one price, or a range of them.

The everyday version is counting. Suppose a jar is known to hold fourteen coins worth Rs 100/- in total, and the coins come in exactly two denominations. Two facts, two unknowns, and the composition is pinned: there is one answer and it can be found. Now allow three denominations. The same two facts are still true, they still rule out most compositions, and they no longer pin one. The question has moved from one with an answer to one with a range of answers, and nothing about the two facts got weaker. The second theorem is that arithmetic and nothing more: as many independent instruments as outcomes pins the weights, and one instrument short of that leaves a range.

Why is each result an equivalence rather than a one way street?

Both results are stated with a double arrow, and the double arrow is not decoration. An equivalenceTwo conditions that stand or fall together, so establishing either one establishes the other. is two implications bundled: the left gives the right, and the right gives the left. Running both ways is what makes these results usable from either end, and it is exactly the property that gets lost when somebody quotes them as a slogan.

The first result read rightwards starts from the assumption that there is no arbitrage and concludes that a pricing measure exists. The rightward reading licenses the whole apparatus of pricing by expectation: it says the object the computation needs is there to be computed with.

Read leftwards, the same result starts from a pricing measure exhibited by construction, on a lattice or in a model, and concludes that the market has no arbitrage in it. Exhibiting one set of weights is a finite job, and checking every possible trade for arbitrage is not, so the leftward direction is the one that does the work in practice. The leftward direction is how anybody ever verifies a model is arbitrage free: build one measure, and the theorem does the rest.

The same holds for the second result. Rightwards: assume every payoff can be built, conclude the measure is the only one. Leftwards: show that two different measures both work, conclude that some payoff cannot be built. The second direction is how incompleteness is usually demonstrated, and it is precisely what the worked instance below does, by exhibiting a whole interval of measures.

What happens in the balanced case, two outcomes against two instruments?

Time to make all of this arithmetic. One step, one year, the standard process starting at Rs 100/-. Over the step it moves to one of two values: up to Rs 122.140276/- or down to Rs 81.873075/-. Cash held at the risk-free rate grows from 1 to 1.051271 over the same step. The process and cash are the only two instruments in existence here.

The pricing measure here is a single number: the weight on the up outcome, with the rest going to the down outcome. The condition it has to satisfy is the one in the first result, that the discounted average of the two outcomes is today's value. The condition is one equation in one unknown, and one equation in one unknown has one answer.

Now hand over a payoff, as numbers, with no explanation of what produced it. The payoff pays Rs 22.140276/- if the process finishes at Rs 122.140276/-, and nothing if it finishes at Rs 81.873075/-. Those two numbers are the entire specification. The argument never needs to know what contract produced them: a payoff is a list of numbers attached to outcomes, and the machinery prices lists.

Both results are satisfied here, and it is worth saying which does which. The first is satisfied because a measure exists: the number 0.577493 sits strictly between nothing and one, so it is a legitimate set of weights and no arbitrage is available. The second is satisfied because it is the only such number: the equation had one solution, so every payoff on this lattice can be built and every payoff has one price. The comfort of the balanced case comes entirely from the count of outcomes matching the count of instruments.

What changes when a third outcome appears and nothing else does?

Change one thing. The process can now finish at Rs 122.140276/-, at Rs 100/-, or at Rs 81.873075/-. The starting value has not moved. The rate has not moved. The up and down values have not moved. No new instrument has been introduced: there is still the process and there is still cash, and that is all. One outcome has been added to the list of things that can happen.

So the answer to the prediction is that no arbitrage survives untouched. A pricing measure still exists: put 0.40 on the up outcome, 0.394284 on the middle and 0.205716 on the down, and check it. All three weights are positive, and their discounted average is exactly Rs 100/-. A measure exists, so by the first result there is no arbitrage. The first theorem is completely undisturbed by the third outcome, and only the second one breaks.

Why does one extra outcome leave a range instead of a price?

Count the equations. There are now three unknown weights. There are two conditions on them: they sum to one, and the discounted average of the three outcomes returns today's value. Three unknowns, two equations, and one degree of freedom left over. Every value of that free parameter which keeps all three weights positive is a legitimate pricing measure, and there is a continuous interval of them.

The two equations do impose real structure, and it is worth seeing exactly what. Solving them shows that the weights move together in a fixed proportion: every unit of weight added to the up outcome forces exactly 1.221403 units onto the down outcome, and the pair of them is taken out of the middle weight. The middle weight therefore falls by 2.221403 units. The ratio of 1.221403 is not a rounding artefact. The up outcome overshoots today's value and the down outcome undershoots it in exactly that proportion, so the ratio is the up factor itself.

Which means the interval ends where one of the three weights runs out. Push the up weight down and the down weight vanishes first, at an up weight of 0.231574. Push the up weight up and the middle weight vanishes first, at 0.577493. Outside those two points a weight would have to be negative, and a negative weight is not a set of odds at all. The permitted weights form a closed interval from 0.231574 to 0.577493, and no argument from no arbitrage picks a point inside it.

And because the price of the payoff is a straight multiple of the up weight, the interval of measures becomes an interval of prices. The payoff pays only at the up outcome, so its price is the discount factor times the up weight times Rs 22.140276/-, or Rs 21.060482/- for every unit of up weight. Run the up weight across its permitted interval and the price runs from Rs 4.877058/- to Rs 12.162285/-, a price rangeThe interval of prices that no arbitrage permits when more than one pricing measure exists. Rs 7.285227/- wide on a payoff that a moment ago had one answer.

Where do the two ends of the range come from?

Both ends are exact, and neither is a coincidence. A range whose ends look like arbitrary numbers reads as a mess. A range whose ends can be derived reads as structure.

Take the top end first. At an up weight of 0.577493 the middle outcome carries a weight of exactly nothing. A weight of nothing on an outcome means the outcome plays no part in any average taken under that measure, so the three outcome model is doing arithmetic identical to the two outcome model, with the same two surviving weights of 0.577493 and 0.422507. A weight of nothing on an outcome is a collapseA model reducing exactly to a simpler one when a weight or a parameter goes to zero.: the larger model becomes the smaller one, not approximately but identically. The collapse is why the top of the range is the two outcome price to the last decimal.

Now the bottom end, the prettier of the two. At an up weight of 0.231574 the down outcome carries nothing, and the whole weight is shared between the up outcome and the middle one. Work through what the price becomes and the up factor cancels out completely, leaving a quantity with no lattice in it at all.

The bottom of the range is Rs 100/- times one less 0.951229, and that identity holds whatever the up factor is. The bottom of the range is therefore a structural fact about the lattice rather than an artefact of these particular numbers. Changing the volatility, and with it the up and down values, moves the top of the range while the bottom stays exactly where it is. The two ends are answering different questions: the top is about the shape of the lattice, and the bottom is only about the rate and the horizon.

What do the two results not promise?

More than most readers expect, and this is where the misuse lives. Taken one at a time, each denial is a direct reading of the two statements themselves, not an extra caveat bolted on.

1. Neither states what the measure isThe first result says a pricing measure exists. It does not produce one, name one, or narrow the search for one. On a lattice with three nodes it can be solved for by hand. On a model with a continuum of outcomes and several driving factors, finding it is the entire job of work, and the theorem has contributed a licence to look rather than a place to look.
   Existence, not construction.
2. Neither states whether a given market is completeThe second result says the measure is unique exactly when every payoff can be built. It does not say which side of that equivalence a given market sits on. Deciding whether the instruments span the outcomes is a modelling question for the analyst, and the theorem states only what follows once that question has been answered.
   A link between two conditions, not a test.
3. Neither picks a price inside the rangeWhen the measure is not unique, no arbitrage rules out everything outside the interval and nothing inside it. Picking a point requires an argument of a different kind entirely, about preferences, about hedging error, about which risks somebody is willing to carry. Both results are silent on all of it.
   The interval is the whole of their output.
4. Neither says anything about whether a model is any goodThe theorems operate inside whatever model was written down. A model that is arbitrage free and complete can still be a poor description of anything, and the results will price payoffs in it with perfect internal consistency regardless.
   Internal consistency, not external truth.

The everyday version, again from counting. Being told that the drawer contains a matching pair of socks is a statement about the drawer, and it is genuinely useful: it establishes that the search is not hopeless. The statement is not the pair. Someone who reads it as the pair will plan their morning badly. Both results are existence resultsStatements that something exists, carrying no method whatever for finding it., and an existence result is a licence to search rather than the end of the search.

What goes wrong when a theorem is quoted as though it produced something?

How does someone building a valuation actually use these two results?

As a scoping test, run before any arithmetic and again whenever the model changes. The test takes three questions and about ten minutes, and it decides what the deliverable can honestly be.

1. Can one set of pricing weights be written down?Not the right one, just one that is positive on every possible outcome and returns today's value on average after discounting. If it can, the first result establishes that the model is arbitrage free, verified by exhibiting an object rather than by checking every trade. If it cannot, the model has an arbitrage in it and every price it produces is meaningless.
   On the three outcome lattice: 0.40, 0.394284, 0.205716, all positive, discounted average Rs 100/-.
2. Count the outcomes against the instrumentsAs many independent instruments as outcomes and the weights are pinned. One short and they are not. On the two outcome lattice the count is two against two and the answer is Rs 12.162285/-. On the three outcome lattice it is three against two, and no amount of care changes that.
   The count decides the shape of the answer before any arithmetic is done.
3. Decide what is being delivered before computing itIf the counts balance, the deliverable is a number. If they do not, the deliverable is an interval plus a written argument for the point chosen inside it, and that argument is a separate task with its own effort attached to it.
   Rs 4.877058/- to Rs 12.162285/- is a complete and honest answer. A single number pulled from inside it, without the argument, is not.

Why are these two results the spine of everything that follows?

Because everything later in this subject is one of these two statements applied to a harder model. The claim is not a figure of speech, and the mapping follows.

Pricing by discounted expectation under a different set of weights is the first result, used leftwards: the measure was constructed, so the model is arbitrage free, so the expectation is a price. The change of measure machinery exists to construct that object, and the theorem is what makes constructing it worth the effort. Every lattice, every partial differential equation, every numerical scheme in this subject is a method for computing an average under weights whose existence the first result licensed.

Every hedging argument is the second result, used rightwards or leftwards depending on which end is in hand. Showing that a payoff can be replicated shows that the price is unique. Showing that two measures both price the model shows that some payoff cannot be replicated. The whole distinction between a market where hedging removes risk and a market where it only reduces it sits in that one equivalence.

And every model that adds a source of randomness without adding an instrument, among them the models for a moving volatility and the models that let a path jump, lands in the same place the three outcome lattice landed: existence intact, uniqueness gone, an interval instead of a point. The three outcome lattice above is the smallest possible working example of every incomplete model that comes later, and it can be computed by hand in under a minute.