European and American Options: Why Early Exercise Changes Everything

The right to exercise early adds value only when acting sooner can beat waiting. The standard process pays nothing out, so acting sooner never beats waiting on the call and sometimes does on the put. So one early exercise premium here is exactly nil and the other is 0.540094, close to a tenth of what that contract is worth.

Two contracts. One process. One strike. One horizon. The only thing separating them is a single clause about when the holder is permitted to act, and that one clause turns out to be worth nothing at all on one of them and a substantial amount on the other. Both figures are computed below rather than asserted. The interesting part of the pair is not that they differ but that one of them is exactly zero and stays exactly zero however carefully it is examined.

Here is the everyday shape of the question, and it has nothing to do with markets. Somebody leaves a refundable deposit of Rs 100/- at a counter and is told they may collect it at the end of the year. Somebody else leaves the identical Rs 100/- and is told they may collect it at any time, up to and including the end of the year. The second arrangement clearly permits more. Whether that extra permission is worth anything depends entirely on whether there is ever a moment when collecting early beats leaving it there, and that is a question to be answered rather than assumed.

A permission is worth exactly what it allows that would not otherwise have been done, and on one of the two contracts here the answer to that is nothing whatsoever. The argument that establishes it comes first, then the arithmetic for the contract where the same argument does not close.

What does the right to exercise early actually add?

Start with the two arrangements written as valuation problems rather than as descriptions. A European styleExercisable only at the horizon, so there is no decision to make before then. contract has no decision in it at all: the horizon arrives and whatever the payoff function returns is what arrives with it. Its value today is one discounted expectation under the risk-neutral measure Q and there is nothing to choose. An American styleExercisable at any moment up to the horizon, so the holder carries a decision as well as a payoff. contract carries a decision at every moment, and its value is the best that decision can be made to do.

The difference between a search and a single number is not cosmetic. The second problem is a search over strategies and the first is a single number. But notice immediately what the search contains. One of the available strategies is "wait until the horizon and act then", and that strategy is precisely the European arrangement. So the search cannot come out below the single number, ever, on any process, for any payoff function. The most it can do is fail to improve on it.

The gap between those two quantities has a name. The early exercise premiumThe extra value the right to act before the horizon carries, which can be exactly nil. is the American value less the European value on an otherwise identical contract. The premium is never negative, for the reason just given, and the two contracts below sit at its two extremes: exactly nothing, and something too large to ignore.

So what does acting early cost? Two things, and both of them are given up the moment the holder acts. The first is the remaining time valueWhat is given up by acting now rather than later: the chance the position improves over the time that is left.. Time value is the chance that the rest of the year moves the position in the holder's favour. The second is the interest on any amount that has to be paid at the moment of acting. Paying Rs 100/- today instead of a year from today costs the year's interest on it. Against those two, acting gains one thing: whatever is available right now, plus the interest on any amount that is received rather than paid.

The clause about amounts received rather than paid is not padding, and the entire asymmetry between the two contracts lives in it. On one of the two contracts the strike is paid out by the holder. On the other it is received by the holder. The time value term points the same way on both contracts, but the interest term points in opposite directions. So one contract has two reasons never to act early and the other has one reason for and one against.

Why can exercising the call never be better than waiting?

Take the first of the two contracts, the one struck at Rs 100/- on the standard process with a one year horizon. The claim is not that acting early is usually a poor idea, or that it is a poor idea for these particular numbers. The claim is that it is never better, at any level of the process, at any moment before the horizon, and the argument that establishes it does not require computing anything.

Here is the argument in three lines. The standard process pays nothing out, so under the risk-neutral measure Q its expected level at the horizon is \(S_t e^{r(T-t)}\), given what is known at time \(t\). The payoff at the horizon is that quantity floored at zero, so it is never below \(S_T - K\). Taking the discounted expectation of both sides, the floor yields an inequality: the European value at time \(t\) is at least \(S_t - Ke^{-r(T-t)}\).

The margin by which waiting wins is not vague: it is exactly the interest on the strike over the time that is left, or \(K\left(1-e^{-r(T-t)}\right)\). A full year from the horizon on these numbers that is Rs 4.877058/-. One step from the horizon on the fifty step lattice used later, where a step is 0.02 of a year, it is Rs 0.099950/-. The margin shrinks as the horizon approaches, exactly as it should, and it never reaches zero until the horizon does.

Set against the two things being given up, this contract has no case at all. Acting early on the call pays the strike out. So the interest term and the time value term both argue against it, and there is nothing on the other side of the comparison. The everyday version is a bill that may be settled early at the same amount. Settling early is never the better move, not because settling is bad, but because the amount does not shrink and the money would otherwise have been sitting somewhere earning until the day it was due.

What makes the put different, in one sentence?

One sentence, and it really is one: on the put the strike comes in rather than goes out, so acting early brings that amount forward instead of pushing it out, and the interest that argued against acting on the call now argues for it.

The reversal of direction on the strike is enough to break the argument just given. The same three lines yield the counterpart bound: the European value on the put is at least \(Ke^{-r(T-t)} - S_t\). The relevant comparison is with what acting now would hand over, namely \(K - S_t\). The floor sits below that quantity, by the same Rs 4.877058/- a year out. The inequality that closed the case on the call points the wrong way here, and a bound pointing the wrong way proves nothing at all. The failed bound does not show that acting early is better, only that the easy argument has run out and the question is now open.

There is a second way to see the same room opening up, and it is worth carrying because it is the tidier one. The very best the horizon can deliver on the put is the strike itself, and the strike has to be brought back a year, so the most the European arrangement can ever be worth is the discounted strike, Rs 95.122942/-. The holder of the American arrangement can take the strike today, so the most that arrangement can be worth is the strike itself, Rs 100/-. The two ceilings differ by Rs 4.877058/-, and every rupee of early exercise premium on a put has to live inside that gap.

Does it actually live there? On these numbers, yes, and here is a single instance that settles it without any argument at all. Put the process at Rs 60/- with a year to run. Acting now hands over Rs 40.000000/-. Waiting, valued on the fifty step lattice, is worth Rs 35.170103/-. Waiting is worth less than acting, by Rs 4.829897/-, so at that level the holder of the American arrangement acts and the American value is Rs 40.000000/- exactly. No inequality was needed; the two numbers simply came out that way round.

The everyday version is the refundable deposit from the opening. The counter will hand back Rs 100/- on request. Asking today means the Rs 100/- is in the depositor's hands from today and can sit and earn. Asking at the end of the year means the counter had the use of it for the year. Nothing about the deposit changed; the only thing that changed is who holds the money in the meantime, and that is worth precisely the interest.

How is the extra value computed when no formula for it exists?

There is no closed form for the American arrangement on the put. The absence of a formula is not a gap waiting to be filled by a cleverer derivation, but a consequence of the problem's shape. The European problem is one expectation, and expectations of that kind can be written down. The American problem is an optimal stoppingChoosing the best moment to act, which is the extra problem the right to act early creates. problem, a search over every rule for deciding when to act using only what is known at the time, and searches of that kind do not collapse into a formula.

So it is computed instead, and the way it is computed is worth understanding even though the machinery itself is set out under the binomial model. Build a lattice of levels the process can reach. At the horizon the value at every terminal point is whatever the payoff function returns there, with no decision involved. Then work backwards. At every earlier point, compute what waiting is worth, compute what acting is worth, and take the larger of the two. Working back like that is backward inductionWorking from the horizon back to today, taking the better of two choices at each point on the lattice., and it is the only route to the number.

Watch it happen on a lattice small enough to check by hand. Two steps over the year, so each step is six months. The up factor is 1.151910, the down factor is its reciprocal 0.868123, the weight on the up move is 0.553908 and the discount over one step is 0.975310.

At the horizon the three levels are Rs 132.689644/-, Rs 100/- exactly and Rs 75.363832/-. The put pays nothing at the first two and Rs 24.636168/- at the third. Step back six months. At the upper nodeOne point on the lattice, at one time and one level of the process. the level is Rs 115.190991/-, waiting is worth nothing and acting is worth nothing, so the two arrangements agree. At the lower node the level is Rs 86.812345/-, waiting is worth Rs 10.718647/- and acting is worth Rs 13.187655/-. Acting wins there, by Rs 2.469009/-, and that single node is the entire source of the premium on this lattice.

Carry both back to today. The European arrangement discounts the two waiting values and returns Rs 4.663444/-. The American arrangement discounts the two larger values and returns Rs 5.737654/-. The Rs 1.074211/- between them came from one node and nowhere else, where acting beat waiting by Rs 2.469009/-, weighted by 0.446092 and discounted by 0.975310. Multiplying those three together returns Rs 1.074211/-, a check worth doing because it shows the premium is a sum over decisions rather than a correction factor.

Now refine the lattice and something happens that is easy to misread. The premium on the put does not hold steady: it runs Rs 1.074211/- at two steps, Rs 0.789335/- at four, Rs 0.609760/- at twelve and Rs 0.540094/- at fifty. The premium falls, and it falls a long way, by half over that range.

The instinct is to say a coarse lattice offers too few moments to act and therefore understates the right. The arithmetic says otherwise, and the reason is worth having. A two step lattice has only three terminal outcomes and badly understates how spread out the horizon really is, so both values come out too low. But the American value is propped up by the exercise floor at the single intermediate date, a crude but generous substitute for the spread the lattice is missing. So the European value is the one that is badly wrong, and the difference between them is inflated.

Refining fixes the European value far faster than it moves the American one, and that is why the premium comes down. From two steps to fifty the European value climbs by Rs 0.870190/-, from Rs 4.663444/- to Rs 5.533634/-. The American value climbs by only Rs 0.336074/-, from Rs 5.737654/- to Rs 6.073728/-. The difference between those two rises is Rs 0.534116/-, exactly the fall in the premium to the last decimal the two subtractions can carry. The premium continues to ease slowly beyond fifty steps. Fifty is where the worked instance sits, so fifty is where the arithmetic here stops.

What do the two premiums look like set side by side?

Everything to this point has been argument. Here is the worked instance, and the discipline of it is that nothing at all differs between the two columns except which contract is being held. The same standard process starting at Rs 100/-, the same volatility of 20 per cent a year, the same risk-free rate of 5 per cent, the same strike of Rs 100/-, the same one year horizon and the same fifty step lattice with an up factor of 1.028688, a down factor of 0.972112, a weight of 0.510614 on the up move and a discount of 0.999000 over each step of 0.02 years.

The premium on the call is 0.00000000 to eight decimal places, and the eight decimal places are the point: it is not a small number, it is the number nothing at all. At every one of the 1,275 nodes where a decision is available the comparison came out in favour of waiting, and the maximum returned the waiting branch untouched. So the two values in that column are not merely close, they are the identical quantity. A recursion that never once takes the other branch reproduces the European recursion exactly.

The put column tells the opposite story with the same machinery. At 514 of those same 1,275 decisions acting beat waiting, and every one of those contributes to the Rs 0.540094/- that separates the two rows. The premium is 8.89 per cent of what the contract with the right is worth and 9.76 per cent of what the contract without it is worth, so calling it close to a tenth is fair in either direction.

One arithmetic check before moving on rules out the lattice as a suspect. On the same lattice, at every one of the four step counts, the two European values differ by exactly 4.877058, the starting value of Rs 100/- less the strike discounted back over the year. The agreement holds to ten decimal places at two steps and at fifty. The relation between the two European values is settled under put-call parity and serves here only as a check that the lattice treats both contracts identically, and the check passes. Whatever produced the asymmetry, it was not the lattice.

What does the asymmetry teach about pricing anything unfamiliar?

The comparison pays off at this point. An unfamiliar contract arrives with a clause in it granting some right, and its worth is what has to be settled. The two clauses here have the identical shape and their answers are 0.00000000 and 0.540094, so no answer follows from the shape of a clause alone.

The habit the asymmetry teaches has three parts, and none of them requires software. The first is to establish which way the interest runs. If exercising the right means paying an amount sooner than it would otherwise be paid, interest argues against acting; if it means receiving an amount sooner, interest argues for it. The single question of direction separated the two contracts completely. The second is whether waiting carries a floor that already beats acting. On the call it did, by exactly the interest on the strike, and the matter was closed without a computation. The third, when the floor does not close it, is to compute, with an answer of nothing a live possibility.

1. Find the direction the interest runsDoes acting early bring an amount in or send an amount out? On the call the strike goes out and interest argues against acting. On the put it comes in and interest argues for it. Same clause, opposite sign, and this question costs nothing to ask.
2. Look for a floor under waitingWaiting is worth at least something, and sometimes that floor already sits above what acting hands over. When it does, the matter is settled by argument alone and the margin has a formula: on the call it is the interest on the strike over the remaining life, Rs 4.877058/- a year out.
3. If the floor does not close it, compute and accept the answerThe put's floor pointed the wrong way, so the question stayed open and only backward induction on a lattice could settle it at Rs 0.540094/-. There is no closed form waiting to be found, and the figure moves with the refinement of the lattice, so quote the lattice alongside the number.
4. Never read a nil as a failure of the computationA right whose exercise is never optimal has a premium of exactly nothing, and that is as legitimate an answer as any other. Check whether the comparison is near before assuming precision is the problem: here the narrowest margin was Rs 0.099950/- and it never approached zero at any of the 1,275 decisions.

There is one more lesson in the pair and it is about assumptions rather than arithmetic. The whole argument for the call rested on the standard process paying nothing out. The no-payout condition was written down as an assumption at the start, and the assumption is what carried the result. An answer of exactly nothing is usually resting on exactly one assumption, and knowing which one is more useful than the answer. Somebody who has only the figure 0.00000000 knows nothing about when it would stop being true; somebody who has the argument knows precisely which sentence to check.

The everyday closing image is two people holding what look like identical passes, each with an extra clause printed at the bottom permitting early use. One person's clause is worth something because there are days when using it early genuinely helps. The other person's clause permits something that is never worth doing, so it might as well not be printed. Neither of them can tell which is which by reading the clause. Both have to look at what using the clause early would cost them, and that cost is the whole of the comparison.

The early exercise premium is a comparison between two computed quantities, so it holds identically everywhere and nowhere in particular. No regulator sets it, no exchange publishes it and no jurisdiction, threshold, rate or period from any rulebook alters it. Where a contract or settlement convention matters, the exchange or the clearing corporation is the place to confirm it.

The standard process and both contracts worked here are invented.
Educational material. Not advice on any investment, tax, budget or market position.