The Exercise Boundary: When Early Exercise Becomes Optimal

The exercise boundary is the level, at each moment, at or below which acting now is worth more than waiting. The boundary is a curve rather than a number. The curve climbs toward the strike as the horizon approaches, and it must arrive there exactly when no time is left. No formula returns it. The curve is computed alongside the price rather than read off one.

Under European versus American options the two premiums were computed rather than asserted, and they came out at 0.00000000 and 0.540094 on one process with one strike and one horizon. Those two figures settled how much the permission to act early is worth. The premiums left the harder question standing: where. At which levels, and at which moments, does acting actually win? A premium is one number attached to today. The answer to where is an object with a shape.

The everyday version is exactly the same problem and it costs nothing to hold in mind. Somebody stands at a stop waiting for a bus, free to set off walking at any moment. Setting off ends the wait: once walking, there is no un-walking and rejoining the queue. Whether walking beats waiting is obviously not a fixed rule such as walk after nine minutes. The choice depends on how much longer remains before the walker needs to be somewhere. Early in the evening, with hours to spare, waiting is easy to justify. Five minutes before the appointed time, almost any further delay makes walking the better move. The rule separating walking from waiting is not a number, it is a number that changes as the clock runs, and that is precisely what an exercise boundary is.

What is the exercise boundary, as an object?

The picture has two axes. Along the bottom runs time, from today at the left to the horizon at the right. Up the side runs the level of the standard process. The standard process is the single invented traded quantity used throughout this subject area, written S with a time subscript and starting at Rs 100/- exactly. Every point in that plane is a pair: a moment, and a level the process might be at when that moment arrives. The holder of a contract permitting action at any time faces exactly one comparison at every one of those points, and the comparison has only two outcomes.

So the plane splits in two. One set holds the points at which the contract is worth strictly more than what acting would hand over. That set is the continuation regionThe moments and levels at which holding on is worth more than acting, so the sensible move is to do nothing.. The other set holds the points at which acting is at least as good. That set is the stopping regionThe moments and levels at which acting immediately is at least as good as holding on.. The comparison is between two real numbers, and one of them is either larger than the other or it is not. So every point in the plane belongs to one set or the other, with no gaps and no overlaps.

The exercise boundary is not a third region: it is the frontier between those two, and on the contract worked here it is a single curve running left to right across the plane. A boundary quoted as a single figure has been collapsed from two dimensions down to one, and the dimension thrown away is the one carrying all the movement.

Two things about that definition deserve saying out loud. Both are easy to slide past. The first is that the value function appears on the right hand side of the definition of the boundary. The boundary is defined in terms of the price. Neither can be found without the other, and the whole later argument about why no formula exists is already visible in that dependency. The second concerns the supremum. The stopping region at a moment is a set of levels rather than one level, so the definition has to take a top rather than an equality. On the contract worked here that set runs from zero up to the boundary, so the supremum is a clean top edge. A clean top edge is a property of this contract rather than a rule about every contract.

Where does the boundary sit on the contract that carries a premium?

Here is the worked instance, with every figure recomputed rather than carried over. The contract is the one struck at Rs 100/- on the standard process with a one year horizon, priced on a fifty step lattice, so a step is 0.02 of a year and there are 1,275 separate decision points in the computation. At each of those points the backward passWorking from the horizon back toward today, which produces the value and the decision at every point together. compares what acting hands over against what holding on is worth, and the boundary at a moment is simply the highest level at which acting won.

The third column, read downward, shows the entire content of this guide: the boundary starts more than twenty rupees below the strike and finishes touching it. The boundary sits below the strike at every moment before the horizon, never above it, and it never crosses. Notice that the fourth column is just the strike less the third. Acting at the boundary hands over the difference between the strike and the level. Notice too that the fifth column shrinks toward nothing. The fourth and fifth columns are the two halves of the argument, and the section below takes them apart.

Three honesty notes belong here. A table of four numbers looks more solid than it is, and correct distrust of it serves a reader better than misplaced trust.

The first is that a fifty step lattice does not offer a continuous range of levels. At any step the process can only be at one of a discrete ladder of levels, spaced by the up factor of 1.028688 and its reciprocal, and the boundary reading is always one rung of that ladder. The rungs immediately below the strike are Rs 97.211198/-, Rs 94.500171/-, Rs 91.864749/-, Rs 89.302823/-, Rs 86.812345/-, Rs 84.391320/-, Rs 82.037814/- and Rs 79.749942/-, and every reading in the table above is one of them. The boundary is not being measured to six decimals because it is known that precisely; it is quoted to six decimals because it is exactly a rung, and the rungs are exact.

The second note is about the shape of the ladder, and it produces something that looks at first like an error. The levels available at a step alternate: even numbered steps sit on one set of rungs and odd numbered steps sit on the set interleaved between them. Read the boundary step by step and it therefore jags, falling by one rung and rising by two, eighteen times over the forty three steps at which it is resolved at all. Read only the even steps and it rises without a single fall, through Rs 79.749942/-, Rs 84.391320/-, Rs 89.302823/- and Rs 94.500171/-. Read only the odd steps and it rises without a single fall too, through Rs 82.037814/-, Rs 86.812345/-, Rs 91.864749/- and Rs 97.211198/-. The jag is the grid, not the contract, and quietly smoothing it would show something the computation did not produce.

The third note is the one most worth carrying. For the first seven steps of the lattice, the first 0.14 of the year, the boundary cannot be read at all. Not because acting is never worthwhile that early, but because the lowest rung the grid reaches by then is Rs 84.391320/-, and the boundary is below it. The computation has nothing to report because it has nowhere low enough to look. A coarse grid can hide a frontier simply by not having any points on the far side of it, and the same reasoning near the horizon is why the last reading, Rs 97.211198/-, is the top rung under the strike rather than a genuine separation: by then the boundary and the strike are closer together than the grid can tell apart.

The last detail in the picture is worth pausing on, and it is the cleanest single demonstration of the whole idea. The level Rs 84.391320/- is a rung the grid reaches at every even step. With four fifths of the year still to run, holding on at that level is worth more than acting. With two fifths of the year still to run, at exactly the same level, acting is worth more than holding on. Nothing about the process changed. Nothing about the contract changed. The only thing that moved was the calendar, and it moved the decision.

What actually decides whether acting now beats waiting?

Everything above described the boundary. Nothing above explained it. The explanation lives in a single comparison made at a single point, repeated 1,275 times, and with that comparison in view the shape of the curve stops being a fact to memorise and becomes something that could have been predicted.

At one point in the plane the holder has two quantities in front of them. Acting hands over what the payoff function returns at the level standing there, immediately and finally: it is an absorbing decisionA choice that ends the arrangement, so everything that would have happened afterwards is forfeited., and everything after that instant is forfeited. Holding on hands over the value of still being in the arrangement one step later, discounted back. The larger of the two is the value of the arrangement at that point, and which of the two was larger is the decision recorded at that point.

Now open up the second branch. The movement all comes from there. Ask what holding on actually keeps hold of. Two separate things, and they do not point the same way.

The first is the chance that the rest of the time still to run moves the position further in the holder's favour. The chance is worth something strictly positive whenever any time remains and the process still moves, and it is worth exactly nothing once no time remains. Call it what it is: the value of not yet having committed. Not having committed is the only reason waiting is ever attractive on a position already worth something, and its worth shrinks continuously as the horizon approaches.

The second is the interest on the strike, and this one has a sign that depends entirely on which direction the strike travels. On the contract worked here, acting brings the strike in. Waiting therefore postpones receiving Rs 100/-. The postponement costs the holder the interest on that sum over whatever time is still to run. The cost is Rs 3.921056/- with four fifths of the year left, Rs 1.980133/- with two fifths left, Rs 0.498752/- with a tenth left and Rs 0.099950/- with one fiftieth left. On this contract the interest argues for acting and the remaining time argues for waiting, and the boundary is exactly the level at which those two arguments balance.

The everyday version is a refundable deposit collectable at any moment the holder likes. Collecting it puts the money in hand sooner, and money in hand sooner is worth the interest. Leaving it where it is keeps open whatever chance there was that the arrangement improves. Nobody sensible collects on the first day and nobody sensible waits to the last if the arrangement has clearly stopped improving, and the moment they should collect depends on how much of the arrangement is left to run.

One number, Rs 0.099950/-, does more work here than anything else. The strike earns exactly that over one step of this grid, and it is the entire reason the two contracts compared under European versus American options came out at 0.00000000 and 0.540094. Where the strike goes out, that Rs 0.099950/- is added to the case for waiting, and the case for waiting was already ahead: across all 1,275 decision points the smallest margin in favour of holding on came out at exactly Rs 0.099950/-, at a level of Rs 128.9892/- one step from the horizon, and it never once turned over. Where the strike comes in, the same Rs 0.099950/- is added to the case for acting, and the case for acting only has to beat what is left of the time value. Deep enough below the strike, where the position can hardly improve, it does, and 514 of the 1,275 points come out as places to act.

Watch that balance tip at one fixed level. The table below holds the level at Rs 84.391320/- and lets only the calendar move. Acting hands over the same amount at every row. The level never changes. The worth of waiting falls steadily down the rows: less and less time is left for the position to improve. Somewhere between three fifths of the year remaining and a shade under it, the falling quantity crosses the flat one.

The crossing in that fourth column is the boundary arriving at Rs 84.391320/-, and it is the same event the earlier picture drew as a dot moving from the green region into the red one. The crossing also explains why the boundary reading in the first table changes to Rs 84.391320/- exactly when it does. Nothing was chosen; the arithmetic tipped.

Why does the boundary climb as the horizon approaches?

With the two quantities the decision is made of now in hand, the shape of the curve follows from how each of them behaves as time runs out. Both of them shrink to nothing at the horizon. The shrinking is not the interesting part. The interesting part is that they shrink at completely different speeds, and the boundary climbs because one of them collapses while the other merely fades.

Take the interest on the strike first, as the simpler of the two. Over a remaining stretch of time it is the strike less the strike discounted back, and for a small stretch that is very close to the strike times the rate times the time. The interest falls in proportion to the time left. Halving the time remaining roughly halves it. From Rs 3.921056/- at four fifths of a year it goes to Rs 1.980133/- at two fifths, very nearly half, and then to Rs 0.498752/- at a tenth and Rs 0.099950/- at a fiftieth. Nothing dramatic: a straight, steady fade.

Now take the other quantity, what waiting keeps. At a level well below the strike, what waiting keeps is the chance the position improves further over the time still to run, and the size of that chance depends on how far the process would have to travel relative to how far it can plausibly travel in the time available. The distance is fixed. The plausible travel shrinks with the square root of the time remaining, so as the horizon closes the required move goes from ordinary to unlikely to essentially impossible, and the value of that chance does not fade, it collapses. At a level of Rs 84.391320/-, a shade under sixteen rupees below the strike, that collapse is what runs the table below.

The last column of that table is the exact difference between the two quantities before either was rounded for display. Subtracting the printed figures can therefore leave a discrepancy of one unit in the sixth decimal. The two middle columns run as a race. At 0.88 of a year to run, what waiting keeps is ahead by fifteen paise and the sensible move is to hold on. Both quantities then fall, but the second column falls by 2.562275 over the stretch shown while the third falls by only 2.324471, and that difference of about a quarter of a rupee is enough to reverse the order. By 0.56 of a year to run the third column is in front and the sensible move has flipped, permanently, at that level.

Repeated at every level, that crossing gives the whole curve. At a level far below the strike the chance of further improvement is already small, so the crossing happens early and the level is in the acting region for most of the year. At a level just below the strike the chance of improvement stays substantial almost to the end, so the crossing happens very late. Lined up in order, the crossings trace a curve that starts low and climbs. The staircase drawn in the figure above is exactly that curve. The boundary climbs because the crossing moment gets later the closer the level sits to the strike, and for no other reason.

Here is the everyday version of the same race, and it is one everybody has run. Somebody queues for something with a fixed walk-away alternative that never improves and never worsens. Early on there is plenty of time for the queue to move, so the queue is worth staying in. As the deadline closes in, the chance the queue delivers falls away far faster than the value of walking away changes, and at some point walking wins. Somebody standing further back in the queue reaches that point sooner than somebody near the front. Sorted by position, those points trace a boundary.

Where must the boundary end up at the horizon?

One point on this curve can be written down without computing anything at all. Everything else here came out of a lattice and that one point did not, so it is worth separating carefully from the rest. At the horizon the boundary is the strike exactly, and that is forced by the terms rather than produced by any computation.

The argument takes one line. At the horizon there is no decision left to make. Whatever the payoff function returns is what the holder gets, so the value of the arrangement at the horizon is the payoff itself at every level, with nothing added for the possibility of anything further. There is nothing left to wait through, so waiting is worth precisely nothing. Acting is therefore at least as good at every level where acting hands over anything positive at all. On this contract that means every level below the strike, and acting is no better at any level above, where acting hands over nothing. The frontier therefore sits at the strike.