Interest-Rate Derivatives: Why They Need Their Own Models

A price may wander off and stay there. A rate may not: something works against every departure, so its level returns toward a target and the uncertainty about it stops widening instead of growing forever. One rate must also value an amount due at any date rather than one date. Here four invented numbers produce nine values.

Everything earlier in this subject area has been built on one quantity that behaves like a price. The standard process starts at Rs 100/-, drifts at 8 per cent a year, carries a volatility of 20 per cent a year, and lives in a world with one rate of 5 per cent a year, continuously compounded, over a horizon of one year. Every result so far has been a statement about that quantity or about a function of it.

Now the rate itself becomes the thing being modelled, and almost nothing carries across. The machinery carries across completely and the behaviour does not, and separating those two facts is the entire job of this guide. The chain rule still applies. The measure change still applies. The pricing argument still applies. The failure is in the assumption underneath the standard process about how the quantity moves, and that single failure is enough to require a separate subject.

Two demands make it separate. A price is one number that may go anywhere; a rate is one number that must stay inside a band. A price is one number that means one thing; a rate is one number that has to imply a value at every horizon at once. Neither demand appears anywhere in the standard process, so neither is met by anything built on it.

The four rate parameters below are chosen rather than estimated, and the curve they produce is a computed consequence of them rather than a reading taken from anywhere. A curve is the one object a reader most readily assumes was observed. The difference between a computed curve and an observed one therefore matters more here than almost anywhere else in the subject.

Why can a rate not be modelled the way a price is?

The mathematics here is a translation of an everyday picture rather than a replacement for it. Start with the everyday version. Put a ball on a long flat floor and give it a push. The ball goes, and where it stops has nothing to do with where it started; push it again and it goes further. Nothing in the room is working to bring it home. Now set a thermostat in that room to a temperature. The air is knocked around all day by doors opening and sunlight moving across the wall, so the reading wanders, but something is quietly working against every departure. The further the reading strays, the harder that something pulls.

A price is the ball and a rate is the thermostat, and every difference in this guide is a consequence of that one contrast. The standard process has no thermostat in it anywhere. Written down, its logarithm drifts at a constant rate and is knocked about by a Brownian term, and there is no term whose job is to notice that the level has strayed and pull it back. The omission is not an oversight. For a quantity that can compound indefinitely, having no home is the correct assumption.

The consequence shows up in the spread. Under the standard process, the spread of the logarithm of the level widens as the square root of the horizon and never stops widening. Push the horizon out and it keeps opening: at one year 0.200000, at four years 0.400000, at twenty-five years 1.000000, at a hundred years 2.000000. There is no number it approaches. Mean reversionBeing pulled back toward a level, which a rate does and a price does not. The further the quantity strays, the stronger the pull that works against the departure. is exactly the ingredient that is missing, and a model without it cannot be talked into producing it.

The limit is the whole objection. A model whose spread has no ceiling is saying that, given long enough, any level whatsoever becomes ordinary. For a quantity that compounds, that is a defensible thing to say. For a rate it is not, and a model that says it will happily produce a distribution in which a rate of 40 per cent and a rate of minus 30 per cent are both unremarkable, purely because the horizon was long.

What are the three differences, taken one at a time?

The contrast above is one idea, but it separates into three distinct demands, and they are worth taking apart because different parts of this reading order answer different ones. The first is the pull. The second is the band. The third is the simultaneity, and it is the one that surprises people. Randomness has nothing to do with it at all.

The first difference: the pull, and how fast it acts

The rate model used throughout this reading order starts at 5 per cent and is pulled toward a long-run levelWhere the pull is directed. Six per cent in the parameters used throughout this reading order, chosen rather than measured. of 6 per cent, with a speed of reversionHow fast the pull acts. Half a year to the minus one here, setting a half-life of 1.386294 years for any gap to the long-run level. of 0.5 a year and a rate volatility of 1 percentage point a year. The four numbers are the whole specification, and every result below is a consequence of them.

The pull does not act all at once and it does not act at a constant rate. The pull closes a fixed proportion of whatever gap remains. The gap therefore decays rather than shrinking by a fixed amount each year. Starting a hundred basis points below the long-run level, the expected rate is 5.393469 per cent after one year, 5.632121 per cent after two, 5.917915 per cent after five, and 5.993262 per cent after ten. The single most quotable number here is the half-life: at these parameters the pull closes exactly half the remaining gap in 1.386294 years, at which moment the expected rate is 5.500000 per cent exactly.

The second difference: the band the rate stays inside

The pull has a second consequence, and it is the more important of the two. Because every departure is worked against, the uncertainty about where the rate will be stops accumulating. Push the horizon out far enough and the spread stops growing entirely, settling on a number the model never exceeds. The settled figure is the stationary spreadA spread that settles rather than growing. At these parameters it is exactly 0.010000, reached to six decimal places by ten years., and at these parameters it is exactly 0.010000, or one percentage point.

The exactness is not luck, and the arithmetic behind it is worth seeing. The rate volatility is 1 percentage point a year and the speed is 0.5 a year, so the stationary spread is 0.01 divided by the square root of twice 0.5. The square root of one is one, so the answer is 0.010000 to as many decimals as anyone cares to write. The parameters were chosen so this figure lands on a number anyone can check mentally, and a figure anyone can check is the right one to anchor the difference on.

The last sentence in the block above is the hinge of the argument. The two formulas are not rivals from different worlds. One is the other with the pull switched off. Everything separating a rate model from a price model is contained in a single parameter, and the moment that parameter is zero, the rate model stops being a rate model and inherits every behaviour the price model has.

One honesty note before the control below. The two panels make one wrong conclusion very easy to draw. The two spreads are not in the same units and the comparison is not about which number is larger. One is in rate units, where 0.010000 means one percentage point; the other is in logarithm units, where 0.200000 means roughly twenty per cent proportionally. The comparison being made is entirely about shape: one of these two quantities approaches a ceiling and the other has none, and that difference of shape is the whole reason a rate needs its own model.

The third difference: one rate, many maturities, all at once

The third demand has nothing to do with randomness and it is the one most often skipped. A price is one number that answers one question: what is this worth now. A rate is one number that has to answer a whole row of questions at once: what is an amount due in six months worth, what is an amount due in three years worth, what is an amount due in thirty years worth. One quantity, many answers, and all of them have to move together whenever the quantity does.

Here is the everyday version. A thermostat setting is one number, but from it and a model of how a room warms, a predicted temperature can be stated for every hour of the coming day. The eight o'clock prediction cannot be set independently of the nine o'clock one; both come from the same dial and the same model of the room, so turning the dial moves every hour of the forecast together. A rate model is that dial and the row of maturities is that forecast.

What is a short-rate model, and what does it produce?

A short-rate modelA model of one rate from which a value at every maturity follows. Nothing beyond that one rate and its parameters is supplied. is the answer to the third demand. Exactly one quantity is modelled, the rate applying over the very next instant, and everything else follows as a consequence. Nothing is supplied for six months separately from thirty years. The value of an amount due at any maturity is what comes out of taking the modelled rate, following it along its own path, accumulating what it does to money over that path, and taking the expectation under the pricing measure.

Two things follow immediately. The first is that the whole set of values is a consequence of the parameters rather than a set of inputs, so it is not something that can be adjusted one maturity at a time. The second is that the expectation is over paths, not over endpoints. The spread of the rate therefore matters, and not just its expected level. A path that dips and returns accumulates a different shrinking factor from one that stays flat at the average, even when both end in the same place.

What do these four numbers actually produce?

A claim about what a model produces is worth nothing until the output has been seen. Time to make the claim concrete. The four invented numbers are a starting rate of 5 per cent, a long-run level of 6 per cent, a speed of 0.5 a year and a rate volatility of 1 percentage point. Nothing else is supplied. No table of values is fitted to anything, no maturity is adjusted by hand and no observation of any kind enters.

Here is what those four numbers produce. Nine maturities, nine values, each one the expectation above evaluated at that maturity and nothing more. The values are quoted per one unit due, so a value of 0.949216 means Rs 94.9216/- today for Rs 100/- due in one year. Nine maturities from four numbers, with nothing supplied in between, is the claim of this whole reading order made concrete.

A note on precision, because a reader who checks will notice. Every value is carried at full precision inside the arithmetic and shown here to six decimal places, and the rates in the last column are computed from the unrounded values rather than from the six figures printed beside them. Recomputing a rate from the printed value will therefore agree to five decimal places and can differ in the sixth.

Now the number that justifies the whole subject. At the one year maturity, the model can be set against the flat 5 per cent world assumed throughout this subject area earlier. A flat 5 per cent, continuously compounded, gives 0.951229. The model gives 0.949216. Read back as rates, the flat world gives 5.000000 per cent while the model gives 5.211896 per cent. The gap is 0.211896 percentage points, and it is a computed consequence of the pull and the spread rather than an assertion about anything.

Notice what produced that gap. Two effects worked on it rather than one, and the two worked in opposite directions. The rate is pulled upward toward 6 per cent, and the pull pushes the value of the unit down and the rate read back from it up. The rate also carries a spread, and because the shrinking factor is a curved function of the accumulated rate, spread on its own pushes the value up and the rate read back down. At one year the pull wins by 0.211896 percentage points. Both effects are absent from a flat rate. A flat rate therefore cannot produce this number at all.

What does one rate have to produce that one price does not?

A price produces a number. A rate produces a shape, and the shape has a name: the term structureHow a quantity varies with maturity. One modelled rate has to produce a whole term structure, not a single reading.. Reading the nine values above as rates rather than as values, they rise from 5.114856 per cent at six months to 5.915333 per cent at thirty years, and that rise was not put in anywhere. The rise fell out of a starting rate below the long-run level and a pull working on it.

Two later subjects hold the name of this shape and its uses. The collection of values across every maturity is the discount curveThe collection of present values across every maturity. The discount curve has its own heading later in this reading order.. The rates read back from those values are zero rates. All that is needed from either here is the fact that one modelled quantity produced both.

One behaviour in the shape catches people, and pinning it down now saves trouble later. The rate read back climbs toward the long-run level and never reaches it. Its limit at these parameters is 0.059800, not 0.060000, and even the thirty year reading is only 0.059153. The shortfall of 0.000200 is not an error, and it exists because the spread of the rate pushes values up and therefore pushes the rates read back down, permanently. Exactly why that happens is set out under the zero rate. What matters here is only that it happens, and that it is a consequence rather than an adjustment.

Who actually needs a whole shape from one quantity?

The demand for a term structure is not an academic taste. Think of a household with school fees due in March of each of the next seven years. Every one of those seven amounts has to be turned into a figure today, and the person doing it cannot invent seven independent rates and hope they hang together. Seven independent numbers will contradict each other the moment anyone checks that the three year answer is consistent with the two year and the four year ones sitting either side of it. One quantity producing all seven is not a simplification, it is the only arrangement in which the seven answers cannot disagree with each other.

The same logic is what anyone modelling a rate is doing at a keyboard, and it produces three habits worth carrying away. First, the parameters are the artefact, not the shape: a modeller who hands over a set of values has handed over an output, and the question to ask is which four numbers produced it. Second, the long horizon is the test: any two rate models can be made to agree over six months, so a check that does not push the horizon out has not tested the pull, which is what distinguishes them. Third, a stated shape without its parameters cannot be reproduced, in the same way that a measurement without its instrument cannot be reproduced.

There is a fourth habit, and it is the hardest to keep. Any drawn shape raises one thing to settle first: was the shape computed or was it observed. The shapes here were computed, from four chosen numbers, so the behaviour has something to be read off. A shape that came from somewhere else entirely, and was then fitted to something, is a different object with different properties, and fitting is covered in a later reading order.

What does this reading order build, and in what order?

Six things carry the rest of the argument, and each is set out under its own subject below.

1. The pull, on its own termsWhy a rate is modelled as returning to a level at all, rather than as wandering. The pull is used here and its half-life of 1.386294 years quoted; mean reversion is about the idea itself.
2. The primitiveThe zero-coupon bondThe primitive of rate modelling, a claim to one amount at one date and nothing in between. Set out under its own heading later in this reading order., which is what the nine values above actually are. The nine are used here as outputs, with no property of them quoted beyond their arithmetic.
3. The process, formalisedThe mean reverting process itself, written down properly, carrying the names of Uhlenbeck and Ornstein, 1930. Only two of its consequences are quoted here, the expected level and the spread; the equation and its solution are covered there.
4. The collection of valuesThe discount curve, assembled across every maturity rather than at the nine sampled here.
5. The rates read backThe zero rate, the reading with no intermediate amount, used here as a measuring device and defined there.
6. The comparison at the endModelling one rate and letting everything follow, set against modelling the whole shape directly. The comparison comes last and needs all five above.

What do these four numbers not do?

Two points are worth stating plainly. Both are places where a reader could take more from the figures than they support. The first concerns fitting. The four parameters were chosen so that the results land on numbers a reader can check, not estimated from anything, and no procedure in this reading order takes an observation and returns a parameter. Fitting a model to observed values, and the ways that can quietly go wrong, is covered in a later reading order.

The second is that the shape drawn above is an output and not an input. The shape was not read off anything. The shape exists because four chosen numbers were pushed through an expectation, and if any of the four is changed, all nine values change together and a different shape comes out. A shape that was computed and a shape that was observed look identical on a chart and are completely different objects, and confusing the two is the single most common way a reader is misled by a picture like the one above.

The standard process and the four rate parameters are invented.
Educational material. Not advice on any investment, tax, budget or market position.