Monte Carlo Pricing: Simulating Paths to a Price

Monte Carlo pricing swaps an average taken over every outcome the process could reach for an average taken over a finite handful of drawn ones. The swap is legitimate because a sample mean settles onto the true mean as the count of draws rises, and it is slow because that settling is governed by a square root. The method hands back a range, not a number.

Accuracy here improves with the square root of the path count. Buying a hundred times the accuracy therefore means paying for ten thousand times the work, and that one fact decides where the method gets used and where it does not. Everything that follows is a consequence of that relationship, worked through on the standard process with numbers that can be checked.

What does Monte Carlo pricing actually do?

Think about a sack of rice at a grain shop. Nobody weighs every grain to work out the average grain. The shopkeeper takes a scoop, weighs it, counts the grains and divides. The answer is not the true average grain weight, and everyone involved knows it is not, but it is close enough and it took a minute instead of a lifetime. If the shopkeeper wants a better answer, the only lever available is a bigger scoop.

The Monte Carlo methodReplacing an average over every outcome with an average over finitely many draws. is that scoop, applied to a price. The price of a claim under the risk-neutral measure Q is an expectation. An expectation is an average over every outcome the process could reach, weighted by how likely each one is, and that average is an integral over a continuum. The method replaces it with an average over a finite handful of drawn outcomes.

The substitution is the whole method, and everything difficult about it follows from the fact that the two sides of the arrow are not equal. The left side is one number. The right side is a random variable. Two different sets of draws go into two runs, so the method run twice gives two different answers. A method that returns a different answer each time it is run is not returning a number. The method returns a range instead, and that range has always been its honest output.

The last row of that picture carries the point. The band at ten lakh paths, blown up a hundred times, is exactly as wide as the band at a hundred paths seen plainly. Going from one to the other took ten thousand times as many paths. The magnification is not a defect in the drawing. The arithmetic of the method forces it, and what follows is why it cannot be avoided.

Why does averaging a finite number of draws work at all?

There is nothing obviously legitimate about replacing a true average with a scoop. The justification is a single property of averages, and it has two halves that are often run together and should not be.

The first half is that the estimator is centred correctly. The average of a finite set of draws has, as its own expectation, the exact quantity being estimated. The centring holds at every path count, including one. A single pathOne drawn outcome, of which the method uses many. already gives an answer that is right on average. Being right on average sounds impressive, and it is also compatible with being badly wrong every single time.

The second half is that the spread of the estimator shrinks as the count grows. Being centred correctly is worth nothing without it. Put together, they say the sample average settles on the true average as the count is increased without limit.

Being centred and being convergent are separate properties and the method needs both, so a small path count is not merely a rough version of a large one but a genuinely different situation. At a hundred paths the answer is centred on the truth and spread out enough to be useless. At ten lakh paths it is centred on the truth and tight. Nothing about the centring changed between them.

The limits of the property matter just as much. Neither half says any particular run is close. Both statements are about the limit and about the average behaviour, and a statement about a limit says nothing binding about any finite case actually run. The gap between the limit and the run in hand is where the failure described further on lives.

How to Set Up a Monte Carlo Pricing Illustration

Everything above is general. Here the method is put on the standard process, the single invented traded quantity used throughout this subject area and never a company. The starting value is Rs 100/-, volatility is 20 per cent a year, the risk-free rate is 5 per cent continuously compounded, and the horizon is one year. The contract is the at-the-money one, strike Rs 100/-, and what it pays is taken as known. The closed form value against which everything is checked is Rs 10.450584/-.

The quantity being averaged is the discounted payoffThe quantity being averaged, whose spread is 14.719404 here.. Its spread governs everything that follows, so the first thing to establish about the discounted payoff is not its average but its spread. On this contract, under Q, that spread is 14.719404. The spread was computed from the distribution of the discounted payoff itself, not measured off any set of draws, and it is exact to the six figures shown.

Look at the size of it. The thing being averaged has a spread of 14.719404 around an average of 10.450584. The quantity being averaged is more than one and a third times as variable as it is large, and that ratio, 1.408477, is a property of the contract and the process rather than anything the method chose. Change the number of paths and it does not move. Change the strike and it does.

The shape explains the size. Under Q, the chance this contract finishes in the money is 0.559618, so 0.440382 of all the probability sits on the single value zero. Every one of those outcomes contributes a payoff of nothing. The remaining 0.559618 has to carry the whole of the average, and does so by reaching a long way out. A distribution that is mostly a block at zero and partly a long tail is about as awkward a thing to average as could be handed to anyone, and its spread is correspondingly large.

Here is the full illustration set out, and every row of it is computed from the formula rather than measured from any set of draws.

The bottom row deserves an honest reading. Ten lakh paths is a substantial computation, and the answer that comes out of it is still uncertain in the second decimal place of a price of Rs 10.450584/-. No line of that table was sampled. Every row comes from the formula, so the same numbers appear on every reload.

How fast does the accuracy improve as paths are added?

The spread of the sample average is the spread of the thing being averaged divided by the square root of how many things went into the average. The division by a square root is the entire character of the method, and it is worth writing down before anything else is said about it.

The consequence is a fixed exchange rate between effort and accuracy. Ten times the paths gives the square root of ten times the accuracy. The factor is 3.162278, and it holds exactly at every scale, neither improving as the path count rises nor degrading. The ratio 3.162278 is not an approximation or a rule of thumb, it is exact, and it is the same factor whether the move is from a hundred paths to a thousand or from a lakh to ten lakh.

An everyday version: pacing out the length of a wall. Two people pacing and averaging are better than one, but not twice as good, and getting an answer four times as tight needs sixteen people rather than four. The convergence rateHow fast the approximation improves, one over the square root of the paths here. is the same one over the square root of the count in both settings, and it is the reason the method is described as slow even by people who use it every day.

What does that scaling cost?

Turned round, the relationship reads the way anyone actually asks it. The path count is not usually chosen first, with the accuracy discovered afterwards. An acceptable accuracy is settled on first and the path count is discovered afterwards, and the discovery is regularly unpleasant.

On this contract the demand schedule reads as follows, every figure computed from the expression above. A standard error of 1 needs 217 paths. A standard error of 0.1 needs 21,667. A standard error of 0.01 needs 21,66,609. A standard error of 0.001 needs 21,66,60,857 paths, more than twenty one crore, for three decimal places on a single price of a single contract on a process with one source of randomness.

The cost of accuracy in this method rises with the square of the accuracy, and no arrangement of the same idea escapes it. Anyone choosing the method needs that fact before any other. Going from a hundred paths to ten lakh is ten thousand times the work for a hundred times the accuracy, and at the end of it the answer is still uncertain by 0.014719 against a true Rs 10.450584/-.

Squares are the honest reason the method has a reputation for being expensive. No single path is costly. The paths needed to sharpen an answer arrive in squares, and squares get large quickly. Square root scalingAccuracy improving with the square root of the effort, which is this method's character. is generous at the start and brutal at the end, and most of the pain lands exactly where a practitioner wants a clean answer.

What is the standard error, and what is it not?

The standard errorThe spread of the answers the method would give, not the distance from the truth. is the most useful number the method produces and the most frequently misread. The standard error is the standard deviation of the estimator: a description of how far apart the answers would be spread if the whole procedure were run again and again, each run with its own fresh set of draws.

The standard error is not the error. The error on any particular run is the distance between the number that run produced and the true price. The method never sees that distance: a method able to see the true price would never have needed to run at all. The standard error describes a population of runs. The error describes the single run in hand. Standard error and error are different quantities, and only one of them is knowable from inside the calculation.

Notice the shape of that sentence. The probability is a statement about a procedure, made before the procedure is run. Once a particular run has produced a particular number, that number either is or is not within one standard error of the truth, and no probability attaches to it any more. The standard error is a promise about the method, made in advance, and it stops applying the moment an answer is in hand.

Why is the calculator here computed rather than sampled?

The control above draws no paths. The control has no random number generator in it at all. Every reading it produces comes from the standard error expression, evaluated at whatever path count the slider is sitting on.

A worked example is something a reader checks a calculation against. Numbers produced by drawing would change the default reading on every reload, and no reader could ever confirm that the worked figures agree with each other. Worse, a treatment teaching that the method is uncertain would be demonstrating that uncertainty in the one place where the reader most needs the illustration to be deterministicReturning the same answer every run, which this method does not.. The method samples and this calculator does not, and that separation is what allows exact figures to be stated about a method whose whole character is that it cannot.

There is a second reason, and it is the more interesting one. The quantities at issue here, the standard error, the half-width and the scaling factor, are not learned by drawing. All three are properties of the distribution, computed exactly, and drawing would give only noisy estimates of numbers already known in closed form. Sampling would replace an exact figure with an approximation of that same figure, for no gain, and call it a demonstration.

When is this method the right one, and when is it not?

Everything so far argues against the method. Its accuracy is the worst of the four numerical approaches at any given effort, it returns a range rather than a number, and the cost of tightening the range rises with the square. The method should have been abandoned decades ago. Instead it survived, and the reason is a single property that has nothing to do with accuracy.

The standard error expression contains the spread of the payoff and the number of paths. The expression does not contain the number of quantities being modelled. Add a second source of randomness, or a fifth, and the expression is unchanged. The work per path rises, roughly in proportion, but the number of paths needed for a given accuracy does not move at all. The method is almost indifferent to dimensionThe number of quantities being modelled, which this method is unusually indifferent to., and that indifference is the entire reason it survives alongside methods that beat it comfortably on a single quantity.

A method that lays a grid over the space is in the opposite position. A hundred points along one quantity is a hundred points. The count multiplies rather than adds, so a hundred points along each of five quantities is a thousand crore points. Multiplying growth is what makes a grid unusable past three or four quantities, and it is why a problem with several sources of randomness has, in practice, one method available to it.

So the choice is not made on accuracy at all. The choice is made on whether the other methods can reach the problem. Where they can, they are usually better. Where they cannot, this one is what remains, and its owner accepts the square root and the range that comes with it because the alternative is no number.

Where does the Lattice Method sit against this one?

The Lattice Method is set out under tree methods rather than here, but the comparison is worth making now because it sharpens what is distinctive about drawing paths. Both approaches produce a price for the same contract on the same process, and both are checked against the same closed form of Rs 10.450584/-. The two differ in what a refinement buys.

Refining a lattice means adding steps, and the locked lattice on the standard process moves 12.16, then 9.54, then 9.97, then 10.29 as the steps go from one to two to four to twelve, against a closed form of Rs 10.450584/-. The lattice price overshoots, undershoots and oscillates inward. Every one of those figures is a single definite number: the same lattice run twice gives the same answer twice, so the notion of a standard error does not arise. The error is real but it is not random.

The two methods fail in opposite ways: the lattice gives an answer that is repeatable and biased, and the simulation gives an answer that is unbiased and not repeatable. That is why the standard error belongs to one and not the other, and why the reading habit taught here has no counterpart on a lattice. A lattice asked for its uncertainty has none to give. The silence is a comfort right up to the moment the answer is wrong.

How does a working analyst decide the number of paths?

Not by picking a round number. The order of the decision runs backwards from the use the price is being put to, and it goes in three steps that are worth spelling out because they are almost never written down.

Step one is to name a tolerance. How far can this price be wrong before the decision it feeds changes? For a quote being shown to a counterparty, that is often a fraction of the smallest quotable increment. For a valuation feeding an internal report, it may be a few paise. The tolerance is a business judgement, not a mathematical one, and it has to be made first because everything else follows from it.

Step two is to compute the path count that tolerance demands, from the expression above. On this contract a tolerance of five paise on the standard error needs 86,665 paths, and a tolerance of one paisa needs 21,66,609. If that number is affordable, the work proceeds. If it is not, step three begins.

Step three is to accept that the accuracy cannot be bought by paths alone and to go looking for a way to shrink the spread of the thing being averaged rather than the count of things averaged. The route exists and it is real, and variance reduction is where it is set out. The skill worth having is recognising the moment the path count stops being the answer. An analyst who responds to an unaffordable path count by simply running fewer paths and quoting the result without its band has converted a computation problem into a communication failure.

A household version of the same discipline: deciding how carefully to weigh out spices before cooking. Nobody uses a laboratory balance for salt, and nobody eyeballs saffron. The precision is chosen from what the error would cost, not from what precision is available, and the same ordering applies to a price.