The Black-Scholes PDE: How It Is Derived and What It Assumes

The Black-Scholes equation says the value of a contract on the process satisfies a relation between its sensitivity to time, its sensitivity to the level, and its curvature in the level. The equation is derived by building a position whose randomness cancels and insisting that what remains earns the risk-free rate. The drift never appears in it, and that absence is the whole point.

Every ingredient of the derivation has already been met. The chain rule for random processes was settled earlier in this subject area, and so was the pricing argument that refuses to let a position cost nothing and gain for certain. Putting the two together is all that remains. The result is one line of notation that has been rewritten, extended and argued over for half a century, and the surprising thing about the line is not what it contains but what it leaves out.

Here is the everyday version before any notation arrives, and it involves nothing more exotic than a moving staircase. A person on an escalator that is running upward, walking down it at exactly the speed it is climbing, stays level with the wall. Now a question about that arrangement: does where the walker ends up depend on how fast the escalator runs? The answer does not depend on the speed, and the reason has nothing to do with approximation. The machine was matched step for step, so the machine's speed left the answer entirely. Speed it up, slow it down, run it at any rate at all; the walker is still level with the same tile.

The derivation that follows is that escalator, and the drift of the process is the speed of the machine. A position is arranged so that it moves with the process step for step. Once it does, the rate at which the process is expected to climb drops out of the answer completely, not approximately, and everything that survives can be written down as a single relation. The surviving relation is the equation.

How is the equation derived, in a teaching setting?

The derivation has five steps, and four of them are bookkeeping. Only one of them is an idea. Most treatments race through the algebra and leave a reader unable to say afterwards which line carried the weight, so each step below is taken slowly, with what just happened and why it was allowed stated alongside it. The reasons alone, read without the notation, still carry the whole derivation.

Start with what is on the table. There is the standard process, invented for this subject area, written S with a time subscript, starting at Rs 100/- and moving by proportional increments with a drift of 8 per cent a year and a volatility of 20 per cent a year. There is cash, growing at the risk-free rate of 5 per cent a year, continuously compounded. And there is a contract on the process, whose value at any moment depends on where the process is and how much time is left. Call that value V, a function of time and of the level. Nothing about what the contract pays is being taught here; the contract arrives already known, and it is being used purely as a function whose shape the mathematics is about.

How to Derive a Black-Scholes PDE in a Teaching Setting

1. Build a position out of the contract and the process.The position holds the contract and gives up some quantity of the process. The quantity given up is the hedge quantity. Naming it can wait one more step. The whole trick is that the quantity can be chosen afterwards.
   Why this is allowed: nothing has been priced yet. The step is bookkeeping about what is being held, and it makes no claim about what anything costs.
2. Expand the value of the contract using the chain rule for random processes.The value is a function of two things that move, so its change has three sources: the passage of time, the movement of the level, and the curvature of the value acting on the variance of the level. The third term is the one ordinary calculus does not have, and it is there because a Brownian path accumulates squared increments that refuse to vanish.
   Why this is allowed: Ito's lemma requires the value to be twice differentiable in the level and once in time, and it is. Ito's lemma was settled earlier in this subject area and is not re-derived here.
3. Choose the hedge quantity so that the random term cancels exactly.Both the contract and the process carry the same random driver. In the combined position that driver appears with a coefficient, and the coefficient is one equation in one unknown. Set the hedge quantity equal to the sensitivity of the value to the level, and the coefficient is nought.
   Why this is allowed: it is a choice, not an assumption. Any quantity at all may be held, and this is the quantity that makes the randomness go away.
4. Force what is left to earn the risk-free rate.The position now carries no random term over the next instant, so its value over that instant is known. A known amount that could be had for a known outlay must grow at the risk-free rate, or a position costing nothing that never loses and sometimes gains is available.
   Why this is allowed: no-arbitrage, settled earlier in this subject area. Step four is the only step on the list that is an idea rather than an operation.
5. Collect the terms and divide through.Two expressions for the change in the same position are now written down. Set them equal, cancel the time increment that multiplies every surviving term, and rearrange until the derivatives sit on one side.
   Why this is allowed: it is algebra. Nothing new enters, nothing is discarded, and no term is called small.

Now the same five steps in notation, one block at a time, with the prose kept between them rather than after them. The first block is only the position, and it says what is being held and nothing else.

Nothing has been claimed. Writing the position down on its own makes the absence of a claim visible: a reader who suspects sleight of hand can see that the first line contains no assertion at all. The first line is a description of a holding.

The second block expands how the contract value changes over a small interval. The expansion is the chain rule for random processes, settled earlier in this subject area and used here rather than proved. The curvatureThe second sensitivity of a value to the level it depends on, which is what randomness acts on to produce the extra term. term is the one an ordinary chain rule would not carry.

Substituting the movement of the process into that expansion splits the change into a part that multiplies the time increment and a part that multiplies the random driver. Giving up a quantity of the process gives up a proportional share of each part, so the combined position inherits both. Written out, the coefficient of the random driver in the combined position is the sensitivity of the value to the level, less the hedge quantity, all multiplied by the volatility and the level.

Cancelling the randomnessCombining holdings so that the random term in the combined position disappears exactly, leaving something whose next move is known. is the entire derivation, and everything after it is rearrangement. A number makes the word exactly mean something. At the locked parameters the sensitivity of the value to the level is 0.636831, so the random term in the contract alone carries a coefficient of 0.20 times Rs 100/- times 0.636831, and the product is Rs 12.736613/- a year. The hedge leg carries the same Rs 12.736613/- with the opposite sign. The two legs are the same expression, so the difference is nought to every decimal place worth writing.

The remainder now has no random term at the next instant, so its value an instant from now is known. Step four says what a known amount must earn.

The forced returnWhat a position carrying no randomness has to earn over the next instant, fixed by no-arbitrage rather than chosen. is where the assumption enters, and it is worth naming it as an assumption rather than a fact. Nobody measured that a hedged position earns the risk-free rate. The rate is imposed, on the grounds that the alternative is an arrangement the pricing argument has already ruled out. Every consequence that follows inherits that imposition.

What does each term of the equation actually do?

Setting the two expressions for the change in the hedged position equal, cancelling the time increment and rearranging gives one line. The line is the Black-Scholes equationThe relation that the value of a contract on the process must satisfy at every level and every time before the horizon., named for Black, Scholes and Merton, whose 1973 papers set it out. The equation holds at every level and at every time strictly before the horizon.

Read left to right, the four terms have four jobs, and it helps to give each a plain name before any number arrives. The first term is what the value loses simply because time passed and nothing happened. The second is what the value gains because the level, under the forced return, is treated as growing at the risk-free rate. The third is what the curvature earns from variance, the same term the chain rule for random processes introduced. And the right side is what the whole holding must earn if it is to be consistent with cash. The equation is a balance sheet for an instant: three sources of change on the left, one required total on the right.

Because every one of those four terms is a number once the locked parameters are fixed, the balance can be checked rather than admired. At the at-the-money contract on the standard process, at Rs 100/- with one year left, the four readings are these, and they add up.

The two sides agree to ten decimal places, and they agree because the closed form the equation admits was substituted straight back into the equation. The agreement is not a coincidence and not a check of arithmetic luck. Agreement is what solving an equation means, and having the numbers set out turns an abstract relation into something a reader can audit in an afternoon.

What is the Underlying Price, and what does moving it do?

The Underlying PriceThe level of the process the contract is written on, at the moment the value is being read. is the level of the process at the moment the value is read, and in the equation it is the variable S. Being pedantic about one thing here pays for itself later. In the derivation, S was a process with a time subscript and a path. In the equation, S is a variable free to take any positive number at all. The equation does not know about paths. The equation is a statement about a surface: for every level and every time, one value.

The shift from a path to a variable is the reason the equation generalises where a lattice does not, and also the reason readers get confused about what they are looking at. Feeding Rs 110/- into a calculator does not say the process went to Rs 110/-. The calculator is reading a different point on the same surface.

Moving the level does two things at once, and both are visible in the equation. The level moves the value directly through the sensitivity term. The level also enters the curvature term as a square, so a higher level makes the curvature contribution larger. On the locked contract at Rs 100/- the sensitivity to the level is 0.636831. A small rise in the level therefore lifts the value by roughly sixty-four paise for each rupee. The reading is local and it does not stay put, and the curvature term exists exactly because it does not.

Every step of Rs 10/- in the level buys more value than the step before it, and that widening is the curvature the equation carries as its third term. The last column shows the second sensitivity without one ever being computed: Rs 3.231802/-, then Rs 5.359362/-, then Rs 7.212370/-, then Rs 8.506090/-. A relation with no curvature term would have produced four equal numbers in that column. The last column here does not, and the failure to be equal is exactly what the third term of the equation measures.

What is Time to Expiry, and why does it enter twice?

Time to ExpiryHow much time remains between now and the horizon at which the contract settles. is how much time remains between now and the horizon, and it is the one input here that enters the problem in two separate places. Readers who miss the second place end up puzzled about why an equation in time needs anything else specified.

The first place is inside the equation itself, as the sensitivity of the value to the passage of time. The first term carries it, reading minus Rs 6.414028/- a year on the locked contract, or about minus 1.76 paise a day. The reading says that if nothing else moves, tomorrow's value is lower than today's.

The second place is not in the equation at all, but in the condition attached to the equation at the horizon. A partial differential equation (PDE) on its own has an enormous number of solutions; what makes it produce one answer is being told what the value must equal at a particular time and at the edges of the level. The equation describes how a value may change; only the terminal and boundary conditions say which value it is.

The direction of time in that first condition is worth noting. The condition fixes the value at the end and asks the equation to run the answer backwards to today. An instinct trained in physics finds the arrangement unusual. A condition in physics is normally given at the start. Here the known thing is the ending, and the unknown thing is now.

The everyday version is a lift with a fixed arrival. Given that the lift must be at the ground floor at six o'clock, and given the rule governing how it may move between floors, those two facts work backwards to where it must be at five. The rule alone, with no arrival, settles nothing about five o'clock at all. The equation is the rule. The condition at the horizon is the arrival.

What do the volatility and the rate each do?

Take them one at a time. The volatility and the rate enter the equation at different places and do different work. The volatility appears only in the third term, multiplied by the curvature. The rate appears twice, once in the second term and once on the right side, and those two appearances partly offset each other. The offset is why the rate moves the value less than a first glance suggests.

Start with volatility, the input the whole apparatus exists to handle. Raise it and the value rises, on the locked contract, from Rs 10.450584/- at 20 per cent to Rs 14.231255/- at 30 per cent. Lower it and the value falls. Take it all the way to nought and something clean happens: the third term of the equation disappears entirely, the equation stops being a diffusion, and the value collapses to the level less the discounted strike, Rs 4.877058/-. The collapse is a check that can be run, and it appears again below as the third of six.

Push the volatility the other way and a different limit appears. At 100 per cent the reading is Rs 39.840162/-, at 200 per cent it is Rs 69.057470/-, at 500 per cent it is Rs 98.788779/- and at 1,000 per cent it is Rs 99.999944/-. The value climbs toward Rs 100/-, the level itself, and never passes it. Volatility can make a contract on the process worth almost as much as the process, and never more, and any calculator that reports more than the level has failed.

The rate is the quieter of the two. Move it from 5 per cent to 10 per cent and the value moves from Rs 10.450584/- to Rs 13.269677/-. Move it down to nought and the value falls to Rs 7.965567/-. The value rises with the rate because a higher rate discounts the strike harder, and the strike is the amount handed over rather than received. The sensitivity at the locked parameters is 0.532325 for each percentage point, a modest number beside a value of Rs 10.450584/-.

The everyday version of the rate is a deposit and a bill. If a fixed sum is owed a year from now, and the rate rises, the amount that has to be set aside today falls. The strike is that fixed sum. Nothing more exotic is happening.

Why does the drift disappear, which is the surprising part?

Look back at the equation and hunt for the driftThe growth rate of the process under the physical measure P, which does not appear in the equation at all.. The drift is not there. The parameter that says how fast the process is expected to climb, the single number most people would guess matters most, does not appear in the relation that fixes the value. The absence is not a simplification and not an omission for tidiness. The absence is a result.

Here is where it went. In step two the drift entered the expansion multiplied by the sensitivity of the value to the level. In step one the position gave up a quantity of the process, and that leg carried the drift multiplied by the hedge quantity. In step three the hedge quantity was set equal to that same sensitivity. So the two drift terms were made identical and opposite by the choice, and they cancelled with the random term, in the same stroke, for the same reason.

Put the numbers in. At the locked drift of 8 per cent the contract leg carries Rs 5.094645/- a year and the hedge leg carries the same Rs 5.094645/- with the opposite sign. At a drift of 20 per cent both legs read Rs 12.736613/-. At a drift of 50 per cent both read Rs 31.841533/-. At a drift of nought both read nothing. The two legs are the same expression written twice, so their difference is nought at every drift, not small at plausible ones.

The control has just shown the whole point of the derivation. The chance of the contract finishing above Rs 100/- under the physical measure P moved from 0.460172 to 0.815940 as the control travelled. The odds nearly doubled. The value did not move by a paisa. The two facts are not in tension: the value was never a statement about the odds. The value was a statement about what building the payoff costs, and the cost was fixed by an argument that discarded the odds in step three.

What does the equation assume, listed rather than implied?

Assumptions that stay implied cannot be argued with, and an assumption nobody can argue with is not doing any honest work. So here they are as a list of five statements, each of which is a claim about the world rather than a piece of mathematics, and each of which somebody could reasonably dispute.

A sixth assumption sits further back, in the description of the process rather than in the derivation, and it is the one most often forgotten. The process moves by proportional increments driven by Brownian motion, and a process that moves that way never jumps. The cancellation was built on the level moving through every intermediate point. If the level can gap from Rs 100/- to Rs 80/- with no intervening values, the hedge quantity chosen an instant earlier cancels nothing. The whole argument depends on the process being continuous, and a jump defeats it not by making the answer inaccurate but by removing the step that produced the answer.

None of these six is a defect. The six are the price of getting one number instead of a range, and every one of them is a place where a later model in this subject area begins its work. What matters is that the six are statements, that they are listed, and that a disagreement with the equation can be traced to one of them rather than left as a vague unease.

What solution does the equation admit once it is pinned down?

Deriving an equation and solving it are two different jobs, and the solving is covered separately. The six checks below all lean on what that second job produces, so the closed form is worth setting out here. With the terminal condition at the horizon and the two boundary conditions in the level, the equation admits a single closed form, and Black, Scholes and Merton set it out in 1973.

Two things about that expression are worth pointing at. First, the drift is absent from the expression, as it must be: the drift was absent from the equation the expression solves. Second, the parameters of the standard process were chosen so that the rate plus half the variance rate is 0.07 exactly and the volatility multiplied by the square root of the horizon is 0.20 exactly, and the two intermediate quantities therefore land on 0.350000 and 0.150000 exactly at the locked parameters. The landing is a construction, not a coincidence, and it exists so that a reader can check the arithmetic without a calculator.

How is a calculator built from this equation checked?

Checking a calculator is the part a reader is most likely to use, and it works because the mathematics supplies a set of readings that are forced rather than chosen. A validation checkA reproducible number a calculator must return, so that a single disagreement condemns it. is a number the calculator has no freedom about. If it returns something else, the calculator is wrong, and no amount of plausibility in its other output rescues it.

How to Validate a Black-Scholes Calculator

1. Feed it the locked parameters and read the value.Level Rs 100/-, strike Rs 100/-, rate 5 per cent, volatility 20 per cent, one year. The value must be Rs 10.450584/-, and the two intermediate quantities must read 0.350000 and 0.150000 exactly.
   If the intermediates are right and the value is wrong, the fault is in the distribution function. If the intermediates are wrong, the fault is upstream of it.
2. Check that parity holds between the two contracts.The at-the-money contract less its counterpart must equal Rs 4.877058/-, and the level less the discounted strike must equal the same Rs 4.877058/-. The two figures must agree to six decimal places.
   The parity check catches a wrong discount factor faster than anything else. The discount factor appears on both sides and cancels only if it is right.
3. Set the volatility to nought.The value must collapse to the level less the discounted strike, Rs 4.877058/-. Nothing random remains, so nothing uncertain is being valued.
   Many implementations divide by the volatility and fall over here rather than returning the limit. A calculator that returns an error at zero volatility has failed the check.
4. Set the volatility very large.The value must climb toward the level and never pass it. At 500 per cent the reading is Rs 98.788779/- and at 1,000 per cent it is Rs 99.999944/-.
   A value above Rs 100/- means the contract is worth more than the process it is written on, and the boundary condition forbids that outright.
5. Set the time remaining to nought.The value must equal the payoff. At this strike and this level the payoff is nothing, so the calculator must return nothing rather than an error or a residual.
   The terminal condition is the one input that fixes which solution is in view, so a calculator that does not honour it is solving a different problem.
6. Read the sensitivity to the level.The sensitivity must sit between nought and one, and at the locked parameters it must read 0.636831. Anything outside that interval condemns the calculator.
   The interval is forced by the boundary conditions: the value goes to nothing at a level of nought and grows one for one at very high levels, so the slope between cannot leave that range.

Who reaches for this equation, and what do they do with it?

Almost nobody who uses this result is deriving it. People use it to check things, and the checking splits into three jobs that look different but are the same job underneath.

The first is somebody handed a number and asked whether it can be trusted. A value arrives from a spreadsheet or a script, and the question is not whether the number is attractive but whether the instrument that produced it is sound. The six checks above are exactly the tool for that, and they take about ten minutes. The person running them does not need to understand the derivation at all; they need to know which readings are forced.

The second is somebody who has to explain a disagreement. Two implementations return different values for the same inputs, and somebody has to say which is wrong and where. Because the equation is a balance between four terms, the disagreement can be localised: compute each term separately and see which one differs. On the locked contract those four readings are minus Rs 6.414028/-, plus Rs 3.184153/-, plus Rs 3.752403/- and plus Rs 0.522529/-, and a discrepancy will sit in one of them rather than being spread evenly across all four.

The third is somebody choosing which assumption to relax. Every extension of this result in the rest of this subject area begins by taking one of the six listed statements and replacing it. A volatility that moves rather than staying constant leads to one class of model. A process that can jump leads to another. A second rate for borrowing leads to a band rather than a number. Knowing the list of assumptions is what lets a reader place any later model on a map rather than meeting it as a new invention.

The household version of all three is a kitchen scale, and it is worth stating plainly because it stops a reader treating the output as more than it is. A scale reports a reading. The reading is not the weight of the object; it is what the object weighs according to this scale with this zero setting. Checking that the scale reads nothing with an empty pan, and reads a known amount with a known weight on it, is precisely what the six checks do. Neither exercise says anything about the object. Both say whether the instrument can be believed.

What is covered separately?

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