Delta Hedging in Theory: The Argument Behind the Price

The delta hedging argument establishes a price. Hold the contract against a quantity of the standard process chosen so the combined position has no first order response to the level, and what remains carries no randomness, so no-arbitrage forces its rate of return and the price follows. Delta and the hedge ratio are that one quantity named from two sides. The position must be adjusted because that quantity moves with the level.

A two-pan balance sits in the corner of every school laboratory, and it is the cleanest picture of the delta hedging argument. An unknown object goes on the left pan and brass weights are added to the right until the pointer stops moving. The moment the pointer stops, a weight has been established. The balance has not established that anybody is going to stand there holding the pans level for the rest of the afternoon. The balancing was scaffoldingSomething used to establish a result and then taken away, leaving the result standing on its own.. Once the object is lifted off, the pans swing wildly, and the weight that was measured is still the weight.

The delta hedging argument is that balance, and the price is the reading on it. Almost every difficulty a reader has with this argument comes from mistaking the scaffolding for the building: from reading a construction whose only job is to produce a number as though it were an instruction about what somebody must do. The scaffolding and the building come apart cleanly, and separating them is the whole of the difficulty.

The sensitivities themselves arrive already settled, and are covered separately. The contract also arrives already known, with what the contract pays covered separately. The contract appears only as a function whose response to the level is the quantity the argument needs.

What does the delta hedging argument establish?

The argument has three moves and only the first one contains a choice. Move one: form a combined position out of the contract and some quantity of the standard process, and leave that quantity unnamed for the moment. Move two: notice that the randomness in the combined position arrives multiplied by a single coefficient, and that the unnamed quantity sits inside exactly that coefficient, so there is one value of it that sets the coefficient to nought. Move three: point out that whatever is left over now carries no randomness at all, and that a position carrying no randomness has its rate of return settled by no-arbitrage rather than by preference, taste or forecast.

The third move is where the price comes from, and the kind of statement it makes is worth being precise about. The no-arbitrage step is not a measurement. Nobody went out and observed that positions with no randomness in them grow at the risk-free rate. The rate is imposed, on the grounds that any other rate would describe an arrangement the pricing framework has already refused to admit. So what the argument produces is a conditional statement: given that such arrangements are refused, the price is this one and no other. The conditional form is not a weakness. A condition can be inspected instead of a conclusion argued over, and inspection is what makes the result checkable.

The quantity is still free at this stage, and that is the reason the argument works at all. If the quantity had to be known in advance, the argument would need something it has not got. Because it does not, the algebra can be left to say what the quantity has to be, and the value it has to be turns out to be a quantity that already has a name.

Delta vs Hedge Ratio: are they one number or two?

Delta and the hedge ratio are one number. The two words are two vantage points on the same derivative, and the reason both words exist is that two different people needed to name the same thing for two different purposes.

Start from the contract's side. DeltaThe first order response of the contract's value to the level, meaning the part of the value change that is proportional to the move in the level. answers a question about the contract on its own: if the level moves by a small amount, how much of that move shows up in the contract's value? Delta is a property of the contract, computed by differentiating the value function with respect to the level, and it exists whether or not anybody ever forms a position against it.

Now start from the position's side. The hedge ratioThe quantity of the process held against the contract so that the combined position has no first order response to the level. The hedge ratio is the same number as delta. answers a question about a combination: how many units of the standard process must sit against the contract so that the combination stops responding to the level to first order? The hedge ratio is a property of a position, and it is a quantity somebody would have to go and choose.

The two questions are genuinely different. The reason they have the same answer is not a coincidence and it is not a definition. The identity is forced. A unit of the process responds to the level by exactly one, so the first order responseThe part of a change that is proportional to the move in the level, with everything of higher order set aside. of the combination is the contract's response less the quantity held. Setting that difference to nought has one solution, and the solution is the contract's response itself.

One derivative, two vantage points, one number, and there is nothing further to reconcile between them. On the at-the-money contract on the standard process, struck at Rs 100/- with one year to run, that number is 0.636831. Ask the contract's side what happens when the level moves and the answer is 0.636831. Ask the position's side how much of the process has to sit against it and the answer is 0.636831.

Delta Hedging vs Replicating Portfolio: how do the two arguments relate?

Delta hedging and replication are the same argument, run in two settings, and the difference between the settings is the whole of the difference between the answers.

A replicating portfolioA holding of the process and cash, built so that it reproduces the contract's payoff exactly at every reachable outcome, in a setting with a finite number of steps. argument works in a world with a finite number of steps. Over one step the level can arrive at a small number of places, so a holding of the process plus a cash amount can be written down, with the pair required to reproduce the contract's value at every one of those places. The requirement is a system of equations with as many rows as there are places to arrive at, and where the rows and the unknowns match it has a solution. The two arrangements deliver the same thing everywhere, so once the system is solved, the cost of assembling that holding today is the contract's price.

The delta hedging argument works in a world with no smallest step. There is no list of places the level can arrive at over the next instant, so there is no system of equations to solve. A derivative stands in its place, and the derivative does the same job the system of equations did: it identifies the one quantity that makes the two arrangements agree, not at a finite list of outcomes, but to first order over an instant. The price then comes not from the cost of assembly but from the rate of return the leftover is forced to earn.

Notice what that expression is. The expression is a slope measured across a chord, from one end of the step to the other, and the chord has a width of Rs 40.267200/- because the level can land at Rs 122.1403/- or at Rs 81.8731/-. The continuous argument uses the slope of the tangent at the starting point instead. A chord across a curved function and a tangent at one end of it are different slopes, and that is the entire reason the two numbers differ.

Neither number is wrong, and this is worth saying flatly because it is where readers most often go looking for an error. The one step holding of 0.549834 is the exactly correct answer to the question a one step world poses, and it reproduces the contract's value at both outcomes to every decimal place worth writing. The continuous quantity of 0.636831 is the exactly correct answer to the question a continuous world poses. The two holdings differ because the questions differ, not because one calculation went astray.

One turns into the other under refinement. When the construction is refined so that instead of one step over the year there are two, then four, then twelve, then fifty, and the holding at the start is recomputed each time, the holding climbs toward 0.636831, and it gets most of the way there in the first refinement.

One honest warning about that table: a reader who takes it as a smooth climb will be caught out. The five step counts were chosen to match the construction that is already published for this contract, and along that particular ladder the holding rises at every rung. Insert the odd step counts and it stops being a climb: at three steps the holding is 0.614086 and at five it is 0.623875, both below the four step figure of 0.629164. The approach to the limit oscillates between odd and even refinements rather than rising smoothly, exactly as the price of the same contract does, and a picture drawn as a smooth rise would be drawing something the arithmetic does not do. Convergence is a statement about the limit, not a promise that each rung sits above the one before it.

Why does the position have to be adjusted at all?

Because the quantity that removes the response is itself a function of the level, and the level moves. The moving level is the whole reason, and everything below makes it concrete.

Here is the everyday version. A cyclist going up a hill picks a gear that matches the gradient at the bottom. The gradient at the bottom is a fact about one point on the road, not about the road. Two hundred metres up, the slope has changed and the chosen gear no longer matches it. Nothing went wrong with the choice; the choice was correct where it was made and correct nowhere else. How often the gear has to change depends on how fast the gradient changes. The rate at which a gradient changes is a different property of the road from the gradient itself.

CurvatureHow delta itself changes as the level moves. Curvature is the second derivative of the contract's value with respect to the level, and it decides how fast a fixed holding goes stale. is that second property. Delta has a derivative of its own with respect to the level, and the size of that derivative is the rate at which a holding chosen at one level stops being the right holding at another. On the at-the-money contract on the standard process the curvature reads 0.018762 per rupee squared, and that number is settled separately in the treatment of the sensitivities.

The curvature can be checked against the thing it claims to measure without any calculus. Delta at Rs 99/- is 0.617815 and delta at Rs 101/- is 0.655330. The change across those two rupees is 0.037515, so the change per rupee is 0.018757. The curvature quoted at Rs 100/- is 0.018762. The two do not match exactly, and they should not: one is a slope at a single point and the other is a slope measured across a two rupee span of a bending function. Over a span that short, agreement to four decimal places is what to expect.

Now push the level further and the fixed holding stops being nearly right. Freeze the quantity at 0.636831 and move the level to Rs 120/-, where the contract's own response has risen to 0.896455. The combination is now short of the offset by 0.259624 per rupee, and that leftover response is not a rounding error: it is more than a quarter of a unit of exposure that the frozen holding is no longer covering. Move the level down to Rs 80/- instead, where the response has fallen to 0.221922, and the frozen holding overshoots by 0.414909 in the other direction.

What do the two arguments give on the locked contract?

Everything above lands on one contract: the at-the-money contract struck at Rs 100/- with one year to run, on the standard process that starts at Rs 100/-, carries a volatility of 20 per cent a year and sits against a risk-free rate of 5 per cent a year continuously compounded.

Two lines in that table deserve to be read together rather than separately. The first two rows carry the identical number, and that is the answer to the naming question: one derivative, two vantage points. The last row carries a different number, and that is the answer to the settings question: a one step world poses a slightly different problem and gets a slightly different answer. A reader who expects all three to agree has merged two questions that are not the same question, and a reader who expects the first two to differ has split one question that is not two.

The three readings worth carrying away from that control are these. At Rs 70/- the hedge quantity is 0.075875, at Rs 100/- it is 0.636831 and at Rs 140/- it is 0.978941. The one step replicating holding sits at 0.549834 and never moves. Neither the moving quantity nor the fixed one is the wrong answer; they belong to two different settings, and only the moving one belongs to the continuous setting.

What does the argument assume that an activity does not get?

Four things, and the list is what separates the argument from any activity resembling it. Naming them is not a disclaimer tacked on at the end. The argument's conclusion is conditional on exactly these four and on nothing else, so the list is the substance.

1. Continuous adjustmentThe quantity held is changed at every instant, with no smallest gap between one change and the next. There is no first moment after the present one at which the holding is still yesterday's holding.
2. Free tradingChanging the quantity costs nothing at all. No spread, no fee, no impact on the level from the act of changing, and no smallest size that can be traded.
3. One borrowing rateCash can be borrowed and lent at the same rate, in any amount, in either direction. There is one rate of 5 per cent a year in the argument and it applies to both directions.
4. Constant volatilityThe volatility of the standard process is 20 per cent a year at every level and at every moment, and it is known now rather than estimated from anything.

Continuous adjustmentChanging the holding at every instant, with no smallest gap between one change and the next. The argument assumes it; nothing else does. is the one that does the most work, and it is worth seeing what it buys with a number rather than an adjective. Over a step of finite width, the position that was built to carry no randomness carries something after all. The size of the leftover is set by the curvature and by the gap between the squared move that actually happened and the squared move the volatility allowed for.

Put the locked path through it and the leftover stops being abstract. Over the first month the level goes from Rs 100/- to Rs 97.64/-, a relative move of minus 0.023600, whose square is 0.00055696. The squared move allowed for over a month is 0.003333. Half the curvature multiplied by the level squared is 93.810087, so the leftover comes out at 93.810087 multiplied by minus 0.00277637, which is minus 0.260452 in rupees. The leftover is the price of a monthly step rather than a continuous one, and the argument does not pay it because the argument does not take steps.

What does the argument establish, and what does it not?

The argument establishes one thing: if such arrangements are refused, the contract has this price and no other. The result is strong, and strong precisely because it does not depend on anybody's view about where the level is going. The drift of the standard process, 8 per cent a year, appears nowhere in the price. The argument removed the level's randomness before ever asking what the level was expected to do, so two people who disagree completely about that drift agree exactly about the price.

The argument does not establish that anybody holds the combined position. Nor does it establish that holding the position would be cheap, or possible, or wise. Nor does it establish that the four assumptions describe anything. And it does not establish that the price would be different if somebody tried and failed to carry out the offsetting. The balance in the laboratory makes the point: the reading is a weight, and the weight does not change once the pans are no longer held level.

The gap between what the argument establishes and what people take it to establish is the single most useful thing to carry away, and it is a gap about status rather than about arithmetic. Nothing in the algebra is subtle. The subtlety lies in what kind of claim comes out of the other end.

Who actually leans on this argument, and what does it settle for them?

Somebody reviewing a valuation is the clearest case, and what they get from this argument is a licence to stop arguing about one particular thing. The argument fixes everything else, so when two readings of the same contract disagree, the disagreement has to live in an input. The drift is not an input. Anybody's view of where the level is heading is not an input. So a disagreement about the price is a disagreement about the level, the strike, the horizon, the rate or the volatility, and four of those five are read off a specification or a calendar. The narrowing turns a long argument into a short one, and knowing where a disagreement can and cannot live is most of the work of settling it.

The second use is diagnostic rather than settling. If a valuation is quoted together with the offsetting quantity, the two have to be consistent with each other: the quantity has to be the derivative of that valuation with respect to the level. A quantity that does not sit as the slope of the valuation being quoted is a sign that the two came from different places, and it is a cheap check to run because it needs only the valuation at two nearby levels. Delta at Rs 99/- and at Rs 101/- differ by 0.037515, and any quoted quantity that misses the midpoint of that span by a wide margin is worth a question.

The third use is knowing what a difficulty means. Somebody who reports that keeping an offsetting position aligned is expensive is reporting something true about an activity and nothing whatever about the price. Somebody who reports that a valuation and its quoted quantity are inconsistent is reporting something about the valuation. The two reports look similar and mean entirely different things, and the delta hedging argument is what tells them apart.

What is covered elsewhere?

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