- 1 Introduction to Chart to Scalar Theory
- 2 Geometry
- 3 Flat Market Simulations
- 4 Pricing Options
- 5 Single Solution
- 6 References
Introduction to Chart to Scalar Theory
Chart to Scalar Theory says there is a correspondence or relation between discrete buy and sell orders in the stock market volume and stock market geometry on the boundary. The boundary represents the charts which we can see. If you peel it back, there are buy and sell orders which are contained in the volume. The idea of the scalar is to encode the properties of the stock market into a scalar which represents a constraint of a hypothetical stock chart constrained by boundary conditions.
The motivation behind Chart to Scalar is to convert stock charts and trading volume into a mathematical framework to better understand how the stock market works, and make predictions. This has applications such as measuring capital inflows, market impact, simulating how external events such as insider trading affect how a stock will trade in the expectation that the market is flat or rising, explaining how crashes occur, and pricing options.
The name Chart to Scalar arises from a variational problem that involves finding a hypothetical RHS (right hand side) chart or function that represents the path of a stock constrained by a scalar value that encodes information about the LHS (left hand side) chart, volume, geometry and other characteristics of the charts and external events, in addition to the constraints of the boundary conditions and endpoints.
where is a function of minimum length
There are many models that try to explain how the markets work from a wide variety of research areas, such as behavior economics, behavior finance, fundamental valuation, but there are fewer papers that envisage the stock market itself as a closed physical system. Dilip Abreu and Markus K. Brunnermeier(2003) discusses in the context of behavior finance how presence of rational arbitrageurs can create persistent bubbles. A paper by Bouchaud (1998) posits a non linear Langevin equation as a model for stock market fluctuations and crashes. Racorean (2014) proposes a model that encapsulates all of the trading activity of a group of stocks as a high dimensional polygon. C Zhang (2010) appropriates classical physics establish the Schrödinger equation for a stock price. Hsinan Hsu (2001) applies the kinematic and kinetic theories of physics to derive price behavior equations for the stock markets.
In Chart to Scalar, the stock market is simplified to 1-dimensional 'universe' with two particles; a buy and a sell order and the action of the system is restricted in such a way where along the x axis there is only allowed to be one y intercept (obviously because stocks cannot go 'back' in time). The variational equation outputs either a monotonically increasing/decreasing concave, linear, or convex RHS (right hand side) curve of how the stock should theoretically trade given the values in the equilibrium equation . If , the equilibrium equation can be interpreted to mean that the amount of money flowing into a stock on the LHS must equal the amount that flows out on the RHS. Or if , the amount of money that flows into the stock on the RHS must equal the amount that flowed out on the LHS.
Rather than looking at forces, we reduce the market to static scalars that contain information about the market. The equation of motion is the path of least action that encloses the area of the scalar. This is a Lagrangian approach instead of a Newtonian one. Second, we introduce LHS 'resistance' which is encoded into the scalar. This resistance is analogous to the resistance of gravity or friction in Newtonian mechanics or a energy barrier in quantum mechanics.
The first section covers the notation, the formulas that arise out of the buy/sell order annihilation process, and some examples. Then the model is extended to include a geometric component to provide an additional degree of freedom when needed. The second part will cover simulations and data extraction. In the third part of the paper an option ricing formula is derived via Chart to Scalar and the results are compared to Black Scholes. The inclusion of volume and trade size could explain phenomena observed in real live stocks that isn't accounted for in conventional option pricing models. This is important because traditional models (such as Black-Scholes and Binomial) have many significant limitations which Chart to Scalar may be able to resolve, such as the inability to account for historical market movements (Baggett & Thompson (2006)) and their frequent overpricing of options, with the overpricing increasing with the time to maturity. (Hull & White (1987))  In our paper, we conclude that Chart to Scalar options grow at a slower rate than Black Scholes options , and secondly, that rare events according to Black-Scholes are more common in Chart to Scalar.
1. The tendency of a stock to rise or fall is determined by it's volume of the RHS (right hand side) of the stock graph functional relative to the volume of LHS (left hand side) functional, in addition to other variables in the stock market equilibrium equation.
2. Buy and sell orders and their interaction are the propagators of price change.
3. In a restricted 1-d trading space, stocks trade along a geodesic, swathing out convex, linear, or concave curves as the solution of a variational equation constrained by the equilibrium equation.
4. Most equations are scale invariant meaning that adjusting time frames by a scaling factor will yield the same results.
Trading Space and Notation
The trading space is a quasi two-dimensional space that consists of two charts. We call it quasi-two dimensional because backward motion is prohibited and hence all curves cannot have more than one y intercept. One chart is on LHS (left-hand side) and the second on the RHS (right-hand side) and are adjoined at some point, denoted as . The resistance and support forces denoted by occupies the LHS. A horizontal line drawn through will intersect the LHS curve at where .
The curve is either concave, convex or a line. Depending on and additional variables like buy or sell orders, a RHS curve is produced, which is either concave, convex or linear.
On the LHS, for either a convex , concave curve or line is signed either + 1 or -1 denoted by . A positive sign means that the price on the LHS is rising. This is a support curve. A resistance curve has a negative sign. The RHS always has the opposite sign as the LHS or . If the RHS is rising it will run up against the negatively signed LHS resistance curve. If the RHS is falling it will fall against the positively signed LHS support curve, as shown in the appended diagram. (the shaded region represents volume)
Negative values are not permitted
The perimeter trading space:
For the parameters are reversed:
Imagine a diagonal line that connects the points for a LHS curve and for a RHS curve . For either LHS or RHS, a concave curve cannot exceed this line and convex curve cannot fall below it.
With the new notation, can be re-written as . As a rule of thumb, must always be positive, for any problem. must be chosen in such a way that this holds.
In Chart to Scalar theory, the interaction of buy orders and sell orders is the propagator is price change. This is expressed by the simple system of equations:
Where is some function inputs price and returns volume. And is RHS volume and is the average order size.
The challenge is finding . In the subsequent sections, we'll try to derive a relationship between price and volume using simple geometric shapes such as triangles, lines, and arcs to represent the paths of stocks. An alternative derivation of using an order book can be found here: User:Optionpricing
As a stock is trading in some time frame there exists a value called an energy level that is the product of two or three components: a slope/time component , a volume component and geometric component , which is added for more complex problems and will be discussed in further detail later. The LHS (left-hand side) values are denoted by subscript such as and RHS by such as . The RHS product is ; the LHS is . Setting them equal, we have the equilibrium equation: .
In this trading space a buy and sell order of equal price and quantity will instantly annihilate producing no net energy (represented as a flat line on a chart). An excess of buy orders will cause a price rise appearing as a rising slope on a chart or . However, the actual price of the stock need not matter. Instead, the ratio of buy orders to total orders does. Thus a stock trading at 10 cents will experience the same percentage change in price as one trading at $10 if the ratios of buy and sell orders are the same. We're looking for a function where where is a scaling factor; the solution is . All else (volume, time) being equal, a stock rising from $90 to $100 should have the same properties as one rising from $9 to $10.
We also need some way to define the x component of a slope in a manner that is time or scale invariant. Define the variable that relate the time duration of events on the LHS to the RHS. The actual units of time are not important. For the LHS, the default time is . This serves as our reference point. For example, if then it means the time or duration of the LHS of the chart is the same as the RHS.
We define as a function that ranges between 0 and 1 and measures the percentage of volume 'buys' or 'sells' remaining after buy and sell order annihilation has occurred, and is dependent on the rate of change of a stock in some time frame or . can be interpreted as some function of or .
For the RHS, we need to define both in terms of the ratio buy & sell orders and the rate of change.
In terms of rate of change of price over the 'base' or time duration (where ) and :
and in terms of buy and sell orders:
Intuitively this makes sense. If then or no change in price. If vastly exceeds then increases.
Because the total number of buy and sell orders is then .
Thus can be rewritten as:
We're looking for a function that converges to 1 as the ratio of increases, and 0 if . The example below suffices:
Define The total number of orders is equal to the volume divided by the average order size. and are recovered from the equation above:
Consider there are 5 total orders: three buys and two sells in some infinitesimal trading space. The two buys and sells will cancel and one buy order will be left. Hence, . If then all the buys and sells have cancelled and there is none left over, . If then because all of the volume is buy orders.
We can convert charts into buy and sell orders and vice versa through the relation:
Putting these values back into and and are obtained.
is our time/slope reference triangle:
The buy and sell order equations:
is added to account for negative RHS slopes, so to preserve symmetry between the buys and sells.
Suppose a stock rises from $50 to $51 in a one day period with a volume of . How much volume would be required for the stock to fall back to $50 instantaneously? is irrelevant.
The time duration of events on the RHS is infinitely small relative to the LHS. Because and , we have
Setting up the equilibrium equation
The solution shares need to be sold.
Part 2: How many buy and sell orders?
With (the LHS is rising), when we plug into the buy order equation we get . This answer is interpreted to mean that there are zero buy orders as the stock falls from $51 to $50 in an infinitesimal time frame. . Thus we have 100% sell orders for the volume, for any .
A stock has risen in from $10 to $11 in 30 trading days with an arbitrary daily volume. How much volume is requires for the stock to fall back to $10 in one trading day?
The solution is 2.3 which means that it requires 2.3 times the daily volume for the stock to fall back to $10 in a one day trading period. This agrees with empirical observations of how stocks and indexes can erase months or weeks worth of gains in single day or a couple days during a crash. An increase of 130% of the daily volume or 7.8% of the total LHS volume is sufficient for the stock to forfeit a month worth of gains.
Part 2: How many buys and sell orders?
When we plug into the buy order equation we get This answer is interpreted to mean that 20.5% of the RHS volume consists of buy orders and 79.5% sell orders, for any . Also note how
This price/volume dynamic could explain why flash crashes occur and how preventing them would be difficult because a relatively little amount of high density volume is required for a stock experience a rapid decline.
For certain problems the linearized versions of and doesn't give enough degrees of freedom, hence a geometric component is introduced. measures efficiency of a path . is a value between 0 and 1 that measures how much a stock path defects from a strait line, with 1 being a line. A line is the most efficient path and has a , but in some other circumstances such as modeling insider selling then a different path is stipulated that may be convex or concave shaped. To see why an extra variable is needed consider we have our buy order equation with
Consider the equilibrium equation
If we wish to simulate churning we have:
(Churning volume is volume that is evenly split between buy and sell orders)
Obviously, the equilibrium equation doesn't hold, but introducing a new variable bypasses this.
As grows the path becomes less efficient as indicated by falling . By defining to be a functional path, we can give the output as a solution of a variational equation as the path of least action as constrained by the equilibrium equation. This path may not be linear, but concave or convex depending on our criteria. The path is how the stock should hypothetically trade on the RHS with the introduction of constraints, such as extra buy or sell orders or churning volume.
The buy & sell order equations include
As , then and . As more churning volume is added, the number of buy orders approaches half the total volume . Either an inefficient path or a small results in an even admixture of buy and sell orders.
Define as a function that ranges between 0 and 1 depending on how much the area under the curve or differs from the area under the right-triangle adjoining the points (LHS) or (RHS), and approaches zero for increasing deviation or 1 for a diagonal line.
An expression of or for concave curves:
The convex curves:
where are convex curves
If are expressed as a column vector, the complementary convex curves are found by using a transformation matrix to reflect about the line adjoining for the LHS and for the RHS. These new curves have the same as the original curves about the endpoints of the region.
For example, let's assume our RHS space is a square with the coordinates and we have a concave curve . We have: and for we also have
The following table determines is you use a convex or a concave curve
Convex LHS Convex RHS
Convex LHS Convex RHS
Concave LHS Concave RHS
Concave LHS Concave RHS
Invariance of 
Crucially, exhibits scale invariance. Imagine that a chart occupying a computer screen is re-sized in such a way that the curves or appear stretched, compressed, or shifted. Another person on a different computer terminal looks at the same stock with a different time frame. Both should arrive at the same value of .
Begin with the transformation:
The limits of integration are compressed and and are the same. Define . Via integration by substitution obtain:
Plugging in the transformed values of and into 1.0 the 'a' variable cancels out and we're left with the original equation.
For example, consider a concave RHS chart given by the potion of the equation where . We know because the RHS is rising. Also: , , , , . Plugging everything into 1.0 gives
Now consider the the person viewing the graph wants a longer time frame on the same screen. will be compressed on the x axis by some factor . Substituting gives . The transformed values of are now because the time frame is compressed. As expected and are unchanged because all we're doing is making the graph compressed, not changing the starting and final price. Plugging these new values into 1.0 still gives .
Invariance of Volume
Let's assume is bounded between and where . There also exists a LHS volume function between the aforementioned boundaries. For simplicity, will be linear. The total volume between and will be denoted as . Hence, we can express as a proportional relationship .
Suppose 'Tom' on computer 'A' observes that stock XYZ has carved a convex chart shape by bounded by . 'Mark' observes this this shape as well but his chart is stretched/resized on his computer and he gets bounded by . . For example, assuming , they will both arrive at:
Or for general volume
Variational equation for 
With , we have the complete equilibrium equation:
The geodesic is the path bounded between the endpoints and that has the shortest length and satisfies the equilibrium equation.
This is a Isoperimetric problem where L is minimized while constrained by and , bounded by , , , and :
Constrained by the equilibrium equation and appropriate boundary conditions
This can be solved using the Euler Formula and Lagrange multipliers, although the boundary conditions makes the problem difficult and closed form solutions often don't exist.
Examples computing 
The formulas below assume LHS volume does not vary
For curved , computing can be difficult, but there are some functions such as quadratics and ramps that are easier. Consider with with the boundary , and .
For quadratics of the form:
Upon simplification we have:
Or the concave curve:
After some labor:
Consider we have a LHS ramp function connected by the coordinates where
We have: , and
Flat Market Simulations
In this section, Chart to Scalar will be applied to examples where the expectation is that the market will be flat on the RHS an we wish to simulate how changing or adding variables will affect the hypothetical RHS chart. In a flat market the expected number number of buy & sells on the RHS is equal. Because we have a particularly simple way of writing the sell order equation as . To derive this, begin with the equilibrium and buy order equation:
Note: to simulate a stock sale in flat market we have . The LHS resistor is positively sloped.
The information on the LHS is known ahead of time, but the RHS information is unknown aside from which is evenly split between buy and sell orders (because the market is anticipated to be flat). To make the buy order equation udeful for simulation purposes, we can replace tho unknown RHS variables with the known LHS ones by rearranging the equilibrium equation:
Plugging this into the buy order equation:
As we showed in a previous section, is a function of , giving us a relationship between buy orders and price. Written out completely, we have:
And the complimentary sell order equation:
In the flat market we know hence we have: or as a solution. If we add extra sell orders but leave unchanged, then we have upon solving . This makes sense because we would expect extra sell orders to result in a lower price.
Consider a linear LHS for stock has fallen from $33 to $30 in a 1 month period with a volume of and the RHS is a reflection such that . The LRS/LHS triangles are orientated in such a way that the LHS and RHS vertexes., , , and
It's trivial to show that the equilibrium equation holds. But if we increase volume on the RHS by and we still want the equilibrium equation to hold and we require that , then we must modify . Solving gives To generate the curve use the or equation with the appropriate boundary variables.
We obtain: With the endpoints: and
The is a monotonically increasing curve of minimum length that lies between the boundaries. It's easy to see that when the solution is a line. Increasing results in an convex curve that almost completely hugs the boundaries and while enclosing an area that approaches 30. The diagram below illustrates how increasing results in a concave path
If we let we can compute the proportion of buy orders to total volume with and without the addition of extra volume
Originally, we have or about 52% of the total 10^8 volume are buy orders. If then we have total orders and only 51% of the total volume are buy orders.
Next simulate how insider selling (or a fund selling a specified number of shares) changes the pathway of a stock. The additional sell orders result in a concave/convex curve with a lower final price. The itself has to satisfy both the buy order equation (to factor in the additional sell orders) and the equilibrium equation.
We will use most of the same information in the prior problem but change the coordinates to for the LHS and for the RHS. When plotted the LHS serves as the support triangle and the RHS shows a equal number of buy/seller orders, denoted as a line between its two points. We'll begin by solving for or the final price Let , , , , (irrelevant for this problem) and (to indicate the stock is falling). We'll assume an insider is selling shares.
Because the RHS is initially a line, we have an even admixture of buy & sells and then we add extra sells. And since is linear we have . After some manipulation, we find that Using the buy order equation we have or in expanded form:
Solving gives , or a four percent decline attributed to insider selling.
Solving the equilibrium equation we find . Because is so close to 1 we know the resulting shape is approximately a line with the endpoints and .
An additional application is if a firm or individual shareholder wants to sell to a private investor to raise money. How should the shares be priced? We can use Chart to Scalar to simulate the instantaneous sale of some number of shares required to raise y dollars. If the present price is , the number of shares sold is where is the final price after the shares are sold. Since the transaction is an instantaneous event there are no buy orders on the RHS and hence and obviously since the stock is falling on the RHS from the selling.
For example, consider a stock has risen from $160 to its present price of in 60 trading days with an average daily volume of . The shape of the LHS is linear . If a large shareholder wants to raise $1 billion , calculate how many shares should be sold and at what price?
The volume function:
Solving, we find assuming the shares as sold instantly on the open market and 4.5 million shares sold to raise $1 billion. But some shares will be sold at around $240 and others at $202. Taking the midpoint $221 is a fair value for the private secondary offering.
In summery, we see that the buy order equation tells us the final price of the stock; the equilibrium equation gives us the shape.
Chart to Scalar can be used to show that charts that resemble 'bubbles' are more susceptible to collapse, in agreement with real life results. A bubble on a stock chart typically resembles a parabola. Using the results in section 3, we will compute for our bubble and for an 'anti-bubble'. Then we'll show with some calculus that fewer shares are needed for a stock with a 'bubble' chart to fall, and hence is more vulnerable to collapse than the 'anti-bubble'.
Lets assume that The simplest 'bubble' that passes through the endpoints is . Using methods of linear algebra, it's trivial to show the refection about or 'anti-bubble' is . Both these curves have the same equal to 2/3 when the integrals are evaluated between 0 and 1. Showing that the 'bubble' chart is more vulnerable to collapse is the same as showing that for some Furthermore, using the properties of inverse function we know that . Let and we have and . After some labor we have the following inequality (where the right hand side is the anti-bubble):
Using the binomial theorem on the square root and dividing both sides by we have
Now it's obvious that as approaches zero the anti-bubble is bigger.
If we let with the same boundary conditions as mentioned earlier, we can simulate the behavior as becomes increasingly concave - that is letting approach infinity. Such a curve will appear to hug the boundaries fur sufficiency large .
Using methods given in section 3 and letting , we compute:
Using the binomial theorem and upon simplification we have an infinite series that begins:
Now it's obvious that goes to zero as n becomes increasingly large. What this means is as a stock chart appears increasingly parabolic, it becomes less stable and more prone to falling. This is because of the increasingly small cross-sectional area of between and .
The Chart to Scalar option pricing formula is a consequence of the broader Chart to Scalar theory through an equivalency relation between the statistical property of buy and sell orders and price.
Differences between conventional options pricing models and Chart to Scalar:
1. Volatility-like variable in Chart to Scalar is a combination of variables such as time until expiration, volume/price differentials, and trade size .
2. Normal distribution instead of log-normal.
3. Options prices grow at versus for Black Scholes. Short term options are more expensive and longer term cheaper in Chart to Scalar than Black Scholes. (this is explained in more detail in the final section)
Derivation, Part 1: The Displacement Equivalence Relation
The idea is to establish a relationship between discrete 'steps' and price displacement.
Consider a hypothetical stock chart where a stock has moved some amount (usually a small percentage) over some time duration (often measured in years). It can be visualized as a triangle, with the vertices being and and where is the present price of the stock and the displacement is .
Denote as the time until expiration
Consider a discrete sum of up and down stock orders denoted by
Each 'up' and 'down' represents a 'time unit'. Adding the 'ups' and 'downs' gives the sum of units.
The second part of the fundamental interaction is the difference between 'ups' and 'downs':
This means that the difference between 'ups' and 'downs' gives a function in terms of a new displacement, where .
The pair of linear equations solves for and :
Consider the proportional relation between two displacements, the base one with and our new one,
Solving for gives the needed function in terms , which is plugged into :
When , the stock is unchanged, meaning that the number of 'up' units is equal to the 'down' ones.
What we've done is establish a relationship between displacement of price and 'up' and 'down' units.
'Up' and 'down' units, analogous to tossing a coin, also obey a normal distribution:
There is also for the price.
(this is because if the stock is unchanged, hence meaning that the number of 'up' units is the same as 'down', resulting in no displacement.
We have to find
Because of the equivalence between units and price displacement, the can be solved by setting
From the equivalence:
Rearranging gives the classic result: setting (for a single year and is the fraction of the year)
Derivation, Part 2
Let be linear so that
The LHS resistor force is denoted by a triangle with the vertexes .
The total volume between and is distributed evenly.
Thus the volume between segment and is
Plugging into the buy order equation we have (first order Taylor approximation for is used):
Like above, let Also
The normal distribution is a fundamental solution to the heat equation. This is an initial value problem on (0,∞) with homogeneous Dirichlet boundary conditions.
is a function of and time, hence
So we have (for some constant b)
Because k is a function of t, integration is necessary to solve the PDE
Letting , and restricting the bounds of the integration from to and evaluate the integral to compute the call:
If and are of the form with the time factors multiplied by a constant m, then:
Where is the time until expiration.
Suppose a hypothetical stock has fallen from $40 to $30 in a 6 month period (120 days) with a daily volume of and and . Calculate the $30 call with an expiration of 20 days. Assume no interest.
Since , the call option pricing formula simplifies dramatically and we have .
Chart to Scalar Option Pricing Vs. Black Scholes
Consider the example above where (at-the-money). The call price for Chart to Scalar can be approximated as:
The approximation with Black Scholes for at-the-money calls is:
In both instances, is the number of days until expiration.
The notable difference between Chart to Scalar and conventional option pricing models is that sigma is proportional to instead of .This generates fat tails due to the intrinsic property of the price/volume dynamic, without the need for added fat tail parameters or the market being incomplete.
To test the difference, using the same variables as in the earlier example for the at-the-money call, the strike prices for a variety of times is plotted. The difficulty is converting price displacement into a volatility, but I think seems reasonable based on empirical evidence. This is more volatile than an index fund, which has volatility that fluctuates between .15 to .20.
For the example above, Chart to Scalar (blue) v. Black Sholes (purple) for an at the money call. The vertical axis is the call price. The horizontal is the number of trading days until expiration.
A noted limitation of Black Scholes is that it tends to underestimate the probability of uncommon events, even though these events are observed more frequently in real life. For the example given in this section, if we raise the strike to , reduce the time until expiration to one day , and keep all other variables unchanged, Black Scholes gives a probability of about 1/4000 of the option being exercised, or to put it another way one would have to wait a decade for the underlying to rise 4.7% in a single day. Chart to Scalar, on the other hand, gives a 1/150 probability, which is more realistic.
High Frequency Trading (HFT)
We hypothesise HFT has a stabilizing effect on the market. E. Renshaw (1995) showed that the market is more stable than it used to be. Chart to Scalar could provide a mathematical explanation for why this may be.
Consider which we derived earlier
If the trade size is small relative to , then the variability of moves is smaller. ( for example) . On the other hand, if we increase , ,and by a scaling factor it cancels out and is unchanged. So in the context of this model, HFT is either stabilizing or neutral.
For example, without HFT let’s assume we have:
Now introduce HFT:
We’ve increased the volume of the stock on the LHS (left hand side) and the RHS but the trade size is unchanged because we’re assuming the high frequency trades are no larger than normal trades. Because:
We see that the noise orders have a stabilizing effect.
The limitations of this model is the assumption that HFT is the same as random trades and ignoring possible feedback effects.
Solutions to problems in Chart to Scalar where are not unique, meaning there are two solutions: a concave and a convex one for the RHS (right-hand-side).
The introduction of integrals on the LHS and RHS equilibrium equations, respectively, allow for a single solution. This is because solving the equilibrium equation results in two unique scalar values . Due to the variational principle, the value of d which confers with a longer can be discarded, leaving a path of least distance that adheres to the constraints.
Define as the average price of :
The equilibrium equation with :
The buy order equation:
Sell order equation:
RHS Convex: RHS Concave:
Example 1: Bursting of a Stock Market Bubble
Consider a simulation of a simple stock market market to show how the concavity of the RHS of the bubble bursting must match the LHS of the bubble inflating
The inflation of the bubble on the LHS is a concave curve given by bounded between . The RHS is bounded between . For this simulation, the hypothetical stock rises from to and it falls back to as the bubble deflates. () is chosen to indicate the LHS is rising. The LHS and RHS volume is equal and . Thus the conditions are imposed:
The time symmetry condition: are imposed, meaning that the duration of events on the LHS is equal to the RHS.
For the LHS concave curve of form , the formula is used (making reference to original paper):
Hence, (because ) And
The equilibrium equation which is solved for d (one for the concave RHS and convex RHS):
These are plugged into their respective formulas to calculate the defect. The goal is to show the convex solution has a greater defect than the concave one:
Because of the scale invariance properties of , the above formula reduces to:
letting , we have the infinite series expansion about :
Because the concave has a smaller defect, the path is shorter (closest to a strait line), and hence the concave path is chosen as minimizing the action. Which completes the proof.
This concave-on-concave symmetry agrees with examples in real life of various asset bubbles bursting
Example 2: (buy order equation)
Example: , the x coordinates are the same as the example above, and , and
For this example, the RHS is a concave reflection of the RHS, thus and . To compute the buy and sell orders for the RHS:
For the rest of this summary, The actual choice of does not matter for non-statistical problems.
Example 3: (system of equations)
What if is different? Consider a general case where is only slightly less than . Then it becomes more complicated because is unknown and cannot be assumed to be equal to and becomes a function in terms of instead of just 2/3.
The buy order equation and equilibrium equation must be combined for problems where the and or components are not equal, the result being a system of two equations that is solved for (both for the concave and convex) and (the final price of the stock, for both the concave and convex) The value of and corresponding to the greatest defect is discarded.
The components are as follows:
The inverse of evaluated at :
Both will be specified later.
In the first example, . is the functional form of LHS volume in terms of , whereas is the total volume along the interval
RHS Convex: RHS Concave:
The system of equations are solved for and :
After some labor, has the series approximation about :
As , the solution is . This is because if the number of buy orders is half of the RHS volume, we expect the stock to end unchanged.
Consider a small imbalance: . Let:
Solving fox p and d gives six possible solution pairs, but the only one that logically makes sense
The convex and concave curves enclose roughly the same area, indicating the that resulting LHS path is very close to being linear. Plugging these solutions into their respective convex and concave shows that the defect is very small, roughly 2.5%.
Example 4: deriving the Market impact square-root rule
A formula very similar to the 'square-root' rule  is derived.
Consider the instantaneous sale of stock. For simplicity, let the LHS be linear.
An instantaneous sale means . Therefore,
Define as the lower-end of the stock range and as the present price.
The LHS be visualized as a triangle with the vertices
Since the LHS is linear,
is the final price of the stock after the instantaneous sale is rendered.
Because (the LHS is rising),
Where is the total volume of the LHS between x=0,x=1 (some period of time)
This is obtained by taking the inverse of and finding the proportion of volume that is 'liberated' by the stock falling to . Via the triangle, Set . Then and gives the proportion.
can be approximated as
Set where is the 'impact'
Setting up the equilibrium equation and solving for we have:
The volatility-like variable can be written as: . Hence, we have:
As we would expect, the volatility term is scale invariant, but the impact is proportional to the initial price . If is much small than , we have a greater price range (more volatility).
The term is somewhat arbitrary, but we still have the volatility and square-root impact relation.
Example 5: inflows and outflows
Inflows and outflows for the RHS chart are calculated. However, this is only an approximation; calculating an exact inflow or outflow using a single, closed-form expression is impossible and impractical, but we can get a ballpark estimate. For small increases or decreases in price (), we can use the linear approximation of the natural log and . We have:
If , we have an outflow. If , there is neither an inflow nor outflow
Here, the average price is , which is a linear approximation of the exact average price of . This is good enough for small changes in price.
can be added if the shape resembles a hyperbola, regardless of concavity
Let's assume we want to calculate the dollar inflow for Microsoft for a single trading day. If on the RHS we observe and the shape of the RHS is concave and resembles the function (noticing this satisfies and endpoints and has at ) We have and . Putting it all together, we have an inflow of around $32.6 million while around $303 million changed hands.
- Baggett, L. Scott; Thompson, James; Williams, Edward; Wojciechowski, William (October 2006). "Nobels for nonsense". Journal of Post Keynesian Economics. 29 (1): 3–18.
- Hull, John; White, Alan (June 1987). "The Pricing of Options on Assets with Stochastic Volatilities". Journal of Finance. 42 (2): 281–300.
- Gatheral, Jim (October 2011). "Optimal order execution". Text " JOIM Fall Conference, Boston " ignored (help); Text " http://faculty.baruch.cuny.edu/jgatheral/JOIM2011.pdf " ignored (help)