1. Introduction

In this post I work through the main result in Arora, Barak, Brunnermeier and Ge (2012) which formulates a simple securitization setting with underlying assets (e.g., mortgages) and derivative assets (e.g., CDOs) where:

1. The seller can cheat and create non-random derivatives that will generate lemons cost for the buyer, and…
2. With limited computational power, the buyer will not be able to detect whether or not the seller has cheated.

For a finance-lite discussion of the results, see Dick Lipton’s excellent blog post. I’ve also created a short deck of slides (here).

2. Why Use Derivatives?

I begin by walking through the motivation for using derivatives in the first place via a simple model in the spirit of DeMarzo (2005). In this sort of model, derivatives like CDOs solve an asymmetric information problem between a seller (e.g., a investment bank) and a buyer (e.g., a pension fund).

Suppose that a securities seller owns mortgages which each payout defined as:

(1)  

where is an iid random variable such that with probability and with probability while is Gaussian white noise with distribution . For simplicity, I set and in the analysis below.

The model relies on two main assumptions:

1. The seller knows for each while the buyer does not.
2. The seller would prefer in cash to in mortgages with .

The first assumption defines the information asymmetry. To make the problem interesting, it also has to be the case that the seller cannot credibly reveal his information prior to trading. e.g., he can’t just point out which mortgages are bad apples in an email. The second assumption gives the seller a good faith reason to get rid of the mortgages. This assumption can be interpreted as saying that the seller is less liquid than the buyer and would prefer to have more cash.

If a buyer considered purchasing a “random” collection of mortgages from the seller, he would be worried that the seller would put all of the bad mortgages in this collection so that the buyer would receive a collection of mortgages worth rather than a “fair value” of . In such a world, the buyer would never purchase mortgages from the seller and the seller would be forced to hold onto all of the mortgages which he values at only due to his liquidity concerns. This loss of is the core of the asymmetric information problem.

However, a smart seller can eliminate the asymmetric information problem entirely via careful security design. Suppose that he (a) offers to sell a security with two tranches where the senior tranche has a face value of and (b) holds onto the junior tranche. After agreeing to accept the first in losses by holding onto the junior tranche, the seller essentially converts the remaining payout from the the bundle of mortgages into a riskless asset as since the white noise terms cancel out on average. Thus, by securitizing the mortgages the seller is able to get the “fair” value of his assets in spite of the asymmetric information problem:

(2)  

3. New Framework

In Arora et al. (2012), the seller still wants to sell mortgages which each payoff with probability . However, Arora et al. (2012) adjust this standard framework in three key ways:

1. The seller cannot hold the entire junior tranche.
2. The buyer and seller engage in a zero-sum transaction.
3. The seller creates binary derivative securities (henceforth, CDOs) from these underlying mortgages.

The first assumption means that the seller cannot fix the asymmetric information problem via security design in the same manner as DeMarzo (2012). The second assumption removes any possibility that derivative creation might be socially beneficial. In Arora et al. (2012), the seller gets utility from the amount of money he can extract from the buyer via clever security design. The third and final assumption is the key step that allows computational complexity to enter into model. Specifically, each CDO’s payout now depends on of the mortgages in the entire pool. Each of these new CDOs payout an amount where if no more than of the underlying mortgages default and otherwise with:

(3)  

Given the second assumption, the seller’s assignment of the mortgages to CDOs then defines a bipartite graph as illustrated below:

Now, suppose that the seller has private information that a subset of the mortgages are lemons and guaranteed to payout . Here, private information means that the seller knows which mortgages have a payout of while the buyer is aware that the seller knows this information. To increase his profit, the seller might carefully design this graph using his knowledge of which mortgages are lemons in order to reduce the value of the CDOs relative to their “fair” value of .

Definition (Lemons Cost): Let denote the lemons cost to the buyer if the seller knows a set of the mortgages will payout :

(4)  

where has units of dollars.

Thus, the lemons cost can be thought of as the amount by which a CDO’s value could possibly decline if the seller knew that a set of lemons existed in the entire mortgage pool and tried to cheat the buyer out of as much money as possible. For example, suppose that in the model from the section above the seller didn’t actually know each and every but instead just knew that a subset of the mortgages would have . Then, the fair value of the entire bundle of mortgages would have been so that the lemons cost would have been .

The key insight of this paper is that trying to figure out whether or not the seller has randomly assigned mortgages to CDOs is equivalent to the densest subgraph problem which does not have a polynomial time solution. e.g., if the buyer faces any computational constraints whatsoever, then the seller can foist some lemons costs on the buyer without being detected. More formally, the buyer faces the following problem:

Definition (Densest Subgraph Problem): The densest subgraph problem is to distinguish between the following two distributions, and , over the bipartite graph above:

1. : The seller chooses random mortgages for every CDO.
2. : Let nature select a random set of assets with payout. The seller then selects a subset of CDOs (i.e., boobytrapped CDOs) and a number of lemons to plant in each boobytrapped CDO. For every non-boobytrapped CDO, the seller chooses random mortgages. For every boobytrapped CDO, the seller randomly chooses from the total population of mortgages and mortgages from set of mortgages with known .

If and , then there is no polynomial time algorithm to distinguish between the distributions and .

Roughly speaking, the densest subgraph problem is tantamount to figuring out whether or not the seller planted mortgages with no payout in of the CDOs he issued. The seller then has preferences for maximizing the total amount of lemons costs he can extract from the buyer, subject to the constraint that there is no polynomial time algorithm that the buyer can use to detect his cheating. Note that in the simple DeMarzo (2012) setting, if the seller could commit to issuing a CDO with a “random” collection of mortgages and published the collection of mortgage IDs, the buyer could check whether or not these mortgage IDs were in fact randomly selected using a polynomial time algorithm. The challenge in the Arora et al. (2012) setting comes from the fact that the same mortgage could show up in several CDOs.

Above, I use Landau notation:

Definition (Little- Notation): We say that “ is little- of as approaches ” and write as to mean that:

(5)  

Some simple math reveals that the seller only needs to fix about of the assets in each CDO to affect payout as the expected number of defaults would be while the standard deviation would be . Thus, the lemons cost for a boobytrapped CDO will be where is the increase in the probability that a CDO does not payout due to the seller’s boobytrap and denotes the standard normal distribution. The seller then chooses the number of CDOs to boobytrap, , and the number of lemons to plant in each boobytrapped CDO, , in order to maximize his utility as defined below:

(6)  

Since is concave for and convex if , the optimal choice of is either or for some small constant .

4. Main Result

The main result of the paper is to characterize the seller’s utility given this optimization program:

Theorem: If and , the seller will earn a utility:

(7)  

Thus, if the seller issues roughly the same number of CDOs as there are mortgages, he can each roughly for each additional lemon that he discovers in the underlying pool of mortgages without the buyer being able to figure out whether or not he has cheated an boobytrapped some of the CDOs.

Proof: The strategy of the proof will be to first take any and that satisfy the theorem conditions and proceed to compute the lemons cost for a boobytrapped CDO. Then, use the link between the default rate on mortgages and the default rate on CDOs to compute an upper bound on the lemons cost of a non-boobytrapped CDO.

Suppose that and satisfy the non-polynomial time algorithm bounds. For each boobytrapped CDO , let denote the total number of defaulted mortgages. Since the CDO is boobytrapped mortgages will always default, so the expectation can be computed as:

(8)  

Given that the number of assets in each CDO, , is large enough that the law of large numbers holds, we would then have that:

(9)  

Next, I want to derive a bound on the expected payout of a non-boobytrapped CDO. Suppose that the expected number of defaulted assets for each non-boobytrapped CDO is . We know that has to satisfy the relationship:

(10)  

Both the left and right hand sides of the above equation are the proportion of mortgages that default. The left hand side of the above equation is the proportion of mortgages in the boobytrapped CDOs that default, , plus the proportion of mortgages in the non-boobytrapped CDOs that default, . The right hand side is the number of defaults per securitized mortgage times the average number of times each mortgage is securitized into a CDO. Thus, by solving for we can derive the inequality below:

(11)  

Thus, the probability that the number of defaulted assets in a non-boobytrapped CDO is can be computed as:

(12)  

Thus, the expected number of securities (boobytrapped and non-boobytrapped) that yield no payout is at least which is roughly smaller than the case with no boobytrapping.

1. Introduction

In this post, I review the sparsity based model of bounded rationality introduced in Gabaix (2011) and then extended in Gabaix (2012).

In the baseline framework presented in Gabaix (2011), a boundedly rational agent faces the problem of which action to choose in order to maximize his utility where his optimal action may depend on many different variables. The agent then chooses an action by following a step algorithm: first, he chooses a sparse representation of the world whereby he completely ignores many of the possible variables that might affect his optimal action choice; second, he uses this endogenously chosen sparse representation of the world to choose a boundedly rational action. Gabaix (2012) then shows how to use this sparsity based framework to solve a dynamic programming problem.

2. Illustrative Example

In this section, I work through a simple example showing how to build bounded rationality into a decision maker’s problem in a tractable way. I begin by defining the problem. Consider the manager of a car factory, call him Bill, who gets to choose how many cars the factory should produce. Let denote Bill’s value function in units of given an action over how many cars to produce:

(1)  

Here, the vector denotes a collection of factors that should enter into Bill’s decision process. For instance, he might worry about last month’s demand, , the current US GDP, , the recent increase in the cost of break pads, , and the completion of the new St. Croix River bridge in Minneapolis, MN, . Likewise, the vector denotes how much weight the Bill should place on each of the different elements that might enter into his decision process. Each of the elements are in units that convert decision factors into units of . So, for example, would have units of while would have units of . is a constant with units of which balances the equation.

Next, consider Bill’s optimal and possible sub-optimal action choices. If Bill is completely unconstrained and can pick any whatsoever, he should choose:

(2)  

where has units of . However, suppose that there were some constraints on Bill’s problem and he could not fully adjust his choice of how many cars to produce in response to every little piece of information in the vector . Let denote his choice of how many cars to produce where for most :

(3)  

For instance, if but then Bill will adjust the number of cars he produces in response to a change in the US GDP but not in response to a change in the price of break pads. Thus, Bill’s choice of how many cars to produce can be rewritten as a choice of how much he should weight each potential decision factor .

In order to complete the construction of Bill’s boundedly rational decision problem, I now have to define a loss function for Bill which trades off the benefits of choosing a weighting vector which is closer to the optimal choice against the cognitive costs thinking about all of the nitty-gritty details of the dimensional vector of decision factors . As a benchmark, suppose that there were not cognitive costs. In such a world, Bill would choose as he would suffer a quadratic loss from deviating from this optimal strategy defined by the function below, but no compensating “cognitive” gain from not having to think about how the construction of a bridge in Minneapolis should affect his production decision:

(4)  

where the term is a place holder which helps balance the units in the equation. One method for incorporating cognitive costs into Bill’s decision problem would be to charge him units of utility per unit of emphasis on each decision factor. So, for example, if and Bill increased his production by in GDP growth, then Bill would pay a cognitive cost of where has units of . Below, I formulate this boundedly rational loss function as a function of which I denote :

(5)  

where the term is constant with units in order to balance the equation.

The key idea is that Bill’s decision problem is both convex and sparse only when . Below, I give examples which illustrate this fact:

Example (Quadratic Complexity Costs): First, consider the quadratic case when . Then, taking the partial derivative of with respect to an arbitrarily chosen dimension and setting for simplicity we the optimality condition:

(6)  

This yields an internal optimum for Bill’s choice of :

(7)  

While this is an easy solution to solve for, setting doesn’t yield any sparsity as whenever .

Graphically, in the above example, the boundedly rational agent will choose a weight at the point on the -axis in the figure below where the slope of the solid pink curve is exactly equal to the slope of the dashed black line.

Example (Fixed Complexity Costs): On the other hand, let’s not think about a case where with the convention that , then I would want to again set:

(8)  

However, this problem is no longer convex as the costs to increasing a little bit away from will always outweigh the incremental benefits. Thus, I get a solution:

(9)  

While well posed, this non-convex problem is computationally hard to solve in an extremely severe way as the solution strategy expands combinatorially rather than linearly in the number of variables . e.g., see example in Boyd (2004) describing the regression selector problem.

Graphically, we can see the intuition for the problem posed by non-convexity for in the figure below. We can see for the blue solid line representing the case, an incremental increase in where is just marginally greater than will have a cognitive cost much greater than the benefit; however, for where the increase in the weighting factor will outweigh the benefit. The can be seen as an even more extreme limiting case as the blue line becomes increasingly kinked and eventually becomes a flat line at .

Looking at the previous examples, we can follow the goldie locks logic and see that there is a particular parameterization of which yields both sparsity and convexity… namely, .

Example (Linear Complexity Costs): Finally, consider the case where . Here, we find that:

(10)  

where denotes the sign operator which returns the sign of a real constant. Thus, we have different options for the optimal choice of :

(11)  

Thus, the setting where is both analytically tractable and sparse as any variables with will be ignored by the decision maker.

3. Static Problem Formulation

In the above example, I showed how to embed sparsity into Bill’s decision problem using a linear cost function. However, Bill’s problem was extremely simple in the sense that he had a linear-quadratic value function. Can the intuition developed in this simple example be extended to more complicated example with more elaborate utility functions? In the simple example, I found that there was a clean knife-edge result regarding being the only power coefficient which delivered both sparsity and a tractable solution; however, this result depended on Bill’s utility gain to increasing his weighting on decision factor being linear. e.g., look at the black dashed line in the figures above. Will this same intuition regarding hold for more complicated utility functions?

Gabaix (2011) shows that the intuition indeed holds for a wide variety of utility specification after using an appropriate quadratic approximation of the problem around a reference representation of the world and action . For instance, in Bill’s problem above, this would be like allowing his value function to be non-quadratic and then approximating his problem as quadratic around a reference implying that his default decision is to ignore all variables except for last period’s demand and the current value of GDP and number of cars to produce .

In order to construct this approximation, I need to define objects. First, for an arbitrary value function define the partial derivatives and as:

(12)  

where is negative definite implying that is strictly concave in the neighborhood of . I then use these matrices to define a weighting matrix which captures how much information is lost via the quadratic approximation:

(13)  

This matrix corresponds to the weighting matrix used in GMM to differentially interpret the error terms from each of the equations a la Cochrane (2005). For instance, when digesting a vector of errors from a set of pricing equations in GMM, a large term in the weighting matrix means that a set of coefficients which produce large errors in equations and at the same time will be penalized more heavily. In the quadratic approximation below, will be used to differentially interpret the loadings on different decision factors.

Step 1 (Choose Representation): The agent chooses his optimal sparse representation of the world as the solution to the optimization problem:

(14)  

where captures the cognitive cost of a model and is defined as:

(15)  

In the formulation above, plays the same role as in the motivating example and simply controls the units and scale of the agent’s choice of representations. In general, will not be problem specific in any material sense.

Step 2 (Choose Actions): The agent maximizes over his choice of actions :

(16)  

where the cognitive cost of deviating from the default action is :

(17)  

In the motivating example, Bill did not face Step of the algorithm at all as his choice of car production levels was unconstrained after choosing a representation of the world . This second step allows for physical difficulty of adjusting the action given the agent’s understanding of the decision problem. e.g., consider a decision maker who makes a portfolio decision regarding how to allocate his firm’s wealth. After choosing a representation of the world in Step , he might look at the facts through the lens of this representation and decide that he needs to lengthen the duration of his portfolio; however, since there are many different ways to do this in practice, he may face cognitive costs in executing such a change. Step captures these sort of costs.

In summary, a boundedly rational agent with a preference for sparsity first employs a quadratic approximation of his value function and then uses a linear cost function to price the cognitive cost of the complexity of his model of the world. What’s more, this same linear cognitive cost function can be used in a secondary step to incorporate settings where there are cognitive costs to executing a given action.

4. Application to Dynamic Programming

I conclude by showing how to use these tools to solve for a representative agent’s optimal consumption choice in an infinite horizon economy by treating the terms in a Taylor expansion of the optimal consumption rule around the steady state as increasingly complicated decision factors. Consider a standard, discrete time, consumption based model where a representative agent has power utility with risk aversion parameter and time preference parameter described by the preferences below:

(18)  

Assume for simplicity that there is a single risky asset with return where for simplicity. Then, I can write this representative agent’s problem recursively as follows:

(19)  

where is the agent’s value function given wealth level and is the constant endowment rate. In this world, it is easy to derive that the optimal consumption choice is given by:

(20)  

In the remainder of this example, I assume that even the boundedly rational representative agent can solve this simple problem in closed form. However, when I tweak the problem and allow the true values to be and where the idiosyncratic terms and evolve according to processes below with mean shocks and :

(21)  

the representative agent will want to seek a sparse solution due to cognitive costs. Introducing these idiosyncratic terms yields a more complicated value function with additional state variables when writing the problem recursively:

(22)  

where there will in general be no closed form solution for the optimal choice of .

However, suppose that we took a Taylor expansion of the optimal around the benchmark solution to get:

(23)  

I treat these coefficients asthe boundedly rational representative agent’s . Then, I can define his optimal log consumption choice as:

(24)  

where and are given by the rule:

(25)  

A reasonable choice of this constant might be to choose for as it would have the interpretation that the agent would set whenever taking the interest rate or endowment level into accound would change the standard deviation of log consumption by standard deviations and otherwise . Thus, an sparsity seeking boundedly rational representative agent has a particular rule in mind when figuring out which higher order Taylor terms to ignore.

1. Introduction

In this post, I summarize the econometric method introduced in Analyzing the Spectrum of Asset Returns (JEL 2011) by Yacine Ait-Sahalia and Jean Jacod. From an economic perspective, Delbaen and Schachermayer (1994) (Theorem ) tells us that a (log) stock price follows a semi-martingale if and only if there is no arbitrage where a semi-martingale is defined as follows:

Definition (Semi-Martingale):
A real valued process defined on the filtered probability space is called a semi-martingale if it can be decomposed:

(1)  

where is a local martingale and is an adapted process with locally bounded variation.

What’s more, from a purely statistical perspective, we know that a semi-martingale can be decomposed into the sum of a drift component, a Brownian component and small and large jump components:

(2)  

In the characterization above, which marks the cutoff between large and small jumps is arbitrary, but must be fixed. In this paper, the authors develop a suite of statistical tools based on spectrographic analysis in order to examine a time series properties of high frequency stock data over the time interval and determine whether or not the data has:

1. A Brownian component,
2. A jump component, and
3. (if yes…) A finite number of jumps.

The authors ignore questions about the drift component since it is invisible for all intents and purposes at high frequencies.

2. Rough Idea

The authors’ basic unit of observation is the change in the log stock price over the interval sampled at a frequency of seconds. For, notational convenience, the authors write the change in from observation to observation when sampling at a frequency of as:

(3)  

Since the authors are investigating the volatility and jump behavior of the process , a natural starting point is to look at the sum of the powers of the incremental changes . For instance, if then we’d have .¹ Thus, the authors use the extended concept of variation defined below to decompose movements in into smooth Brownian components and large and small jump components:

(4)  

The power variation function has free parameters which can be adjusted by the econometrician:

1. Power: . As the smaller s get more and more weight in the summation ; whereas, when the larger s get more and more weight. Thus, for the power variation estimator is tuned to the continuous variation in , while for the estimator is tuned to pick up jumps in .
2. Truncation Level: . Truncating large increments will eliminate large jumps from the series. will always contain a finite number of big jumps but may have an infinite number of small jumps.
3. Sampling Frequency: . By sampling at slightly different frequencies and , we can identify the asymptotic behavior of the power variation as . For instance, if , then the power variation will diverge to for a particular power level and truncation level . Alternatively, if this limit is or the power variation with converge to a finite value or to respectively.

3. Examples

To get a feel for how to use the power variation of a series to decompose its movements into Brownian and jump components, I now walk through examples:

Example (Brownian Component Exists): Consider the null hypothesis that the Brownian motion component exists in the stochastic process . If we consider powers , then if the log price process contains a Brownian component the many tiny incremental changes due to Brownian motion will dominate the power variation and it will diverge to . Conversely, without a Brownian component, for some cutoff the power variation of will converge to at exactly the same rate for and for all since only the size of the jumps not their frequency will matter. This intuition yields a test statistic :

(5)  

The asymptotic results for this test as are:

(6)  

Example (Jump Component Exists): Consider the null hypothesis that at least of the jump components exists in the log price process . Then, if we set the power parameter , then as we sample at higher and higher frequencies only the jump components of should matter for the value of the power variation. This yields a test statistic defined below with :

(7)  

The asymptotic results for this test as are:

(8)  

1. Introduction

In this post, I work through Adding and Subtracting Black Scholes, JFE (2011) by Antonio Mele and Dennis Kristensen. This paper develops a method for approximating the price of an asset where no closed form expression exists. In the analysis below, I focus my attention on pricing a European option on a stock when the stock price displays stochastic volatility.

In Section , I describe the approximation method which hinges on the wedge between the infinitesimal generator of the asset price under the true specification (e.g., with stochastic volatility) and a simpler reference specification (e.g., with constant volatility). Then, in Section I conduct some simple numerical exercises to see how the approximation behaves in practice.

2. The Method

Consider a stock with a price in units of ‘s at time in units of ‘s whose evolution under the risk neutral measure is described by the equations below:

(1)  

Here, is the risk free rate in units of , is the instantaneous variance of the stock price in units of , is the mean reversion rate of the variance of the stock price in units of , is the mean of the variance of the stock price in units of and both and are constants parameterizing the volatility of the variance of the stock price where has units of and is unitless.

Let be the payout of a European option on this stock with strike price (in ). Then, define the price of this option as follows:

(2)  

As shown in Ait-Sahalia and Kimmel (2007), when there is no closed form solution for . The authors approximate by computing the difference between the infinitesimal generators of and where is the price of a European option on a stock whose volatility is constant and set to .

Definition (Infinitesimal Generator): Let be a time homogeneous Ito diffusion in . The infinitesimal generator of is defined as:

(3)  

Source: Oksendal (2003)

The infinitesimal generator characterizes how a stochastic process would look different if we peered an instant into the future. The theorem below gives a method for computing the infinitesimal generator.

Theorem (Infinitesimal Generator of an Ito Process): Let be the Ito diffusion in :

(4)  

If with a compact support then:

(5)  

Source: Oksendal (2003)

To tie this mathematical construction back into the financial application of options pricing, note that in a world with no financial frictions a la Black and Scholes (1973) the instantaneous change in the price of an asset must exactly equal the interest payment earned by depositing the current price of the asset in a bank for the next instant. Thus, in generator form the second order partial differential equation pinning down the price of a European option on a stock whose price has constant volatility can be written as follows:

(6)  

where the generator can be characterized as:

Proposition (Infinitesimal Generator given Constant Variance): The generator for the price of a European option on a stock with constant volatility is given by:

(7)  

Proof: If a stock price displays constant volatility , I can write as follows:

(8)  

where is the instantaneous volatility of the stock price in units of . Plugging this differential equation into the formula above yield:

(9)  

I can conduct a similar exercise to characterize the price of a European option on a stock with stochastic volatility. For this asset, I know that:

(10)  

What’s more, I can use the same theorem to characterize the infinitesimal generator as:

Proposition (Infinitesimal Generator given Stochastic Variance): The generator for the price of a European option on a stock with stochastic volatility is given by:

(11)  

Proof: Applying Ito’s lemma to and defining for shorthand yields an expression for as:

(12)  

Collecting terms yields expressions for and which given .

The authors’ key insight is that applying the infinitesimal generator to the difference yields the expression:

(13)  

given the boundary condition that for all and . This wedge given by has the natural interpretation of the hedging cost incurred by a trader continuously hedging the risk of the European option with stochastic volatility using the incorrect constant volatility model. Thus, we can write the true price as the Black and Scholes price plus an error correction term:

(14)  

This is a particularly nice formulation as each of the error terms is a function of only the Black and Scholes (1973) formulation with constant volatility.

3. Numerical Results

To put this approximation method to work, use a Taylor expansion around :

(15)  

Each of the expansion terms can be expressed recursively as follows:

(16)  

The Black and Scholes (1973) options price given constant volatility can be expressed as:

(17)  

Thus, expanding out the first approximating terms yields:

(18)  

In order to simplify calculations, I take to kill off the term in my numerical analysis. Below I replicate Figure from the Kristensen and Mele (2011) for and plot the mispricing as a fraction of the Black and Scholes (1973) price with for . The strike price is , the time to maturity is year, the remaining parameter values are , , , and .

The code for the simulations can be found on www.SageMath.org. The simulation represents estimates from iterations at price points between and .

1. Introduction

In this post I outline the Buckingham Theorem which shows how to use dimensional analysis to compute answers to seemingly intractable physical problems. For instance, in Geoffrey Taylor used the theorem to work out the energy payload released by the 1945 Trinity test atomic explosion in New Mexico simply by looking at slow motion video records. My main source for this post is Bluman and Kumei (1989).

2. Basic Framework

The Buckingham Theorem concerns physical problems with the following form: There is a variable of interest, , which is some unknown function of different physical quantities .

(1)  

Each of these physical quantities is composed of measurements in only fundamental dimensions labeled . Thus, I can define a dimension operator which gives the dimensions of an arbitrary variable and write its output as:

(2)  

So, for example, if is measuring pressure on the surface of a table, I could write where , , and .

Definition (Dimensionless Quantity):
An arbitrary variable is dimensionless if:

(3)  

3. Main Result

Now, I show how to reformulate this problem and apply linear algebra to the dimensional exponents to derive a characterization of the solution to as a function of dimensionless quantities. First, I define as an matrix of dimensional exponents for and as the vector of dimensional exponents for :

(4)  

Next, we know that if has full rank, then there will solutions to the system of equations . Define as this matrix of solutions.

(5)  

Finally, define as the solution to system of equations :

(6)  

With these objects in hand, I can now state the Buckingham Theorem.

Proposition (Buckingham ):
A physical system with fundamental dimensions can be restated as:

(7)  

where , is an unknown function and are dimensionless parameters constructed from the physical parameters using equations of the form below:

(8)  

4. An Example

I now give an example of how to employ this theorem by working out the nuclear payload example from the introduction. For more information on this example, take a look at this blog post. Suppose that an atomic blast has a shock wave radius of with variables:

1. : Energery released by the explosion,
2. : Time elapsed since the explosion took place,
3. : Initial density, and
4. : Initial pressure.

For this problem and with fundamental dimensions of length , mass , and time yielding the dimensional matrix written below:

(9)  

The energy (force) released by the bomb is the amount of mass accelerating through a unit square on the surface of the blast wave. Since , we have only dimensionless constant which can be computed as the solution to the system of equations :

(10)  

Setting the scalar free parameter , I can write as:

(11)  

The dimension of the shock wave radius is in units of distance yielding the dimension matrix below:

(12)  

The system of equations has unknowns:

(13)  

Thus, the vector will thus be defined up to a single free parameter :

(14)  

Using the formula and tuning , I can compute as:

(15)  

Combining all of these results yields a formulation for in terms of a constant and an unknown function of the dimensionless quantity :

(16)  

Taylor expanding around where yields a formulation where with scaling constant given by:

(17)  

Setting , the plot of vs yielded an accurate fit where the predicted values fall on the solid line and the (declassified) empirically observed values are denoted by ‘s in the plot below: