Chain pricing


Goods and services are produced from other goods and services. Production chains may be long and entangled. They may have constant parameters like risk, interest-rate, taxes, etc. If price model of intermediate products contains such parameters as multipliers then end product contains them multiplied by itself many times. This creates nonlinearity and lead to arbitrage opportunity.

Price chain

Consider assets A, B, C with exchange rates P^{BA}_t (one unit of B costs P^{BA}_t units of A) and P^{CB}_t:

P^{CB}_t= P^{BA}_t=P_t (1.1)

Assume we borrow two units of B at moment 0. Than after time T we have to return the following amount of B:

V^B_T = 2\cdot e^{r \cdot T} (1.2)

Instead of storing B, we exchange one unit of B to A and one unit of B to C. Amounts of A and C are as follows:

P_0 and \frac{1}{P_0} (1.3)

Assets A, B and C may have risk-free interest rates. It means their amount may increase while we hold them. Of course, it may be equal to zero depending on nature of asset. Risk-free interest rate r is one for all three assets A, B and C. Otherwise, (1.1) does not hold (if you buy something with higher interest rate, price of this asset should become lower in time to reflect your excess return comparing to investment in something with lower interest rate).

After time period T we have following amounts of A and C:

P_0\cdot e^{r \cdot T} and \frac{1}{P_0}\cdot e^{r \cdot T} (1.4)

Than we exchange A and C back to B. Amount of B:

\frac{P_0}{P_T}\cdot e^{r \cdot T} + \frac{P_T}{P_0}\cdot e^{r \cdot T} (1.5)

If price P_T changes then (1.5) is higher than (1.2) for every P_T \neq P_0. The only extremum of (1.5) is P_T = P_0.


Fig.1.1 (1.5) for P_0=1 and r=0

It means that one can borrow some amount of B, invest in A and C and have a risk-free profit (risk-free return exceeding premium for borrowing B). Theoretically,  infinite. This is arbitrage opportunity (riskless profit, “free lunch”). Consequently, in an ideal world (efficient market) such situation where (1.1) holds has to be impossible. But we do not live in an ideal world.

Technological, natural, behavioral and many other aspects affect prices. It means that if there are, for example, technological or economical reasons that lead to inefficient (1.1) prices then such reasons are also inefficient.

Hypothetically, it may appear in different cases:

  1. Money market. If you are able to borrow money at one rate for different time periods. Actually, in the efficient state the longer period is the lower risk-free rate should be.
  2. Transportation. Goods move from point A to point B, then from B to point C. Transportation costs, risks and risk-free return apply causing increase of price.
  3. Production. Goods and services are not created from nothing. They are produced from other goods and services and may be used to produce goods and services. It can be a long chain.

Chain pricing

Let’s consider simplified transportation or production case. Asset B price derive from price of asset A (P), expenses factor (e_p), risk factor (c_r) and risk-free rate (r):

P^B = P^A\cdot e_p \cdot c_r\cdot e^{r\cdot t} (2.1)

Price of asset A and expenses (fares, taxes, etc.) are your investments. After period T you expect to have risk-free return increased by a risk factor (risk to lose a product or your cost of insurance). If price of B is lower you invest in something else. If it is higher many other investors enter the same business and lower the price.

We can also add constant expenses to the model. However, expenses are barely can be constant. Every company optimizes them depending on income. So, here we continue with one-input model. Model with two inputs will be considered later.

There is the second company that buys asset B and transform it into asset C using the same process (2.1) (production or transportation). Both companies operate in the same conditions, to which they are perfectly fit. Their pricing model include the same parameters. Price of asset C:

P^C = P^B\cdot e_p\cdot c_r\cdot e^{r\cdot t} (2.2)


P^C = P^A\cdot (e_p\cdot c_r\cdot e^{r\cdot t})^2 (2.3)

Exchange rated between C and B, B and A:

P^{CB}=e_p\cdot c_r\cdot e^{r\cdot t} (2.4)

P^{BA} = e_p\cdot c_r\cdot e^{r\cdot t} (2.5)

(2.4) and (2.5) are equal as in (1.1).

If expenses e_p, risk factor c_r or risk-free rate r change it will cause changes in prices (2.1) and (2.2 ) that satisfy (1.1). Consequently, pricing (2.1) and (2.2) should not exist. But why? Prices of B and C are market prices. They cannot be higher or lower. Otherwise, they are overvalued or undervalued causing risk-free return differ from available on a market value that leads to arbitrage. So, arbitrage is possible when prices are as in (2.1) and (2.2) and when they are not.

General case

Actually, we deal with portfolio with the following value:

V(x)=a\cdot x + \frac{b}{x} (3.1)

It cannot be arbitrage-free because of nonlinearity. All you need to do is to choose such a and b that:

\frac{d}{dx}\cdot V(x)=0 (3.2)

In previous cases we have used asset B as numeraire. Another way of using nonlinearity and making risk-free profit here is to create a portfolio without changing numeraire:

V=a\cdot P^B + b\cdot P^C = a\cdot P^A\cdot P_t + b\cdot P^A \cdot (P_t)^2 (3.3)

Nonlinearity in (3.3) creates an arbitrage opportunity.

Until now, we used model with one input. In general case, there are many inputs. They can be materials, labor, equipment, rent, etc.

Untitled Diagram.png

Fig. 3.1 Production chains

(2.1) transforms to:

P^B = \sum^{}_{i} P^{A_i}\cdot x^{A_i}_u \cdot x_c (3.4)

x^{A_i}_u is a parameter unique to every input.

x_c is a parameter of chain, it is constant for every part of a chain (or a subchain). It can combine interest rate, taxes, fares, risk factor, etc. The important thing is this parameter is applied as a multiplier in every piece of a chain (or a subchain). It gets powered by itself. That is why polynomial occurs.

As as special case let’s consider production model with two inputs: product from chain and constant expenses E_c. Then (2.1) changes to

P^B = (P^A+E_c)\cdot x_c (3.5)

x_c=e_p \cdot c_r\cdot e^{r\cdot t}(3.6)

Price of asset C in this case:

P^C = (P^A+E_c)\cdot x_c^2+E_c\cdot x_c (3.7)

Portfolio consist of assets A, B and C:

$latex V = a_1 \cdot P^A + a_2 \cdot P^B + a_3 \cdot P^C (3.8)

It is a polynomial with three chain parameters e_p, c_r, e^{r\cdot t}.

Consider a chain consisting of many products (not only A, B and C). Actually, it may be a combination of different chains creating a complex labyrinth or something like a spider web. Every piece is an asset that you can add to a portfolio:

V = \sum^{}_{i=A, B, C...}a_i \cdot P^i = \sum^{}_{j=0,1...}b_j \cdot (x_c)^j (3.9)

It is a polynomial. By selecting a_i we can create arbitrage portfolio unless b_j \neq 0 for j \geq 2. As it was shown above at least in some special cases arbitrage opportunity exists.

If a chain consist of subchains with their own chain parameters then (3.9) becomes multivariate polynomial.

Considering all aforementioned we can formulate a hypothesis: if there are chain parameters-multipliers arbitrage opportunity exist (maybe expect some special cases).


Arbitrage opportunity cannot always exist because it allows unlimited risk-free profit and somebody have to pay for this (be a source). So, to find conditions of arbitrage we need to follow the money.

One explanation is b_j = 0 for j \geq 2 in (3.9). It may be because of correlation between parameters. For example, in (3.7) may be such E_c=f(x_c) that (3.7) becomes linear. However, physical nature of parameters should prevent such dependencies. Companies should optimize their expenses in order to raise profit and to to be competitive.

Another explanation involve consumer behavior. Assume transportation risk changes. Transport companies adopt prices respectively. But in average at the end of a chain price rises because of nonlinearity. Consumer has to pay higher price. Everybody according to the pricing model is not suffering from risk changes except consumers. They become less wealthy in general. We can suppose they limit arbitrage mentioned arbitrage opportunity.

End consumers are interesting and very special participants of a market. They buy and sell goods and services not to make a profit (as other participants), but to consume. Their pricing model is not based on return they want to make, but individual consumer value of products. If no other explanation is found then we should create pricing model that combine both mentioned types of pricing models.

Thoughts about relativity of prices.

What can be said by now is that dependencies between prices are complex and may be nonlinear. They reflect processes of the real world (technological, natural, etc.) like risks, taxes, interest rates. Nonlinearity, in turn, may lead to arbitrage opportunity. Many prices may not be efficient and allow arbitrage opportunity if we look at them together with other connected prices. To be efficient dependencies, prices and maybe processes have to be different.

Practical issues

First, we need to research existing product chains and find chain parameters that influence prices and create nonlinearities.

Price is determined by supply and demand. There can be many factors in a price model, only few of them create nonlinearities. Those that do not create nonlinear behavior create noise. There can be different strategies for noise cancellation. Following are  thoughts about approaches that could be used.

First, some noise can be filtered using numeraire change. If you look at (1.5) you will see that it does not contain P^A. This strategy always wins independently from P^A, and risk-free profit can be exchanged to every other numeraire.

Another way is hedging and diversification.

One of the best, but hard to implement, ways would be creation of instruments with certain behavior that follow simple models. For example, products that strictly follow (2.1) or (3.4) behavior.

Two parameters are the most obvious among others: risk and interest rate. Price of a risky asset increases over time. It can be debt, investment in startup or something else. When you invest in something risky you expect a discount. Your investment equals to expected income in the future multiplied by e^{-r\cdot t}, r reflects risk and interest rate. Consider you have investments in securities or projects on different stages (with different t). Relationship between costs of such investments is nonlinear.

There are also a lot of other parameters.


Relativity of prices


Here I present a hypothesis that price efficiency is a relative conception. Efficiency depends on what you want to maximize. On efficient market prices are optimized to the goal. For example, price should not allow “free lunch”, i.e. making more than risk-free profit without taking a risk. If price is higher then demand rises and price goes down. If price is lower then demand falls and price goes up. But what if different people have different goals? If you want to make more euro then you have increased expectations of return from investments into USA companies comparing to EU companies. Moreover, is amount of currency what do you want the most? Maybe you want to consume more goods and services instead of currency? If so then in theory there are strategies that allow you to earn what you want.

It should be noted that all written below correspond with arbitrage theory.

It is all about the probabilities

Consider a security which payoff at moment T H_T depends on some event at that moment. If there is only one scenario of future then price of a security is equal to discounted payoff at moment T:

H_0^A=P^A\cdot H_T^A   (1)

P^A is a discount factor, H_T is a payoff, A is a numeraire.

If there are different scenarios but derivative pays off only in one of them (and zero otherwise) then coefficient should be added to the previous equation:

H_0^A=P^A\cdot H_T^A\cdot p^A   (2)

This security can be used to construct other securities with a complex payoff:

H_0^A=P^A\cdot \sum_{i=1}^n(H_T^A(i)\cdot p_i^A)   (3)

Scenarios can be following: “price on underlying asset is equal to X or is within range I”, “next president will be Mr.T” or other events.

As we can see, p_i^A has a role of scenario i probability. If p_i^A differs from the probability that we think is real then we expect return to be above the risk-free rate for asset A.

Let’s take a look from a different point of view and change numeraire from A to B – we are interested in maximizing asset B rather than A. Then

H_0^B=P^B\cdot \sum_{i=1}^n(H_T^B(i)\cdot p_i^B)   (4)

But there is an exchange rate between A and B. Payoffs and securities can be exchanged:

H_T^B(i)=H_T^A(i)\cdot N^AB_T(i)   (5)

H_0^B=H_0^A\cdot N^AB_0 (6)

N^AB_t(i) is an exchange rate between asset A and B at the moment t.

To hold (3), (4), (5) and (6) p^A(i) and p^B(i) have to be connected:

p^B=p^A\cdot \frac{P^A}{P^B}\cdot \frac{1}{N^AB_T^B}\cdot \frac{1}{N^AB_0^B}   (7)

If N^AB_T(i) is not constant then pi^A and pi^B are different. So, they are not real probabilities. Real probabilities that reflect situation in the real world are not equal to pi^A, pi^B or both. Consequently, it is possible to construct a portfolio with average return exceeding the risk-free rate for asset A, B or both.

How to use this

Excess average return exist if value of numeraire changes. Every price is a ratio: value of asset divided by value of numeraire. Assume you hold a long position. If numeraire drops by 90% then price will be 10 times more then it was. It numeraire goes up (+90%) then price will be only 2 times (almost) less. In average you win.

You may ask why would you need such prize if numeraire falls? But this question is more complex than it may look like.

You can construct a strategy that almost always wins in the long run and has a limited loss. With 90% probability you win one unit of domestic currency, with 10% – lose the same one unit. You lose if value of domestic currency rises significantly. Loss of one unit of domestic currency is equal to loss of many more units of other currencies. If it does not matter to you then you are ok. For example, if your income from other activities is not affected by domestic currency rise you may win even in this situation, because imported goods become cheaper.

Another option takes place if consumer value of two assets are equal to you no matter how their prices change. You can use them both as numeraire. There are strategies that guarantee you excess return at least in one numeraire.

Average return depends on what numeraire you and others use (asset you want to maximize). Generally speaking, we all are different and interested in maximizing different assets (e.g. currencies, goods, services). We work harder to consume more goods and services. But value of goods and services is individual. One may want an island, another – plane. Our goals are different. Consequently, excess average return is a normal thing.

It seems that there should be only one numeraire in the world with “normal” average return. Every price is an exchange rate – a ratio of values of two assets. Price is a relative thing. We measure value of a single asset by comparing it with other assets (e.g., with one chosen as a numeraire). If we imagine some extremely stable asset with no interest rate then it could be used as a universal measure. If this asset is numeraire then its average return have to be equal to zero. Arbitrage opportunity exist otherwise.

Arbitrage conditions

Arbitrage opportunity may occur if dependencies between prices are not efficient. There are limitations on N^AB_T(i). For example, the sum of p^X(i) has to be equal to one for every numeraire X:

\sum_{i=0}^{n} p^X(i)=1   (8)

Arbitrage opportunity exist otherwise. p^X(i) should be probabilities in mathematical sense, but they may be not if market decides so.

In the example above (90%/10% game), you may want to buy shares of importers. They win from domestic currency rise. You may try to use their stock to compensate loss. But there should be no arbitrage opportunities on an efficient market. Consequently, value of importers should behave in such a way to make this strategy not working. The same situation is with insurance (of currency rise) and hedging. But do they behave in such way in the real world?

Dependencies between assets may be nonlinear and complex. This, in turn, may lead to potential inefficiencies and corresponding opportunities. Let’s take a look on equations (7) and (8). Only such N^AB_T(i) is efficient that do not brake (8). Are all dependencies between prices efficient? I don’t think so. Dependencies between prices reflect real world processes (technological, natural, etc). Such things as changes in interest rates, referendums, election results, natural disasters and even season change affect many prices simultaneously. Most companies are connected to many other companies and form long chains. It is like a three-dimensional spider’s web.

Here is an investigation of dependencies between prices or “chain pricing” effect.

Let’s conduct an experiment

If we have positive average return and make many independent operations then we diversify result and transform positive average into almost certain profit – the probability of profit tends to one. Let’s demonstrate this. We have two assets: A and B with prices x^A_j and x^B_j, respectively. Let

x^A_0=x^B_0=1   (9)

Let ratios of two consequent elements be normally distributed:

x^A_{j+1}/x^A_j\sim N(1,\sigma^2)   (10)

x^B_{j+1}/x^B_j\sim N(1,\sigma^2)   (11)

\sigma is a standard deviation.

Numeraire is asset A. So we deal with price

S_j=\frac{x^B_j}{x^A_j}   (12)

Mean value for any long position is positive. Let V_j be a value of portfolio at the moment (iteration)$latex  j$. V_0=1. Every iteration we calculate position – amount of asset B that we hold from j until j+1 moment:

pos_j=\frac{c}{S_j} \cdot V_j   (13)

c is a coefficient. Division by S_j is added for diversification purposes (to make individual results independent of price). All positions are long. The maximum loss is the initial value V_0.

Value V_{j+1} is as following:

V_{j+1}=V_j + pos_j \cdot (S_{j+1}-S_j)   (14)

Modeling of 100 cases gives following results


Fig.1 Value of portfolio


Fig.2. Value of portfolio (logarithmic scale)

Figures look good, but no arbitrage opportunities exist here. At least, not with random data. As the first experiment with real data, I decided to test exchange rates between 150 stocks (first 150 from NYSE alphabetical list) and Dow Jones Industrial Average  (using SPDR Dow Jones Industrial Average ETF – DIA) with a little bit adapted strategy. Numeraire is USD.


Fig.3. Real data

This is a very simple strategy that does not consider real limitations of trading like transaction costs. However, it shows that dependencies between prices are complex and contain opportunities that can be used. I think the reason is that processes in the real economy – technological, political, natural and others, do not always follow financial market laws, but they should.