# Summary

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. Then 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)

or

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

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).

# Interpretation

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.

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.