Chain pricing


There are many examples of prices moving in the same way. It is normal because assets are related and there are events that affect many of them simultaneously. But what about efficiency of this relationship? What if response of one asset to price change in another is convex? For example, price model of asset A contains multiplication of asset B price and some parameter that move in the same way as asset A. If this is the case then it is possible to create profitable portfolios (even arbitrage portfolios).

In an ideal world it has to be impossible. But we do not live in an ideal world. Actually, the more I look the more I find examples of such type of relationship. They exist because so are economical relationships. Moreover, it looks like processes in real economy (technological, natural, political, etc) that create such behavior know nothing about arbitrage, free lunch, efficiency and related conceptions. It seems it is pretty fundamental.

Theoretical question. What would be the world without these opportunities? If they are not efficient because of arbitrage then someday they will disappear.
Practical question. How to make use of them?


When national currency falls significantly, some of local companies nominated in this currency also become cheaper and vice versa. Stock price measured in a foreign currency is stock price in a national currency multiplied by the exchange rate. In this case both multipliers change in the same direction. Consequently, this chain effect makes stock price to be non-linearly (convexly) dependent from the currency. Portfolio consisting of assets with linear (currency) and convex (stock) returns is not arbitrage-free.

The same thing is with debt instruments with different maturities. Yields for different maturities usually move in the same direction. Multiplication also takes place here.

In general, such chains can be found almost everywhere. Diverse technological chains contain the same effect. Price of a product is expenses multiplied by parameters reflecting risks, interest rate, profit factor, taxes, etc. But expenses include prices of other products, which, in turn, may have the same multipliers (for example, in case of systematic risk/international beta in CAPM). Production chains may be long and entangled. Price of an end product may contain parameters multiplied by themselves many times.

From the theoretical point of view in an ideal world (with efficient market) such situation where arbitrage opportunity exist has to be impossible. Including described chain effect. But from the practical point of view we do not live in an ideal world. Diverse technological, natural, behavioral and many other aspects affect assets and create different levels of nonlinearity. It can be used to create profitable portfolios. Mathematically it is possible that long-term bond yields were independent from short-term. But they are related due to underlying processes (risks and interest rates).

The convexity

Let us see how it works. Assume we have two assets A and B with prices P^A_t and

$P^B_t=P^A_t \cdot c$ (1.1)

where c=f(P^A_t).

Let P^A_t and c move in the same direction. When P^A_t rises, c does the same and vice versa. Consequently, P^B_t = f(P^A_t) is convex function because of multiplication.

We create portfolio with the following value:

V=a \cdot P^A_t + b \cdot P^B_t (1.2)

It is convex with minimum at some point P_min. By choosing a and b we can make P_min=P^A_0. In this case portfolio is always profitable and there is arbitrage opportunity. Coefficients can be chosen from the following equality:

\frac{d}{dP^A_t}\cdot V(P^A_0)=0 (1.3)

For example, it may look like


Fig.1.1 Non-linear portfolio

In real life price models are complex and contain many parameters. Relationships between prices are more complex. But still some components of price models may follow (1.1) behavior.

Debt and risky investments

Discounting of future cash flow is an exponential process that depends on time period before maturity and interest rate:

P=e^{r_{t_1, t_3} \cdot (t_3 - t_1)} (2.1)

It has the following important property:

e^{r_{t_1, t_3} \cdot (t_3 - t_1)}=e^{r_{t_1, t_2} \cdot (t_2 - t_1)} \cdot e^{r_{t_2, t_3} \cdot (t_3 - t_2)} (2.2)

r_{t_1, t_2} and r_{t_2, t_3} do not have to be equal all the time, but still they usually move in the same direction. If this is the case then it is possible to create arbitrage portfolio from (2.1) and (2.2).

Following figures depict how 30-year Treasury Bills yields (y-axis) are related to 5-year Treasury Bills yields (x-axis).


Fig.2.1 30-years T-Bills yields (y) and 5-year T-Bill (x) from 03/15/2010 to 03/13/2020. Data from


Fig.2.2 30-years T-Bills yields (y) and 5-year T-Bill (x) from 03/15/2010 to 06/25/2013. Data from


Fig.2.3 30-years T-Bills yields (y) and 5-year T-Bill (x) from 06/26/2013 to 10/05/2016. Data from


Fig.2.4 30-years T-Bills yields (y) and 5-year T-Bill (x) from 10/06/2016 to 03/13/2020. Data from


Fig.2.5 30-years T-Bills yields and 5-year T-Bill from 03/15/2010 to 03/13/2020. Data from

As you can see, in average, yields move in the same direction.

Chain pricing

To produce goods and provide services companies use other goods and services. They buy from suppliers and transform (add value) to create their product.

Let asset B price contains price of asset A (P^A), proportional expenses (e^B), constant expenses (E^B), risks (c^B) and risk-free rate (r^B):

P^B = (P^A+E^B)\cdot x^B (3.1)

where x^B=e^B \cdot c^B\cdot e^{r^B\cdot t}

Price of asset A and expenses (fares, taxes, etc.) are investments. After period T one expect to have risk-free return increased by a risk factor (or cost of insurance).

There is the second company that buys asset B and transform it into asset C using the same process (3.1) (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+E^C)\cdot x^C (3.2)

P^C = (P^A+E^B)\cdot x^B \cdot x^C+E^C\cdot x^C (3.3)

if x^B and x^C change in the same direction then (3.3) is convex. Portfolio consisting of P^B and P^C is not arbitrage-free.

Practical issues

The main issue is that real life pricing is far more complex than described above. Price is determined by supply and demand. There are many parameters influencing prices, only few of them can create nonlinearities. Necessary condition that “two multipliers in a price model move in the same direction” usually holds in average, but not for every price change. But still, more or less hidden, it exists. It is a matter of financial engineering to make efficient use of them. From the practical point of view those parameters that do not create nonlinear behavior create noise. But there are strategies for noise cancellation.

Hedging and diversification could help. Influence of linear parameters can be removed by adding to portfolio proper instruments. We can also use derivatives like exotic options.

Different instruments reflect chain effect differently. For example, if one invest in bonds then return depends on implied interest rate and time (time to maturity decreases as he holds a security). Instead, he can buy a forward contract that price depend only on interest rate.

The best way would be creation of instruments with certain behavior that follow simple model (1.1). Prices may contain described convexity, but they also contain noise. However, this convexity seems to be very natural as it reflects normal processes in economy. Consequently, it is possible to create instrument that reflect this effect and are free of noise.


We keep in mind that theoretical efficient market and real one are not identical. But to what extent? It seems that described above chain effect is not efficient because it creates opportunities that should not exist. But it reflect processes in real world: technological, natural, political, etc. It means that efficient market might not be what we think of it.

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 out what limits arbitrage we need to follow the money.

There are different types of market actors: manufacturers, resellers, investors, etc. They all do their business for profit. If someone have excess profit then another one have loss. No one wants to lose money to make others wealthier. This moves market to equilibrium and arbitrage-free state.

This logic is related to everybody on a market except end consumers. They buy things not for profit but to consume. Their pricing model is not based on return they want to make, but individual consumer value of products they buy. If no other explanation is found then end consumers should be the source for described above opportunities. It should be them who is able to limit such opportunities.

Thoughts about relativity of prices.


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