Chain pricing


There are many examples of related prices that move in the same way. It is normal because many assets are related (due to technological, behavioral, natural, political reasons) and some events affect many of them simultaneously. But what about the efficiency of these relationships? Latter may be very complex and nonlinear. The idea is simple – to find related pairs of assets that look good separately, but considered together, contain opportunities.

For example, what if the relationship leads to convex dependence between prices? If the first price goes up, the second price responds with a strong upward movement. If the first price goes down, the second responds with a weak downward movement. In this case, it is possible to create arbitrage portfolios – that always win. Even if the relationship is not strong and exists on average or in certain situations, it is possible to make use of it.

In an ideal world, it has to be impossible. But we do not live in an ideal world. 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 the real economy (technological, natural, political, etc) that create such behavior know nothing about arbitrage, free lunch, efficiency, and related conceptions. It seems to be pretty fundamental.

There are a lot of nonlinear usable dependencies between prices. Using financial engineering we can create instruments and portfolios to make profitable use of them.

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 better use of them?

The following is a hypothesis based on observations and modeling of economical processes. I see reasons why it may be hard to observe but I do not see reasons why it could be impossible to use.


When a national currency falls significantly, some of the local companies nominated in this currency also become cheaper and vice versa. Share price measured in a foreign currency is share 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 share price to be non-linearly (convexly) dependent on the currency. A portfolio consisting of assets with linear (currency) and convex (stock) returns is not arbitrage-free.

The same thing is to debt instruments with different maturities. The price of a long-term debt instrument can be seen as a multiplication of the price of a short-term debt instrument and a discount for maturity change. The discount reflects the expected short-term rate for that period. Observed and expected short-term rates usually move in the same direction.

In general, such chains can be found almost everywhere. Diverse technological chains contain the same effect. The price of a product is expenses multiplied by parameters reflecting risks, interest rates, profit factors, taxes, etc. But expenses include the prices of other products. There are situations (for example, if the systematic risk/international beta changes) that affect both multipliers. Production chains may be long and entangled. The price of an end product may contain parameters multiplied by themselves many times.

The convexity

First, let us see how it works in theory. Assume we have two assets A and B with prices P^A_t and

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

Let P^A_t and c(P^A_t) 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 and relation.

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 often more complex than (1.1). But still, some components of price models may follow (1.1). I believe it is possible to extract and enhance such behavior.

Debt and risky investments

Discounting of future cash flow is an exponential process that depends on time period before maturity and interest rate.  It can be also seen as a multiplication of expected annual returns and other ways, but all models describe the same nature of time value of money. The value today of receiving one unit of currency years in the future:

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)

Interest rate r_{t_1, t_2} and expected interest rate r_{t_2, t_3} do not have to be equal all the time, term structure of interest rates can change, but still they usually move in the same direction. They reflect risk-free interest rate and risks. If something serious happens we understand that we live in a new reality and both short-term and long-term risks change.

If this is the case then it is possible to create an arbitrage portfolio from (2.1) and (2.2) in the same way as in (1.2).

The following figures depict how 30-year Treasury security yields (y-axis) are related to 5-year Treasury security yields (x-axis).


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


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


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


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


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

As you can see, on 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 products. Thus production chains are created. If something happens that affects all parts of a chain then the effect is increasing as it goes through this chain. It affects the first company, then the second company becomes affected by the first company and by the event.

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 transforms 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 includes 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. A portfolio consisting of P^B and P^C is not arbitrage-free.

Many parameters affect prices in a chain way. For example, risks, taxes, oil prices, global events, etc.

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, not all of them can create nonlinearities. The necessary condition that “two multipliers in a price model move in the same direction” usually holds on 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 it. From the practical point of view, those parameters that do not create convex behavior create noise. But hopefully, there are strategies for noise cancellation.

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

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

The best way would be the creation of instruments with certain behavior that follow a simple model (like mentioned above). For example, something that only depends on a risk with constant and simple term structure. The convexity seems to be very natural as it reflects normal processes in the economy. Consequently, it is possible to create instruments that reflect this effect and are free of noise.


We keep in mind that theoretical efficient market and the 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 reflects processes in the real world: technological, natural, political, etc. It means that an efficient market might not be what we think of it.

Arbitrage opportunity cannot always exist because it allows unlimited risk-free profit and somebody has to pay for this (be a source of money). 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 has excess profit then another one has a loss. No one wants to lose money to make others wealthier. This moves the 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 the return they want to make, but the individual consumer value of the products they buy. If no other explanation is found then end consumers should be the source for described above opportunities. It should be they who can to limit such opportunities.

Thoughts about the relativity of prices.

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