Chain pricing

Summary

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?

Examples

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

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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).

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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 finance.yahoo.com

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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 finance.yahoo.com

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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 finance.yahoo.com

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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 finance.yahoo.com

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

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.

Interpretation

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.

 

Relativity of prices

Summary

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

rplot1_end

Fig.1 Value of portfolio

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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.

rplot3_eng

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.