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.

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