## Taking advantage of inefficiencies in dependencies between prices. Theoretical basis and first promising results

Dependencies between prices are complex and contain inefficiencies, i.e. opportunities.

There are reasons why different dependent prices behave not in an efficient way. One them is a dualism in prices’ nature. On the one hand, they have to follow financial rules from the arbitrage pricing theory (with some limitations). On the other hand, they reflect complex economic relationships between companies, countries, resources, individuals, etc. How well these two hands are coordinated is a good (and may be profitable) question.

In series of posts, I want to share my results about the potential of using mentioned dualism. Loss of coordination means opportunities. The existence of opportunities means inefficiency. This is interesting for a theorist. Vice versa is also true, and this side is attractable for a practitioner. Some opportunities should be stable, some not, but still present for a long time.

Next, there will be a little bit theory followed by experiment. It should be noted that all written below correspond to arbitrage theory.

# It is all about the probabilities

Implied in prices information about future is a cornerstone of quantitative analysis. There is one interesting property – we can derive different information depending on factors unconnected to the asset that we analyze.

If there is only one scenario of future then a price of a security is equal to discounted payoff at moment T:

$H_0^A=P^A\cdot H_T^A$   (1)

Price and payoff are in asset A.

If there are different scenarios but security pays off only in one of them 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 some asset is equal to X/is in the interval I”, “the next president will be Mr.T” or other events.

If there are enough securities with different $H_T^A(i)$ then we can determine $p_i^A$. As we can see $p_i^A$ has a role of scenario probability. If $latex p_i^A$ differs from the probability that we think is real then we expect return above the risk-free rate for asset A.

Let’s take a look from different point of view, and change asset A to B – we are interested in 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)

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)

In certain scenario $k$ payoff is $H_T^A(k)$ units of asset A or latex H_T^B(k)$units of asset B. For example, payoff depends on the exchange rate between EUR and USD, and in one case payoff is in EUR, in another case – in USD. Then $p^A(i)$ and $p^B(i)$ are 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. However, there is also a real probability that reflects the situation in the world, and it is not equal to $pi^A$, $pi^B$ or both. Consequently, it is possible to construct a portfolio with a mean value of return exceeding the risk-free rate for asset A, B or both (taken as numeraire). Diversification may be useful and improve results. $N^AB_T(i)$ is not constant in cases when an impact of an event on different assets is not equal. This situation takes place very often. Actually, it is possible to construct mentioned above portfolio for every pair of assets with a nonconstant exchange rate. There are also more complex cases and I think they are prevailing. There are two different cases depending on event’s impact on different assets and efficiency of dependencies between assets. ## No arbitrage opportunities exist and it is ok The mean value is above the risk-free interest rate but there are no arbitrage opportunities. You can construct a strategy that almost always wins in the long run (mean value tends to infinity) and has a limited loss. But rare losses in the asset you maximize (numeraire) coincide with the significant increase of this asset’s value. Losses can be high in another asset. If it does not matter then you are ok. For example, numeraire is your domestic currency. With 90% probability you win 1$, with 10% – lose 1$. In latter case value of the domestic currency rises significantly, but your income and expenses are not affected. Moreover, imported goods become cheaper. Mean value depends on what asset you and others are trying to maximize. Generally speaking, we all are different and interested in different assets (e.g. currencies). Consequently, excess mean is a normal thing. It seems that there should be only one asset in the world mean value equal to zero. ## Arbitrage opportunities exist and it is splendid There are strong limitations on $N^AB_T(i)$. For example, the sum of probabilities have to be equal to one for every asset X: $\sum_{i=0}^{n} p^X(i)=1$ (8) Arbitrage is possible otherwise. If you are able to compensate rare losses using other operations then arbitrage becomes possible and some market inefficiency takes place. In the example with currency, you may want to buy shares of importers. Their prices are increasing when a value of the domestic currency is rising. Dependencies between assets are often nonlinear and complex. For this reason, they contain potential inefficiencies and corresponding opportunities. Take a look once again on the equations (7) and (8). Are all dependencies between prices efficient? I don’t think so. How many assets are affected by changes in interest rates, referendums, election results or even season change? Most companies are connected to many other companies and form long chains. It is like spider’s web, but three-dimensional. # Let’s conduct an experiment If we have positive mean and make many independent operations then we diversify result and transform positive mean 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 value of portfolio at the moment (iteration) j. $V_0=1$. Every iteration we determine amount of asset B that we hold from j until j+1 moment: $pos_j=\frac{c}{S_j} \cdot V_j$ (13) $c is a coefficient. Division by$latex 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 determined 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. Then results were exchanged to USD.

Fig.3. Real data

This is a very simple strategy that does not consider real limitations of trading like transaction costs (for now). 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.

I will continue an investigation of described above opportunities. It is interesting to take a closer look at limitations on dependencies between prices. I plan to improve the strategy in order to make it closer to practical application.