The economic singularity - Equilibrium of sustained arbitrage leading to securities with a value tending towards infinity

Abstract:

Subsidies can be used to create an arbitrage. The resultant arbitrage does not reach the typical equilibrium in which the prices in the two markets equalise. The equilibrium point at which the arbitrage is eradicated is when the return from executing the arbitrage is equal to the risk free rate of return. The price of the security or good in market 1 tends towards infinity as the speed of transactions viewed as the number of transactions per year increases.

Introduction:

In The impact of a subsidy on an arbitrage involving goods with perfect cross elasticity, a mechanism involving the use of a subsidy to create an arbitrage was explored. In the mechanism, the arbitrage is maintained as the price difference between the two markets is not eradicated. As a result, an equilibrium is not reached. In the following paper, we aim to determine the point at which an equilibrium is reached then proceed to use this idea to determine the pricing of the securities or goods sold in the market.

Method:

Diagram 1:

Diagram 1 shows the interaction between Market 1 and Market 2. The producer A produces good j for a cost of x. Producer A then sells good j in Market 1 at price y to Consumer B thus making a profit of y - x. Consumer B becomes a producer in Market 2 as they sell good j in Market 2 at price z (y + y - x) to Consumer C. Consumer C then receives z - y = y - x (a subsidy) from Producer A at a given point in the future thus Consumer C purchases good j at y while Consumer B receives z for good j in Market 2.

As an arbitrage exists between Market 1 and Market 2, rational participants would purchase good j in Market 1 and sell it in Market 2. This would result in an increase in demand in Market 1 and a rightward shift in the demand curve resulting in an increase in price in Market 1. However, as the price in Market 1 rises, a second arbitrage opportunity results. As Consumer C purchased good j in Market 2 for z and will receive z - y at a given point in the future, they essentially purchased good j for price y. Thus if the price of good j in Market 1 is greater than y, Consumer C can sell good j in Market 1 then receive z - y (the subsidy) from Producer A at a given time in the future and to make a risk free profit.

In an arbitrage, an equilibrium will be reached when the prices in the two involved markets are equal thus removing the possibility of making a risk free profit. To equilibrium point occurs when the guaranteed return from the above mechanism; the difference between market 1 and market 2 is equal to the risk free rate of return. That is, the equilibrium is reached when one can either invest a given amount of money in market 1 or in a risk free asset such as a government bond and the returns would be identical. This in turn will provide the value of the security or good being sold in market 1.

Example 1:

• Assume the risk free rate of return is 5% per annum

• Equilibrium will be reached when the return when one purchases market 1 and sells in market 2 is equal to 5% per annum.

Formula:

y = xn/i

• y = cost of security/good in market 1

• x = difference in price between market 1 and market 2

• n = number of trades between market 1 and market 2 per year

• i = risk free interest rate

Results:

Example 1:

• Assume the risk free rate of return is 5% per annum

• Equilibrium will be reached when the return when one purchases market 1 and sells in market 2 is equal to 5% per annum.

Formula:

• y = xn/i

Table 1: Value of contract in Market 1 and annual return when the number of trades per year is varied

Table 2: Amount required to invest into government bonds to equal the return from investing in market 1

The principal invested in a government bond that yields a return that is equal to the difference between market 1 and 2 is the value of the contract is market 1.

Discussion:

An equilibrium is reached when the value of the contract in market 1 is equal to the amount of money that when invested in a government bond will yield the difference between market 1 and 2 in the same amount of time it takes for the profit between market 1 and 2 to be realised. As the number of possible trades per year increases, the value of the contract in market 1 increases.

With common use of super computers to conduct financial transactions, trades can occur very quickly. As the speed of trades increase, the value of contracts in market 1 increases and as the speed of trades gets closer to being instantaneous, the theoretical value of the contract will tend towards infinity.

Any security, asset or good can be traded in market 1. As a result, any security, asset or good can have a value tending towards infinity.

Conclusion:

Equilibriums are vital in economics and finance as they indicate the point at which the market is stable. We have shown that the value of contracts in market 1 tend toward infinity as the time taken to make transactions between market 1 and 2 decreases. In addition, equilibrium is reached when the return from purchasing a security or good in market 1 and selling it in market 2 is equal to the return from investing in a risk free asset. Any asset, security or good can be traded within the mechanism meaning anything can have a value tending toward infinity.