Wall
Street New
York City
THE BEHAVIOR OF MARKETS
This article suggests that the trading stock market has some tendency towards
equilibrium. Appropriate portfolio structure, however, is the primary means of
handling volatility.
I do not believe that a human system can ever be analyzed satisfactorily In simple mechanical terms, and that at some point the complexity of the level of abstraction must begin to move towards the complexity of the subject matter. In particular we cannot ultimately avoid economic models which take explicit notice of the nature of the economic system as an information...process, very different from the simple dynamic system of
the physical world.
Kenneth E. Boulding Research in Economics (1958)
This article suggests that the trading stock market has some tendency towards equilibrium. Appropriate portfolio structure, however, is the primary means of handling volatility. I do not believe that a human system can ever be analyzed satisfactorily In simple mechanical terms, and that at some point the complexity of the level of abstraction must begin to move towards the complexity of the subject matter. In particular we cannot ultimately avoid economic models which take explicit notice of the nature of the economic system as an information...process, very different from the simple dynamic system of the physical world. Kenneth E. Boulding Research in Economics(1958) |
In a recent magazine article, a hedge fund manager wrote that on a bad trading day, he would break "...furniture and monitors and keyboards (the keys flying) and coffee mugs...."
We ask two questions: Since much stock trading is momentum inspired, how does the stock market result in economic efficiency? Furthermore, what is the likely behavior of a market comprised of value investors and momentum investors? The following comments first upon the relevance of economic theory to market behavior and discusses a complexly mathematical article written by J. Doyne Farmer of the Santa Fe Institute.
The concept of equilibrium, the state to which economic systems tend, defines standard economics. At economic equilibrium, the profits of each individual market participant and that of the economic system is maximized because:
1. At the market clearing price, the amount supplied equals the amount demanded.
2. Given the law of diminishing returns, all resources have been utilized to the points where their marginal costs equals their marginal revenues, and no further profits are possible.
Most crucially, it is furthermore assumed that information is perfect and that consumer preferences and the state of technology remain unchanged. Textbook economics is not literally applicable in a dynamic environment. Economics is useful for describing the structure of the system, tallying the risks and weighing those against the positives by judgment. This is why economists disagree. Tracy Clark says, "What economics is good for is not so much saying, 'We're going to have a recession starting next Thursday,' or that 'GDP will be precisely 3.26% next year. ' What economics is useful for is saying, 'Here are the risks, here's what to watch, here's what could go wrong based on what we know about the economy.' "
How is realized return determined in financial markets? To try to determine the price where supply and demand will intersect, we turn to Mr. Farmer's article titled, "Market Force, Ecology, and Evolution." (The author has revised this paper for publication; the following discussion pertains to the original.)
The Santa Fe Institute studies complexity, that is the behavior of non-equilibrium systems comprised of a large number of agents obeying simple and local rules. The Institute has been criticized for its reliance upon computer simulations and therefore the lack of empirical content in its studies. Nonetheless, to try to predict the path of a real market is to predict how a system evolves over time, at dates and levels certain. This can only be done by computer simulation.
We think this study is interesting because rather than assuming a simple and unrealistic uniformity among investors, it describes a dynamic of how investors with different strategies contend in markets and the result. We suggest that those whose utilities do not include mathematical formulations skip to the conclusion. For those who choose to proceed, the following describes observed market prices as the result of a complex dynamic between fundamentally driven value investors and momentum investors.
Mr. Farmer begins by writing equations describing the behaviors of three kinds of agents: the market maker, value investors, and momentum investors.
The
Market Maker
The market maker uses an algorithm to calculate the price of an order, (w):
P t+1 = f(P t,w)
where: P t is the current price.
Function
(f) is an increasing function of the order size; the larger the buy order, the
higher the price must be to clear the market. This order has an impact upon the
market approximated by the equation:
p t+1 ~ w/lambda + p t
where: p t = log P t. For the sake of simplicity, all the
following price and value calculations
are carried out in terms of logarithms.
w is the order size.
lambda is the market's liquidity.
this is mainly of concern to market
makers and traders.
In the case of a whole market comprised of a market maker and many trading decisions, the future price of a security will be, summing the transactions:
n
(1) p t+1 = p t + (1/lambda) * SUM [w (i) (p t, p t-1, ...I t)] + N t+1 i = 1
where: lambda, as before, is market liquidity.
i is the ith market transaction.
w is the order size.
(p t ,p t-1 ,...I t) is an investment
strategy that utilizes the information available
in the economy, I t. This is very well stated.
N t+1 is a random term; it can include
unanticipated events.
The behavior of a market, even a simple one, is complicated. The following describes the market dynamics of value and momentum trading strategies based upon (I t), the information available in the economy at time (t).
Value Investors
Utilizing the investment strategy (p t, p t-1,...I t), value investors make a common assessment of value (V t) in relation to price.* They buy stock if the price is less than the estimated value and sell stock if the price exceeds this value. The amount of stock held is approximated by an equation related to the perceived mispricing:
X t+1 = c(v t - p t)
where: X t+1 is the amount of stock held.
c is a constant proportional to trading capital.
v t = log V t.
The
change in the logarithm of price, caused by a market consisting only of value
investors and the market maker is:
|
change p t+1 = c/lambda * (change v t
- change p t) + N t+1 |
This price change has no explicit dependence upon the consensus value, V t , or log V t , but only upon the directional change in this consensus; thus stock purchases according to the formula above will not make market prices approach value. This equation furthermore shows that stock prices in such a market will, in fact exhibit a negative dependency between adjacent data points. For every market driven trade that causes prices to approach value, there is a corresponding exit trade that drives prices away.
In a more complicated case, which is included in the overall market model, prices will approach value if the expected price movement exceeds transaction costs by a certain threshold. Nonetheless, in the words of the author, even under those conditions "mispricings...can persist for thousands of iterations." Value convergence occurs but slowly.
*If there is a range of perceived values, say occurring within a range of (1:5), the amount of excess volatility increases only slightly. The following computer simulation assumes that investors perceive a range of values.
Momentum
Investors
Momentum investors buy stocks if the price is going up and sell stocks if the price is going down. Momentum investors hold stocks according to the formula:
X t+1 = c(p t - p t-theta)
where: theta is the time scale used to measure a
trend.