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My goal is provide come concrete quantitative direction around the themes touched by “mathematical finance” as well as those brought-up by Mandelbrot.
Random Observations
1. Obviously, mathematical finance has ignored Mandelbrot so-far and thus there’s little reason to suppose that it will take up now.
2. The appeal of mathematical finance has two parts.
a. An ability to produce tool which correspond to concrete processes of economics – i.e. tools which involve measuring specific supplies and demands.
b. An ability to produce concretely computable results using the same tools as mathematical physics.
These appeal both for their justifiability but also for their “slickness”, their ability to seem like the product of “rock scientists” and thus seem infallible.
3. The notion of market as a random walk neglects entirely the idea of “information diffusion”.
a. Relatively few investors would invest in company whose soundness was not demonstrated, directly or indirectly. Thus no one would have to invest in those companies having the quality of “pure” randomness (or more accurately the market-model collapses under such assumptions).
b. The price paid for a stock cannot actually be reduced to the price quoted by an average – instead there a supply curve and any investor wishing to sell many stocks would those face a lower price and any investor wishing to buy many stocks would face a higher price. Moreover, the situation of stocks being held by many separate, relatively small investor is exceptional – the stock market is generally concentrated in ownership. Moreover, the users of mathematical finance are large corporations, “hedge funds” and such who stand in the highly concentrated realm.
c. The Brownian model neglects the tendency of asset prices to change with broader economic trends – the “martingale” is a device which breaks down even in the assumption of uniform growth. et*B(t) is not B(t). et is a convenient scaling factor but it assumes uncritically not only that an economy that will be growing at an exponential rate but that utility functions that factor this in uniformly and so-forth.
4. We can look at “toy models” of information through markets by starting with shares which have an uncertain payoff at a definite time in the future and where a predictor can be bought by an investor for a fixed price. Here, those who “follow the market” can be seen as benefiting parasitically from the investment of those who have definite information about the investment. If the market also has randomness in it, the market can indeed rationally follow this randomness on occasion.
a. This gives rise to a paradox in the abstract – if the market’s behavior can be deduced and investors can profit from parasitizing off public information, then more investors will join the market. Yet once enough investor join this market, it becomes saturated and no investor can benefit and thus no investor has an incentive to invest, causing the market to go back to the condition of being dominated by those with special information (starting the process over again). If we assume a mix of insight, of willingness to research, etc. then this can indeed be something of a cycle though not of the usual sort.
b. In anycase, another paradox in the abstract is that the more investors can use the market as a source of information, the less incentive they have to engage in research – instead they follow the market’s apparent beliefs. And in this
5. We’ve noted these paradoxes become models of cycles of activity if we assume some “friction” in the operations of the markets. This can be used in such model but there are paradoxes within such models.
a. Since things depend on the structure of the market, the markets will quickly eliminate this friction and those change the nature of the model.
b. It is difficult to determine what scale the various changes happen. Here is where Mandelbrot’s insights on the scaling of price fluctuation are extremely useful. The “scaling distribution” deals with many apparently paradoxical phenomena.
6. The usual reasoning of financial mathematics is that the markets operate fluidly to eliminate the possibility of price arbitrage. Yet these same models create a model of “risk arbitrage” which seems to just as much guarantee “something for nothing” (higher returns with lower risk). Moreover, these models fail to model the process of a market compensating for the opportunity of risk arbitrage by saturating those markets. We can imagine interesting equations here. If bi(t) is the distribution of i separate indices, each with a random distribution, we would argue that the market would tend to create a situation where åAibi(t) is not fundamentally “superior” to any of the bi(t)s taken separately. In the abstract, it is easy to see how this would happen. Concretely looking at possible distributions for bi(t), it is trickier but this certainly a source of interest and may yield a particular class of distributions (perhaps we can develop a class of distributions which minimize |V(bi,bj)/ V(bi)|. This isn’t saying that “diversification” is useless but simply that the claims of its usefulness most automatically be compensated for by the market.
7.
Hyman Minsky describes the instability of the economy as a
whole influencing the price of asserts in a complex manner. The rate of total
economic growth is a relative unknown – it rises and falls with smaller or
larger fluctuations. The profits of some companies rise according to this while
others are less directly affected. Rises and falls in the market itself act as
part of the total demand and supply equation so we have a complex process. And this gives rise to many instances of
what Minsky terms “Ponzi Finance”. The
irony of Ponzi finance is that while it is unsustainable, it is more appealing
than other forms of finance when taken as a “black box” – only when you “open
the box” does the machine seem a bad investment indeed. Indeed, the quality of
Ponzi mimics perfectly the firm growth model in that a Ponzi scheme grows
exponentially until it dies completely. And by this token, if a market consists
of Ponzi finance items, we can expect a fractional distribution of stock
prices. Whether a stock “becomes” Ponzi through a market drift or starts out
that way doesn’t matter here either. A stock becomes Ponzi when it sustains its
payout rate despite an economy which indicates less of a rate. Thus its rate is
more appealing than the economy as a whole. Again – if the whole story were
known, this would make the stock the least desirable possible. But with only a
little known, the stock is the most desirable. So our question is modeling the
diffusion of the information that XYZ Corporation is actually a Ponzi scheme.
It is worth mentioning that the profitability of XYZ may depend on the
particular economic flows, such as the price and volume of click-throughs or
the price of grain, which in-turn vary in a more or less predictable way in
relation to economic change. Yet there is sufficient complexity in this
dependence that the same market-of-information effects prevail as in (4). This
means that there is a paradoxical away from actually researching the situation
can happen at various points. And the “dispersion” of a Ponzi scheme itself
should be looked at more critically. We can safely assume that when the scheme
“sees the light of day” that it will vanish. But this isn’t simply because each
participant now finds out that it was a fraud but because each participant
believes that each other participant will now be heading for the doors. Many
kinds of schemes could be seen as cooperatives games between brokers to screw
the small investor. And indeed such activity doesn’t have to intent – it could
simply involve a sequence of market maneuvers which brokers have been
habituated.
Indeed, if the value of company is seen as a differential between flows – price
and demand of its goods and supplies as well as its own productivity, then its
value is closer to being a multiple of various random variables.
8.
Let us imagine a game in which everyone in a country must
“invest” in either “red” or “green” bonds on a given day. Whichever bond
receives the most investors “wins” and all the money invested in both bonds
will go to the investors in the winning bond. (particular considerations).
General cooperative/competitive games are what we call game of “information
transduction” – each player’s activity sends information about their general
behavior and the sum of the tendencies to a general behavior is irreducibly
linked to the state of the game; Unlike a zero-sum, there is no strategy that
can be chosen blindly. The evolutionary framework has been introduced to make
these games fully solvable and has the characteristics we’ve mentioned. Games
of buying and using information mentioned in (4) fit within this very general
framework. Also, this framework is ultimately too – too general, any kind of
interaction can be modeled yet the model is fairly unsolvable.
a. What will be key for our approach is to give a scaling-aspect to such embedded information. The “normal” formulation of such games makes information about the structure of a game absolutely dependent on the scale that is observed.
9. If a distribution is “non-Martindale”, it follows that there IS a sum of values åAibt which has an expected positive value – there is a wining strategy. But if the distribution is not stationary, then this distribution is also changing and thus we have not violated the “something for nothing” rule. We are assuming a dynamic where a winning strategy can be deduced – but this opportunity naturally is always closing up as the market continues to evolve and folks take advantage of the strategy.
10. The frame of “utility functions” is interesting. The standard utility function always has an absolute scale. The amount of a trader makes won’t come back and influence the scale of the utility function. But really, we can assume various kinds of inherent scaling utilities – exponential utility where each factor is worth another point. But also, we can look at utility factors varying with time – how much is one willing to lose in a hour, in a day, in a week and so-forth.
11. An assumption of the rational market is that there is as much money to be made betting against a factor as betting for it. This means that those who see an unaccountably high value will be as likely to sell as those who see an unaccountably low level will be to buy. This is clearly untrue and if it were true is would lead to the collapse of markets. Shorting a stock is a less profitable than buying it, options sell are not traded on individual stocks and all option have spreads. The flood of money coming into a bull market goes to investment, to “real” productive economic activity and can then be recycle back to the market. This situation prejudices the information coming into a market, and it tends to make the accuracy of the information less. But also it explains a particular kind of prejudice.
12. Information roughly has “scale” and “level”, there are vague statements about where we will next week in the economy. There are statements about correlation between different economic variables in general and there are homilies like “the stock market has been the most profitable form of investment for the last fifty years”. Any part of this entire fabric of information has a time-scale which is what period it is most relevant to and an abstraction level, which is the area it relates to as well as what kind of relationship it is describing. By our previous reasoning, information around opportunities for stock prices to increase is uniformly more valuable than information about possibilities for stock prices to decrease.
13. The equation we would make would essentially involve an “equality of likelihood” between different scales of information.