Energy Trading: Asian Options.


Consider a risky asset (stock, commodity, a unit of energy) with the price S(t), where t ∈ [0, T], for a given T > 0. Consider an option with the payoff

Fu = Φ(u(·), S(·)) —– (1)

This payoff depends on a control process u(·) that is selected by an option holder from a certain class of admissible controls U. The mapping Φ : U × S → R is given; S is the set of paths of S(t). All processes from U has to be adapted to the current information flow, i.e., adapted to some filtration Ft that describes this information flow. We call the corresponding options controlled options.

For simplicity, we assume that all options give the right on the corresponding payoff of the amount Fu in cash rather than the right to buy or sell stock or commodities.

Consider a risky asset with the price S(t). Let T > 0 be given, and let g : R → R and f : R × [0, T] → R be some functions. Consider an option with the payoff at time T

Fu = g(∫0u(t) f (S(t), t)dt) —– (2)

Here u(t) is the control process that is selected by the option holder. The process u(t) has to be adapted to the filtration Ft describing the information flow. In addition, it has to be selected such that

0T u(t)dt = 1

A possible modification is the option with the payoff

Fu = ∫0T u(t) f(S(t), t)dt + (1 – ∫0T u(t)dt) f(S(T), T)

In this case, the unused u(t) are accumulated and used at the terminal time. Let us consider some examples of possible selection of f and g. We denote x+ = max(0, x)

Important special cases are the options with g(x) = x, g(x) = (x − k)+, g(x) = (K − x)+,

g(x) = min(M, x), where M > 0 is the cap for benefits, and with

f(x, t) = x, f(x, t) = (x − K)+, f(x, t) = (K − x)+ —– (3)


f(x, t) = er(T−t)(x − K)+, f(x, t) = er(T−t)(K − x)+ —– (4)

where K > 0 is given and where r > 0 is the risk-free rate. Options (3) correspond to the case when the payments are made at current time t ∈ [0, T], and options (4) correspond to the case when the payment is made at terminal time T. This takes into account accumulation of interest up to time T on any payoff.

The option with payoff (2) with f(x, t) ≡ x represents a generalization of Asian option where the weight u(t) is selected by the holder. It needs to be noted that an Asian option , which is also called an average option, is an option whose payoff depends on the average price of the underlying asset over a certain period of time as opposed to at maturity. The option with payoff (2) with g(x) ≡ x represents a limit version of the multi-exercise options, when the distribution of exercise time approaches a continuous distribution. An additional restriction on |u(t)| ≤ const would represent the continuous analog of the requirement for multi-exercise options that exercise times must be on some distance from each other. For an analog of the model without this condition, strategies may approach delta-functions.

These options can be used, for instance, for energy trading with u(t) representing the quantity of energy purchased at time t for the fixed price K when the market price is above K. In this case, the option represents a modification of the multi-exercise call option with continuously distributed payoff time. For this model, the total amount of energy that can be purchased is limited per option. Therefore, the option holder may prefer to postpone the purchase if she expects better opportunities in future.


Pareto Optimality

There are some solutions. (“If you don’t give a solution, you are part of the problem”). Most important: Human wealth should be set as the only goal in society and economy. Liberalism is ruinous for humans, while it may be optimal for fitter entities. Nobody is out there to take away the money of others without working for it. In a way of ‘revenge’ or ‘envy’, (basically justifying laziness) taking away the hard-work earnings of others. No way. Nobody wants it. Thinking that yours can be the only way a rational person can think. Anybody not ‘winning’ the game is a ‘loser’. Some of us, actually, do not even want to enter the game.

Yet – the big dilemma – that money-grabbing mentality is essential for the economy. Without it we would be equally doomed. But, what we will see now is that you’ll will lose every last penny either way, even without divine intervention.

Having said that, the solution is to take away the money. Seeing that the system is not stable and accumulates the capital on a big pile, disconnected from humans, mathematically there are two solutions:

1) Put all the capital in the hands of people. If profit is made M’-M, this profit falls to the hands of the people that caused it. This seems fair, and mathematically stable. However, how the wealth is then distributed? That would be the task of politicians, and history has shown that they are a worse pest than capital. Politicians, actually, always wind up representing the capital. No country in the world ever managed to avoid it.

2) Let the system be as it is, which is great for giving people incentives to work and develop things, but at the end of the year, redistribute the wealth to follow an ideal curve that optimizes both wealth and increments of wealth.

The latter is an interesting idea. Also since it does not need rigorous restructuring of society, something that would only be possible after a total collapse of civilization. While unavoidable in the system we have, it would be better to act pro-actively and do something before it happens. Moreover, since money is air – or worse, vacuum – there is actually nothing that is ‘taken away’. Money is just a right to consume and can thus be redistributed at will if there is a just cause to do so. In normal cases this euphemistic word ‘redistribution’ amounts to theft and undermines incentives for work and production and thus causes poverty. Yet, if it can be shown to actually increase incentives to work, and thus increase overall wealth, it would need no further justification.

We set out to calculate this idea. However, it turned out to give quite remarkable results. Basically, the optimal distribution is slavery. Let us present them here. Let’s look at the distribution of wealth. Figure below shows a curve of wealth per person, with the richest conventionally placed at the right and the poor on the left, to result in what is in mathematics called a monotonously-increasing function. This virtual country has 10 million inhabitants and a certain wealth that ranges from nearly nothing to millions, but it can easily be mapped to any country.


Figure 1: Absolute wealth distribution function

As the overall wealth increases, it condenses over time at the right side of the curve. Left unchecked, the curve would become ever-more skew, ending eventually in a straight horizontal line at zero up to the last uttermost right point, where it shoots up to an astronomical value. The integral of the curve (total wealth/capital M) always increases, but it eventually goes to one person. Here it is intrinsically assumed that wealth, actually, is still connected to people and not, as it in fact is, becomes independent of people, becomes ‘capital’ autonomously by itself. If independent of people, this wealth can anyway be without any form of remorse whatsoever be confiscated and redistributed. Ergo, only the system where all the wealth is owned by people is needed to be studied.

A more interesting figure is the fractional distribution of wealth, with the normalized wealth w(x) plotted as a function of normalized population x (that thus runs from 0 to 1). Once again with the richest plotted on the right. See Figure below.


Figure 2: Relative wealth distribution functions: ‘ideal communist’ (dotted line. constant distribution), ‘ideal capitalist’ (one person owns all, dashed line) and ‘ideal’ functions (work-incentive optimized, solid line).

Every person x in this figure feels an incentive to work harder, because it wants to overtake his/her right-side neighbor and move to the right on the curve. We can define an incentive i(x) for work for person x as the derivative of the curve, divided by the curve itself (a person will work harder proportional to the relative increase in wealth)

i(x) = dw(x)/dx/w(x) —– (1)

A ‘communistic’ (in the negative connotation) distribution is that everybody earns equally, that means that w(x) is constant, with the constant being one

‘ideal’ communist: w(x) = 1.

and nobody has an incentive to work, i(x) = 0 ∀ x. However, in a utopic capitalist world, as shown, the distribution is ‘all on a big pile’. This is what mathematicians call a delta-function

‘ideal’ capitalist: w(x) = δ(x − 1),

and once again, the incentive is zero for all people, i(x) = 0. If you work, or don’t work, you get nothing. Except one person who, working or not, gets everything.

Thus, there is somewhere an ‘ideal curve’ w(x) that optimizes the sum of incentives I defined as the integral of i(x) over x.

I = ∫01i(x)dx = ∫01(dw(x)/dx)/w(x) dx = ∫x=0x=1dw(x)/w(x) = ln[w(x)]|x=0x=1 —– (2)

Which function w is that? Boundary conditions are

1. The total wealth is normalized: The integral of w(x) over x from 0 to 1 is unity.

01w(x)dx = 1 —– (3)

2. Everybody has a at least a minimal income, defined as the survival minimum. (A concept that actually many societies implement). We can call this w0, defined as a percentage of the total wealth, to make the calculation easy (every year this parameter can be reevaluated, for instance when the total wealth increased, but not the minimum wealth needed to survive). Thus, w(0) = w0.

The curve also has an intrinsic parameter wmax. This represents the scale of the figure, and is the result of the other boundary conditions and therefore not really a parameter as such. The function basically has two parameters, minimal subsistence level w0 and skewness b.

As an example, we can try an exponentially-rising function with offset that starts by being forced to pass through the points (0, w0) and (1, wmax):

w(x) = w0 + (wmax − w0)(ebx −1)/(eb − 1) —– (4)

An example of such a function is given in the above Figure. To analytically determine which function is ideal is very complicated, but it can easily be simulated in a genetic algorithm way. In this, we start with a given distribution and make random mutations to it. If the total incentive for work goes up, we keep that new distribution. If not, we go back to the previous distribution.

The results are shown in the figure 3 below for a 30-person population, with w0 = 10% of average (w0 = 1/300 = 0.33%).


Figure 3: Genetic algorithm results for the distribution of wealth (w) and incentive to work (i) in a liberal system where everybody only has money (wealth) as incentive. 

Depending on the starting distribution, the system winds up in different optima. If we start with a communistic distribution of figure 2, we wind up with a situation in which the distribution stays homogeneous ‘everybody equal’, with the exception of two people. A ‘slave’ earns the minimum wages and does nearly all the work, and a ‘party official’ that does not do much, but gets a large part of the wealth. Everybody else is equally poor (total incentive/production equal to 21), w = 1/30 = 10w0, with most people doing nothing, nor being encouraged to do anything. The other situation we find when we start with a random distribution or linear increasing distribution. The final situation is shown in situation 2 of the figure 3. It is equal to everybody getting minimum wealth, w0, except the ‘banker’ who gets 90% (270 times more than minimum), while nobody is doing anything, except, curiously, the penultimate person, which we can call the ‘wheedler’, for cajoling the banker into giving him money. The total wealth is higher (156), but the average person gets less, w0.

Note that this isn’t necessarily an evolution of the distribution of wealth over time. Instead, it is a final, stable, distribution calculated with an evolutionary (‘genetic’) algorithm. Moreover, this analysis can be made within a country, analyzing the distribution of wealth between people of the same country, as well as between countries.

We thus find that a liberal system, moreover one in which people are motivated by the relative wealth increase they might attain, winds up with most of the wealth accumulated by one person who not necessarily does any work. This is then consistent with the tendency of liberal capitalist societies to have indeed the capital and wealth accumulate in a single point, and consistent with Marx’s theories that predict it as well. A singularity of distribution of wealth is what you get in a liberal capitalist society where personal wealth is the only driving force of people. Which is ironic, in a way, because by going only for personal wealth, nobody gets any of it, except the big leader. It is a form of Prisoner’s Dilemma.

Banking? There isn’t much to it than this anyways.

Don’t go by the innocuous sounding title, for this is at its wit alternative. The modus operandi (Oh!, how much I feel like saying modis operandi in honor of the you Indians know who?) for accumulation of wealth to parts of ‘the system’ (which, for historic reasons, we call ‘capitalists’) is banking. The ‘capitalists’ (defined as those that skim the surplus labor of others) accumulate it through the banking system. That is nearly an empty statement, since wealth = money. That is, money is the means of increasing wealth and thus one represents the other. If capitalists skim surplus labor, it means that they skim surplus money. Money is linked to (only!) banks, and thus, accumulation is in the banks.

If interest is charged, borrowers will go bankrupt. This idea can be extended. If interest is charged, all money is accumulated in banks. Or, better to say, a larger and larger fraction of money is accumulated in the banks, and kept in financial institutions. The accumulation of wealth is accumulation of money in and by banks. It can only be interesting to see whom the money belongs to.

By the way, these institutions, the capitalists naturally wanting to part with as little as possible from this money, are often in fiscal paradises. Famous are The Cayman Islands, The Bahamas, The Seychelles, etc. With the accumulated money the physical property is bought. Once again, this is an empty statement. Money represents buying power (to buy more wealth). For instance buying the means-of-production (the Marxian mathematical equation raises its head again, MoP), such as land, factories, people’s houses (which will then be rented to them; more money). Etc.

Also, a tiny fraction of the money is squandered. It is what normally draws most attention. Oil sheiks that drive golden cars, bunga-bunga parties etc. That, however, is rather insignificant, this way of re-injecting money into the system. Mostly money is used to increase capital. That is why it is an obvious truth that “When you are rich, you must be extremely stupid to become poor. When you are poor, you must be extremely talented to become rich”. When you are rich, just let the capital work for you; it will have the tendency to increase, even if it increases slower than that of your more talented neighbor.

To accelerate the effect of skimming, means of production (MoP, ‘capital’), are confiscated from everything – countries and individual people – that cannot pay the loan + interest (which is unavoidable). Or bought for a much-below market value price in a way of “Take it or leave it; either give me my money back, which I know there is no way you can, or give me all your possessions and options for confiscation of possessions of future generations as well, i.e., I’ll give you new loans (which you will also not be able to pay back, I know, but that way I’ll manage to forever take everything you will ever produce in your life and all generations after you. Slaves, obey your masters!)”

Although not essential (Marx analyzed it not like this), the banking system accelerates the condensation of wealth. It is the modus operandi. Money is accumulated. With that money, capital is bought and then the money is re-confiscated with that newly-bought capital, or by means of new loans, etc. It is a feedback system where all money and capital is condensing on a big pile. Money and capital are synonyms. Note that this pile in not necessarily a set of people. It is just ‘the system’. There is no ‘class struggle’ between rich and poor, where the latter are trying to steal/take-back the money (depending on which side of the alleged theft the person analyzing it is). It is a class struggle of people against ‘the system’.

There is only one stable final distribution: all money/capital belonging to one person or institute, one ‘entity’. That is what is called a ‘singularity’ and the only mathematical function that is stable in this case. It is called a delta-function, or Kronecker-delta function: zero everywhere, except in one point, where it is infinite, with the total integral (total money) equal to unity. In this case: all money on one big pile. All other functions are unstable.

Imagine that there are two brothers that wound up with all the money and the rest of the people are destitute and left without anything. These two brothers will then start lending things to each-other. Since they are doing this in the commercial way (having to give back more than borrowed), one of the brothers will confiscate everything from the other.
Note: There is only one way out of it, namely that the brother ‘feels sorry’ for his sibling and gives him things without anything in return, to compensate for the steady unidirectional flow of wealth….

Financial Entanglement and Complexity Theory. An Adumbration on Financial Crisis.


The complex system approach in finance could be described through the concept of entanglement. The concept of entanglement bears the same features as a definition of a complex system given by a group of physicists working in a field of finance (Stanley et al,). As they defined it – in a complex system all depends upon everything. Just as in the complex system the notion of entanglement is a statement acknowledging interdependence of all the counterparties in financial markets including financial and non-financial corporations, the government and the central bank. How to identify entanglement empirically? Stanley H.E. et al formulated the process of scientific study in finance as a search for patterns. Such a search, going on under the auspices of “econophysics”, could exemplify a thorough analysis of a complex and unstructured assemblage of actual data being finalized in the discovery and experimental validation of an appropriate pattern. On the other side of a spectrum, some patterns underlying the actual processes might be discovered due to synthesizing a vast amount of historical and anecdotal information by applying appropriate reasoning and logical deliberations. The Austrian School of Economic Thought which, in its extreme form, rejects application of any formalized systems, or modeling of any kind, could be viewed as an example. A logical question follows out this comparison: Does there exist any intermediate way of searching for regular patters in finance and economics?

Importantly, patterns could be discovered by developing rather simple models of money and debt interrelationships. Debt cycles were studied extensively by many schools of economic thought (Shiller, Robert J._ Akerlof, George A – Animal Spirits: How Human Psychology Drives the Economy, and Why It Matters for Global Capitalism). The modern financial system worked by spreading risk, promoting economic efficiency and providing cheap capital. It had been formed during the years as bull markets in shares and bonds originated in the early 1990s. These markets were propelled by abundance of money, falling interest rates and new information technology. Financial markets, by combining debt and derivatives, could originate and distribute huge quantities of risky structurized products and sell them to different investors. Meanwhile, financial sector debt, only a tenth of the size of non-financial-sector debt in 1980, became half as big by the beginning of the credit crunch in 2007. As liquidity grew, banks could buy more assets, borrow more against them, and enjoy their value rose. By 2007 financial services were making 40% of America’s corporate profits while employing only 5% of its private sector workers. Thanks to cheap money, banks could have taken on more debt and, by designing complex structurized products, they were able to make their investment more profitable and risky. Securitization facilitating the emergence of the “shadow banking” system foments, simultaneously, bubbles on different segments of a global financial market.

Yet over the past decade this system, or a big part of it, began to lose touch with its ultimate purpose: to reallocate deficit resources in accordance with the social priorities. Instead of writing, managing and trading claims on future cashflows for the rest of the economy, finance became increasingly a game for fees and speculation. Due to disastrously lax regulation, investment banks did not lay aside enough capital in case something went wrong, and, as the crisis began in the middle of 2007, credit markets started to freeze up. Qualitatively, after the spectacular Lehman Brothers disaster in September 2008, laminar flows of financial activity came to an end. Banks began to suffer losses on their holdings of toxic securities and were reluctant to lend to one another that led to shortages of funding system. This only intensified in late 2007 when Nothern Rock, a British mortgage lender, experienced a bank run that started in the money markets. All of a sudden, liquidity became in a short supply, debt was unwound, and investors were forced to sell and write down the assets. For several years, up to now, the market counterparties no longer trust each other. As Walter Bagehot, an authority on bank runs, once wrote:

Every banker knows that if he has to prove that he is worth of credit, however good may be his arguments, in fact his credit is gone.

In an entangled financial system, his axiom should be stretched out to the whole market. And it means, precisely, financial meltdown or the crisis. The most fascinating feature of the post-crisis era on financial markets was the continuation of a ubiquitous liquidity expansion. To fight the market squeeze, all the major central banks have greatly expanded their balance sheets. The latter rose, roughly, from about 10 percent to 25-30 percent of GDP for the appropriate economies. For several years after the credit crunch 2007-09, central banks bought trillions of dollars of toxic and government debts thus increasing, without any precedent in modern history, money issuance. Paradoxically, this enormous credit expansion, though accelerating for several years, has been accompanied by a stagnating and depressed real economy. Yet, until now, central bankers are worried with downside risks and threats of price deflation, mainly. Otherwise, a hectic financial activity that is going on along unbounded credit expansion could be transformed by herding into autocatalytic process that, if being subject to accumulation of a new debt, might drive the entire system at a total collapse. From a financial point of view, this systemic collapse appears to be a natural result of unbounded credit expansion which is ‘supported’ with the zero real resources. Since the wealth of investors, as a whole, becomes nothing but the ‘fool’s gold’, financial process becomes a singular one, and the entire system collapses. In particular, three phases of investors’ behavior – hedge finance, speculation, and the Ponzi game, could be easily identified as a sequence of sub-cycles that unwound ultimately in the total collapse.

Capital as a Symbolic Representation of Power. Nitzan’s and Bichler’s Capital as Power: A Study of Order and Creorder.


The secret to understanding accumulation, lies not in the narrow confines of production and consumption, but in the broader processes and institutions of power. Capital, is neither a material object nor a social relationship embedded in material entities. It is not ‘augmented’ by power. It is, in itself, a symbolic representation of power….

Unlike the elusive liberals, Marxists try to deal with power head on – yet they too end up with a fractured picture. Unable to fit power into Marx’s value analysis, they have split their inquiry into three distinct branches: a neo-Marxian economics that substitutes monopoly for labour values; a cultural analysis whose extreme versions reject the existence of ‘economics’ altogether (and eventually also the existence of any ‘objective’ order); and a state theory that oscillates between two opposite positions – one that prioritizes state power by demoting the ‘laws’ of the economy, and another that endorses the ‘laws’ of the economy by annulling the autonomy of the state. Gradually, each of these branches has developed its own orthodoxies, academic bureaucracies and barriers. And as the fractures have deepened, the capitalist totality that Marx was so keen on uncovering has dissipated….

The commodified structure of capitalism, Marx argues, is anchored in the labour process: the accumulation of capital is denominated in prices; prices reflect labour values; and labour values are determined by the productive labour time necessary to make the commodities. This sequence is intuitively appealing and politically motivating, but it runs into logical and empirical impossibilities at every step of the way. First, it is impossible to differentiate productive from unproductive labour. Second, even if we knew what productive labour was, there would still be no way of knowing how much productive labour goes into a given commodity, and therefore no way of knowing the labour value of that commodity and the amount of surplus value it embodies. And finally, even if labour values were known, there would be no consistent way to convert them into prices and surplus value into profit. So, in the end, Marxism cannot explain the prices of commodities – not in detail and not even approximately. And without a theory of prices, there can be no theory of profit and accumulation and therefore no theory of capitalism….

Modern capitalists are removed from production: they are absentee owners. Their ownership, says Veblen, doesn’t contribute to industry; it merely controls it for profitable ends. And since the owners are absent from industry, the only way for them to exact their profit is by ‘sabotaging’ industry. From this viewpoint, the accumulation of capital is the manifestation not of productive contribution but of organized power.

To be sure, the process by which capitalists ‘translate’ qualitatively different power processes into quantitatively unified measures of earnings and capitalization isn’t very ‘objective’. Filtered through the conventional assessments of accountants and the future speculations of investors, the conversion is deeply inter-subjective. But it is also very real, extremely imposing and, as we shall see, surprisingly well-defined.

These insights can be extended into a broader metaphor of a ‘social hologram’: a framework that integrates the resonating productive interactions of industry with the dissonant power limitations of business. These hologramic spectacles allow us to theorize the power underpinnings of accumulation, explore their historical evolution and understand the ways in which various forms of power are imprinted on and instituted in the corporation…..

Business enterprise diverts and limits industry instead of boosting it; that ‘business as usual’ needs and implies strategic limitation; that most firms are not passive price takers but active price makers, and that their autonomy makes ‘pure’ economics indeterminate; that the ‘normal rate of return’, just like the ancient rate of interest, is a manifestation not of productive yield but of organized power; that the corporation emerged not to enhance productivity but to contain it; that equity and debt have little to do with material wealth and everything to do with systemic power; and, finally, that there is little point talking about the deviations and distortions of ‘financial capital’ simply because there is no ‘productive capital’ to deviate from and distort.

Jonathan Nitzan, Shimshon Bichler- Capital as Power:_ A Study of Order and Creorder