Financial Models


Alexander Liss

12/09/97; 11/15/98

Preface *

Basic Concepts *

Economic Value *

Liquid Assets *

Money *

Credit *

Basic Volatility *

Uncertainty *

Reaction on the Change *

Stock Market *

Wealth *

Measurement of the Economic Value *

Compensation for the Economic Value *

Economic Value of Business *

Measurement of Wealth *

Main Properties of Wealth *

Models and Market *

The Economic Value in Financial Models *

Main Equation *

Compensation Components *

Market Invariant *

Uncertainty and Risk *

Variability *

Cushion *

Measuring the Cushion *

Modeling Uncertainty *

Model's Imprecision *

Insurance Premium *

Application of Uncertainty Modeling *

Economic Value of the Business *

Price of the Business *

Publicly Traded Company *

Price of Stock *

Employee's Options *

Short-term Investment *

Wealth *

Measuring Wealth *

Present Value of Future Wealth *

Liquid Assets *

General Properties of Liquid Assets *

US Currency *

Basics *

Tools of Control *

Possible Future of Monetary System *

Special Demand on Treasuries *

Shortage of Liquid Assets *

The Economic Value of Liquidity *

Market Dynamics *

Market Oscillation *

Natural Phenomenon *

Oscillation of Economic Value and Prices *

Models *

Management of Oscillations *

Types of Market Activities *

Influence of Computer Trading *

Employee Options *

Model for Trading *

Short-term Investment *

"Technical Analysis" and Positive Feedback *

Dynamics of Defaults *

Stream of Defaults *

Defaults and Market Cycle *

Forecast of Economic Value of Publicly Traded Company *

Derivatives *

Definition *

Economic Value of Underlying Asset *

Price of Underlying Asset *

Influence on Other Prices *

Market Tools of Pricing *

Spectrum of Goals *

Our Tasks *

Cash Flow Instruments *

Definition *

The Economic Value *

Major Sectors of the Market *

Interest Rate Structure *

Profit from Planning *

Account Funding *

Cash Management *

Cash and Commodity Optimization *

Stochastic Nature of Market and Forecast *

Momentarily and Statistical Arbitrage *

Uncertainty of Forecast *

Collective Forecast *

Work Horse Stochastic Process *

F-process *

M-process *

Primary Parameters *

Choice *

Usage *

Nonlinear Nature of Market *

Place of Collective Forecast *

Liquidity *

Currency Exchange *

Monetary System *

Complexity of the Market *



The extensive use and advance of technological ideas became possible only because of the development of the methods of computation, which moved the use of technology from the art, where success largely depends on the artist's ability, to the area of predictable and repeatable designs. Similar development is needed in the methods of economic and financial management. However, here we face an unusual problem.

The technology deals with nature, which laws are stable, even when they are unknown. The development in the technological research builds up on previous developments, and as a result, we have an accumulated knowledge, where "break-through" ideas are shared and tested many times in the practical applications.

This is not true with economic and financial models. They also try to grasp the laws of the development of the society, economy and finance, but their object of study is changing fast, and the results of the study make it change even faster. The better the society knows this area the more new ways of development are explored. The factors, which influence was negligible yesterday come into play today. Models, which were perfectly adequate yesterday, do not reflect new major forces, and do not give proper results.

The major source of the concepts, ideas and models for the technology is the world of academia. This social structure lives by its own rules, studies whatever is its subject of study as a way to satisfy the natural human curiosity, and as a side effect, almost accidentally, produces the knowledge needed for the technology.

There is no mechanism in academia, which can keep people interested in research, which results will be outdated a few years later. But this is exactly what we need - models which are right today, and the system, which updates them, that tomorrow we have other models, which are right then. Hence we need a dual system, one part, which develops concepts, and in addition, the perpetually updated system of models suited for today's needs of Market decision-makers.

The entire society will benefit from this system. However small investors and traders will be first and immediate beneficiaries.

The main effect of models - they reduce uncertainty. The effect is twofold. First, models extract major factors affecting decisions, they build a foundation for reasonable decision-making. Second, models give tools of computation, which increase precision of decision-making, and allow the development of the Market in the areas previously too fuzzy and inaccessible.

Reduction of the level of uncertainty has direct economic effects. High level of uncertainty causes high decision's risk. High risk can be offset only by high level of reserves. If level of reserves is low, then possible losses can push a decision-maker out of the Market, forcing acceptance of losses as final. Hence, in the world of high uncertainty deep pockets have more investment and trading opportunities, than small guys, which can't afford to loose much. The reduction of uncertainty leads to a wider participation of investors in the Market.

The question of applicability of the particular model to describe particular situation is difficult even for engineering models. It is many times more difficult for the economic and financial models, because models, which are good to describe yesterday's situation, can be not applicable to today's situation. New models, good for today's situation, can be not applicable to yesterday's situation. This makes systematization of models even more important. In the system of models, there are special methods, which allow checking new model in the system against other models, which we used before understand better and trust more.

We concentrate on quantitative models. But to describe them, we have to start with concepts. Scientific and technological concepts are well defined and well studied. Many concepts needed for the quantitative financial models need a more precise definition, or one definition has to be chosen among many different definitions. Concepts change slower than models, but they change also. The most important change in concept happens when it becomes clear, that instead of one concept we have to use two or more other concepts, which can be a basis of a series of models with easy interpretation and measurement. We offer here a few examples of such development.

There is one important approach to financial modeling, which we favor, and in which we differ from many currently used approaches. We explain it here explicitly.

There is always a desire, in part of designers of financial models, to deal with something stable, with kind of Laws of Market, similar to Laws of Nature. While these models are important, there are other models dealing with the Market participant's forecast and goals.

We prefer embracing this process, instead of ignoring it. We make only one reasonable assumption, that Market participants make optimal decisions in their situation according to their goals. We assume that they make the best efforts to forecast future and estimate mistakes of such forecast. We assume that, because we are sure that this behavior is the only responsible behavior of the Market participant. This is the only way the Market as a system can establish prices.

Hence, forecast and the specifics of decision-making in many cases are explicitly present in our models.

This unusual approach often leads to unusual form of models and unusual interpretation of their parameters. However, in many cases, they are actually very close to existing popular models.

It is important to have a variety of models. Simple models we understand well, know boundaries of their applicability, and we can get e quick estimate with their help. After we can employ more sophisticated models to get better precision. The result we have to check against a few other models, to be sure, that factors neglected in the model do not affect much the result in particular situation we are in. This dictates the way we describe our models. We start with simple model, which deals with a few factors, and proceed with corrections-improvements of the model, creating more sophisticated and potentially more precise one.

We do not insist that our approach to modeling is the only right approach. Many choices we had made here were dictated by our goal to arrive to one coherent system of models. Other approaches are not only possible, they are desirable, because when one system of models hits the bounds of its applicability, we need the other approaches, the other ideas to solve the problem.


Basic Concepts

Economic Value

Economic Value is something created by society and distributed between some members of society. For example whey was grown by a farmer, was sold to a backer, was backed into bread; bread was sold and was consumed by some group of people. This is a set of activities, where the Economic Value was created and distributed between participants. The participants are - the owner of the land, the farmer, producers of the tools the farmer used, providers of transportation, the backer, his workers, the bank (which lent the money to the baker), the store owner (which sells the bread) and people who consumed bread. How much of the Economic Value was created depends on circumstances. If the process is hard, there is no other way to get the result, and there is not enough whey stored, then the Economic Value is big. If there is abundance of whey, that people do not need more - then the Economic Value is small, even if it was difficult to produce it.

The Economic Value of product or service depends on circumstances and can change with time for the same unit of product, or for the same type of product created in different times.

Consumer is an important participant in the Economic Value creation. The consumer extracts the Economic Value; this requires special skills and training. An interesting example is old art, antique and collectibles. Mostly the consumer creates the Economic Value of this kind of products.

We have following basic classes of the sources of Economic Value:

The Economic Value of natural resource is in the beginning of a process, which leads to the final consumption or use of product or service. Some resources are clearly limited, other are not (in our time span), as (sometimes) a water from the river.

Business is similar to the river - business keeps creating the Economic Value of product "from nothing", because of the society's creativity. This is very important idea, with many implications, however its discussion is beyond the scope.

Consumer adds the Economic Value with the consumption skills. The consumed Economic Value is in the end of the process.

In the process of the creation of the Economic Value society changes (usually destroys) its habitat. Some social groups move to other place leaving the Nature to deal with it. The other groups control this process in a way that economic activity leads to acceptable and even desirable changes in habitat; usually this is possible only for the rich societies. Again, this is very important idea also, and its discussion is beyond the scope also.

Prices are tools of the distribution of the Economic Value between all participants of its creation and consumption and protectors of the current social structure - the owners of resources, government, etc. From economic point of view there is no "fair" or "unfair" prices, including prices of work done (wages). This distinction comes from other point of view, which takes in consideration the direction of development of society as a whole.


Liquid Assets

If we want to distribute the Economic Value among Market participants, we need easily exchangeable assets - liquid assets. The long term Economic Value of this kind of assets is not important, because they change hands fast, and small corrections reflecting their changing value can be easily made at each transaction. The main source of the Economic Value of liquid assets is their exchangeability. Usually liquid assets carry some additional Economic Value, for example if gold is used as liquid asset, it is valuable as a material used in the jewelry and some technical applications, if the debt obligation is used as liquid asset, it is useful as a tool of borrowing, etc.



Money is liquid asset which main purpose is to serve as liquid asset. This is clear with paper money, which does not have the Economic Value beyond this one. The money's ability to serve as liquid asset is created by the special social structures, for example by laws, which oblige courts to recognize it as a valid repayment of debt. Usually the Economic Value of the money unit is changing, but this does not impede the money's ability to serve in transactions.

We measure many economic characteristics with money. This is a secondary use of money. Money is not created for that. This kind of measurement works fine when we want to compare economic characteristics in the same moment of time (because of equalizing work of the Market), and it does not work well for different moments of time. For this measurement, we need a calibration of our measurement tool - time discount.

There are two major factors in time discount.

First is well known - inflation (deflation). If k is a rate of inflation we multiply payment in the future moment t: P(t) by exp(-k*t) and arrive to Present Value P.V[P(t)] of this payment:

P.V[P(t)] = exp(-k*t)*P(t).

This is a fine calibration, except the precision of measurement of the rate of inflation is low and the rate is changing.

The other factor is rarely discussed, but it is important nevertheless and its importance is growing. Even in the absence of inflation one who accepts future payment in exchange for current asset or service carries the risk. Future situation is unknown: it is quite possible that that it is more difficult (more expensive) to satisfy our desires in the future than now. Even our desires might change in the future. This factor has to be taken in consideration also.



Credit is a way of fast moving of assets, and lending is a special business crucial to the very existence of the Market.

The lender charges the payment for the right to use his assets temporarily. The payment is stretched during the period of assets use or it can be made in the end of the period, when assets are returned. The lender creates the Economic Value by finding a combination of social structure, business opportunity and peoples' character, where the borrowed money works properly on the Market. This Economic Value is divided between the borrower, which puts assets to use, and the lender, who gets his share, which includes its cost (lender needs to check initially and monitor later the creditworthiness of borrower), its profit, and its risk insurance.

The ratio of the profit to the lent assets (for a short term) is a basic characteristic of the Market. The society protects lending; hence lending is a very special business. Its risk is lower than in other businesses, and this ratio is lower also. If the lender is in the business of lending, then his assets (his wealth) grow exponentially, as it should be for the active market participant. However, the speed of this exponential growth is lower than in other businesses.


Basic Volatility

Price volatility is in the very essence of the Market. Degree of this volatility depends on particular trade mechanisms involved. For example, the presence of market-makers for particular stock in the stock market reduces stock's volatility, and the opposite is true - the possibility of situation that there is no trade, while there are potential buyers and sellers, leads to higher price volatility.

This is a well-known phenomenon under the name "price discovery".

This volatility, which we call the Basic Volatility, is a subject of study of the Market as a whole. For us there is nothing to look for in it, we simply accept its existence and measure its value. We cannot explain changes of the price, which occur in the boundaries of Basic Volatility.

The simple model of the effects of Basic Volatility of the price is a normally distributed random value, which has to be added to the logarithm of price. These random additions are independent for different prices, their mean value is zero and their variance is the same for all assets traded in similar conditions.

There is no sense in defining the Basic Volatility for the rarely traded assets.

While the structure of particular sector of the Market - the mechanisms, which the Market employs, stays the same, and the general Market conditions are the same, we have reason to believe that the level of Basic Volatility stays the same. However as soon the Market enters the situation where neither current experience nor current models lead to successful decisions, market-participants use substantially different decision-making tools and the level of Basic Volatility grows. As a result, the Market, as a whole, experiments with different hypothesis until it finds the acceptable set of methods and ideas. Then, the level of Basic Volatility lowers to the level usual for stable times.

The level of Basic Volatility we measure with the variance of the random value, which we use to model the effects of this volatility - the random variation of the logarithm of price.



The Economic Value creation and distribution is stretched in time. There is an uncertainty in these processes, and this uncertainty is factored into decisions of the participants of the process. For example, one would not participate in the process, if one is not sure, with acceptable level of certainty, that one can cover his expenses and get the profit. Acceptable level of certainty means an insurance premium, which one expects for his participation.

The concept of uncertainty comes up in many different modeling situations - when we describe:


Reaction on the Change

In the first estimate, market participants react on small changes of the Economic Value or the price of asset with the strength of reaction, which is proportional to the ratio of the value difference to the initial level of this value. There are limitations to applicability of this rule - this change should not endanger the ability of the decision-maker to satisfy what he perceives as his basic needs.


Stock Market

Publicly traded company is a combination of a manufacturing or service company and a financial company. An issue of stock, using the stock to buy other companies, repurchases of the stock, granting options to employees - all this is financial activity, all this creates the Economic Value by itself, in addition to the Economic Value created by the main company's activity. Shares of company's stock are liquid assets, which have the Economic Value by themselves as other liquid assets.

There are specifics of the market of shares of stock - it is organized in a special way, which make stock more liquid and restricts the trade in the same time. This places special requirement on financial models - they have to reflect the specifics of stock market functioning (explicitly or implicitly).

Shares of stock are special standard business contracts, which have many different properties. Each property of this contract affects stock pricing.

The behavior of the stock price depends on the way the particular stock market is organized, where the stock is listed.

We point here a few major factors:

First, if there is a market maker for the particular stock, its price changes smoothly, if there is no market maker - it is possible to have periods, when no trades are executed.

Second, there is a lag between the moment the order is placed and it is executed. This lag is different for different groups of market participants. For example for automatic computer trading this lag is very short. For brokers trading on their own account it is longer, but relatively short also. For investors who have to place orders over the telephone, this lag can be big. The shorter is the lag the bigger are trading possibilities.

The Market behavior depends on the ratio of the trading volumes of different groups of market participants. When this ratio changes rapidly, as in the case of introduction of computer trading, the stock prices can be affected.

Shorter lag gives the possibility to explore the trading variants unavailable before. Hence, smart use of computer trading should be able to deliver higher profits without additional risk or additional pressure on the stock market.



Wealth is the ability to get products and services (the potential ability). The concept has sense only in the market-based society, which delivers goods and services when demand arises.

We distinguish a few Sources of the Wealth:

Natural Resources is an obvious source of wealth of the society as a whole, however who benefits from them personally, or which social group benefits more, or even which country benefits more is a separate question. The Market and social structures (as ownership) work together in the process of distribution of wealth related to this source.

Business Inventions, which create combinations of "natural" and social structures desired by members of society, have their origin in the creativity of society. Who benefits more from this creativity - one who made the creation known (the inventor), or one who made it a product (businessmen, employees), or one who took risk and financed the whole enterprise (investor), depends on circumstances, social structures, etc.

A goal of the business is creation of wealth for customers, investors, businessmen and employees. The ability to create wealth is the Economic Value of the business. There are some additional components of its Economic Value related to creation of stable social structures, improvement of knowledge of its employees, etc.

Trade makes goods and services available, hence the degree of development of trade affects directly the level of wealth of society and each of its member.


Measurement of the Economic Value

There is no tool, which allows measurement of the Economic Value as money. With all variability of money, money is a universal concept related to the Market as whole. The attempts to measure the Economic Value with some "natural" units (production time spent, heat producing ability of the fuel, etc.) did not produce any useful quantitative models - sooner or later we need to describe the compromise between different characteristics reflecting the Economic Value. Money is a tool to describe this compromise.

The best possible tool of the Economic Value measurement is a free Market. Hence, we have to use the system of prices to measure the Economic Value. The Market moves resources where they needed the most. This gives a possibility to measure the Economic Value.

The Economic Value of product we can measure adding the Economic Value of natural resources used to create the product and all Economic Values added by the participants including trade, transportation, advertising. Because of the equalizing work of the Market, this Economic Value is about the same in different chains of creation of the same product.

The Economic Value of natural resource we define for the unit of measurement of it. The Market takes care of moving the natural resource from the place Nature has it to any place it is needed even across borders of counters. This equalizes resources of the same type, regardless their location. The Economic Value of the unit of natural resource is the same in all chains of product creation, which use it.

This Economic Value is changing in time, usually slowly.


Compensation for the Economic Value

The price charged for natural resources by their owner, is a compensation for the participation in the Economic Value creation.

The price charged by the business for the participation in process is a compensation for the Economic Value creation also.

The final consumer participates in the Economic Value creation also, and he has his compensation - a consumed Economic Value. The consumer has a double participation in the Economic Value creation - one as the Economic Value adding party with the consumption skills, and the other as a pure consumer. The Economic Value adding part needs to be compensated separately and pure consumption part separately.

Each participant in the Economic Value creation has expenses: owner of resources has to extract resources, the business has its processing expenses, and the consumer pays for the product and has some other expenses.

It is obvious, that the difference between compensation and expenses is positive and substantial - otherwise why someone would participate in this process of creation of Economic Value.

Compensation is about always lower than the Economic Value created. The price for the resources is lower than their Economic Value. The payment given to the Economic Value creators covers their expenses and brings some profit but still it is lower than the created Economic Value. As a result, the final customer pays the price for the product and consumes the Economic Value, which is higher than the paid price.

As we see, this "not fair" compensation drives the Market, and without the Market, there is no Economic Value to be compensated for.


Economic Value of Business

Business as something that can be bought or sold hence something that can satisfy someone's desires has an Economic Value. This is a special Economic Value, because the desires associated with it are different from desires associated with consumables, however it is close to such products as tools.

The Economic Value of the business mostly is its ability to generate wealth. By the nature of the Market, all this is possible only when the business adds the Economic Value in the process of creation of products and services. The Economic Value of the business we define for the time unit.

A defining characteristic of the Economic Value of the business is the amount of compensation the business gets for the increasing of the Economic Value of product in the time unit.

Measurement of Wealth

In quantitative models, we need to measure the level of wealth. As in the case of the Economic Value, there is no better tool to measure it than money. However, we need to make proper corrections (for the inflation, etc.), because we are interested here in the buying power of money.

The ability to satisfy the desires starts after the needs are satisfied. Hence when we measure the Wealth, we have to subtract what is needed to satisfy the person's (family's, society's) needs from the available resources.

How fast the resources can be made available to satisfy the desires is the other factor, which has to be considered. For example, illiquid assets should be discounted, if we want to equate them with liquid assets (as if the loan is taken with the illiquid assets as collateral).

The other discount has to be made to reflect the premium to pay in some societies or some circumstances to deliver the desirable product or service in time.

The measurement of the level of wealth is important, for example for proper description of the goals of investment.


Main Properties of Wealth

There are a few empirical facts about the level of wealth.

In the first estimate, market participants react to small changes of wealth with the strength of reaction, which is proportional to the ratio of the wealth difference to the initial wealth level.

More precisely, this reaction gets weaker as the basic level grows. This is an important observation, because it says how we should build goal functions, etc.

In the first estimate, the Market has a tendency of producing the exponential growth of wealth of its society, if the natural barriers or social turmoil does not stop it.

More precisely, the growth is super-exponential. This corresponds well with the first observation.


Models and Market

We have two types of models.

One type gives us the estimates (of prices) based on information available to us. We use these models to make our own Market decisions.

The other type gives us description of the actual Market behavior - the real movement of interest rates, stock prices, etc. We use these models to improve the models, which we use to make decisions. This type of models reduces uncertainty.

The Market is a combination of the organization, which automatically monitors the state of economy and reflects it in prices, and conscious efforts of the specialists in the doing the same.

The main tool of the Market as organization is the stochastic adaptation. All prices are subject to perpetual small random changes, and the Market perpetually estimates if this randomly changed system of prices is adequate. As a result, the Market has an adequate to current situation system of average prices. The difference between an average price and particular concrete price is the price volatility, needed for the normal functioning of the Market.

The modeling, in which we are involved here, is a part of the other side of Market, its conscious side. If we are able to formulate models, which reflect adequately the current state of the Market, then these models are reinforced by the Market, because market participants make their decisions based on them.


The Economic Value in Financial Models

We had defined the Economic Value - a concept and a parameter of financial models, here we will proceed with the study of it.


Main Equation

All participants of the creation of Economic Value get more than they spend - otherwise they would not participate. Hence, if someone is buying in the process, he pays less than the part of Economic Value he gets, and someone who is selling in the process gets higher price than the Economic Value he had contributed (with his work and knowledge). This is a driving force behind the movement of the product across the society in the process of the product creation (including advertisement, trade, transportation, etc.). It looks as if something is created from nothing.

The first impression is wrong, as one could expect. The additional hidden part of the Economic Value is Natural resources used to create it, the ingenuity of society, which generates new ideas and casts them into products and tools of their creation and the skills of consumers who can consume products and services useless in less developed society. This is a source of "profit", which each participant takes. The profit, which happens because the effect (the distributed portion of the Economic Value) is higher than the cost (the sum of what was paid for it and efforts spent).

The price, which the consumer pays, is equal to the sum of all compensations (except if there is some intangible compensation as an improvement of the production skills). To make equation uniform, we can pretend that the consumer "pays" himself the compensation for the efforts, knowledge and skills, which add the product Economic Value. The difference between the Economic Value extracted by the consumer and the price paid is big enough to keep the consumer interested - the consumer is a crucial decisive element in this process.

In other words, the difference between the consumed Economic Value and the price paid by consumer is a sum of natural resources and uncompensated Economic Values created on each step of the product creation. As one can see there is no an equation, from which the Economic Value can be directly computed from (average) prices, because we do not measure the consumed Economic Value. The Market does not help in the computing of the consumed Economic Value - the prices are tools of distribution and they work fine without the measured consumed Economic Value. But in some models we need to measure the consumed Economic Value, for example when we try to predict the future of the new product, or simply to set the price for the new product.

Measurement of the consumed Economic Value can be done with a set of natural characteristics (weight, productivity, etc.) and the special model, which translates the values of these natural characteristics into money value of the Economic Value of the product. This is a specific model for each product.


Compensation Components

The compensation for the participation in the Economic Value creation has a few distinctive components. Two components are obvious - the cost and the profit. The third one is often forgotten - the insurance premium. We all act in an uncertain world, and successful among us always have a Cushion to maneuver in the case the things do not go as planed. Hence, successful market participants build into the required compensation the premium for risk taken. The degree of the risk is different for different sectors of economy and the premium differs correspondingly.


Market Invariant

Free Market equalizes profit on all steps: if there is an area of better application of resources, then more resources are moved to this area until the level playing field achieved.

It is important to define what it means equalization of profit. Definitely, it does not mean the same monetary value of the profit. Most likely, this is a ratio of the profit to the cost, which stays the same. But in many cases we need more precise model to find the actual values produced by this equalization.

Similar reasoning brings us to the conclusion, that the Market equalizes the degree of compensation for the Economic Value created by businesses. We can define this degree as a ratio of the profit to the value: the Economic Value minus cost and minus insurance premium. The business does not need to make more efforts to increase the uncompensated Economic Value beyond what is customary for the Market as a whole - it would rather move to the other area of activity where it is easier to make a profit.

Note that this relation allows the direct computation of the Economic Value for the known products with known insurance premiums.

The Market equalizes the degree of compensation for different natural resources also. This works similar to the compensation for the created Economic Value - if the compensation differ, then there is an initiative to move capital. The value, which stays the same, is a ratio of the profit to the cost.

The cost here includes the payment for the locking-in capital, however here it is more obvious, because the capital is locked-in for the longer terms. The insurance premium usually is higher here also. Note that the degree of uncompensated Economic Value is irrelevant here.

The equalizing ability of the real Market can not be so perfect as we assume here; hence, more precise models have to take in consideration the variation of characteristics, which we assume fixed here. In reality, there are barriers and delays, which make the movement of capital difficult.


Uncertainty and Risk

Uncertainty, as a characteristic of the situation, and risk, as a characteristic of the decision, are complex concepts, which require a careful analysis in each particular case. Often, the variability is confused with uncertainty.



While variability often is a good characteristic of uncertainty, there are kinds of variability, which are irrelevant to the uncertainty of the decision-making situation and to the risk. All depends on the modus of operation of the system, which makes the decision.

If there are different prices, which one has to pay for the same product in different stores, then it takes some shopping to find the good price (if the average price does not change fast while we are shopping). If, in the average, the stock price stays the same through some time, one can place the order and sell high or buy low, if one has time to wait while the price volatility brings the price to the proper relation to the preset boundary. In this cases the variability causes the time-price trade-off - if we are in a hurry we pay more for the convenience. In this case, the uncertainty of the price value is an uncertainty of the movement in the time of the average price - we cannot eliminate it with the shopping, we can loose the opportunity to buy at current good price.



The handling of uncertainty defines many characteristics of business and its level of profitability. It is important to describe specific ways it is done by the business and find out particular classes of uncertainty the business deal with. However, there are some common ways of handling uncertainty in the business decision-making. The business makes a lot of decisions, and there is risk involved with each of them. Each decision moves business assets, holdings, ability to make decisions, etc. up or down. Hence, we have these accumulated values moving up and down.

The business always has a Cushion in these values - it can allow them to go down in anticipation of the future to turn even higher up. However, there is a limit how much down these accumulated values can go. This limit determines the level of risk the business can take - risk in general and risk in each small decision.

If the business has "deep pockets" - a substantial Cushion of resources, then it can take higher level of risk. It means that this business simply has more opportunities; hence, it can command higher profit levels. Hence, the level of the Cushion, which the business has, is an important characteristic of the business.


Measuring the Cushion

The Cushion is not a frozen set of resources - this is a set of resources, which risk of loosing is mach lover, than the level of risk in the main activity. The profit related to the Cushion we do not analyze separately, but as a part of the business profit.

In the models, we often define the Cushion probabilistically: "There is available with the probability P(C) a Cushion resource of the price C".


Modeling Uncertainty

There are many types of uncertainty we would like to reflect in our models - different types often require different models. In financial models, modeling uncertainty is one of the most difficult tasks. Each break through in this area brings changes in the way the financial market functions, as it had happened with the introduction of models, which allow more precise computation of prices of options.

Because the concepts of uncertainty and risk are so complex, the only models we can use are simple models, which allow a good interpretation and which lead to accumulation of the substantial level of experience in their usage.

In the same time a good description of uncertainty in the particular situation leads to new strategies or new financial products or systems, which can counteract the described uncertainty. Then new models are needed, which describe this new situation caused by new products and systems.

To be able to follow-up with new models fast, it is prudent to build the system of models based on universal building blocks of the models, which can be used in new models.


Model's Imprecision

The uncertainty of the future affects our models, it is important to build into the model the estimate of the model's mistakes. This is similar to what is done with estimates of mistakes in engineering. This is extremely important in financial models, where mistakes are big. If the model's mistakes are too big, it is better to make decision without the model than with such a model. This seems obvious, but it is not a common practice in financial modeling yet.


Insurance Premium

The important goal of the modeling uncertainty in financial models is computation of the "insurance premium": the part of compensation for Economic Value creation or a part of the price, which counterweights the risk taken.

The main idea in our models is using the probability theory to make this premium better interpretable and arrive to some parameters, which do not change fast.

We present a disaster, which we want to avoid, as a situation, where a certain parameter decreases beyond a certain boundary. We compute the probability of this event with the help of our model, and we demand this probability to be lower than some level, which we set for this type of the risk. This is similar to models used in reliability theory.


Application of Uncertainty Modeling

It is important to not overestimate the results, which can be achieved with the modeling of uncertainty. Most of all these models are needed to avoid bad decisions. This is already a lot. Anyone who invests in the stock market knows how much effort and time it takes to achieve a small improvement in investment results, and how fast all this can evaporate if the Market turns in the wrong direction. Better models help avoiding bad Market situations - they reduce risk. It means that small investors can expect investment results currently achievable only by big ones. It means that the Market as whole can be more stable because of more reasonable behavior of investors.

This tool should be very useful in the regular trading, because it allows monitoring of safe boundaries of the short-term holdings.


Economic Value of the Business


Price of the Business

If one buys a business, one is involved in some modeling already. There is an expected flow of profit from the business (this is a forecast, a model), and there is the level of risk involved when one runs this business (this is another model). This level of risk shows how much Cushion of assets is needed to survive not anticipated turns of events in the future; also, it determines the risk insurance premium. There are a few businesses available to buy, including some lending businesses. Comparing their wealth creating abilities and the needed Cushion one establishes the price of the business. Because there are many potential buyers of the business and businesses change hands relatively often, business' price does not depend on the particular buyer's limitations (lack of expertise in some areas, etc.). The Market works as equalizer.

The real profit generated by the business is profit computed as usual minus the risk insurance premium. The real assets of the business are assets computed as usual minus the Cushion. Now the price is a sum of real assets and an amount, which is needed to generate this real profit. If the Cushion is big, then it can produce a temptation to break the business and sell all assets.

Obviously, the real situation is much more complicated, but this is a way to take in consideration the business uncertainty when the price is calculated.

Publicly Traded Company

Market capitalization (the price of all shares outstanding) is not a business' price. When the business buys the publicly traded company and pays cash for it, the price paid is usually higher than market capitalization. Stock is a special way of financing the company's business activity.

Investors chose the company to invest their assets and they create the Economic Value this way. They deserve the compensation. According to the structure of this activity, the risk involved in this case is higher than in case of lending. Hence, investor's risk insurance premium and profit have to be higher, than in case of lending. How mach higher depends on the average investor's Cushion - how much loses the investor can sustain without getting out of the game, i.e. without losing the opportunity to get his money back with the Market.

The publicly traded company creates in turn an additional Economic Value by making investors' shares liquid assets (some more some less). Now the investor gets from the company payments for his assets and the Economic Value created by the company - liquid assets, which can participate in many different transactions. Now the investor's required profit can be lower.

The picture gets more complicated as soon the other property of the investment gets into play - the claim on company assets. Now, if the company is sold, the investor has the right on predefined share of the sale price (after expenses). This brings into play the price of the company. The investor estimates the time to the moment the company is to be sold and company's future price, discounts it properly in time and gets the discount to the income from the investment he is ready to accept. If the discount is big, then he is ready to accept the discount to the claim on the company's Economic Value.

The Economic Value of the company is changing, and public companies work hard increasing it, hence the estimated price of it is changing. In addition, the company buys back shares and issues new shares changing the share of the company the investor can theoretically claim.

These are underlying factors, which affect the company's share price.


Price of Stock

We put together the reasoning of previous paragraph in the different form.

First, we estimate the price of the business in the time of it suggested sale - we need to forecast the business growth and take the lower estimate of it. After we have to discount in time that value.

Second, we have to estimate the future cash flow (if any), generated by the investment, take the lower bound of the estimate and discount it in time.

Third, we need the investment risk estimate and from it the estimate of the Cushion and investment insurance premium. The premium we need to discount in time (as cash flow) and get the current value estimate.

Now we add the company's price estimate and income estimate and subtract the insurance premium - this is a price of stock (market capitalization) from the point of view of long-term investor.

It is interesting, that investors are ready to take really ling view with the company, which has a dominant position in the area, where the Market expands. Investors are ready to accept the negative profit and even negative revenue stream for such company for a few years.


Employee's Options

The company's liability presented by employee's options is "open-ended". It depends on the price of the stock. When we evaluating the price of the business to estimate the price of the stock we need the price of the stock. This could be a bad circle, but there is a power of equation to the rescue. We present the price of the stock as a variable and construct the equation, from which we compute it.


Short-term Investment

Short-term investors and long-term investors operate on the same stock market, hence short-term investors directly affect the price of the stock.

Short-term investor is interested in the change of the stock price in the short period of time - in the speed of the change. Short-term investor runs its risk and needs the insurance premium and a Cushion to operate persistently on the Market. Also, he has other competing ways to use the Market to accumulate his wealth, for example bond investment. Hence, short-term investor does not invest in the stock, which grows with the speed below some level. The periods of time when the stock goes down or simply does not grow fast enough, short-term investors does not invest, generally.

If the stock price grows with the speed higher than average, then it attracts many investors and it grows even faster. However, while it grows, the difference between the current price and the price, which long-term investors are willing to pay, or the price they can get in event of the sale of the company, grows also. This development causes the level of risk and the premium, required to offset it, to grow also. In some moment, there is no way for the price to grow any more and, therefore, there is no any short-term buyer of the stock any more. The stock dramatically drops until it reaches the level attractive for long-term buyers, which pick it up. If there is a market-maker for this stock the drop is visible, if there is not - there is no trades for a while until sellers understand to lower the price to the needed level.

Here emerges a specific pattern of the movement of the stock price - steady growth for a while (short-term investors) and a sharp drop (to the level of long-term investors), and again steady growth and a sharp drop. This pattern is well visible, if the company business grows much faster, than average. If this growth rate is close to the average, then random Market fluctuations obscure the picture. If it is lower, than average - only long-term investors are present.

When the pattern is known it is not a random volatility any more - investors can adjust to it.

Smart companies, which business grow faster than average, use this market phenomenon to issue additional stock as means of financing their activity and pay back the debt, or grant stock options to their employees. This is done without substantial deprivation of the stock price - they only decrease stock volatility. Often this leads to better company performance and the business grows even faster with all consequences.



Measuring Wealth

We use money values to measure Wealth. Money is a tool of the exchange, where transaction takes short period of time. From this point of view, the change of relative value of money in time is not important. However, from the modeling perspective, where the change in time is a major concern, this is a key question. To make money the tool of Wealth measurement we need to normalize money values related to different moments of time - a familiar concept, for example presentation of the prices in 1970 dollars.

This is a specific normalization procedure. First we have a base moment t0, and we compare money values (prices) in moments t1 and t2: V(t1) and V(t2) in t0 money. We introduce normalization coefficients N(t1,t0) and N(t2,t0) and compare values

N(t1,t0)*V(t1) and N(t2,t0)*V(t2).

For example, we can estimate the amount of money the family of four has to spend in moments t0, t1 and t2 to feel comfortable. From the family point of view, these amounts deliver the same wealth in different moments. From this we can compute coefficients N(t1,t0) and N(t2,t0).

In the models related to the society as a whole we have to work with average values, hence we need average values of normalization coefficient N(t,t0). We take in consideration other types of families, and also needs of businesses. Also we have to take a substantial period of time [ta,tb] where t is from that period, estimate cumulative expenses in this period and from this - the average (for the period) expense in the moment t.

The ways all this averaging is done reflect in essence our definition of the Wealth as economic parameter. Different averaging means different definition. With different definitions of the parameter, we get different sets of models - "parallel universes". Usually, it is difficult to find which definition is better, however there is no practical way of reconciling results to which we arrived with different definitions. It means, that we need to choose one of definitions and stick to it. If we find the definition insufficient, we need to compute again all models where this parameter is used. Also, we have to acquire the experience

The common sense suggests that N(t0,t0)=1, N(t,t0)>0, and if t0<t1<t2 then N(t2,t0)=N(t2,t1)*N(t1,t0).

It is convenient to have exponential presentation of N(t,t0):




exp(-n(t2,t1))* exp(-n(t1,t0))=

exp(-[n(t2,t1)+ n(t1,t0)]).

For the forecasts we often assume the linear change of n(t,t0)=n0*(t-t0). This is what we have in traditional models of inflation.

We are ready to introduce in our model an economic parameter Wealth. Normalization coefficients give us relation between parameter Wealth and parameter Price.

We have a basket of economic values - assets, sources of income, etc. How does the wealth, which this basket delivers, changes in the time? We have a reason to believe that small relative changes in the wealth we perceive as equivalent, regardless to the basic level of the wealth. This means that we can describe the process of the change of the logarithm of parameter Wealth as F-process. This is very important, because now we can assume that normalization coefficient n(t,t0) is F-process.

When we use the forecast of n(t,t0) we need to take in consideration mistakes of this forecast, and our best way of describing them is presenting them as F-process.


Present Value of Future Wealth

We use the concept of Present Value to compare financial characteristics in different moments of time, mostly to compare the quality of different decisions. There is a considerable controversy surrounds this issue, hence, we need to come up with the strict definition, which we will use all over our models.

We base our definition on the concept of exponential growth of wealth generated by the Market. If we limited our observation to some sector of the Market and we limit the period of time, we can assume that the speed of the growth of wealth delivered by this sector in this period of time is constant. We take two moments in time t and t0, t>t0, and compare wealth delivered by this Market sector in these moments W(t) and W(t0):

W(t)/W(t0)=exp(m*(t-t0) ),

m is constant speed of the wealth creation.

Such defined value m is the market invariant. This is the speed of the wealth creation, which the participant in this sector of the Market anticipates of getting.

Now we compute Present Value for the moment t0 of the value V(t) in the moment t in two steps. First we translate monetary values into Wealth values with normalization coefficient. After that we multiply by

exp(-m*(t-t0) ),

and we get the number, which reflects inflation and standard activity of the Market. This number we can compare with present monetary values directly.


Liquid Assets

The business conduct requires assets which can be used as payment, and which transfer can be done in a short time. The time of transfer is a main characteristic of these assets, their price can fluctuate, and they can be valuable still. This type of asset - Liquid Asset (LA) is known as a mobile economy is known. First some commodity (gold, silver, fur, etc.) was used as LA. In the developed economies the intangibles - promises to pay, became an asset, the asset, which relies heavily on the social stability. It served well as LA, which is not intended to stay long with one holder.


General Properties of Liquid Assets

LA, which transfer takes longer, suppose to worth relatively less than fast transferable one, as if we hold currency for the period of the delay. Hence the portfolio of treasuries is not a good LA, however freely traded shares of it can be one.

The changing price of LA does present some inconvenience, but LA is used in the economic transactions, which profit is much bigger than any possible loss from this price change. In addition, LA is usually held for a short time before it is passed along - this limits the actual influence of the LA value fluctuation.

Creators of LA could charge a price for the convenience of the fast assets transfer, and it was a steady business - LA creation. Banks took this business over, creating Bank Notes, which were traded at discount to their face value, but the discount was smaller than it would be, if Bank Notes were not used as LA.

The governments took over this business from banks, creating legal tender paper money (has to be accepted as a payment of the debt at the face value in the event the payment is enforced by the court).

The falling relative value of the currency to some other non-perishable commodity is a natural consequence of this system. It requires efforts to produce these highly liquid assets and it is natural for the system to charge for it. It means that inflation is a natural element of this system and conscious perpetual efforts are needed to keep it below some level.

Nowadays ordinary citizens with the checking account create this kind of an asset when they write a check. Endorsed check can be used as money.

Some LA converts itself into other LA (and possibly, disappears after some period of time). These are bonds, commercial papers, etc., which make regular payments and possibly a big payment in the end of their life. As a result, we have a cash flow: a few different LA in the different moments of time instead of original one. Presumably, the value of the original LA is equal to the value of its cash flow discounted for the delayed payments and possibility not receiving them. The value of this LA is changing in time.

Bank deposits (which pay interest, but it takes some time to transfer assets), and cash in hand (which does not pay interest, but can be transferred immediately) are LA, available to us because of the high level of development of the social structure, which supports these LA.

Moneylenders, liquefy assets. They take temporary hold on some valuables and give in exchange LA - usually a bank deposit, for a price. This price is the promise of cash flow. If this price is not paid properly, the lender is satisfied from the valuables with a standard procedure. This satisfaction can include ruined borrower’s reputation. Sometimes this is the only satisfaction available. Note, that the business reputation always was a kind of business asset, and especially now with the introduction of credit cards.

LA creators and moneylenders collectively affect the parity between LA and other assets - a buying power of LA. This parity is a very delicate matter, and it can be stable, or have a stable speed of change only in a stable social and economic situation. When the society is in troubles, the economy tends to go back to simple forms of exchange of valuables, and the very notion of valuables is changing - basic needs become more important than other.


US Currency


Currently the system, which creates the basic LA - currency in US consists of Treasury, Federal Reserve Bank, and its member banks, which have funds and deposits in the Federal Reserve Bank. Federal Reserve Bank buys treasuries on the secondary market and pays for it with deposits. Banks can get dollars in exchange for their deposits (Feds print dollars). The currency (dollars) is a legal tender.

This system was designed during the Civil War as a temporary measure to pay for the government’s debt, and stayed since. The government in the time of social distress can stretch economic system to the limit and delude currency, redistributing economic pains between holders of currency or bank deposits.

Now banks do not issue Bank Notes in a bearer form (freely exchangeable without owner registration in books), because it was made expensive for them. Actually, dollar is the only currency in US.


Tools of Control

Feds conduct the auction, which determines what kind of cash flow the new treasury security has to produce for its price. This is a powerful tool Feds use to influence the relative value of LA of different maturity. It is not completely arbitrary, because market participants have a lot of freedom to use imbalances in the treasuries interest rate to their advantage (and the government disadvantage). Sometimes these imbalances are introduced purposely to uplift depressed economy (hidden money infusion into economy by the government).

There is an additional tool of control, which Feds use to control bank’s behavior. In some conditions banks can borrow funds (set aside reserves they have to have to conduct a given amount of the business), and they pay an interest set by Feds. This interest affects bank's decisions of buying or borrowing deposits. The amount of currency the bank needs daily fluctuates, and the bank needs some form of assurance that it is able to get it in time. If bank has to buy deposits instead of borrowing them, the bank needs a Cushion - some form of liquid assets available at short notice, most likely a portfolio of Treasuries. This Cushion generates lower rate of return than other bank's businesses; hence, the bank has to compensate with higher interest on loans. Hence raising interest rate, which banks charge each other for overnight borrowing, Feds effectively raise the loans interest rate.


Possible Future of Monetary System

The growing economy needs growing amounts of LA. These amounts are determined by the relative amount of transactions. The main source of LA is Treasuries (in private hands or held by Federal Reserve). This amount has to grow accordingly. If the government issues too much of the debt, then we have inflation, too little - we have deflation. With this monetary system, the government cannot stop borrowing money. However it can borrow money at such low rate that borrowing does not lead to negative effects to the wealth of the state. This balance is misunderstood by many.

With low interest borrowing, the government has an advantage of lower uncertainty and better planning. This allows effective usage of money, which offsets the interest.

In the event of low budget deficit (or no deficit at all), the Treasuries interest rates have to move independently from interest rates on bonds issued by the businesses or local and state governments.

Again, even in the absence of the budget deficit the US government has to continue to borrow money for two reasons:

The excess of money can be used to reduce taxes (gradually) and create reserves (invested).

If money is invested, the government has a cash machine - it borrows at interest rate much lower than the rate of return on investment. This can happens because actually US monetary system is a way of hidden taxation by its design. It was designed to pay off huge debt of the government. When the debt is paid off (the debt generated by the government during the Cold War), we can use this system as a supplement of taxation or even instead of taxation.


Special Demand on Treasuries

There are some hidden factors, which affect the Treasuries interest rates. One of them is the demand on Treasuries from the issuers of the local and state municipal bonds.

Municipal bonds pay lower interest rate than Treasuries, and municipalities do not pay federal taxes. It would be possible to create a cash machine - issue municipal bonds and use proceeds to buy Treasuries, if not for regulations. (These regulations were introduced when such cash machines were actually created). However, in spite of regulations there is still a room to profit from this spread, at least to pay the expenses related to the issue itself. This creates a special demand on Treasuries, which is not determined by the Market, but by arcane system of regulations. This demand lowers the interest Treasuries have to pay (indirectly benefiting federal government), and makes the estimation of it complicated.

Derivatives and trades induced by the derivatives market are the other source of special demand on Treasuries.

Special demand on Treasuries changes the Treasuries interest rate structure, its interest rate curve cannot be used directly as a basis for pricing of other financial instruments, it has to be adjusted for the effects of this special demand.


Shortage of Liquid Assets

Booming trade produces pressure on Liquid Assets and even can lead to the shortage of LA and deflation. The situation is not new. In the old times, when precious metals and coins made with them (species) were used as LA and as means of insurance (they were dug in the ground) shortage of LA was usual and caused high interest rate. Shortage of LA laid the ground for the special business of LA creation. Now special promises supported by special guaranties are widely used as Liquid Assets. The guaranties - collateral, are in short supply, and society perpetually invents new forms of them. Real estate is an old form of collateral, personal credit, which stays behind credit card business - is a new form.

Current fast development of the global Market creates a challenge to the existing system of the Liquid Assets. If there is collateral, creation of the new Liquid Asset is relatively inexpensive. However, society had developed only relatively few forms of collateral, which come in short supply time after time. This creates overproduction of the assets, which can be used as collateral, as real estate, or lowers requirements for the collateral, as lower requirements for the personal credit. Both tendencies are unsafe and lead to high and dangerous volatility of the Market.

There are safe approaches to this problem also, as

The development of new forms of collateral is a long and expensive process, the example - introduction of credit cards.

One of the possible forms of inexpensive conversion of low liquidity LA into high liquidity LA, which any corporate Treasurer can appreciate, is a portfolio of Treasuries kept in the bank by the corporation. The bank can lend money against this portfolio at lower interest rate. Also, there are possible different standard forms of "swaps" of parts of portfolio of securities, which bypass selling and buying procedures and need for intermediate Liquid Assets.

Widely accepted and inexpensive Electronic Payment System or Electronic Fund Transfer can reduce the need for the Liquid Assets in two directions. It reduces the holding period of highly liquid LA and it facilitates prompt payment in all transactions, which reduces part of the collateral unavailable for the lending, because it has to guaranty transactions.

We expect fast development in these directions in the future.


The Economic Value of Liquidity

When an asset is liquid, it has an additional value. This can be measured. For example, when the stock is accepted for trading in the major stock market its value grows and correspondingly its price grows. This change in price can be detected. From the other hand, a debt instrument, for example bond, during its life can enter the situation, when it is not reasonable to sell it, but it is better to wait to its maturity; when one needs cash, it is better to borrow it, than to sell the bond. This situation happens when the bond pays low coupon, current interest rate is high, and money is needed for a short period. This is known "liquidity crunch". This fact - the possibility of the bond to enter a situation when it is illiquid, causes special discount of the bond's price. This discount is a measure of the Economic Value of liquidity.


Market Dynamics


Market Oscillation

Nothing concerns market participants more than the possibility of the Market recession, depression or crash.

Among market participants, there is a strong belief that oscillations of the Market are the normal way of the Market functioning. Hence these oscillations should not be prevented but managed.


Natural Phenomenon

The object of our study - the Market, is separated for the purpose of the study from the world. We can do it because there are characteristics of the object, which are stable. This makes it possible to perceive the Market as something independent. "Stable characteristics" is something that oscillates with the small "amplitude" around some "mean" state - nothing in the real word is beyond the change. This "mean" state is what we perceive as the Market.

Hence, we have to expect the oscillation of market characteristics. Absence of such oscillation we have to perceive as a sign of low precision of measurement of these characteristics.

The periods of the Market expansion and contraction are as natural as it can be. We have to study the degree of this oscillation, mechanisms of its managing, and details of it, which allow its modeling and forecast. Also, we have to study the boundaries, inside which this oscillation does not cause major reorganization of the Market and society.

The difficulties of the modeling of these oscillations come from the place of the Market in the modern market-based society. The Market acts in the area, where there is little study - it acts on the edge of the knowledge of society.

By its nature, the Market is statically unstable system - it can exist only in the dynamic. It always has imbalances of limited degree.

When possibility to produce a lot of goods was discovered, the natural Market imbalance - supply exceeds demand, was exaggerated and the Market contracted to restore the acceptable level of this imbalance. Now businesses know to watch this level and plan production according to the potential demand. Also, Feds regulate it on the global level.

Recently the possibilities of the global economy were discovered and a natural Market imbalance - the level of investment exceeds the level of its justification, was exaggerated and the Market contracted to restore acceptable level of this imbalance.

In general, it is in the nature of the Market to expand into unknown area and learn the limits of this area by doing. There is no other way to learn it - there is no theory, there is no experience. In the way, expansion always goes beyond acceptable limits and the Market cannot function properly - each time different set of limits of the Market functionality is hit. Then the Market contracts in this particular area, where excessive expansion had happened. This sets the stage for the next cycle of expansion in the same area, but much more cautious - market participants have already some knowledge here. Consecutive cycles of expansion and contraction - the process of learning of limits of this new area, are milder and more manageable.

Because the Market connects its elements, the expansion in one particular area, where the experimentation is going on, causes expansion in many other areas, the contraction causes contraction in many other areas also.

The real problems start, when this expansion-contraction endangers the core of the Market - financial system, monetary system, and the supply of goods and services needed for satisfaction of basic needs of population.


Oscillation of Economic Value and Prices

An oscillation of the Market causes an oscillation of any value measured with money. If we measure something with natural characteristics - the results of such measurement do not change from the Market crash overseas. However, the usefulness of something, usefulness as a product or as a business, the ability to exchange it for something else or to get profitable production from the business strongly depends on the state of Market as whole. Hence, when the Market expands the particular value as a rule grows, and when it contracts - it diminishes.

This is very inconvenient property for modeling, but it comes with the convenience of measuring value and wealth with money.

Prices, as facilitators of the exchange of the Economic Value, do not follow directly the oscillation of the Economic Value, but they oscillate also. The reason is presence among goods and services some goods and services, which satisfy basic needs. Their value does not change much with the change of the Market conditions. The prices of other assets rise and fall relatively to prices of this type of assets with the oscillation of Market.



Models, which we presented above, do not take in the consideration the Market oscillation. This immediately limits the area of their applicability. Some of our models have room for correction-improvement, but it is really difficult to find reliable data about degree of oscillation of the Economic Value and periods of this oscillation. However the work in this direction should be done, and we will present some models in this area.


Management of Oscillations

Management of Market oscillations can be done only with force external to the Market, usually this is a kind of government intervention. This should be an activity in contra-phase with the movement of Market. It does not need to be expensive, because it can accumulate "resources" in one leg of the cycle and spend them on the other leg.

Current Feds' policy is a good example of such management. In the time of Market expansion, Feds raise interest rate, thus slowing the speed of expansion and preventing the Market imbalances from reaching high level. In the way, Feds accumulate the possibility to act on the next leg of the cycle. In the time of Market contraction (when there is natural reaction of the Market on the growing risk of defaults is raising interest rate), Feds act in opposite direction lowering interest rate. As a result, the period of contraction is shorter and not so dip than it could be without such management.


Types of Market Activities

The Market exists because of the diversity of goals and means of market participants - the transaction is completed only when both sides benefit from it. However this plurality is not homogeneous - there are distinctive groups of market participants and the Dynamics of Market depend on their relative activity. When this relative activity is changing, because new tools are used on the Market as computer trading or Internet brokerages, or because there is a drastic change in political situation the Dynamics of Market changes. Some of these changes can be predicted.

One of the most important divisions of Market activities is according to expected holding period of the bought security. We saw already that long-term investor is willing to pay higher price than short-term investor. This produces a "spread" and natural movement of the price between these two boundaries. The character of this movement depends on the relative number of trades of long-term investors and short-term investors. Increased relative quantity of short-term operations caused by the computer trading and Internet brokerages should increase a natural Market volatility - the Basic Volatility.

Traders and investors affect the Market differently. The main source of the profit for the trader is market volatility - all forms of it. The combined work of traders reduces volatility and makes investments possible and convenient. There are different kinds of trading: sometimes the trader holds the securities for relatively long time, sometimes only for the short time. These different kinds of trading play on different kinds of market volatility and correspondingly reduce volatility.

Presence on the Market of the big quantity of short-term trading activity leads to the fast Market reaction to the business news (otherwise, this reaction is stretched in time). This lead to a specific pattern of the price movement - long periods of the price change with relatively small volatility and short periods of rapid price change from one level to the other.


Influence of Computer Trading

Introduction of computer trading has far reaching consequences for the financial market in general and for the financial modeling in particular. Computer Trading was an exotic feature accessible only to big firms for awhile, but now we are moving fast to the moment when individual investors are able to employ it.

The important consequence of computer trading - it changes the pattern of holding of securities, it introduces very short periods of holding and makes this type of holding a substantial share of all holding. As we saw, this affects the Market volatility.

If Computer Trading is allowed only during certain hours (as in NYSE), then we effectively have two markets - without Computer Trading and with it. This affects some coefficients in our formulae.

Source of Computer Trading profitability is a set of new tools, which allow making a decision and a trade in much shorter intervals than before. It means we can go after Market imbalances we could not go before. This is a source of the new wealth. Hence, we have to expect further shortening of the average holding period.

Initially Computer Trading increases the Market volatility. One reason is lack of diversity in the approaches to Computer Trading, which generates unwelcome synchronization of traders' reactions and hence increased the Market imbalance. The other reason is experimentation in the new area, where traders go after profits, which cannot be achieved with Computer Trading. Eventually, when the new experience get accumulated, Computer Trading has to decrease volatility. Computer Trading will go after its part of imbalance and do not try to go after traditional part (which is of longer periods by its nature, and cannot be presented as a sum of big amounts of short-term imbalances). Eventually Computer Trading should reduces shot-term volatility.

With the introduction of Computer Trading prices have to have stable states - values they stay for a while and rapidly change to the other stable state. Rapid changes of prices can look scary, because it reminds the Market behavior in the time of the Market crashes, but this how it should be when short period volatility is reduced by the Computer Trading.


Employee Options

Employee Options do not affect directly the market capitalization but affect the value of particular investment in the stock of the company. Because, as soon the price of the stock is above given value, the company has to allocate resources to buy stock promised by the options (in reality, the situation is somewhat more complicated, but this description is enough for our analysis).

This arrangement makes the model for the price of the stock depending on stock value - below given level it is computed with one formula and above with the other, where price of the stock grows slower than the value of the company.

Incidentally, it produces an interesting effect on the stock price volatility: if the estimate of the value of the company decreases with some speed, then the price of the stock decreases with corresponding speed. However, if the price decreases below given level the speed of its decreasing drops because the company does not pay for the options in this case and bigger share of the company corresponds to the stock held by the investor. This works as "price insurance". The price volatility in this case is substantially asymmetric. This observation has serious implications for the analysis of the movement of the stock price for the modeling and for the investment decisions.


Model for Trading

Traders use short-term supply-demand imbalances, they buy where supply is higher and demand lower, and sell shortly after where demand is higher and supply lower. This is a general nature of the trade in commodity market or in financial market. The result of the work of traders is a more balanced Market and lower diversity in space and volatility in time of prices. Because it is so easy for the trader to move from one sector of the Market to the other and their goals are so close, the competition among traders is high and there is no sharing of information.

Transactions of the traders are short-term, each has beginning (opening to the risk) and the end (closing - assets are in the low risk form). The normal reaction of the trader on the high uncertainty situation - restrain from the trade, wait until situation is more certain. Their source of revenue is a natural variation of short-term Market imbalances, which occurs in the stable Market situation. If they go beyond this source, they act not as traders, but as other classes of market participants and need appropriate tools. This is as if the wheat wholesaler decides to open a bakery.

Often traders work with borrowed assets. This restricts their activity even further, because they have to limit their transactions to low risk, short-term and relatively high profit transactions.

The knowledge of "fair price" - an average price of the product they are trading (commodity or security) has relatively low effect on their decisions. This is a main difference between the traders and investors.

Investors watch for the events, which create temporary imbalances in the Market and predict short-term reactions of the Market on these events. To do so they need to know factors, which affect Market imbalances and affect the process of market stabilization after it was thrown out of balance by the event.

Psychological factors, while practically irrelevant for the investor, are the major group of factors for the trader. Any time there is a "herd effect" in the Market - market participants act in concert because they rely on each other opinion, the major short-term Market imbalance is introduced, creating the profit potential for the trader. As soon traders step in contradiction to the "herd", they stabilize the Market (and benefit from it).

Stable price oscillates around average price all the time. Traders buy at the lower point of oscillation and sell at high point. If the period of oscillation is small, they can make many trades in the given period of time and benefit more. If the amplitude of oscillation is high, they can benefit more from each trade.

If the volume of trades increases the period of the Market oscillation becomes shorter and the amplitude becomes smaller.

We want to build a simple model, which describes the oscillation of the stable price. We can build it the way similar to the building of the model of the oscillation of the spring.

First, we need to move to variables, which addition is easily psychologically interpretable. We know this already; this is a logarithm of the price. If p0 is the average value of the logarithm of the price and p(t) is its value in the moment t, then

d(t) = p(t) - p0

is the price imbalance.

The value d(t) is changing according to the trade balance b(t) in the moment t - amount of shares bought minus amount of shares sold in the moment t:

(d(t+dt) - d(t))/dt = C1*b(t),


dt is a small time increment and

C1 is a constant defined by the Market.

If we use function derivatives, then we can write

d’(t) = C1*b(t)


d’’(t) = C1*b’(t),

where b’(t) is the speed of the change of trade balance.

As a rough model, we assume that

b’(t) = (b(t+dt) - b(t))/dt = - C2*d(t),

i.e. that there is a pressure on the trade balance which is proportional to the price imbalance and pushing the price to the balanced state. Characteristic C2 can depend on the product and on the Market.

Hence as a rough model of the stable price dynamics we have

d’’(t) = - c * d(t),

where c = C1*C2 is a characteristic of the trading of particular product on a particular market.

The solution of this differential equation is

d(t) = a * sin( z * t + f),


coefficient z is defined by the coefficient c only,

amplitude a and phase f are specific to the movement of particular price in the given period when we assume the price to be stable.

We arrived to the model of the normal oscillation of the price around average price. When the average price

P0(t) = exp(p0(t))

is moving, the oscillation is moving with it:

P(t) = exp(p(t)) = P0(t) * exp( a * sin( z * t + f) ).

More precise model should be a kind of stochastic oscillation.

The movement of the average price often is not important for the traders, because it is much slower than oscillation and trading operation is usually completed during one or two cycles of oscillation.

There is a risk associated with trading activity - how high is the probability of unusual development of the stock (news), which causes traders to take unusual steps, for example close their position in a hurry. This risk determines how high should be rate of return for the trading of particular product. This rate of return determines the pattern of the activity of traders on the Market and affects price oscillation.

If there is higher activity of traders, then there is higher pressure on the price imbalance and coefficient C2 is bigger. This leads to higher frequency of oscillation.

If the amplitude of oscillation is too high, there are traders who are willing to take less profit at lower risk. This lowers the amplitude to the level, which is determined by the risk the traders take with the particular product.

The oscillation starts with the jolt produced by the news. If this news affect many companies, the price oscillation of their stock has similar phase in the first moments after the jolt. If the periods of these oscillations are similar, the stocks move in synch.

On the Market, there is another group of traders, who trade based on technical analysis. They buy when the Market starts to buy and sell when the Market starts to sell. They can make profit because they react faster. Their activity remarkably affects the oscillation of the price. First of all the shape of oscillation changes - it rises and drops faster with relatively stable states in between. Second, the duration of this stable state, which should be about equal to half of the period of oscillation, becomes random number, because small random events on the Market push the price from this stable state. The entire oscillation becomes random oscillation and the period of oscillation becomes shorter. It removes the patterns in the collective price movement; this makes the Market more effective, because it can test more different random hypothesis about price combinations. Shorter period of oscillation also brings additional income opportunities for traders.

Trading can be combined with short-term investment when the trader-investor uses the analysis of price oscillation and holds the stock for quite a few cycles of oscillation and utilizes the movement of the average price also. This kind of investment is practically indistinguishable from the trading. It is done also when the price is stable - there is no substantial news related to the Market as whole or to the particular stock.


Short-term Investment

Short-term investors step in where traders back off - in anticipation of news. They can do it because of the better than average analysis of the situation (or because of the access to the information unavailable to others, but this is illegal in the stock market). Hence, only professionals can be successful in the business of short-term investment.

Short-term investment is done in transitional periods of the price, when it moves from one stable state to the other. In transitional period, the participation of traders is limited and the amplitude and period of oscillation gets higher. The transitional period is prolonged by the period where traders step in gradually and stabilize the frequency and amplitude of the price oscillation. The end of this period is mostly fuzzy. Sometimes the beginning is fuzzy also because of the early dissemination of the news through non-official channels (sometimes illegal). The amplitude of the oscillation in the transitional period can be comparable with the jump of the price level around which the oscillation occurs. Hence the beginning and especially the end of the short-term investment cannot be well timed, and this produces additional price instability caused by short-term investors themselves.

The usual pattern of the short-term investment and affected movement of the price is as follows.

At the news (or before) there is an activity of knowledgeable short-term investors. This initially moves the price in the direction opposite to one it is heading, which is only helpful for the investors. Eventually price moves where it should go and the herd of investors catches up with their investments - they make decisions based not on their analysis but on the Market behavior. They produce a Market imbalance and the price overshoots its target. Smart investors close their positions (take profit) - price goes in the opposite direction. The herd of investors rushes to close their positions, create a Market imbalance and the price overshoots its target in the opposite direction. Traders step in and gradually stabilize the price oscillation.


"Technical Analysis" and Positive Feedback

Technical Analysis is a way of conducting investment based on the analysis of the Market behavior. It is popular because it is simple, has a flare of scientific analysis and it is absolutely wrong. While trading requires analysis of Market dynamics, investment requires the Economic Value analysis.

Technical Analysts rely on the flawed argument that all relevant information is factored into the price by the Market and there is no sense in doing it again. This argument is from the same family as the argument of the "effective market": there are no price imbalances because market solves them. May be for someone outside the Market, who looks at the Market for general theoretical analysis this can be a good working hypothesis. However, for the market participant this point of view is at best useless and at worst costly.

Market participants are ones who create the balance and who incorporate the news into the price by their very activity. While the input of particular participant can be small, the influences of the bad theory shared by many participants can be huge.

We saw already how widely accepted Technical Analysis (in sophisticated or primitive forms) creates positive feedback and magnifies the price oscillation. Fortunately the adepts of Technical Analysis have to pay with their own money for their mistakes - this cools heads fast. It is a duty of the market participants to work for the finding the price, which delivers the balance. It is interesting how neglecting of this duty by adepts of Technical Analysis and attempt to piggyback on the work of others backfires.


Dynamics of Defaults


Stream of Defaults

There are a few cases, when we need to monitor and the stream of defaults and estimate the future behavior of this stream:

It is interesting that similar problem exists in non-financial area - monitoring and estimation:

Usually there are a few obvious statistical characteristics which are used to monitor defaults, mostly they are of the nature of ratio of number of defaults to the number of events; they are estimated directly from the observation. Their change in time is plotted, and the dynamic of the change in the defaults is derived from this.

There are two main reasons why this approach is not sufficient.

First, we work with small probabilities and small amounts of events; hence, we have a big margin of error in our estimates. In the same time, we need to make important decisions based on this limited information.

Second, characteristics, which we described above, depend on some factors, which have nothing to do with the causes of defaults, and when these factors change our characteristics change giving the impression of changed situation with causes of defaults. These factors are quantities of events in observation and the dynamic of their change.

In the case of production, the dynamic of the change in the volume of produced product affects our characteristics. When the volume grows this characteristic is smaller, when it decreases it is bigger with the same nature of the defects. In the case of loans, with the same quality of the loans, and the same probability of the $1 of the loan to default, when amount of loans grows, our characteristics are smaller, and when this amount decreases, it is bigger.

Consider the situation, when executive of the lending business get under the pressure when characteristics of the quality of lending worsens only because the amount of loans decreases (there is no enough opportunity to make loans of good quality). The way to improve these characteristics is to increase the amount of loans in expense of their quality. Actually, the executive would be forced to sacrifice the quality of lending only to make characteristics, which suppose to measure this quality, look good.

There is a simple approach, which allows solving these problems. We start with the more formalized description of the situation - the model, and we will use this model to achieve better understanding of the situation and more precise results.

We have a stream of creation events (creation of the product, originating the loan, etc.); for each creation event, we have the termination event. Some termination events are abnormal - defaults. Defaults have their causes hidden in the moment of creation.

From the experience, we divide creation events on classes, that hidden causes of the default of the creation events from the same class are about the same. We have a set of n such classes 1, …, n.

For each class j we find the amount of creation events in the moment t0:


Also we determine the probability that default occur in the moment t0+T for the creation event from the class j in the moment t0:


We believe that the function f(j,T) is pretty stable because the causes of the defaults stay the same and the situation stays the same. In the most of practical cases we assume that f(j,T) is of the form

f(j,T) = L(j)*exp(-L(j)*T),

where L(j) is a small number.

In this case the probability of default during the period from t0, to t0+T is


In the moment t the quantity of creation events V(t) is a sum of all amounts v(j,t) for all classes j=1, …, n.

In the same moment t the defaults occur for the creation events, which happen in different moments in the past. In the moment t0 < t and the class j we had v(j,t0) creation events, which default in the moment t with probability f(j,t-t0) (which usually is L(j)*exp(-L(j)*(t-t0)) ).

Some creation events were followed by normal termination events, we have to exclude them. We have


of creation events which were not terminated in the normal way yet.

The product


is an estimate of quantity of defaults attributed to the class j and creation moment t0. If we sum these products for all j and all t0 up we get the estimate of the quantity of defaults


Our default characteristic is


Now we can analyze how dynamics of the v(j,t), which have nothing to do with the causes of the defaults, affect this characteristic. We will analyze the simple case:

Value V(t) = v(1,t) is changing linear when t grows. Value D(t) stabilizes when t grows. Hence, if V(t) grows, then D(t)/V(t) falls, and if V(t) decreases, then D(t)/V(t) grows. Obviously, this change of the characteristic D(t)/V(t) has nothing to do with the quality of the creation events.

It is easy to show in the case of two classes that certain dynamics of the creation events can produce the situation when the quality of creation events get worse while the characteristic D(t)/V(t) gets better.

Conclusion: In the situation where the quantities of creation evens are changing and the ratios between different classes of creation events are changing it is important to use in decision-making together with aggregated characteristics of defaults (D(t)/V(t)) more differentiated characteristics:

Particularly in the case of analysis of loan defaults it is important to see the dynamic of lending by the lending risk class. This is especially important in times of changing credit structure, for example when consumers reduce their borrowing.

We can use above model to increase the precision of our estimates. Usually we have to rely on statistics only, when we estimate characteristics of default. However with the model presented above we can add to it some additional information - knowledge of the dynamics of the changes in quantities of the creation events, knowledge of classification of creation events and results of study of the defaults for each particular class. This knowledge effectively translates itself, with the help of the model, into more precise estimate. The gain in precision can be substantial.


Defaults and Market Cycle

Market Cycle affects the stream of defaults and the structure of business activity simultaneously.

In the time of the expansion of the Market, there is lower level of defaults on debt and business defaults, because new opportunities allow finding acceptable solutions even when some serious mistakes were made. Also, there is easier to find credit, which allows the exploration of new opportunities.

In the time of the Market contraction there are less opportunities, less loans available, there is no business expansion, which compensate for some mistakes and even past mistakes catch up with market participants. Also interest rates are rising, because lenders want to compensate for higher risk levels, and this limits opportunities further.

As a result, there is a clear cycle in the stream of defaults of any kind, which follows the Market Cycle.

Interest rates and the factoring in the possibilities of default into the price of company stock follow this cycle. The prudent behavior would be to incorporate this information about cyclic activity of the Market into decision making, but only a handful of market participants does it; the rest follows the cycle.

Contra-phase activity of Feds (they justify it with the maintenance of price stability, but the result is exactly that stabilizing activity) creates uniform conditions, which force market participants behave more "cautiously" - higher interest rates, when default data justifies lower, and vice versa.

The rest should be done with proper modeling of the Market Cycle.

There are a few problems with modeling the effects of Market Cycle. The Market Cycle is highly irregular, and in many cases there is no possibility to model across it assuming that the Market Cycle is one additional volatility factor. Too many factors are changing when the Market goes from expansion to contraction. Many models, which work well on one leg of the cycle, do not work on the other.


Forecast of Economic Value of Publicly Traded Company

It should be a simple method of the rough forecast of the Economic Value of publicly traded company, otherwise we are doomed to rely on "oracles"-stock analysts with their buy-hold recommendations without quantitative justification. We present here one such simple method.

In the analysis of privately held company we can concentrate on the revenue produced by the company and its expenses, we can work with standard accounting characteristics. Publicly traded company has to deliver something additional - current growth and assurance of future growth. This is especially true now, when tax law discourages dividends.

We start with the presentation of the publicly traded company as a company with some idealized structure. The real company does not need to have this structure, but we need this presentation as a model.

The publicly traded company has three basic components:

A few companies actually have this structure. Examples are Berkshire Hathaway and General Electric.

With this presentation, we can treat the underlying businesses as we treat private companies, which do not need to expand. Actual expansion of underlying business we interpret as an investment of the "umbrella". And we can judge this investment against other possible investments.

The presentation of underlying businesses is idealized also - we might employ a few different presentations to check our results.

Some companies actually have the division on relatively independent underlying businesses. Even departments, which exclusively serve other departments in the company, are presented as independent businesses. We do not care much about actual division, however we have to pay attention to it because we need to find information in the standard reports, which we can use in conjunction with our model.

The most difficult part is the estimate of the Economic Value of two other components: an "umbrella" and the balancing team. From the sharp reaction of the Market on changes in these components of the company, we know that they are important.

An "umbrella" has to be analyzed as a financial company. It has cash inflow - profit of businesses, and cash outflow - dividends, investment in the renovation and expansion of underlying businesses, acquisition of new businesses, management of employee benefits, issue of additional stock and stock buy-backs, borrowing and issue of bonds, etc. It is hard to estimate the price of this company, because it depends on the Market conditions, but it is possible to estimate the Economic Value of it with tools of accounting applied to financial company and evaluation of results of its activity.

For example, if the company borrows money, we should compare the interest rate it pays to what other similar companies pay. If it buys back stock, then we have to treat it as an investment, and compare it to other available investments, etc.

The evaluation of the team of executives, who balance the simple goals and rigid structure of businesses and dynamic social and market situation is difficult and has to be done on the basis concrete situation the company is in. The team, which is good for the turn around, can be ill suited for the steady expansion of the company. Before we assign the monetary value to this team, we estimate its abilities in:

its abilities in creation of:

and the strength of its connections with:

and most of all - the ability of maintaining the dynamic of the relentless expansion of the company.

These factors are not additional requirements - they are basic economic factors in the market-based society. However these factors are managed by the special small group of the company employees and they are practically out of control of the rest of employees, hence, there is little study in this area, and little general understanding. Still even a small amount of understanding, which we can gain by the quantitative estimates of these factors, can substantially improve the estimate of the Economic Value of the company.





Derivatives are financial contracts, which are defined in terms of delivery of underlying asset sometimes in the future or delivery of cash determined by the future price of underlying asset. It could be a price of underlying asset in a particular future moment, as in the case of European options, or an average price for the future period, as in the case of Asian options, and so on.

Some such contracts are perpetually readjusted; for example, futures margin account is readjusted daily.

Many such contracts are standardized and regulated; this increases their trading volume, liquidity. The process of delivery of underlying assets in the case of standard contracts is well managed, and often can be delegated for the small fee. Hence, in many cases it is reasonable to assume cash settlement, even when the asset is actually delivered.

"Private" derivative contracts often have neither liquidity nor special mechanism enforcing their delivery. They have to rely on proprietary systems guarding against defaults.


Economic Value of Underlying Asset

Derivatives are special financial products. As with other products the main justification for their existence is desire of market participants. It is not really important why people want them. As long we have desire and products, which satisfy it, we have a sector of market, which contributes to overall wealth of society. Hence, introduction of derivatives increases wealth of society. (Note that wealth is an economic concept, even introduction of lottery tickets on the market increases wealth at the expense of other important characteristics).

This is an important general observation, because from it we can deduct that the value of the asset underlying the derivative increases with the introduction of the corresponding derivative. This is true for the commodity and for the business, in the case its stock or bonds is used as an underlying asset.


Price of Underlying Asset

The movement of the price of underlying asset depends on the patterns of trading of corresponding derivatives, because derivative trading affects demand-supply structure of underlying asset and it directly affects the value "premium" created by the introduction of derivative.

The volatility of the price in this case is affected by the original volatility, volatility of the "premium" and shifting trading patterns on asset itself and on derivatives. These factors increase volatility of the price of underlying asset with introduction of derivatives. However, prices of derivatives are closely related to prices of the underlying asset and the Market has more trades to experiment with, hence it needs lower volatility to establish the price.


Influence on Other Prices

Introduction of derivatives on the market leads to the higher degree of flexibility and this is their main attractiveness. They are relatively inexpensive tools of portfolio management and they are used to exploit minor inefficiencies of market, expressed in price correlation. The more they are used this way the less correlation is left. Hence, they affect prices of assets, which are not their underlying assets. However, the effect is rather positive from the market perspective.


Market Tools of Pricing

The price of derivatives is established by the market with the familiar mechanism of "stochastic optimization" - based on small movements up and down, on the Basic Volatility. However the price of derivatives is closely related to the price of underlying asset - this is why it is called derivative. Hence the price of the underlying asset and the prices of derivatives, are determined together: after each small movement of the prices, market participants use information on asset prices and derivative prices together to evaluate this move. This should reduce the Basic Volatility.

In addition, derivatives brought to the Market sophisticated modeling, which allows better extraction of information from the market experimentation with prices. This should reduce the Basic Volatility even further.

From the other hand patterns of derivatives trading shift more readily, especially in times of market uncertainty. This increases the degree of uncertainty in the future value of prices of the asset and derivatives.


Spectrum of Goals

Derivatives are used with many different goals. Some of these goals even can destabilize market, and there are strict regulations, which limit this type of the use of derivatives.

On one side of the spectrum of goals is the use of derivatives as insurance, on the other - the use of them for virtual trading of underlying asset, speculation.

Derivatives as tools of insurance are intended to be exercised in appropriate conditions and they are often held until expiration. Their value is a value of insurance contract. The writer of such contract hedges his risk by writing many of such contracts, where gains (contract prices) compensate for losses. The buyer of such contract reduces his uncertainty, and gains from better planning.

Derivatives as tools of trading are held for a short time - they substitute the trading of underlying asset. Their price is a part of a premium for the asset liquidity.

Which goal is dominating the market affects the price of derivative.

In some cases, it is easy to determine. If derivative has low liquidity, it is most likely is used as insurance. If it has a high level of liquidity, it is most likely that this level was created by the mass of speculators. The derivative with underlying asset, which is consumable commodity, most likely has shifting trading patterns during its life. In the beginning, it can be bought for speculation and in the end as insurance. This should affect the price and the models.

Derivatives allow "reinsurance" - writers of the contracts insure themselves by buying some other contracts, or buying contracts later. This makes derivatives contracts writing more flexible business than ordinary insurance contract writing.

However, we can perceive the contracts writers as a group as one insurer. We can analyze this "virtual" insurer - its gains, loses and its Cushion. This gives us additional information about factors affecting prices of derivatives.

Unlike ordinary insurance contracts, derivative contracts come in pairs - when one generates loss the other generates gain. This makes writing derivatives contracts easier.

Private derivative contracts are mostly used as insurance - they are rarely traded. The originators of these contracts (usually financial institutions) use both methods to hedge their risk. They write a set of contracts, which mutually compensate each other risk on the side of contract writer, and they maintain the Cushion to compensate for the credit risk (if some of their counterparts renege on obligation).


Our Tasks

As market participants involved with derivatives, we have a few types of tasks.

First is an estimate of market price of derivatives. This is a kind of general analysis of the state of the Market.

Next is our own estimate of the worth of particular derivative for our goals. This estimate should be based on the analysis of underlying asset and the state of the Market. The results of this analysis can be presented in the form of the price bounds, which can trigger profitable transaction.

This information can be used to make a decision about potential transaction, including its timing.


Cash Flow Instruments



The concept of Cash Flow can be used to describe a wide class of financial products and portfolios of financial products.

It is defined as follows. One party promises the delivery to the other party of monetary values or other liquid assets according to the schedule. This promise is enclosed in the contract, which includes sellers guaranties in the form of assets and Cash Flows which are taken over by the buyer in the case of seller's default to satisfy seller's contractual obligations. Because the system of Cash Flow contracts is so interdependent, contracts' reliability requirements are high and the law heavily protects the entire system.

The simplest example is borrowing money - the borrower is a Cash Flow seller and the lender is a buyer. The bond is a similar example.

Often, fixed income Cash Flow is used in the complex financial instruments (starting with escrow accounts). If Cash Flow is used as a support in the complex financial instrument, then the question we ask is how well it fits into this instrument with amounts and times when they suppose to be delivered.

Cash Flow stream is a set of pairs-components

(V0, t0), (V1, t1), ..., (Vn, tn),

in the moment ti value Vi is scheduled (Cash Flow can include a few streams of values of different type, hence, generally, Vi can be a vector).

Each component (Vi, ti) can be interpreted as a zero coupon bond maturity ti and face value Vi. The difference, however, is in the probabilities of defaults. In a Cash Flow instrument the loss events related to different components can be interdependent.

In spite of this, if we know how to price zero coupon bonds, we can apply this knowledge to pricing this portfolio of Cash Flow components.


The Economic Value

We want to evaluate the Economic Value of Cash Flow instrument. The concept is applicable only to traded instruments.

There are two major factors, which determine it. One is the current estimate of the buying power of cash in the future. The other is the degree of liquidity of the instrument.

The future buying power of money is uncertain, and we forecast it with the stochastic process. Hence, the future buying power of each Cash Flow component is a random number.

With the time, when the period to payment nears, the Economic Value of Cash Flow instrument grows, because degree of uncertainty reduces. This is a basis for the liquidity of the Cash Flow instrument.

From the other hand, when interest rate grows, it is difficult or impossible to sell the instrument without loss, and the holder of it has to wait until maturity (or more favorable interest rates). This phenomenon limits liquidity.

Market provides many alternative Cash Flow instruments. Zero coupon Treasuries are among them and they are the least risky.

We can get an immediate estimate of the current Economic Value of Cash Flow component, by presenting it as such bond. The price of it corresponds to the Economic Value. (If we want to arrive to the price we have to take in the consideration possibilities of default).

More precise model, which does not rely on the prices of Treasuries, has to rely on the inflation rate, and discount for the loss of liquidity.


Major Sectors of the Market

The prices on Cash Flow instruments are defined by the complex interplay between different sectors of the Market, where these instruments are traded.

One is short-term. Market participants here are much more concerned with the reliability of the investment, than its rate of return. Interest rate here is influenced greatly by the Feds' decisions.

Other group (lenders) buys instruments for their cash flow. Some buy them as means of secure investment, other to place them in the escrow account, etc. This group buys instruments mostly at issue and mostly holds them to maturity. Nevertheless this group sometimes has to sell instruments, which were initially intended to be held to maturity; hence, it wants some assurance that this can be done without big loss. They want discount for the loss of liquidity.

The third group (traders) buys instruments in anticipation of an opportunity to sell them at profit before maturity.

The price of the instrument is established in interplay between these market sectors. For example, if traders have available bond of given maturity with big initial interest, which was bought awhile ago, they will compete on secondary market with a new issue of debt with low current interest.


Interest Rate Structure

The concept of interest rate is very complex. It is convenient for the rough estimates and very difficult to formalize, when we need it as a part of a model.

There are so many different interest rates related to borrowing between banks, Treasuries, mortgages, municipal bonds, corporate bonds, rates banks pay for the deposits, etc. They all are affected in a similar way by the market conditions. In the same time, there is a substantial degree of independence in their relative movement. Some relations between them are obvious; for example, bonds with lower credit rating have to pay higher interest rates than bonds with higher credit ratings.

From the other hand, interest rates of Treasuries can be influenced greatly by the Feds activity and by the Treasuries derivatives market. Hence, prices of Treasuries, at least sometimes, should move independently from prices of corporate bonds.

It is difficult to decide what is a benchmark interest rate.


Profit from Planning

The uncertainty of the future requires from business keeping itself flexible, able to react on unpredicted events. In particular, it requires reserves of liquid assets. However each time some planning is possible there is a possibility to save money. We describe here how even limited information about the future cash inflow and outflow can be incorporated into financial optimization models and increase profit.

One of the difficulties in the use of optimization models in finance is the choice of the goal function. The theory developed in the Part I and Part II gives us a clear guidance in this issue.


Account Funding

We have specified cash needs according a given schedule. We have to come up with money up front, but we can invest money in a bond portfolio, and use its cash flow to cover known cash outflow.

The core part of this model is a set of constrains, which reflect portfolio as a whole.

We have a set of moments (dates) d when the cash outflow is planed, and when initial investment is made.

There is an accumulated amount of cash produced by the bond a up to the moment d. This accumulated cash includes coupon and payment at maturity.

In the moment of initial investment, we incur cost, which is determined by the bond price and quantity of bonds bought.

We calculate accumulated cash outflow in the moment d, and we demand, that the difference between accumulated cash produced by the portfolio and accumulated cash outflow should not be negative - accumulated balance constraints.

Now we minimize the cost of the portfolio.

This is a linear optimization problem, however with integer variables.

If we have already a bond portfolio in the account, but our cash outflow is increased and we need to add funding, then we have a similar optimization problem. Only now:


Cash Management

A business has an estimate of cash inflow and cash outflow. Excess of cash is invested in (short-term) fixed income securities. The main goal of cash management is keeping investment secure and liquid, the profit is secondary. However there is a possibility of increasing the profit while having money available when it is needed.

Each time we have new estimate of cash inflow and cash outflow we have to adjust our portfolio - sell some securities and buy the other. This transaction has its cost, and it produces the change in the portfolio cash flow (which can be negative). Sometimes it is better to borrow than to sell securities and to repay the loan when enough cash flow is accumulated from the portfolio.

Often, we do not know exactly future cash flow, and we want to stay on the safe side and not jeopardize the portfolio liquidity.

All these factors can be captured in one optimization model, which has to be used every time the new information about cash flow is coming.

For each moment d we define a constraint: the accumulated cash produced by the portfolio plus the accumulated cash inflow minus the accumulated cash outflow should not be negative - accumulated balance constraints.

However we have a few scenarios of combinations of cash inflow and cash outflow, and we want the portfolio to fit in any of these scenarios - there is no cash shortfall for any of these scenarios. We can do it. We define accumulated balance constraints for each of these scenarios. This is a sophisticated way of building "worst case scenario".

Borrowing we treat as a special kind of security - with positive cash flow in the initial moment and negative cash flow after. Possible borrowings originate in the moments of cash outflow and they are repaid in one of the moments of cash inflow. This generates a lot of variants of possible borrowings, but this is manageable. Note, that we have to know in advance the interest rate the lender will charge us for the loan originated somewhere in the future, if we want to incorporate future borrowing as a part of cash management strategy. This means we need special contracts with the lender.

We treat the situation as investment, and we maximize the present value of the change in the cash flow generated by the portfolio minus the cost of the transaction. This goal function is not linear.

The optimization has to show us what to sell, what to buy, how to reschedule repayments of the debt and what to borrow anew.



Cash and Commodity Optimization

We need to buy some commodities for our business and we produce commodities. We want to reduce the business uncertainty: we want to lock today into the future price of the commodity we need for our business or the commodity we produce.

We can do it in different ways. We can buy forward contract for the needed date. We can buy call options with expirations dates in some date before needed date, and convert them into forward contract for the needed date in the options expiration dates.

The projected commodity needs or projected production can change, even what we thought as sure knowledge about future can change. Hence, we need the ability to adjust the portfolio of forward contracts and options depending on the available information at the time of the decision.

Options allow us flexibility. If later on we get information that there is not enough money to buy a needed contract, or we can make money with some intermediate investment, or price on commodity is changed substantially, we roll over some options based on information available in that time.

We do not want to participate in speculation; we trade contracts and options only as a part of roll over operation or when commodity requirements are changed. We trade to adjust to the changing market situation and changing business situation. On the way we can expect profit, because we can catch the best for us combination of prices among changing in time commodities prices

Each trade changes the balance of commodities prices fixed with our forward contracts and options. This change can be positive for the business or negative. It is positive, if our requirements stay the same, and it can be negative, if our requirements change.

Commodity requirement is the future date and amount. We have two types of commodity requirements.

Basic requirements are requirements we know now with high degree of certainty. Basic requirements we cover with forward contracts or options, if there is not enough cash for the forward contracts.

Uncertain requirements are an estimate of additional requirements, which is changing as new information is coming. Uncertain requirements we cover entirely with options.

There are different combinations of forward contracts and options, and we want to find the best one according to information we have today.

We combine commodity decisions with Cash Management decisions because:

Now we have cash portfolio (securities and borrowing, cash inflow and outflow) and commodity portfolio (forward contracts, options and commodity requirements), which we optimize together.

On regular basis, we revise this combined portfolio:

We build a model for this case as an extension of Cash Management model.

Additional constraints:

Additional variables:

Now, transaction includes trades of forward contracts and options. These trades add to the cost of the transaction (which can be negative if the trade is profitable itself). Also, each trade changes the balance of fixed future prices of commodities. We have to compute the present value of this change.

As a result, the goal function is a sum of two goal functions. One for the cash portfolio - we know it from the previous model. The other - for the commodity portfolio: present value of the changes of the balance of fixed future commodities prices minus the cost of the transaction.


Stochastic Nature of Market and Forecast


Momentarily and Statistical Arbitrage

When we deal with actively traded assets, we can safely assume, that by the nature of the Market all possibilities to make money with the set of simultaneous trades quickly disappear. Imbalances appear randomly here and there, but with the help of traders they are eliminated - this is their work.

There are two major types of such trades (types of arbitrage).

One is when the particular imbalance is spotted and transaction occurs, which leads to sure trader's profit (if there is no defaults).

The other is set as a system of trades by the rule, which leads to catching imbalances in average and leads to steady profit.

This gives us an important principle of modeling the prices: in average prices do not allow arbitrage.

This shows that we have to model the Market behavior as a stochastic process and apply to this model this statistical rule. This alone should give us a lot of information about prices.

With the concept of Basic Volatility, we only had touched the possibilities of stochastic modeling of the Market. We should expand it as far as we can.

However, in the time of market turmoil, the delicate mechanism of the arbitrage does not work well and it is usually partially suspended for a while. We should remember, that in this moment, the assumptions on which we base some of our models do not hold any more, and models do not deliver accurate results.


Uncertainty of Forecast

The forecast of the Economic Value or price carries uncertainty. It is very important to estimate the degree of this uncertainty. Stochastic processes give us ready tools to model this uncertainty. They have characteristics, which we can use as measure of uncertainty.

We can argument that forecast of the Economic Value can presented as a stochastic process. First, the Market equalizes the Economic Value, and this process can be described with stochastic process. Second, the Economic Value of the business is built by the sequence of business decisions, some right, some wrong, and this process is a good candidate for the stochastic modeling.


Collective Forecast

It is obvious that one investor uses forecast for investment decisions. However, we can present collective activity of market participants as a collective forecast.

This forecast is not taken lightly by and cannot be influenced easily, because it is made from the mass of difficult and responsible investments and trading decisions.

This Collective Forecast is an underlying concept of many models. In many cases, we use models, which are not applicable to the modeling of the movement of prices, but which are good for the presentation of forecast. These models reflect collective decision-making process of market participants.



Work Horse Stochastic Process

We present here a simple stochastic process and ways of application of it, which we consistently use as a module of our models, where we need to describe the Market behavior or uncertainty. The advantage of this process is its simplicity. It is used already to estimate price of derivatives. We hope this model will stay in use in different variants a long time, that the experience of its use will accumulate, and some sort of general familiarity with it will help in finding the proper use of it.



The generalization of the Brown's process is a "homogeneous in time additive Markov's process with continuous realizations". We call it F-process (it is called also generalized Wiener process).

If we can make an assumption that the increment of the value v(t) of some parameter in the period [t,t+s] (this increment is a random value) -


is determined by s only and does not depend on parameter's value or on t, or on the previous process' history, we have a simple stochastic process, which is used in our models.

This assumption leads to:

1) the increment

{v(t+s) - v(t)}

is a normally distributed random value,

2) its mean value is proportional to s

M{v(t+s) - v(t)} = m * s

3) its variance is proportional to s also

Var{v(t+s) - v(t)} = D * s

4) the density of distribution function of {v(t+s) - v(t)} (the transition function) is

f(x) = P{ v(t+s) - v(t) = x } =

exp(-[x-m*s]2/[D*s]) / [2*p *D*s]1/2.

This process is a convenient modeling tool. We carefully choose parameters, which process of their change can be described with this process.

This stochastic process lays in the foundation of Black-Scholes model, and we think that a successful use of this model is an additional justification of our choice.

This process is close to a Brownian process, which describes the movement of the small particle in the liquid, the movement, which is induced by the thermal movement of the liquid's molecules.



The further generalization of the Brownian process is an "additive Markov's process with continuous realizations". We call it M-process.

If we can make an assumption that the increment of the value v(t) of some parameter in the period [t,t+s] (the random value)


does not depend on parameter's value or on previous process' history, we have another stochastic process, which is used in our models.

This assumption leads to:

there are functions m(s) and D(s),

D(s+s1) ³ D(s),

that the increment

{v(t+s) - v(t)}

is a normally distributed random value with mean value

M{v(t+s) - v(t)} = m(s),

and its variance

Var{v(t+s) - v(t)} = D(s),


and the transition function is

f(x) = P{ v(t+s) - v(t) = x } =

exp(-[x-(m(t+s)-m(t))]2/D(s)) / [2*p *D(s)]1/2.

This process is a convenient modeling tool, when functions m(s) and D(s) can be determined from the other models or experiments.


Primary Parameters


This is obvious that the great majority of financial parameters do not exhibit the behavior, which F-process or M-process describes. For example, the price can not be negative when F-process can. However, the properties of these processes are very good, and it worth the effort finding the parameters, which conform to requirements of these stochastic processes. These new artificial parameters we will use as a basis of our models - we will call them primary parameters. The parameters we are interested with, we will recalculate from these special parameters - we will call them secondary parameters.

In any models, especially in stochastic models, the choice of primary parameters, for which we make assumption about their properties, is very important.

We try to find parameter, which can be presented as Markov's process. In this stochastic process the value of parameter in the moment t+s might depend on the value of parameter in the moment t, but does not depend on the "history" (on any of its values in the moments before t). Even more - we would like to find a parameter, which increment - a difference between the value of it in the moment t+s and its value in the moment t does not depend on this value (M-process). In this case, only "external" factors command changes of the parameter, and we can apply theorems of the corresponding stochastic process behavior.

Here we make a few basic assumptions about financial and economic parameters, based on observations made above.

We assume that market participants are sensitive to relative value of their investment increment (ratio to the value). This assumption leads to the choice of the primary parameter as logarithm of this value. We apply this observation to prices and to the Economic Value.

Hence, as far as we deal with relatively small changes in investment value, we can use as primary parameters

p = ln(price), and, therefore, price = exp(p),


v = ln(Economic Value), and, therefore, Economic Value = exp(v).

This is already an important result.

We can get an idea of the boundaries of applicability of our model. We can use this model as long the participant's react to the ratio to the basic value. We saw that this would not be true, if the subject is not investment, but money needed to satisfy basic needs. Also, we saw that the strength of the reaction really depends on the basic value, and only in the first approximation we can assume, that it does not depend on it.

There is an issue of the unit of measurement - what happens with our presentation when we measure the price say in cents instead of dollars. In our models, we mostly work with the ratio of prices or with the difference of their logarithms. These values do not depend on the chosen unit of measurement. However, to be precise, we have to define primary parameter p like this:

price = $1.00 * exp(p),

and it does not depend on unit of measurement.

Now we have a powerful tool and we will use it consistently.



We use M-process to model the collective decision making process of market participants: forecast of the movement of prices and changes in the Economic Value. This gives us some general understanding of the Market functioning. The statistical characteristics of this process are a way of describing properties of the Market, which are difficult to describe otherwise.

For example, if we know that the value of price in the moment t with good precision, then we can be sure, that in the moment t+s the forecast of logarithm of it can be presented as a random value with symmetric distribution, which is close to normal. However, the forecast of price has a clearly asymmetric distribution, close to log-normal distribution.

This has some profound consequences, which we illustrate with an example. Let us assume that we monitor the change of a primary parameter p(t). In the moment t E(p(t))=0 and we know that there is no reason for this parameter to prefer the movement up to the movement down. We can be sure, that mean value E(p(t+s))=0 also. However, E(exp(p(t+s)) is greater than zero, even when E(exp(p(t))=0. This shows that when we estimate prices, even when we estimate only their mean value, we have to make corrections based on the variance of price distribution.

M-process has too many undefined parameters (actually entire functions m(t) and D(t)). They are inconvenient for the actual computation. Hence, we use F-process as approximation of M-process, (or F-process plus some known function). This approximation is useful as long the period in time is relatively small, and we do not anticipate any sharp changes in the Market during this period.

Characteristic D(t) of the stochastic process, which we use for our modeling, has different meaning in different models.

When we model the forecast of change of the Economic Value of a commodity, it reflects the specifics of the Market as a whole.

When we model the forecast of change of the Economic Value of a company, it reflects in addition the way of company's operation, the forecast of its decision-making process.

Mostly, D(t) reflects uncertainty of Collective Forecast, uncertainty of future.


Nonlinear Nature of Market

One, who is familiar with the technical marvels, definitely can appreciate the "ingenuity" of the stable social structure, which we call Market. The whole variety of electronic devices is based on nonlinear devices, where the electric current output depends nonlinear on its input. This allows all kinds of amplifiers and devices with distinctive stable states and electronic switches from one state to the other.

Similar situation can be observed in the essence of the functioning of Market. There is nonlinear reaction of market participants on the changes of prices of assets. This is a basis for the liquidity, and this is a basis for the accumulating nature of Market, which causes the exponential growth of Wealth of society.


Place of Collective Forecast

The establishment of the Collective Forecast is a result of hard work of society. A lot of creative energy of the society is concentrated in it. It represents perpetually tested consensus; it is similar to the scientific and technological knowledge of society. Openly it is presented in the form of prices.

Collective Forecast can be analyzed better in the terms of logarithms of monetary values - prices, Economic Values, etc. In this presentation, we have symmetric upward and downward potentials. The Market applies nonlinear conversion to the results of Collective Forecast and favors all developments, which can lead to the Market expansion.



How does it happens that the price premium needed to move the asset trough frequent trading is not washed away? When the products are unique each trade is unique, but how does the system work in the case of commodity?

The answer lays in this amazing combination of our reaction to the relative value of the price, and the Basic Volatility, needed to find the right level of the price. This produces the asymmetric volatility of the price. An asymmetric volatility of the price produces price premium to the Economic Value.

Mass production (including mass production of financial instruments) creates a foundation for the statistical arbitrage. Arbitrageurs provide liquidity and they collect this unusual price premium.

Let us assume that the Economic Value of an asset is fixed. The price of this asset fluctuates around the Economic Value. The mean value of the logarithm of the price p, E(p), is equal to the logarithm of the Economic Value v:

E(p) = v.

However, the mean value of the price of asset exp(p), E(exp(p)), is larger than the Economic Value of exp(v):

E(exp(p)) > exp(v).

The difference between two is determined by the Basic Volatility and cannot be driven below some level. This difference is the source of revenue of arbitrageurs.

This reliable stream of revenue of arbitrageurs is recognized by the Market. Arbitrageurs can borrow at very low interest rates.

It is recognized by arbitrageurs also - they do not need to analyze the trends of the movement of Economic Value of asset. They can hold the asset for a short time only, and they can assume that the Economic Value of the asset does not change during this period. The only correction they need to make - the correction for the changing value of money.


Currency Exchange

Currency Exchange shows the relative strength of monetary system. However, Currency Exchange coefficient is affected by many other factors also.


Monetary System

The unit of currency or other liquid asset used as currency has Economic Value. This Economic Value consists of two components:

While the asset is used, the second value is equivalent to free borrowing by the asset issuer.

The asset issuer has to work to make the asset accepted as a tool of exchange. The asset has to be sucked-up back (bought back) to the issuer, when there is less need for it, and it has to be produced, when there is growing need for it.

Generally, the issuer of such asset is in a privileged position, because with the growing Market the need for such asset grows.

The quality of the asset as an exchange tool depends on array of social measures (tradition, law, etc.) and on the responsible behavior of the asset issuer.

No wonder, the governments consistently monopolize currency issue. While US government cannot issue greenbacks at will, it can issue Treasuries at will, which is practically the same.

As a primary parameter, we should take the logarithm of the Economic Value of currency.

The primary parameter for the Currency Exchange coefficient is a difference of the logarithms of corresponding Economic Values. It reflects the relative strength of the social and economic support system for the currencies.

We model the Collective Forecast of Currency Exchange coefficient with M-process.


Complexity of the Market

In the real world we have interconnected by the trading strategies Cash Flow contracts, forward contracts, goods produced in one country (where wages and prices for materials are paid in one currency) and sold in the other (with revenue in the other currency).

When the Market enters new phase in its oscillation, some plans and trading strategies are not suited any more, and should be reconsidered. The process of reorganization of strategies creates new demand-supply structure on currencies. This has nothing to do with the quality of monetary system, but this affects the Currency Exchange coefficient.