# Self-Financing and Dynamically Hedged Portfolio – Robert Merton’s Option Pricing. Thought of the Day 153.0

As an alternative to the riskless hedging approach, Robert Merton derived the option pricing equation via the construction of a self-financing and dynamically hedged portfolio containing the risky asset, option and riskless asset (in the form of money market account). Let QS(t) and QV(t) denote the number of units of asset and option in the portfolio, respectively, and MS(t) and MV(t) denote the currency value of QS(t) units of asset and QV(t) units of option, respectively. The self-financing portfolio is set up with zero initial net investment cost and no additional funds are added or withdrawn afterwards. The additional units acquired for one security in the portfolio is completely financed by the sale of another security in the same portfolio. The portfolio is said to be dynamic since its composition is allowed to change over time. For notational convenience, dropping the subscript t for the asset price process St, the option value process Vt and the standard Brownian process Zt. The portfolio value at time t can be expressed as

Π(t) = MS(t) + MV(t) + M(t) = QS(t)S + QV(t)V + M(t) —– (1)

where M(t) is the currency value of the riskless asset invested in a riskless money market account. Suppose the asset price process is governed by the stochastic differential equation (1) in here, we apply the Ito lemma to obtain the differential of the option value V as:

dV = ∂V/∂t dt + ∂V/∂S dS + σ2/2 S22V/∂S2 dt = (∂V/∂t + μS ∂V/∂S σ2/2 S22V/∂S2)dt + σS ∂V/∂S dZ —– (2)

If we formally write the stochastic dynamics of V as

dV/V = μV dt + σV dZ —– (3)

then μV and σV are given by

μV = (∂V/∂t + ρS ∂V/∂S + σ2/2 S22V/∂S2)/V —– (4)

and

σV = (σS ∂V/∂S)/V —– (5)

The instantaneous currency return dΠ(t) of the above portfolio is attributed to the differential price changes of asset and option and interest accrued, and the differential changes in the amount of asset, option and money market account held. The differential of Π(t) is computed as:

dΠ(t) = [QS(t) dS + QV(t) dV + rM(t) dt] + [S dQS(t) + V dQV(t) + dM(t)] —– (6)

where rM(t)dt gives the interest amount earned from the money market account over dt and dM(t) represents the change in the money market account held due to net currency gained/lost from the sale of the underlying asset and option in the portfolio. And if the portfolio is self-financing, the sum of the last three terms in the above equation is zero. The instantaneous portfolio return dΠ(t) can then be expressed as:

dΠ(t) = QS(t) dS + QV(t) dV + rM(t) dt = MS(t) dS/S + MV(t) dV/V +  rM(t) dt —– (7)

Eliminating M(t) between (1) and (7) and expressing dS/S and dV/V in terms of their stochastic dynamics, we obtain

dΠ(t) = [(μ − r)MS(t) + (μV − r)MV(t)]dt + [σMS(t) + σV MV(t)]dZ —– (8)

How can we make the above self-financing portfolio instantaneously riskless so that its return is non-stochastic? This can be achieved by choosing an appropriate proportion of asset and option according to

σMS(t) + σV MV(t) = σS QS(t) + σS ∂V/∂S QV(t) = 0

that is, the number of units of asset and option in the self-financing portfolio must be in the ratio

QS(t)/QV(t) = -∂V/∂S —– (9)

at all times. The above ratio is time dependent, so continuous readjustment of the portfolio is necessary. We now have a dynamic replicating portfolio that is riskless and requires zero initial net investment, so the non-stochastic portfolio return dΠ(t) must be zero.

(8) becomes

0 = [(μ − r)MS(t) + (μV − r)MV(t)]dt

substituting the ratio factor in the above equation, we get

(μ − r)S ∂V/∂S = (μV − r)V —– (10)

Now substituting μfrom (4) into the above equation, we get the black-Scholes equation for V,

∂V/∂t + σ2/2 S22V/∂S2 + rS ∂V/∂S – rV = 0

Suppose we take QV(t) = −1 in the above dynamically hedged self-financing portfolio, that is, the portfolio always shorts one unit of the option. By the ratio factor, the number of units of risky asset held is always kept at the level of ∂V/∂S units, which is changing continuously over time. To maintain a self-financing hedged portfolio that constantly keeps shorting one unit of the option, we need to have both the underlying asset and the riskfree asset (money market account) in the portfolio. The net cash flow resulting in the buying/selling of the risky asset in the dynamic procedure of maintaining ∂V/∂S units of the risky asset is siphoned to the money market account.

# Cryptocurrency and Efficient Market Hypothesis. Drunken Risibility.

According to the traditional definition, a currency has three main properties: (i) it serves as a medium of exchange, (ii) it is used as a unit of account and (iii) it allows to store value. Along economic history, monies were related to political power. In the beginning, coins were minted in precious metals. Therefore, the value of a coin was intrinsically determined by the value of the metal itself. Later, money was printed in paper bank notes, but its value was linked somewhat to a quantity in gold, guarded in the vault of a central bank. Nation states have been using their political power to regulate the use of currencies and impose one currency (usually the one issued by the same nation state) as legal tender for obligations within their territory. In the twentieth century, a major change took place: abandoning gold standard. The detachment of the currencies (specially the US dollar) from the gold standard meant a recognition that the value of a currency (specially in a world of fractional banking) was not related to its content or representation in gold, but to a broader concept as the confidence in the economy in which such currency is based. In this moment, the value of a currency reflects the best judgment about the monetary policy and the “health” of its economy.

In recent years, a new type of currency, a synthetic one, emerged. We name this new type as “synthetic” because it is not the decision of a nation state, nor represents any underlying asset or tangible wealth source. It appears as a new tradable asset resulting from a private agreement and facilitated by the anonymity of internet. Among this synthetic currencies, Bitcoin (BTC) emerges as the most important one, with a market capitalization of a few hundred million short of \$80 billions.

Bitcoin Price Chart from Bitstamp

There are other cryptocurrencies, based on blockchain technology, such as Litecoin (LTC), Ethereum (ETH), Ripple (XRP). The website https://coinmarketcap.com/currencies/ counts up to 641 of such monies. However, as we can observe in the figure below, Bitcoin represents 89% of the capitalization of the market of all cryptocurrencies.

Cryptocurrencies. Share of market capitalization of each currency.

One open question today is if Bitcoin is in fact a, or may be considered as a, currency. Until now, we cannot observe that Bitcoin fulfills the main properties of a standard currency. It is barely (though increasingly so!) accepted as a medium of exchange (e.g. to buy some products online), it is not used as unit of account (there are no financial statements valued in Bitcoins), and we can hardly believe that, given the great swings in price, anyone can consider Bitcoin as a suitable option to store value. Given these characteristics, Bitcoin could fit as an ideal asset for speculative purposes. There is no underlying asset to relate its value to and there is an open platform to operate round the clock.

Bitcoin returns, sampled every 5 hours.

Speculation has a long history and it seems inherent to capitalism. One common feature of speculative assets in history has been the difficulty in valuation. Tulipmania, the South Sea bubble, and more others, reflect on one side human greedy behavior, and on the other side, the difficulty to set an objective value to an asset. All speculative behaviors were reflected in a super-exponential growth of the time series.

Cryptocurrencies can be seen as the libertarian response to central bank failure to manage financial crises, as the one occurred in 2008. Also cryptocurrencies can bypass national restrictions to international transfers, probably at a cheaper cost. Bitcoin was created by a person or group of persons under the pseudonym Satoshi Nakamoto. The discussion of Bitcoin has several perspectives. The computer science perspective deals with the strengths and weaknesses of blockchain technology. In fact, according to R. Ali et. al., the introduction of a “distributed ledger” is the key innovation. Traditional means of payments (e.g. a credit card), rely on a central clearing house that validate operations, acting as “middleman” between buyer and seller. On contrary, the payment validation system of Bitcoin is decentralized. There is a growing army of miners, who put their computer power at disposal of the network, validating transactions by gathering together blocks, adding them to the ledger and forming a ’block chain’. This work is remunerated by giving the miners Bitcoins, what makes (until now) the validating costs cheaper than in a centralized system. The validation is made by solving some kind of algorithm. With the time solving the algorithm becomes harder, since the whole ledger must be validated. Consequently it takes more time to solve it. Contrary to traditional currencies, the total number of Bitcoins to be issued is beforehand fixed: 21 million. In fact, the issuance rate of Bitcoins is expected to diminish over time. According to Laursen and Kyed, validating the public ledger was initially rewarded with 50 Bitcoins, but the protocol foresaw halving this quantity every four years. At the current pace, the maximum number of Bitcoins will be reached in 2140. Taking into account the decentralized character, Bitcoin transactions seem secure. All transactions are recorded in several computer servers around the world. In order to commit fraud, a person should change and validate (simultaneously) several ledgers, which is almost impossible. Additional, ledgers are public, with encrypted identities of parties, making transactions “pseudonymous, not anonymous”. The legal perspective of Bitcoin is fuzzy. Bitcoin is not issued, nor endorsed by a nation state. It is not an illegal substance. As such, its transaction is not regulated.

In particular, given the nonexistence of saving accounts in Bitcoin, and consequently the absense of a Bitcoin interest rate, precludes the idea of studying the price behavior in relation with cash flows generated by Bitcoins. As a consequence, the underlying dynamics of the price signal, finds the Efficient Market Hypothesis as a theoretical framework. The Efficient Market Hypothesis (EMH) is the cornerstone of financial economics. One of the seminal works on the stochastic dynamics of speculative prices is due to L Bachelier, who in his doctoral thesis developed the first mathematical model concerning the behavior of stock prices. The systematic study of informational efficiency begun in the 1960s, when financial economics was born as a new area within economics. The classical definition due to Eugene Fama (Foundations of Finance_ Portfolio Decisions and Securities Prices 1976-06) says that a market is informationally efficient if it “fully reflects all available information”. Therefore, the key element in assessing efficiency is to determine the appropriate set of information that impels prices. Following Efficient Capital Markets, informational efficiency can be divided into three categories: (i) weak efficiency, if prices reflect the information contained in the past series of prices, (ii) semi-strong efficiency, if prices reflect all public information and (iii) strong efficiency, if prices reflect all public and private information. As a corollary of the EMH, one cannot accept the presence of long memory in financial time series, since its existence would allow a riskless profitable trading strategy. If markets are informationally efficient, arbitrage prevent the possibility of such strategies. If we consider the financial market as a dynamical structure, short term memory can exist (to some extent) without contradicting the EMH. In fact, the presence of some mispriced assets is the necessary stimulus for individuals to trade and reach an (almost) arbitrage free situation. However, the presence of long range memory is at odds with the EMH, because it would allow stable trading rules to beat the market.

The presence of long range dependence in financial time series generates a vivid debate. Whereas the presence of short term memory can stimulate investors to exploit small extra returns, making them disappear, long range correlations poses a challenge to the established financial model. As recognized by Ciaian et. al., Bitcoin price is not driven by macro-financial indicators. Consequently a detailed analysis of the underlying dynamics (Hurst exponent) becomes important to understand its emerging behavior. There are several methods (both parametric and non parametric) to calculate the Hurst exponent, which become a mandatory framework to tackle BTC trading.

# Surplus. What All Could Social Activists Do, But Debate?

The social surplus is a basic concept of classical political economy which has been revived in the post-war period by Paul Baran and Paul Sweezy. They defined it as

.. the difference between what a society produces and the costs of producing it. The size of a surplus is an index of productivity and wealth, and of how much freedom a society has to accomplish whatever goals it may set for itself. The composition of the surplus shows how it uses that freedom: how much it invests in expanding its productive capacity, how much it consumes in various forms, how much it wastes and in what ways.

The surplus can be calculated in alternative ways. One is to estimate the necessary costs of producing the national product, and to deduct the costs from the national product. This raises the conceptual problem of calculating the necessary costs of production. Some of the outlays recorded as costs by firms (such as outlays for superficial product differentiation and advertising) may be unnecessary from the social viewpoint. Hence the determination of the necessary costs is crucial for this first method. A second method is to estimate the various expenditures absorbing the surplus (non-essential consumption, investment etc.) and to add them up.

The re-elaboration of the surplus concept in the post-war period is connected to the evolution of certain features of capitalism. In Monopoly Capital Baran and Sweezy argued that capitalism had made a transition from a competitive phase to a monopolistic phase in the twentieth century. In their view, the concentration of capital in giant corporations enables them to fix prices, in contrast to nineteenth century capitalists who worked under more intense competition. These giant corporations set their sales prices by adding mark-ups to production costs. Such price setting gives the corporations control over the partition of the value added with their workers. Corporations also strive to increase their profits by reducing their production costs. On the macroeconomic plane, the general endeavour to reduce production costs (inclusive of labor costs) tends to raise the share of the surplus in GDP. This rising surplus can be sustained only if it is absorbed. The consumption of capitalists, the consumption of employees in non-productive activities (e.g. superficial product differentiation, advertising, litigation etc.), investment and some part of government expenditure (e.g. public investment, military outlays) are the main outlets for absorbing the surplus.

As almost sixty years have elapsed since the above framework was formulated, it is legitimate to ask: has the increasing ratio of trade to global output impaired the diagnosis of Baran and Sweezy with regard to the monopolization of capital, and with respect to the inclination for the surplus in GDP to increase? Has increasing trade and integration of markets raised competitive pressures so as to restrict the pricing latitude of industrial conglomerates?

The immediate effect of global trade expansion obviously must be to increase overall competition, as greater numbers of firms would come to compete in formerly segregated markets. But a countervailing effect would emerge when large firms with greater financial resources and organizational advantages eliminate smaller firms (as happens when large transnationals take on firms of peripheral countries in opened markets). Another countervailing trend to the competition-enhancing effect of trade expansion is mergers and acquisitions, on which there is evidence in the core countries. A powerful trend increase in the extent of firm level concentration of global markets share could be observed in industries as diverse as aerospace and defence, pharmaceuticals, automobiles, trucks, power equipment, farm equipment, oil and petrochemicals, mining, pulp and paper, brewing, banking, insurance, advertising, and mass media. Indications are that the competition-enhancing effect of trade is balanced (perhaps even overwhelmed) by the monopolizing effect of the centralization of capital, which may sustain the ability of large corporations to control the market prices of their products.

On the other hand, if mergers and acquisitions imply an increase in the average size of the workforce of corporations, this could stimulate a counterbalance to corporate power by higher unionization and worker militancy. However, the increasing mobility of capital, goods and services on the one hand, and unemployment on the other is weakening unionization in the core countries, and making workers accept temporary employment, part-time employment, flexibility in hiring and dismissing, flexible working days and weeks, and flexibility in assigning tasks in the workplace. Increasing flexibility in labor relations shifts various risks related to the product markets and the associated costs from firms onto workers. Enhanced flexibility cannot but boost gross profits. Hence the trend towards increased flexibility in labor practices clearly implies increased surplus generation for given output in individual countries.

The neoliberal global reform agenda also includes measures to increase surplus generation through fiscal and institutional reforms, both in developed and underdeveloped countries. Lowering taxes on corporate profits, capital gains and high incomes; increasing taxes on consumption; raising fees on public services and privatization of these services, of utilities and of social security – all these policies aim at disburdening the high income earners and property owners of contributing to financing essential services for the maintenance of the labor force. These reforms also contribute to increasing the share of surplus in total output.

In brief, in the era of neoliberal policies evidence does not seem to suggest that the tendency for the share of surplus in GDP to rise in individual countries may have waned. If so, what is happening to the surplus generated in international production?

Baran and Sweezy argued that the surplus of underdeveloped countries had been and was being drained away to the centers of the world-system. Their description of core firms‘ overseas activities in Monopoly Capital can be read as a description of offshore outsourcing activities today if one replaces subsidiary with suppliers:

What they [giant multinational corporations] want is monopolistic control over foreign sources of supply and foreign markets, enabling them to buy and sell on specially privileged terms, to shift orders from one subsidiary to another, to favour this country or that depending on which has the most advantageous tax, labour and other policies…

The authors’ view was that imperialism had a two-fold function with respect to the surplus: finding cheap foreign sources of supply (which increases the surplus in the home country), and using other countries‘ markets as outlets (which helps absorb the surplus of the home country). A major motive of transnational companies in their current practice of outsourcing parts of production to underdeveloped countries is to cut production costs, hence to increase gross profits. When the corporation of a core country decides to outsource its production to a peripheral country, or when it shifts its sources of supply of intermediate inputs to a peripheral country, this increases global surplus creation. Global output remains the same, the costs of producing it decline. For the firm, the effect of offshore outsourcing is the same as if it were to reduce its own (in-house) costs of production, or were to outsource to a cheap supplier in the home economy. If the workers in the core country dismissed due to the offshore outsourcing find newly created jobs and continue to produce surplus, then global output increases and surplus creation increases a fortiori. If the workers dismissed due to the outsourcing remain unemployed, then their consumption (provided by family, unemployment benefits etc.) absorbs part of the surplus produced by other workers in employment. Should the supplier in the peripheral country expand her production to meet the order under subcontract, there will also be some increase in surplus creation in the peripheral country. In this case the total increase in surplus may accrue to both countries  economies – in indeterminate proportions.

It is worth noting that the effect of offshore outsourcing on productivity in the core economies is ambiguous. The formula

Productivity = (Sales Revenue – Material Input Cost) / Number of Workers

shows that an increase in material input cost (due to the increase in outsourced inputs) and a reduction of the in-house workforce (due to outsourcing) may ultimately affect the outsourcing firm‘s productivity either way. The gains that motivate firms to outsourcing are not gains in labor productivity (which arguably could legitimize outsourcing from a social viewpoint), but gains in gross profits – i.e. in surplus appropriation.

It emerges that the basic tendencies in the production and growth of the social surplus described by Baran and Sweezy have not changed under globalizing capitalism. New economic policies, corporate strategies and international rules of conduct appear to promote increasing surplus transfers from the periphery to the core of the world-system. In order to lift itself out of destitution the periphery is exhorted to remove restrictions on trade and capital flows, and to compete for advantageous positions in global value chains controlled by transnationals by improving quality, reducing costs, innovating etc. The export-led growth economic strategy compels peripheral producers to individually compete for exportation by repressing wages, and conceding much of the surplus produced to their trade partners in the core countries. Part of the surplus accruing to the periphery is consumed by transnational élites imitating the consumption of the well-to-do in the core societies. On the other hand dollarization, capital flight and official reserve accumulation exert downward pressure (a pressure unrelated to trade balances) on the exchange rate of peripheral currencies. The undervaluation of peripheral currencies, reflected in deteriorating terms of trade, translates into a loss of surplus to the core countries, and reduces the capacity of poor countries to import capital goods from the core. The resulting meager per capita fixed capital formation in the underdeveloped countries bodes grim prospects for the welfare of future generations of working people in the periphery. These trends are maintained by the insertion of millions of workers in Asian hinterlands into global production networks, and by the willingness of peripheral states governed by transnational élites to continue free trade and capital transactions policies, and to accumulate foreign exchange reserves. Africa’s poor populations await their turn to be drawn into the world labor market, to eke out a subsistence and produce a surplus, of which a large part will likely flow to the core.

In order to prevent the drift of the victims of globalizing capitalism to irrational reaction (religious or nationalist fanaticism, clash of civilizations etc.) and to focus their attention on the real issues, social scientists and activists should open to debate the social and economic consequences of the export-led growth idea, all the theories and policies that give precedence to global efficiency over national saving and investment, and the social psychology of consumerism. There is pressing need to promote socio-economic programs based on the principle of self-sufficient and self-reliant national development, wherein the people can decide through democratic procedures how they will dispose the social surplus they produce (how they will distribute it, how much they will save, invest, export) under less pressure from world markets dominated by transnational companies, and with less interefence from international institutions and core states. Within the framework of the capitalist world-system, there is little hope for solving the deep social contradictions the system reproduces. The solution, reason shows, lies outside the logic of the system.

# Fractional Reserve Banking. An Attempt at Demystifying.

FRB is a technique where a bank can lend more money than it has itself available (‘deposited’ by clients). Normally, a ratio is 9:1 is used, money lent vs. the base product of banking.

This base product used to be gold. So, a bank could issue 9 times more ‘bank notes’ (‘rights to gold’) than it had gold in its vault. Imagine, a person comes with a sack of 1 kilo of gold. This person gets a note from the bank saying “you have deposited 1 kilo of gold in my bank. This note can be exchanged for that 1 kilo of gold any time you want”. But it can legally give this same note to 8 more people! 9 notes that promise 1 kilo of gold for every kilo of gold deposited. Banks are masters of promising things they in no way whatsoever can ever fulfill. And, everybody knows it. And, still we trust the banks. It is an amazing mass denial effect. We trust it, because it gives us wealth. This confidence in the system is what is, actually, essential in the economy. Our civilization depends on the low-morality of the system and our unwavering confidence in it. You are allowed to lie even if the lie is totally and utterly obvious and undeniably without a shred of doubt a lie.

In modern times, the gold standard has been abandoned, because it limits the game. Countries with the most advanced financial structures are the richest. Abandoning the gold standard creates enormous wealth. Rich, advanced nations, therefore, have abandoned the gold standard. In modern banks, no longer gold, but money itself is the base. That is, the promissory notes promise promissory notes. It is completely air. Yet, it works, because everybody trusts it’ll work.

Moreover, banks no longer issue bank notes themselves, except the central bank. The ‘real’ money of the central bank is called ‘base money’ (M0 or ‘Tier 1’) and serves as ‘gold’ in modern banks. The ‘bank notes’ from the bank promise bank notes from the central bank.

Banks use this base money no longer to directly print money (bank notes), but something that is equivalent, namely to lend money to their clients by just adding a number on their account. This, once again, works because everybody trusts it works. But is has become even thinner than air. It is equal to vacuum. There is no physical difference whatsoever anymore between having money and not having it. If I have 0 on my account, or 10000000000000000 rupees, I have the same size information on the computer of my bank. The same number of bytes (however many they may be). I just hope that one day a tiny random fluctuation occurs in their computer and sets me the first bit to a ‘1’ (unless it is the ‘sign’ bit, of course!). Nobody would notice, since there is nowhere money disappearing in the world. Simply more vacuum has been created.

But, it gets even worse. This newly created ‘money’ (the number on an account of bank A) can be deposited in other banks (write a cheque, deposit it, or make a bank transfer to bank B). In this other bank B, it can again be used as a base for creating money by adding a number to peoples’ bank account. As long as a certain amount of base money (M0, or ‘Tier 1’) is maintained. As a side mark, note that bankers do not understand the commotion of the people in calling their rewards astronomical, since they know – in contrast to the people that think that money represents earning based on hard work – that money is vacuum. Giving a bonus to the manager in the form of adding a couple of zeros to her account in her own bank is nothing but air. The most flagrant case of self-referential emptiness is the bank that was bought with its own money.

In this way, the money circulating in the economy can be much larger than the base money (of the central bank). And, all this money is completely air. The amount of money in the world is utterly baseless. Since it is air, moreover an air-system that is invented to facilitate the creation of wealth, we can intervene in the system in any way we want, if we see that this intervention is needed to optimize the creation of wealth. Think of it like this: the money and the money system was invented to enable our trade to take place. If we see that money no longer serves us (but we, instead, seem to serve the money) and decide to organize this trade in another way, we can do so without remorse. If we want to confiscate money and redistribute it, this is morally justified if that is what it takes to enable the creation of wealth.

Especially since, as will be shown, there is no justice in the distribution. It is not as if we were going to take away hard-earned money from someone. The money is just accumulated on a big pile. Intervention is adequate, required and justified. Not intervening makes things much worse for everybody.

Important to make this observation: All money thus circulating in the world is borrowed money. Money is nothing less and nothing more than debt. Without lending and borrowing, there is no debt and there is no money. Without money, there is no trade and no economy. Without debt, the economy collapses. The more debt, the bigger the economy. If everybody were to pay back his/her debt, the system would crash.

Anyway, it is technically not possible to pay back the money borrowed. Why? Because of the interest rates.

Interest is the phenomenon that somebody who lends money – or actually whatever other thing – to somebody that borrows it, wants more money back than it gave. This is impossible.

To give you an example. Imagine we have a library, and this library is the only entity in the world that can print books. Imagine it lends books to its customers and after one week, for every book that it lent out, it wants two back. For some customers it may still be possible. I may have somehow got the book from my neighbor (traded it for a DVD movie?), and I can give the two books the library demands for my one book borrowed. But that would just be passing the buck around; now my neighbor has to give back to the library two books, where he has none. This is how our economy works. And, to explain you what the current solution is of our society is that the library says “You don’t have two books? Don’t worry. We make it a new loan. Two books now. Next week you can give us four”. This is the system we have. Printing money (‘books’) is limited to banks (‘libraries’). The rest borrow the money and in no way whatsoever – absolutely out of the question, fat chance, don’t even think about it – is it possible to give back the money borrowed plus the interest, because this extra money simply does not exist, nor can it be created by the borrowers, because that is reserved to the lenders only. Bankrupt, unless these lenders refinance our loans by new loans.

When explaining this to people, they nearly always fervently oppose this idea, because they think that with money new wealth can be created, and thus the loan can be paid back including the interest, namely with the newly created wealth. This, however, is wrong thinking, because wealth and the commodity used in the loan are different things.
Imagine it like this: Imagine I lend society 100 rupees from my bank with 3% interest. The only rupees in circulation, since I am the only bank. Society invests it in tools for mining with which they find a mother lode with 200 million tons of gold. Yet, after one year, I want 103 rupees back. I don’t want gold. I want money! If they cannot give me my rightful money, I will confiscate everything they own. I will offer 2 rupees for all their possessions (do they have a better offer somewhere?!). I’ll just print 2 extra rupees and that’s it. Actually it is not even needed to print new money. I get everything. At the end of the year, I get my 100 rupees back, I get the gold and mining equipment, and they still keep a debt of 1 rupee.

A loan can only be paid back if the borrower can somehow produce the same (!) commodity that is used in the loan, so that it can give back the loan plus the interest. If gold is lent, and the borrower cannot produce gold, he cannot give back the gold plus interest. The borrower will go bankrupt. If, on the other hand, chickens or sacks of grain are borrowed, these chickens or grain can be given back with interest.

Banks are the only ones that can produce money, therefore the borrowers will go bankrupt. Full stop.

To say it in another way. If we have a system where interest is charged on debt, no way whatsoever can all borrowers pay back the money. Somebody has to go bankrupt, unless the game of refinancing goes on forever. This game of state financing can go on forever as long as the economy is growing exponentially. That is, it is growing with constant percentage. The national debt, in terms of a percentage of the gross domestic product (GDP) remains constant, if we continuously refinance and increase the debt, as long as the economy GDP grows steadily too. The moment the economy stagnates, it is game over! Debt will rise quickly. Countries will go bankrupt. (Note that increasing debt is thus the result of a stagnating economy and not the other way around!).

The way the system decides who is going bankrupt, is decided by a feed-back system. The first one that seems to be in trouble has more difficulty refinancing its loans (”You have low credit rating. I fear you will not give me back my books. I want a better risk reward. It is now three books for every book borrowed. Take it or leave it! If you don’t like it, you can always decide to give me my books now and we’ll call it even”).

Thus, some countries will go bankrupt, unless they are allowed to let the debt grow infinitely. If not, sooner or later one of them will go bankrupt. In other words, the average interest rate is always zero. One way or another. If x% interest is charged, about x% go bankrupt. To be more precise, y% of the borrowed money is never returned, compensating for the (100 − y%) that do return it with x% profit. In a mathematical formula: (1 − y/100) × (1 + x/100) = 1, or y = 100x/(100 + x). This percentage goes bankrupt. For example, if 100% interest is charged, 50% goes bankrupt.

To take it to the extreme. If the market is cautious – full of responsible investors – and decides to lend money only to ‘stable’ countries, like Germany, which lately (times are changing indeed) has a very good credit rating from the financial speculators, even these ‘stable’ countries go bankrupt. That is, the weakest of these stable countries. If only money is borrowed to Germany, Germany goes bankrupt. Apart from the technical mathematical certainty that a country can only have a positive trade balance – essential in getting a good credit rating – if another country has a negative trade balance (the sum, being a balance, is always zero). Germany needs countries like Greece as much as it despises them.

Well, in fact, this is not true. A country does not – nay, it cannot – go bankrupt for money borrowing. Not if it is an isolated country with its own currency, being also the currency in which the money is borrowed. It can simply print money. That is because the money is their own currency based on their own economy!!!

# Optimal Hedging…..

Risk management is important in the practices of financial institutions and other corporations. Derivatives are popular instruments to hedge exposures due to currency, interest rate and other market risks. An important step of risk management is to use these derivatives in an optimal way. The most popular derivatives are forwards, options and swaps. They are basic blocks for all sorts of other more complicated derivatives, and should be used prudently. Several parameters need to be determined in the processes of risk management, and it is necessary to investigate the influence of these parameters on the aims of the hedging policies and the possibility of achieving these goals.

The problem of determining the optimal strike price and optimal hedging ratio is considered, where a put option is used to hedge market risk under a constraint of budget. The chosen option is supposed to finish in-the-money at maturity in the, such that the predicted loss of the hedged portfolio is different from the realized loss. The aim of hedging is to minimize the potential loss of investment under a specified level of confidence. In other words, the optimal hedging strategy is to minimize the Value-at-Risk (VaR) under a specified level of risk.

A stock is supposed to be bought at time zero with price S0, and to be sold at time T with uncertain price ST. In order to hedge the market risk of the stock, the company decides to choose one of the available put options written on the same stock with maturity at time τ, where τ is prior and close to T, and the n available put options are specified by their strike prices Ki (i = 1, 2,··· , n). As the prices of different put options are also different, the company needs to determine an optimal hedge ratio h (0 ≤ h ≤ 1) with respect to the chosen strike price. The cost of hedging should be less than or equal to the predetermined hedging budget C. In other words, the company needs to determine the optimal strike price and hedging ratio under the constraint of hedging budget.

Suppose the market price of the stock is S0 at time zero, the hedge ratio is h, the price of the put option is P0, and the riskless interest rate is r. At time T, the time value of the hedging portfolio is

S0erT + hP0erT —– (1)

and the market price of the portfolio is

ST + h(K − Sτ)+ er(T−τ) —– (2)

therefore the loss of the portfolio is

L = (S0erT + hP0erT) − (ST +h(K−Sτ)+ er(T−τ)) —– (3)

where x+ = max(x, 0), which is the payoff function of put option at maturity.

For a given threshold v, the probability that the amount of loss exceeds v is denoted as

α = Prob{L ≥ v} —– (4)

in other words, v is the Value-at-Risk (VaR) at α percentage level. There are several alternative measures of risk, such as CVaR (Conditional Value-at-Risk), ESF (Expected Shortfall), CTE (Conditional Tail Expectation), and other coherent risk measures. The criterion of optimality is to minimize the VaR of the hedging strategy.

The mathematical model of stock price is chosen to be a geometric Brownian motion, i.e.

dSt/St = μdt + σdBt —– (5)

where St is the stock price at time t (0 < t ≤ T), μ and σ are the drift and the volatility of stock price, and Bt is a standard Brownian motion. The solution of the stochastic differential equation is

St = S0 eσBt + (μ−1/2σ2)t —– (6)

where B0 = 0, and St is lognormally distributed.

Proposition:

For a given threshold of loss v, the probability that the loss exceeds v is

Prob {L ≥ v} = E [I{X ≤ c1} FY (g(X) − X)] + E [I{X ≥ c1} FY (c2 − X)] —– (7)

where E[X] is the expectation of random variable X. I{X < c} is the index function of X such that I{X < c} = 1 when {X < c} is true, otherwise I{X < c} = 0. FY (y) is the cumulative distribution function of random variable Y , and

c1 = 1/σ [ln(K/S0) − (μ−1/2σ2)τ] ,

g(X) = 1/σ [(ln (S0 + hP0)erT − h (K − f(X)) er(T−τ) −v)/S0 − (μ − 1/2σ2) T],

f(X) = S0 eσX + (μ−1/2σ2)τ,

c2 = 1/σ [(ln (S0 + hP0) erT − v)/S0 − (μ− 1/2σ2) T

X and Y are both normally distributed, where X ∼ N(0,√τ), Y ∼ N(0,√(T−τ).

For a specified hedging strategy, Q(v) = Prob {L ≥ v} is a decreasing function of v. The VaR under α level can be obtained from equation

Q(v) = α —– (8)

The expectations in Proposition can be calculated with Monte Carlo simulation methods, and the optimal hedging strategy which has the smallest VaR can be obtained from equation (8) by numerical searching methods….

# Is Indian GDP data turning a little too Chinese? Why to be Askance @ India’s Growth Figures?

My take on the statistics:
Well, this is a simple tweaking of the equations that differentiate the growth curve. In short, we have all been a part of exams where 9/10 is different from 99/100, even if just one number distances the actual score from the maximum one could score. On similar lines, the crimes of growth are factored in on growth year/base year. This is mathematical jugglery narrowed in on political ends. Whichever way one looks at the data, some of the indicators are still found lagging the composite growth, thereby dumbing down the economists when the growth curve mandates a pattern recognition.
GDP, when calculated at Factor Cost is related with GDP at Market Price, and written as an equation of the form,
GDP (FC) = GDP (MP) – indirect takes + subsidies
While, Gross Value Added,
GVA (basic prices) = Sum (net of production taxes & subsidies) to GDP (factor cost)
Stamp duties and property taxes make up the production taxes, whereas labour, capital and investment subsidies are the other half. Why is this done? To inflate GDP after it starts representing the GDP of a country in terms of total GVA, i.e. without discounting for depreciation. Moreover, GDP at market price adds taxes and deducts subsidies on products and services to GDP at factor cost. The sum total of the GVA in various economic activities is called the GDP at factor cost. With a change in method and a subsequent change in base year, India has increased or rather expanded its manufacturing base in the sense of capturing it.  This has also enabled the country to include informal sectors, which hitherto had not found its true manifestation. This is mere adherence to standards that become internationalized.
Now, what happens in India’s case is the part subsidies, which has been the fixed denominator for our GDP, unlike most of the developed world, or even the developing economies. So, our GDP hitherto had largely been GDP (FC). After rearranging the equation above, GDP (FC) would have subtraction of the subsidies part, and yield GDP (MP), thus changing the base completely, and giving a large share of the economy as growing, rather than the dismal one predicted in the wake of demonetization. This has been effectuated since 2012 implying that whatever happens after demonetization, the growth period would project only redundant figures. Slip that into the quarterly period, and yes, the new base would indicate a growing economy, as used by the WB/IMF to forecast India growing more than China. So, there is nothing really dastardly an act here, but more about how to integrate the parts into the composite to yell at the world, we are growing.