# Derivative Pricing Theory: Call, Put Options and “Black, Scholes'” Hedged Portfolio.Thought of the Day 152.0

Fischer Black and Myron Scholes revolutionized the pricing theory of options by showing how to hedge continuously the exposure on the short position of an option. Consider the writer of a call option on a risky asset. S/he is exposed to the risk of unlimited liability if the asset price rises above the strike price. To protect the writer’s short position in the call option, s/he should consider purchasing a certain amount of the underlying asset so that the loss in the short position in the call option is offset by the long position in the asset. In this way, the writer is adopting the hedging procedure. A hedged position combines an option with its underlying asset so as to achieve the goal that either the asset compensates the option against loss or otherwise. By adjusting the proportion of the underlying asset and option continuously in a portfolio, Black and Scholes demonstrated that investors can create a riskless hedging portfolio where the risk exposure associated with the stochastic asset price is eliminated. In an efficient market with no riskless arbitrage opportunity, a riskless portfolio must earn an expected rate of return equal to the riskless interest rate.

Black and Scholes made the following assumptions on the financial market.

1. Trading takes place continuously in time.
2. The riskless interest rate r is known and constant over time.
3. The asset pays no dividend.
4. There are no transaction costs in buying or selling the asset or the option, and no taxes.
5. The assets are perfectly divisible.
6. There are no penalties to short selling and the full use of proceeds is permitted.
7. There are no riskless arbitrage opportunities.

The stochastic process of the asset price St is assumed to follow the geometric Brownian motion

dSt/St = μ dt + σ dZt —– (1)

where μ is the expected rate of return, σ is the volatility and Zt is the standard Brownian process. Both μ and σ are assumed to be constant. Consider a portfolio that involves short selling of one unit of a call option and long holding of Δt units of the underlying asset. The portfolio value Π (St, t) at time t is given by

Π = −c + Δt St —– (2)

where c = c(St, t) denotes the call price. Note that Δt changes with time t, reflecting the dynamic nature of hedging. Since c is a stochastic function of St, we apply the Ito lemma to compute its differential as follows:

dc = ∂c/∂t dt + ∂c/∂St dSt + σ2/2 St2 ∂2c/∂St2 dt

such that

-dc + Δt dS= (-∂c/∂t – σ2/2 St2 ∂2c/∂St2)dt + (Δ– ∂c/∂St)dSt

= [-∂c/∂t – σ2/2 St2 ∂2c/∂St+ (Δ– ∂c/∂St)μSt]dt + (Δ– ∂c/∂St)σSdZt

The cumulative financial gain on the portfolio at time t is given by

G(Π (St, t )) = ∫0t -dc + ∫0t Δu dSu

= ∫0t [-∂c/∂u – σ2/2 Su22c/∂Su2 + (Δ– ∂c/∂Su)μSu]du + ∫0t (Δ– ∂c/∂Su)σSdZ—– (3)

The stochastic component of the portfolio gain stems from the last term, ∫0t (Δ– ∂c/∂Su)σSdZu. Suppose we adopt the dynamic hedging strategy by choosing Δu = ∂c/∂Su at all times u < t, then the financial gain becomes deterministic at all times. By virtue of no arbitrage, the financial gain should be the same as the gain from investing on the risk free asset with dynamic position whose value equals -c + Su∂c/∂Su. The deterministic gain from this dynamic position of riskless asset is given by

Mt = ∫0tr(-c + Su∂c/∂Su)du —– (4)

By equating these two deterministic gains, G(Π (St, t)) and Mt, we have

-∂c/∂u – σ2/2 Su22c/∂Su2 = r(-c + Su∂c/∂Su), 0 < u < t

which is satisfied for any asset price S if c(S, t) satisfies the equation

∂c/∂t + σ2/2 S22c/∂S+ rS∂c/∂S – rc = 0 —– (5)

This parabolic partial differential equation is called the Black–Scholes equation. Strangely, the parameter μ, which is the expected rate of return of the asset, does not appear in the equation.

To complete the formulation of the option pricing model, let’s prescribe the auxiliary condition. The terminal payoff at time T of the call with strike price X is translated into the following terminal condition:

c(S, T ) = max(S − X, 0) —– (6)

for the differential equation.

Since both the equation and the auxiliary condition do not contain ρ, one concludes that the call price does not depend on the actual expected rate of return of the asset price. The option pricing model involves five parameters: S, T, X, r and σ. Except for the volatility σ, all others are directly observable parameters. The independence of the pricing model on μ is related to the concept of risk neutrality. In a risk neutral world, investors do not demand extra returns above the riskless interest rate for bearing risks. This is in contrast to usual risk averse investors who would demand extra returns above r for risks borne in their investment portfolios. Apparently, the option is priced as if the rates of return on the underlying asset and the option are both equal to the riskless interest rate. This risk neutral valuation approach is viable if the risks from holding the underlying asset and option are hedgeable.

The governing equation for a put option can be derived similarly and the same Black–Scholes equation is obtained. Let V (S, t) denote the price of a derivative security with dependence on S and t, it can be shown that V is governed by

∂V/∂t + σ2/2 S22V/∂S+ rS∂V/∂S – rV = 0 —– (7)

The price of a particular derivative security is obtained by solving the Black–Scholes equation subject to an appropriate set of auxiliary conditions that model the corresponding contractual specifications in the derivative security.

The original derivation of the governing partial differential equation by Black and Scholes focuses on the financial notion of riskless hedging but misses the precise analysis of the dynamic change in the value of the hedged portfolio. The inconsistencies in their derivation stem from the assumption of keeping the number of units of the underlying asset in the hedged portfolio to be instantaneously constant. They take the differential change of portfolio value Π to be

dΠ =−dc + Δt dSt,

which misses the effect arising from the differential change in Δt. The ability to construct a perfectly hedged portfolio relies on the assumption of continuous trading and continuous asset price path. It has been commonly agreed that the assumed Geometric Brownian process of the asset price may not truly reflect the actual behavior of the asset price process. The asset price may exhibit jumps upon the arrival of a sudden news in the financial market. The interest rate is widely recognized to be fluctuating over time in an irregular manner rather than being constant. For an option on a risky asset, the interest rate appears only in the discount factor so that the assumption of constant/deterministic interest rate is quite acceptable for a short-lived option. The Black–Scholes pricing approach assumes continuous hedging at all times. In the real world of trading with transaction costs, this would lead to infinite transaction costs in the hedging procedure.

# Banking and Lending/Investment. How Monetary Policy Becomes Decisive? Some Branching Rumination.

Among the most notoriously pernicious effects of asset price inflation is that it offers speculators the prospect of gain in excess of the costs of borrowing the money to buy the asset whose price is being inflated. This is how many unstable Ponzi financing structures begin. There are usually strict regulations to prevent or limit banks’ direct investment in financial instruments without any assured residual liquidity, such as equity or common stocks. However, it is less easy to prevent banks from lending to speculative investors, who then use the proceeds of their loans to buy securities or to limit lending secured on financial assets. As long as asset markets are being inflated, such credit expansions also conceal from banks, their shareholders and their regulators the disintermediation that occurs when the banks’ best borrowers, governments and large companies, use bills and company paper instead of bank loans for their short-term financing. As long as the boom proceeds, banks can enjoy the delusion that they can replace the business of governments and large companies with good lending secured on stocks.

In addition to undermining the solvency of the banking system, and distracting commerce and industry with the possibilities of lucrative corporate restructuring, capital market inflation also tends to make monetary policy ineffective. Monetary policy is principally the fixing of reserve requirements, buying and selling short-term paper or bills in the money or inter-bank markets, buying and selling government bonds and fixing short-term interest rates. As noted in the previous section, with capital market inflation there has been a proliferation of short-term financial assets traded in the money markets, as large companies and banks find it cheaper to issue their own paper than to borrow for banks. This disintermediation has extended the range of short-term liquid assets which banks may hold. As a result of this it is no longer possible for central banks, in countries experiencing capital market inflation, to control the overall amount of credit available in the economy: attempts to squeeze the liquidity of banks in order to limit their credit advances by, say, open market operations (selling government bonds) are frustrated by the ease with which banks may restore their liquidity by selling bonds or their holdings of short-term paper or bills. In this situation central banks have been forced to reduce the scope of their monetary policy to the setting of short-term interest rates.

Economists have long believed that monetary policy is effective in controlling price inflation in the economy at large, as opposed to inflation of securities prices. Various rationalizations have been advanced for this efficacy of monetary policy. For the most part they suppose some automatic causal connection between changes in the quantity of money in circulation and changes in prices, although the Austrian School of Economists (here, here, here, and here) tended on occasion to see the connection as being between changes in the rate of interest and changes in prices.

Whatever effect changes in the rate of interest may have on the aggregate of money circulating in the economy, the effect of such changes on prices has to be through the way in which an increase or decrease in the rate of interest causes alterations in expenditure in the economy. Businesses and households are usually hard-headed enough to decide their expenditure and financial commitments in the light of their nominal revenues and cash outflows, which may form their expectations, rather than in accordance with their expectations or optimizing calculations. If the same amount of money continues to be spent in the economy, then there is no effective reason for the business-people setting prices to vary prices. Only if expenditure in markets is rising or falling would retailers and industrialists consider increasing or decreasing prices. Because price expectations are observable directly with difficulty, they may explain everything in general and therefore lack precision in explaining anything in particular. Notwithstanding their effects on all sorts of expectations, interest rate changes affect inflation directly through their effects on expenditure.

The principal expenditure effects of changes in interest rates occur among net debtors in the economy, i.e., economic units whose financial liabilities exceed their financial assets. This is in contrast to net creditors, whose financial assets exceed their liabilities, and who are usually wealthy enough not to have their spending influenced by changes in interest rates. If they do not have sufficient liquid savings out of which to pay the increase in their debt service payments, then net debtors have their expenditure squeezed by having to devote more of their income to debt service payments. The principal net debtors are governments, households with mortgages and companies with large bank loans.

With or without capital market inflation, higher interest rates have never constrained government spending because of the ease with which governments may issue debt. In the case of indebted companies, the degree to which their expenditure is constrained by higher interest rates depends on their degree of indebtedness, the available facilities for additional financing and the liquidity of their assets. As a consequence of capital market inflation, larger companies reduce their borrowing from banks because it becomes cheaper and more convenient to raise even short- term finance in the booming securities markets. This then makes the expenditure of even indebted companies less immediately affected by changes in bank interest rates, because general changes in interest rates cannot affect the rate of discount or interest paid on securities already issued. Increases in short-term interest rates to reduce general price inflation can then be easily evaded by companies financing themselves by issuing longer-term securities, whose interest rates tend to be more stable. Furthermore, with capital market inflation, companies are more likely to be over-capitalized and have excessive financial liabilities, against which companies tend to hold a larger stock of more liquid assets. As inflated financial markets have become more unstable, this has further increased the liquidity preference of large companies. This excess liquidity enables the companies enjoying it to gain higher interest income to offset the higher cost of their borrowing and to maintain their planned spending. Larger companies, with access to capital markets, can afford to issue securities to replenish their liquid reserves.

If capital market inflation reduces the effectiveness of monetary policy against product price inflation, because of the reduced borrowing of companies and the ability of booming asset markets to absorb large quantities of bank credit, interest rate increases have appeared effective in puncturing asset market bubbles in general and capital market inflations in particular. Whether interest rate rises actually can effect an end to capital market inflation depends on how such rises actually affect the capital market. In asset markets, as with anti-inflationary policy in the rest of the economy, such increases are effective when they squeeze the liquidity of indebted economic units by increasing the outflow of cash needed to service debt payments and by discouraging further speculative borrowing. However, they can only be effective in this way if the credit being used to inflate the capital market is short term or is at variable rates of interest determined by the short-term rate.

Keynes’s speculative demand for money is the liquidity preference or demand for short-term securities of rentiers in relation to the yield on long-term securities. Keynes’s speculative motive is ‘a continuous response to gradual changes in the rate of interest’ in which, as interest rates along the whole maturity spectrum decline, there is a shift in rentiers’ portfolio preference toward more liquid assets. Keynes clearly equated a rise in equity (common stock) prices with just such a fall in interest rates. With falling interest rates, the increasing preference of rentiers for short-term financial assets could keep the capital market from excessive inflation.

But the relationship between rates of interest, capital market inflation and liquidity preference is somewhat more complicated. In reality, investors hold liquid assets not only for liquidity, which gives them the option to buy higher-yielding longer-term stocks when their prices fall, but also for yield. This marginalizes Keynes’s speculative motive for liquidity. The motive was based on Keynes’s distinction between what he called ‘speculation’ (investment for capital gain) and ‘enterprise’ (investment long term for income). In our times, the modern rentier is the fund manager investing long term on behalf of pension and insurance funds and competing for returns against other funds managers. An inflow into the capital markets in excess of the financing requirements of firms and governments results in rising prices and turnover of stock. This higher turnover means greater liquidity so that, as long as the capital market is being inflated, the speculative motive for liquidity is more easily satisfied in the market for long-term securities.

Furthermore, capital market inflation adds a premium of expected inflation, or prospective capital gain, to the yield on long-term financial instruments. Hence when the yield decreases, due to an increase in the securities’ market or actual price, the prospective capital gain will not fall in the face of this capital appreciation, but may even increase if it is large or abrupt. Rising short-term interest rates will therefore fail to induce a shift in the liquidity preference of rentiers towards short-term instruments until the central bank pushes these rates of interest above the sum of the prospective capital gain and the market yield on long-term stocks. Only at this point will there be a shift in investors’ preferences, causing capital market inflation to cease, or bursting an asset bubble.

This suggests a new financial instability hypothesis, albeit one that is more modest and more limited in scope and consequence than Minsky’s Financial Instability Hypothesis. During an economic boom, capital market inflation adds a premium of expected capital gain to the market yield on long-term stocks. As long as this yield plus the expected capital gain exceed the rate of interest on short-term securities set by the central bank’s monetary policy, rising short-term interest rates will have no effect on the inflow of funds into the capital market and, if this inflow is greater than the financing requirements of firms and governments, the resulting capital market inflation. Only when the short-term rate of interest exceeds the threshold set by the sum of the prospective capital gain and the yield on long-term stocks will there be a shift in rentiers’ preferences. The increase in liquidity preference will reduce the inflow of funds into the capital market. As the rise in stock prices moderates, the prospective capital gain gets smaller, and may even become negative. The rentiers’ liquidity preference increases further and eventually the stock market crashes, or ceases to be active in stocks of longer maturities.

At this point, the minimal or negative prospective capital gain makes equity or common stocks unattractive to rentiers at any positive yield, until the rate of interest on short-term securities falls below the sum of the prospective capital gain and the market yield on those stocks. When the short-term rate of interest does fall below this threshold, the resulting reduction in rentiers’ liquidity preference revives the capital market. Thus, in between the bursting of speculative bubbles and the resurrection of a dormant capital market, monetary policy has little effect on capital market inflation. Hence it is a poor regulator for ‘squeezing out inflationary expectations’ in the capital market.

# A Monetary Drain due to Excess Liquidity. Why is the RBI Playing Along

And so we thought demonetization was not a success. Let me begin with the Socratic irony to assume that it was indeed a success, albeit not in arresting black money for sure. Yes, the tax net has widened and the cruelty of smashing down the informal sector to smithereens to be replaceable with a formal economy, more in the manner of sucking the former into the latter has been achieved. As far as terror funding is concerned, it is anybody’s guess and so let them be with their imaginations. What none can deny is the surge in deposits and liquidity in the wake of demonetization. But, what one has been consciously, or through an ideological-driven standpoint denying is the fact that demonetization clubbed with the governmental red carpet for foreign direct investment has been an utter failure to attract money into the country. And the reason attributed for the same has been a dip in the economy as a result of the idiosyncratic decision of November 8 added with the conjuring acts of mathematics and statistics in tweaking base years to let go off the reality behind a depleting GDP and project the country as the fastest growing emerging economy in the world. The irony I started off with is defeated here, for none of the claims that the government propaganda machine churns out on the assembly line are in fact anywhere near the truth. But, thats what a propaganda is supposed to doing, else why even call it that, or even call for a successful governance and so on and on (sorry for the Žižekian interjections here).

Assuming the irony still has traces and isn’t vanquished, it is time to move on and look into the effects of what calls for a financial reality-check. Abruptly going vertically through the tiers here, it is recently been talked about in the corridors of financial power that the Reserve Bank of India (RBI) is all set to drain close to 1.5 lakh crore in excess liquidity from the financial system as surging foreign investments forces the central bank to absorb the dollar inflows and sell rupees to cap gains in the local currency. This is really interesting, for the narrative or the discourse is again symptomatic of what the government wants us to believe, and so believe we shall, or shall we? After this brief stopover, chugging off again…Foreign investments into debt and shares have reached a net \$31 billion this year, compared with \$2.7 billion in sales last year, due to factors including India’s low inflation and improving economic growth. This is not merely a leap, but a leap of faith, in this case numerically. Yes, India is suffering from low inflation, but it ain’t deflation, but rather disinflation. There is a method to this maddening reason, if one needs to counter what gets prime time economic news in the media or passes on as Chinese Whispers amongst activists hell-bent on proving the futility of the governmental narrative. There is nothing wrong in the procedure as long as this hell-bent-ness is cooked in proper proportions of reason. But, why call it disinflation and not deflation? A sharp drop in inflation below the Reserve Bank of India’s (RBI’s) 4% target has been driven by only two items – pulses and vegetables. the consumer price index (CPI), excluding pulses and vegetables, rose at the rate of 3.8% in July, much higher than the official headline figure of 2.4% inflation for the month. The re-calculated CPI is based on adjusted weights after excluding pulses and vegetables from the basket of goods and services. The two farm items – pulses and vegetables – have a combined weight of only 8.4% in the consumer price index (CPI) basket. However, they have wielded disproportionate influence over the headline inflation number for more than a year now owing to the sharp volatility in their prices. So, how does it all add up? Prices of pulses and vegetables have fallen significantly this year owing to increased supply amid a normal monsoon last year, as noted by the Economic Survey. The high prices of pulses in the year before and the government’s promises of more effective procurement may have encouraged farmers to produce more last year, resulting in a glut. Demonetisation may have added to farmers’ woes by turning farm markets into buyers’ markets. Thus, there does not seem to be any imminent threat of deflation in India. A more apt characterization of the recent trends in prices may be ‘disinflation’ (a fall in the inflation rate) rather than deflation (falling prices) given that overall inflation, excluding pulses and vegetables, is close to the RBI target of 4%. On the topicality of improving economic growth in the country, this is the bone of contention either weakening or otherwise depending on how the marrow is key up.

Moving on…The strong inflows have sent the rupee up nearly 7 per cent against the dollar and forced the RBI to buy more than \$10 billion in spot market and \$10 billion in forwards this year – which has meant an equivalent infusion in rupees. Those rupee sales have added liquidity into a financial system already flush with cash after a ban on higher-denomination currency in November sparked a surge in bank deposits. Average daily liquidity has risen to around Rs 3 lakh crore, well above the RBI’s goal of around Rs 1 lakh crore, according to traders. That will force the RBI to step up debt sales to remove liquidity and avoid any inflationary impact. Traders estimate the RBI will need to drain Rs 1 lakh crore to Rs 1.4 lakh crore (\$15.7 billion to \$22 billion) after taking into account factors such as festival-related consumer spending that naturally reduce cash in the system. How the RBI drains the cash will thus become an impact factor for bond traders, who have benefitted from a rally in debt markets. The RBI has already drained about Rs 1 lakh crore via one-year bills under a special market stabilisation scheme (MSS), as well as Rs 30,000 crore in longer debt through open market sales. MSS (Market Stabilisation Scheme) securities are issued with the objective of providing the RBI with a stock of securities with which it can intervene in the market for managing liquidity. These securities are issued not to meet the government’s expenditure. The MSS scheme was launched in April 2004 to strengthen the RBI’s ability to conduct exchange rate and monetary management. The bills/bonds issued under MSS have all the attributes of the existing treasury bills and dated securities. These securities will be issued by way of auctions to be conducted by the RBI. The timing of issuance, amount and tenure of such securities will be decided by the RBI. The securities issued under the MSS scheme are matched by an equivalent cash balance held by the government with the RBI. As a result, their issuance will have a negligible impact on the fiscal deficit of the government. It is hoped that the procedure would continue, noting staggered sales in bills, combined with daily reverse repo operations and some long-end sales, would be easily absorbable in markets. The most disruptive fashion would be stepping up open market sales, which tend to focus on longer-ended debt. That may send yields higher and blunt the impact of the central bank’s 25 basis point rate cut in August. The RBI does not provide a timetable of its special debt sales for the year. and if the RBI drains the cash largely through MSS bonds then markets wont get too much impacted. This brings us to close in proving the success story of demonetization as a false beacon, in that with a surge in liquidity, the impact on the market would be negligible if MSS are resorted to culminating in establishing the fact that demonetization clubbed with red-carpeted FDI has had absolutely no nexus in the influx of dollars and thus any propaganda of this resulting as a success story of demonetization is to be seen as purely rhetoric. QED.

# Arbitrage, or Tensors thereof…

What is an arbitrage? Basically it means ”to get something from nothing” and a free lunch after all. More strict definition states the arbitrage as an operational opportunity to make a risk-free profit with a rate of return higher than the risk-free interest rate accured on deposit.

The arbitrage appears in the theory when we consider a curvature of the connection. A rate of excess return for an elementary arbitrage operation (a difference between rate of return for the operation and the risk-free interest rate) is an element of curvature tensor calculated from the connection. It can be understood keeping in mind that a curvature tensor elements are related to a difference between two results of infinitesimal parallel transports performed in different order. In financial terms it means that the curvature tensor elements measure a difference in gains accured from two financial operations with the same initial and final points or, in other words, a gain from an arbitrage operation.

In a certain sense, the rate of excess return for an elementary arbitrage operation is an analogue of the electromagnetic field. In an absence of any uncertanty (or, in other words, in an absense of walks of prices, exchange and interest rates) the only state is realised is the state of zero arbitrage. However, if we place the uncertenty in the game, prices and the rates move and some virtual arbitrage possibilities to get more than less appear. Therefore we can say that the uncertanty play the same role in the developing theory as the quantization did for the quantum gauge theory.

What of “matter” fields then, which interact through the connection. The “matter” fields are money flows fields, which have to be gauged by the connection. Dilatations of money units (which do not change a real wealth) play a role of gauge transformation which eliminates the effect of the dilatation by a proper tune of the connection (interest rate, exchange rates, prices and so on) exactly as the Fisher formula does for the real interest rate in the case of an inflation. The symmetry of the real wealth to a local dilatation of money units (security splits and the like) is the gauge symmetry of the theory.

A theory may contain several types of the “matter” fields which may differ, for example, by a sign of the connection term as it is for positive and negative charges in the electrodynamics. In the financial stage it means different preferances of investors. Investor’s strategy is not always optimal. It is due to partially incomplete information in hands, choice procedure, partially, because of investors’ (or manager’s) internal objectives. Physics of Finance

# Accelerated Capital as an Anathema to the Principles of Communicative Action. A Note Quote on the Reciprocity of Capital and Ethicality of Financial Economics

Markowitz portfolio theory explicitly observes that portfolio managers are not (expected) utility maximisers, as they diversify, and offers the hypothesis that a desire for reward is tempered by a fear of uncertainty. This model concludes that all investors should hold the same portfolio, their individual risk-reward objectives are satisfied by the weighting of this ‘index portfolio’ in comparison to riskless cash in the bank, a point on the capital market line. The slope of the Capital Market Line is the market price of risk, which is an important parameter in arbitrage arguments.

Merton had initially attempted to provide an alternative to Markowitz based on utility maximisation employing stochastic calculus. He was only able to resolve the problem by employing the hedging arguments of Black and Scholes, and in doing so built a model that was based on the absence of arbitrage, free of turpe-lucrum. That the prescriptive statement “it should not be possible to make sure profits”, is a statement explicit in the Efficient Markets Hypothesis and in employing an Arrow security in the context of the Law of One Price. Based on these observations, we conject that the whole paradigm for financial economics is built on the principle of balanced reciprocity. In order to explore this conjecture we shall examine the relationship between commerce and themes in Pragmatic philosophy. Specifically, we highlight Robert Brandom’s (Making It Explicit Reasoning, Representing, and Discursive Commitment) position that there is a pragmatist conception of norms – a notion of primitive correctnesses of performance implicit in practice that precludes and are presupposed by their explicit formulation in rules and principles.

The ‘primitive correctnesses’ of commercial practices was recognised by Aristotle when he investigated the nature of Justice in the context of commerce and then by Olivi when he looked favourably on merchants. It is exhibited in the doux-commerce thesis, compare Fourcade and Healey’s contemporary description of the thesis Commerce teaches ethics mainly through its communicative dimension, that is, by promoting conversations among equals and exchange between strangers, with Putnam’s description of Habermas’ communicative action based on the norm of sincerity, the norm of truth-telling, and the norm of asserting only what is rationally warranted …[and] is contrasted with manipulation (Hilary Putnam The Collapse of the Fact Value Dichotomy and Other Essays)

There are practices (that should be) implicit in commerce that make it an exemplar of communicative action. A further expression of markets as centres of communication is manifested in the Asian description of a market brings to mind Donald Davidson’s (Subjective, Intersubjective, Objective) argument that knowledge is not the product of a bipartite conversations but a tripartite relationship between two speakers and their shared environment. Replacing the negotiation between market agents with an algorithm that delivers a theoretical price replaces ‘knowledge’, generated through communication, with dogma. The problem with the performativity that Donald MacKenzie (An Engine, Not a Camera_ How Financial Models Shape Markets) is concerned with is one of monism. In employing pricing algorithms, the markets cannot perform to something that comes close to ‘true belief’, which can only be identified through communication between sapient humans. This is an almost trivial observation to (successful) market participants, but difficult to appreciate by spectators who seek to attain ‘objective’ knowledge of markets from a distance. To appreciate the relevance to financial crises of the position that ‘true belief’ is about establishing coherence through myriad triangulations centred on an asset rather than relying on a theoretical model.

Shifting gears now, unless the martingale measure is a by-product of a hedging approach, the price given by such martingale measures is not related to the cost of a hedging strategy therefore the meaning of such ‘prices’ is not clear. If the hedging argument cannot be employed, as in the markets studied by Cont and Tankov (Financial Modelling with Jump Processes), there is no conceptual framework supporting the prices obtained from the Fundamental Theorem of Asset Pricing. This lack of meaning can be interpreted as a consequence of the strict fact/value dichotomy in contemporary mathematics that came with the eclipse of Poincaré’s Intuitionism by Hilbert’s Formalism and Bourbaki’s Rationalism. The practical problem of supporting the social norms of market exchange has been replaced by a theoretical problem of developing formal models of markets. These models then legitimate the actions of agents in the market without having to make reference to explicitly normative values.

The Efficient Market Hypothesis is based on the axiom that the market price is determined by the balance between supply and demand, and so an increase in trading facilitates the convergence to equilibrium. If this axiom is replaced by the axiom of reciprocity, the justification for speculative activity in support of efficient markets disappears. In fact, the axiom of reciprocity would de-legitimise ‘true’ arbitrage opportunities, as being unfair. This would not necessarily make the activities of actual market arbitrageurs illicit, since there are rarely strategies that are without the risk of a loss, however, it would place more emphasis on the risks of speculation and inhibit the hubris that has been associated with the prelude to the recent Crisis. These points raise the question of the legitimacy of speculation in the markets. In an attempt to understand this issue Gabrielle and Reuven Brenner identify the three types of market participant. ‘Investors’ are preoccupied with future scarcity and so defer income. Because uncertainty exposes the investor to the risk of loss, investors wish to minimise uncertainty at the cost of potential profits, this is the basis of classical investment theory. ‘Gamblers’ will bet on an outcome taking odds that have been agreed on by society, such as with a sporting bet or in a casino, and relates to de Moivre’s and Montmort’s ‘taming of chance’. ‘Speculators’ bet on a mis-calculation of the odds quoted by society and the reason why speculators are regarded as socially questionable is that they have opinions that are explicitly at odds with the consensus: they are practitioners who rebel against a theoretical ‘Truth’. This is captured in Arjun Appadurai’s argument that the leading agents in modern finance believe in their capacity to channel the workings of chance to win in the games dominated by cultures of control . . . [they] are not those who wish to “tame chance” but those who wish to use chance to animate the otherwise deterministic play of risk [quantifiable uncertainty]”.

In the context of Pragmatism, financial speculators embody pluralism, a concept essential to Pragmatic thinking and an antidote to the problem of radical uncertainty. Appadurai was motivated to study finance by Marcel Mauss’ essay Le Don (The Gift), exploring the moral force behind reciprocity in primitive and archaic societies and goes on to say that the contemporary financial speculator is “betting on the obligation of return”, and this is the fundamental axiom of contemporary finance. David Graeber (Debt The First 5,000 Years) also recognises the fundamental position reciprocity has in finance, but where as Appadurai recognises the importance of reciprocity in the presence of uncertainty, Graeber essentially ignores uncertainty in his analysis that ends with the conclusion that “we don’t ‘all’ have to pay our debts”. In advocating that reciprocity need not be honoured, Graeber is not just challenging contemporary capitalism but also the foundations of the civitas, based on equality and reciprocity. The origins of Graeber’s argument are in the first half of the nineteenth century. In 1836 John Stuart Mill defined political economy as being concerned with [man] solely as a being who desires to possess wealth, and who is capable of judging of the comparative efficacy of means for obtaining that end.

In Principles of Political Economy With Some of Their Applications to Social Philosophy, Mill defended Thomas Malthus’ An Essay on the Principle of Population, which focused on scarcity. Mill was writing at a time when Europe was struck by the Cholera pandemic of 1829–1851 and the famines of 1845–1851 and while Lord Tennyson was describing nature as “red in tooth and claw”. At this time, society’s fear of uncertainty seems to have been replaced by a fear of scarcity, and these standards of objectivity dominated economic thought through the twentieth century. Almost a hundred years after Mill, Lionel Robbins defined economics as “the science which studies human behaviour as a relationship between ends and scarce means which have alternative uses”. Dichotomies emerge in the aftermath of the Cartesian revolution that aims to remove doubt from philosophy. Theory and practice, subject and object, facts and values, means and ends are all separated. In this environment ex cathedra norms, in particular utility (profit) maximisation, encroach on commercial practice.

In order to set boundaries on commercial behaviour motivated by profit maximisation, particularly when market uncertainty returned after the Nixon shock of 1971, society imposes regulations on practice. As a consequence, two competing ethics, functional Consequential ethics guiding market practices and regulatory Deontological ethics attempting stabilise the system, vie for supremacy. It is in this debilitating competition between two essentially theoretical ethical frameworks that we offer an explanation for the Financial Crisis of 2007-2009: profit maximisation, not speculation, is destabilising in the presence of radical uncertainty and regulation cannot keep up with motivated profit maximisers who can justify their actions through abstract mathematical models that bare little resemblance to actual markets. An implication of reorienting financial economics to focus on the markets as centres of ‘communicative action’ is that markets could become self-regulating, in the same way that the legal or medical spheres are self-regulated through professions. This is not a ‘libertarian’ argument based on freeing the Consequential ethic from a Deontological brake. Rather it argues that being a market participant entails restricting norms on the agent such as sincerity and truth telling that support knowledge creation, of asset prices, within a broader objective of social cohesion. This immediately calls into question the legitimacy of algorithmic/high- frequency trading that seems an anathema in regard to the principles of communicative action.

# High Frequency Traders: A Case in Point.

Events on 6th May 2010:

In the ordinary course of business, HFTs use their technological advantage to profit from aggressively removing the last few contracts at the best bid and ask levels and then establishing new best bids and asks at adjacent price levels ahead of an immediacy-demanding customer. As an illustration of this “immediacy absorption” activity, consider the following stylized example, presented in Figure and described below.

Suppose that we observe the central limit order book for a stock index futures contract. The notional value of one stock index futures contract is \$50. The market is very liquid – on average there are hundreds of resting limit orders to buy or sell multiple contracts at either the best bid or the best offer. At some point during the day, due to temporary selling pressure, there is a total of just 100 contracts left at the best bid price of 1000.00. Recognizing that the queue at the best bid is about to be depleted, HFTs submit executable limit orders to aggressively sell a total of 100 contracts, thus completely depleting the queue at the best bid, and very quickly submit sequences of new limit orders to buy a total of 100 contracts at the new best bid price of 999.75, as well as to sell 100 contracts at the new best offer of 1000.00. If the selling pressure continues, then HFTs are able to buy 100 contracts at 999.75 and make a profit of \$1,250 dollars among them. If, however, the selling pressure stops and the new best offer price of 1000.00 attracts buyers, then HFTs would very quickly sell 100 contracts (which are at the very front of the new best offer queue), “scratching” the trade at the same price as they bought, and getting rid of the risky inventory in a few milliseconds.

This type of trading activity reduces, albeit for only a few milliseconds, the latency of a price move. Under normal market conditions, this trading activity somewhat accelerates price changes and adds to the trading volume, but does not result in a significant directional price move. In effect, this activity imparts a small “immediacy absorption” cost on all traders, including the market makers, who are not fast enough to cancel the last remaining orders before an imminent price move.

This activity, however, makes it both costlier and riskier for the slower market makers to maintain continuous market presence. In response to the additional cost and risk, market makers lower their acceptable inventory bounds to levels that are too small to offset temporary liquidity imbalances of any significant size. When the diminished liquidity buffer of the market makers is pierced by a sudden order flow imbalance, they begin to demand a progressively greater compensation for maintaining continuous market presence, and prices start to move directionally. Just as the prices are moving directionally and volatility is elevated, immediacy absorption activity of HFTs can exacerbate a directional price move and amplify volatility. Higher volatility further increases the speed at which the best bid and offer queues are being depleted, inducing HFT algorithms to demand immediacy even more, fueling a spike in trading volume, and making it more costly for the market makers to maintain continuous market presence. This forces more risk averse market makers to withdraw from the market, which results in a full-blown market crash.

Empirically, immediacy absorption activity of the HFTs should manifest itself in the data very differently from the liquidity provision activity of the Market Makers. To establish the presence of these differences in the data, we test the following hypotheses:

Hypothesis H1: HFTs are more likely than Market Makers to aggressively execute the last 100 contracts before a price move in the direction of the trade. Market Makers are more likely than HFTs to have the last 100 resting contracts against which aggressive orders are executed.

Hypothesis H2: HFTs trade aggressively in the direction of the price move. Market Makers get run over by a price move.

Hypothesis H3: Both HFTs and Market Makers scratch trades, but HFTs scratch more.

To statistically test our “immediacy absorption” hypotheses against the “liquidity provision” hypotheses, we divide all of the trades during the 405 minute trading day into two subsets: Aggressive Buy trades and Aggressive Sell trades. Within each subset, we further aggregate multiple aggressive buy or sell transactions resulting from the execution of the same order into Aggressive Buy or Aggressive Sell sequences. The intuition is as follows. Often a specific trade is not a stand alone event, but a part of a sequence of transactions associated with the execution of the same order. For example, an order to aggressively sell 10 contracts may result in four Aggressive Sell transactions: for 2 contracts, 1 contract, 4 contracts, and 3 contracts, respectively, due to the specific sequence of resting bids against which this aggressive sell order was be executed. Using the order ID number, we are able to aggregate these four transactions into one Aggressive Sell sequence for 10 contracts.

Testing Hypothesis H1. Aggressive removal of the last 100 contracts by HFTs; passive provision of the last 100 resting contracts by the Market Makers. Using the Aggressive Buy sequences, we label as a “price increase event” all occurrences of trading sequences in which at least 100 contracts consecutively executed at the same price are followed by some number of contracts at a higher price. To examine indications of low latency, we focus on the the last 100 contracts traded before the price increase and the first 100 contracts at the next higher price (or fewer if the price changes again before 100 contracts are executed). Although we do not look directly at the limit order book data, price increase events are defined to capture occasions where traders use executable buy orders to lift the last remaining offers in the limit order book. Using Aggressive sell trades, we define “price decrease events” symmetrically as occurrences of sequences of trades in which 100 contracts executed at the same price are followed by executions at lower prices. These events are intended to capture occasions where traders use executable sell orders to hit the last few best bids in the limit order book. The results are presented in Table below

For price increase and price decrease events, we calculate each of the six trader categories’ shares of Aggressive and Passive trading volume for the last 100 contracts traded at the “old” price level before the price increase or decrease and the first 100 contracts traded at the “new” price level (or fewer if the number of contracts is less than 100) after the price increase or decrease event.

Table above presents, for the six trader categories, volume shares for the last 100 contracts at the old price and the first 100 contracts at the new price. For comparison, the unconditional shares of aggressive and passive trading volume of each trader category are also reported. Table has four panels covering (A) price increase events on May 3-5, (B) price decrease events on May 3-5, (C) price increase events on May 6, and (D) price decrease events on May 6. In each panel there are six rows of data, one row for each trader category. Relative to panels A and C, the rows for Fundamental Buyers (BUYER) and Fundamental Sellers (SELLER) are reversed in panels B and D to emphasize the symmetry between buying during price increase events and selling during price decrease events. The first two columns report the shares of Aggressive and Passive contract volume for the last 100 contracts before the price change; the next two columns report the shares of Aggressive and Passive volume for up to the next 100 contracts after the price change; and the last two columns report the “unconditional” market shares of Aggressive and Passive sides of all Aggressive buy volume or sell volume. For May 3-5, the data are based on volume pooled across the three days.

Consider panel A, which describes price increase events associated with Aggressive buy trades on May 3-5, 2010. High Frequency Traders participated on the Aggressive side of 34.04% of all aggressive buy volume. Strongly consistent with immediacy absorption hypothesis, the participation rate rises to 57.70% of the Aggressive side of trades on the last 100 contracts of Aggressive buy volume before price increase events and falls to 14.84% of the Aggressive side of trades on the first 100 contracts of Aggressive buy volume after price increase events.

High Frequency Traders participated on the Passive side of 34.33% of all aggressive buy volume. Consistent with hypothesis, the participation rate on the Passive side of Aggressive buy volume falls to 28.72% of the last 100 contracts before a price increase event. It rises to 37.93% of the first 100 contracts after a price increase event.

These results are inconsistent with the idea that high frequency traders behave like textbook market makers, suffering adverse selection losses associated with being picked off by informed traders. Instead, when the price is about to move to a new level, high frequency traders tend to avoid being run over and take the price to the new level with Aggressive trades of their own.

Market Makers follow a noticeably more passive trading strategy than High Frequency Traders. According to panel A, Market Makers are 13.48% of the Passive side of all Aggressive trades, but they are only 7.27% of the Aggressive side of all Aggressive trades. On the last 100 contracts at the old price, Market Makers’ share of volume increases only modestly, from 7.27% to 8.78% of trades. Their share of Passive volume at the old price increases, from 13.48% to 15.80%. These facts are consistent with the interpretation that Market Makers, unlike High Frequency Traders, do engage in a strategy similar to traditional passive market making, buying at the bid price, selling at the offer price, and suffering losses when the price moves against them. These facts are also consistent with the hypothesis that High Frequency Traders have lower latency than Market Makers.

Intuition might suggest that Fundamental Buyers would tend to place the Aggressive trades which move prices up from one tick level to the next. This intuition does not seem to be corroborated by the data. According to panel A, Fundamental Buyers are 21.53% of all Aggressive trades but only 11.61% of the last 100 Aggressive contracts traded at the old price. Instead, Fundamental Buyers increase their share of Aggressive buy volume to 26.17% of the first 100 contracts at the new price.

Taking into account symmetry between buying and selling, panel B shows the results for Aggressive sell trades during May 3-5, 2010, are almost the same as the results for Aggressive buy trades. High Frequency Traders are 34.17% of all Aggressive sell volume, increase their share to 55.20% of the last 100 Aggressive sell contracts at the old price, and decrease their share to 15.04% of the last 100 Aggressive sell contracts at the new price. Market Makers are 7.45% of all Aggressive sell contracts, increase their share to only 8.57% of the last 100 Aggressive sell trades at the old price, and decrease their share to 6.58% of the last 100 Aggressive sell contracts at the new price. Fundamental Sellers’ shares of Aggressive sell trades behave similarly to Fundamental Buyers’ shares of Aggressive Buy trades. Fundamental Sellers are 20.91% of all Aggressive sell contracts, decrease their share to 11.96% of the last 100 Aggressive sell contracts at the old price, and increase their share to 24.87% of the first 100 Aggressive sell contracts at the new price.

The number of price increase and price decrease events increased dramatically on May 6, consistent with the increased volatility of the market on that day. On May 3-5, there were 4100 price increase events and 4062 price decrease events. On May 6 alone, there were 4101 price increase events and 4377 price decrease events. There were therefore approximately three times as many price increase events per day on May 6 as on the three preceding days.

Testing Hypothesis H2. HFTs trade aggressively in the direction of the price move; Market Makers get run over by a price move. To examine this hypothesis, we analyze whether High Frequency Traders use Aggressive trades to trade in the direction of contemporaneous price changes, while Market Makers use Passive trades to trade in the opposite direction from price changes. To this end, we estimate the regression equation

Δyt = α + Φ . Δyt-1 + δ . yt-1 + Σi=120i . Δpt-1 /0.25] + εt

(where yt and Δyt denote inventories and change in inventories of High Frequency Traders for each second of a trading day; t = 0 corresponds to the opening of stock trading on the NYSE at 8:30:00 a.m. CT (9:30:00 ET) and t = 24, 300 denotes the close of Globex at 15:15:00 CT (4:15 p.m. ET); Δpt denotes the price change in index point units between the high-low midpoint of second t-1 and the high-low midpoint of second t. Regressing second-by-second changes in inventory levels of High Frequency Traders on the level of their inventories the previous second, the change in their inventory levels the previous second, the change in prices during the current second, and lagged price changes for each of the previous 20 previous seconds.)

for Passive and Aggressive inventory changes separately.

Table above presents the regression results of the two components of change in holdings on lagged inventory, lagged change in holdings and lagged price changes over one second intervals. Panel A and Panel B report the results for May 3-5 and May 6, respectively. Each panel has four columns, reporting estimated coefficients where the dependent variables are net Aggressive volume (Aggressive buys minus Aggressive sells) by High Frequency Traders (∆AHFT), net Passive volume by High Frequency Traders (∆PHFT), net Aggressive volume by Market Makers (∆AMM), and net Passive volume by Market Makers (∆PMM).

We observe that for lagged inventories (NPHFTt−1), the estimated coefficients for Aggressive and Passive trades by High Frequency Traders are δAHFT = −0.005 (t = −9.55) and δPHFT = −0.001 (t = −3.13), respectively. These coefficient estimates have the interpretation that High Frequency Traders use Aggressive trades to liquidate inventories more intensively than passive trades. In contrast, the results for Market Makers are very different. For lagged inventories (NPMMt−1), the estimated coefficients for Aggressive and Passive volume by Market Makers are δAMM = −0.002 (t = −6.73) and δPMM = −0.002 (t = −5.26), respectively. The similarity of these coefficients estimates has the interpretation that Market Makers favor neither Aggressive trades nor Passive trades when liquidating inventories.

For lagged price changes, coefficient estimates for Aggressive trades by High Frequency Traders and Market Makers are positive and statistically significant at lags 1-4 and lags 1-10, respectively. These results have the interpretation that both High Frequency Traders’ and Market Makers’ trade on recent price momentum, but the trading is compressed into a shorter time frame for High Frequency Traders than for Market Makers.

When High Frequency Traders are net buyers on May 3-5, prices rise by 17% of a tick in the next second. When HFTs execute Aggressively or Passively, prices rise by 20% and 2% of a tick in the next second, respectively. In subsequent seconds, prices in all cases trend downward by about 5% of a tick over the subsequent 19 seconds. For May 3-5, the results are almost symmetric for selling.

The results of this analysis are presented in the table below. Panel A provides results for May 3-5 and panel B for May 6. In each panel, there are five rows of data, one for each trader category. The first three columns report the total number of trades, the total number of immediately scratched trades, and the percentage of trades that are immediately scratched by traders in five categories. For May 3-6, the reported numbers are from the pooled data.

This table presents statistics for immediate trade scratching which measures how many times a trader changes his/her direction of trading in a second aggregated over a day. We define a trade direction change as a buy trade right after a sell trade or vice versa at the same price level in the same second.

This table shows that High Frequency Traders scratched 2.84 % of trades on May 3-5 and 4.26 % on May 6; Market Makers scratched 2.49 % of trades on May 3-5 and 5.53 % of trades on May 6. While the percentages of immediately scratched trades by Market Makers is slightly higher than that for High Frequency Traders on May 6, the percentages for both groups are very similar. The fourth, fifth, and sixth columns of the Table report the mean, standard deviation, and median of the number of scratched trades for the traders in each category.

Although the percentages of scratched trades are similar, the mean number of immediately scratched trades by High Frequency Traders is much greater than for Market Makers: 540.56 per day on May 3-5 and 1610.75 on May 6 for High Frequency Traders versus 13.35 and 72.92 for Market Makers. The differences between High Frequency Traders and Market Makers reflect differences in volume traded. The Table shows that High Frequency Traders and Market Makers scratch a significantly larger percentage of their trades than other trader categories.

# Gauge Theory of Arbitrage, or Financial Markets Resembling Screening in Electrodynamics

When a mispricing appears in a market, market speculators and arbitrageurs rectify the mistake by obtaining a profit from it. In the case of profitable fluctuations they move into profitable assets, leaving comparably less profitable ones. This affects prices in such a way that all assets of similar risk become equally attractive, i.e. the speculators restore the equilibrium. If this process occurs infinitely rapidly, then the market corrects the mispricing instantly and current prices fully reflect all relevant information. In this case one says that the market is efficient. However, clearly it is an idealization and does not hold for small enough times.

The general picture, sketched above, of the restoration of equilibrium in financial markets resembles screening in electrodynamics. Indeed, in the case of electrodynamics, negative charges move into the region of the positive electric field, positive charges get out of the region and thus screen the field. Comparing this with the financial market we can say that a local virtual arbitrage opportunity with a positive excess return plays a role of the positive electric field, speculators in the long position behave as negative charges, whilst the speculators in the short position behave as positive ones. Movements of positive and negative charges screen out a profitable fluctuation and restore the equilibrium so that there is no arbitrage opportunity any more, i.e. the speculators have eliminated the arbitrage opportunity.

The analogy is apparently superficial, but it is not. The analogy emerges naturally in the framework of the Gauge Theory of Arbitrage (GTA). The theory treats a calculation of net present values and asset buying and selling as a parallel transport of money in some curved space, and interpret the interest rate, exchange rates and prices of asset as proper connection components. This structure is exactly equivalent to the geometrical structure underlying the electrodynamics where the components of the vector-potential are connection components responsible for the parallel transport of the charges. The components of the corresponding curvature tensors are the electromagnetic field in the case of electrodynamics and the excess rate of return in case of GTA. The presence of uncertainty is equivalent to the introduction of noise in the electrodynamics, i.e. quantization of the theory. It allows one to map the theory of the capital market onto the theory of quantized gauge field interacting with matter (money flow) fields. The gauge transformations of the matter field correspond to a change of the par value of the asset units which effect is eliminated by a gauge tuning of the prices and rates. Free quantum gauge field dynamics (in the absence of money flows) is described by a geometrical random walk for the assets prices with the log-normal probability distribution. In general case the consideration maps the capital market onto Quantum Electrodynamics where the price walks are affected by money flows.

Electrodynamical model of quasi-efficient financial market