# Incomplete Markets and Calibrations for Coherence with Hedged Portfolios. Thought of the Day 154.0

In complete market models such as the Black-Scholes model, probability does not really matter: the “objective” evolution of the asset is only there to define the set of “impossible” events and serves to specify the class of equivalent measures. Thus, two statistical models P1 ∼ P2 with equivalent measures lead to the same option prices in a complete market setting.

This is not true anymore in incomplete markets: probabilities matter and model specification has to be taken seriously since it will affect hedging decisions. This situation is more realistic but also more challenging and calls for an integrated approach between option pricing methods and statistical modeling. In incomplete markets, not only does probability matter but attitudes to risk also matter: utility based methods explicitly incorporate these into the hedging problem via utility functions. While these methods are focused on hedging with the underlying asset, common practice is to use liquid call/put options to hedge exotic options. In incomplete markets, options are not redundant assets; therefore, if options are available as hedging instruments they can and should be used to improve hedging performance.

While the lack of liquidity in the options market prevents in practice from using dynamic hedges involving options, options are commonly used for static hedging: call options are frequently used for dealing with volatility or convexity exposures and for hedging barrier options.

What are the implications of hedging with options for the choice of a pricing rule? Consider a contingent claim H and assume that we have as hedging instruments a set of benchmark options with prices Ci, i = 1 . . . n and terminal payoffs Hi, i = 1 . . . n. A static hedge of H is a portfolio composed from the options Hi, i = 1 . . . n and the numeraire, in order to match as closely as possible the terminal payoff of H:

H = V0 + ∑i=1n xiHi + ∫0T φdS + ε —– (1)

where ε is an hedging error representing the nonhedgeable risk. Typically Hi are payoffs of call or put options and are not possible to replicate using the underlying so adding them to the hedge portfolio increases the span of hedgeable claims and reduces residual risk.

Consider a pricing rule Q. Assume that EQ[ε] = 0 (otherwise EQ[ε] can be added to V0). Then the claim H is valued under Q as:

e-rTEQ[H] = V0 ∑i=1n xe-rTEQ[Hi] —– (2)

since the stochastic integral term, being a Q-martingale, has zero expectation. On the other hand, the cost of setting up the hedging portfolio is:

V0 + ∑i=1n xCi —– (3)

So the value of the claim given by the pricing rule Q corresponds to the cost of the hedging portfolio if the model prices of the benchmark options Hi correspond to their market prices Ci:

∀i = 1, …, n

e-rTEQ[Hi] = Ci∗ —– (4)

This condition is called calibration, where a pricing rule verifies the calibration of the option prices Ci, i = 1, . . . , n. This condition is necessary to guarantee the coherence between model prices and the cost of hedging with portfolios and if the model is not calibrated then the model price for a claim H may have no relation with the effective cost of hedging it using the available options Hi. If a pricing rule Q is specified in an ad hoc way, the calibration conditions will not be verified, and thus one way to ensure them is to incorporate them as constraints in the choice of the pricing measure Q.

# Financial Fragility in the Margins. Thought of the Day 114.0

If micro-economic crisis is caused by the draining of liquidity from an individual company (or household), macro-economic crisis or instability, in the sense of a reduction in the level of activity in the economy as a whole, is usually associated with an involuntary outflow of funds from companies (or households) as a whole. Macro-economic instability is a ‘real’ economic phenomenon, rather than a monetary contrivance, the sense in which it is used, for example, by the International Monetary Fund to mean price inflation in the non-financial economy. Neo-classical economics has a methodological predilection for attributing all changes in economic activity to relative price changes, specifically the price changes that undoubtedly accompany economic fluctuations. But there is sufficient evidence to indicate that falls in economic activity follow outflows of liquidity from the industrial and commercial company sector. Such outflows then lead to the deflation of economic activity that is the signal feature of economic recession and depression.

The existence of more traders, and the system for avoiding counterparty risk commonly found in most futures market, whereby contracts are made with a central clearing house, introduce sequencing complications which may cause problems: having a central clearing house avoids the possibility that one trader’s default will cause other traders to default on their obligations. But it also denies traders the facility of giving each other credit, and thereby reduces the velocity of circulation of whatever liquidity is in the market. Having to pay all obligations in full to the central clearing house increases the money (or gross inflow) that broking firms and investors have to put into the market as margin payments or on settlement days. This increases the risk that a firm with large net liabilities in the financial futures market will be obliged to realize assets in other markets to meet those liabilities. In this way, the integrity of the market is protected by increasing the effective obligations of all traders, at the expense of potentially unsettling claims on other markets.

This risk is enhanced by the trading of rentiers, or banks and entrepreneurs operating as rentiers, hedging their futures contracts in other financial markets. However, while such incidents generate considerable excitement around the markets at the time of their occurrence, there is little evidence that they could cause involuntary outflows from the corporate sector on such a scale as to produce recession in the real economy. This is because financial futures are still used by few industrial and commercial companies, and their demand for financial derivatives instruments is limited by the relative expense of these instruments and their own exposure to changes in financial parameters (which may more easily be accommodated by holding appropriate stocks of liquid assets, i.e., liquidity preference). Therefore, the future of financial futures depends largely on the interest in them of the contemporary rentiers in pension, insurance and various other forms of investment funds. Their interest, in turn, depends on how those funds approach their ‘maturity’.

However, the decline of pension fund surpluses poses important problems for the main securities markets of the world where insurance and pension funds are now the dominant investors, as well as for more peripheral markets like emerging markets, venture capital and financial futures. A contraction in the net cash inflow of investment funds will be reflected in a reduction in the funds that they are investing, and a greater need to realize assets when a change in investment strategy is undertaken. In the main securities markets of the world, a reduction in the ‘new money’ that pension and insurance funds are putting into those securities markets will slow down the rate of growth of the prices in those markets. How such a fall in the institutions’ net cash inflow will affect the more marginal markets, such as emerging markets, venture capital and financial futures, depends on how institutional portfolios are managed in the period of declining net contributions inflows.

In general, investment managers in their own firms, or as employees of merchant or investment banks, compete to manage institutions’ funds. Such competition is likely to increase as investment funds approach ‘maturity’, i.e., as their cash outflows to investors, pensioners or insurance policyholders, rises faster than their cash inflow from contributions and premiums, so that there are less additional funds to be managed. In principle, this should not affect financial futures markets, in the first instance, since, as argued above, the short-term nature of their instruments and the large proportion in their business of intra-market trade makes them much less dependent on institutional cash inflows. However, this does not mean that they would be unaffected by changes in the portfolio preferences of investment funds in response to lower returns from the main securities markets. Such lower returns make financial investments like financial futures, venture capital and emerging markets, which are more marginal because they are so hazardous, more attractive to normally conservative fund managers. Investment funds typically put out sections of portfolios to specialist fund managers who are awarded contracts to manage a section according to the soundness of their reputation and the returns that they have made hitherto in portfolios under their management. A specialist fund manager reporting high, but not abnormal, profits in a fund devoted to financial futures, is likely to attract correspondingly more funds to manage when returns are lower in the main markets’ securities, even if other investors in financial futures experienced large losses. In this way, the maturing of investment funds could cause an increased inflow of rentier funds into financial futures markets.

An inflow of funds into a financial market entails an increase in liabilities to the rentiers outside the market supplying those funds. Even if profits made in the market as a whole also increase, so too will losses. While brokers commonly seek to hedge their positions within the futures market, rentiers have much greater possibilities of hedging their contracts in another market, where they have assets. An inflow into futures markets means that on any settlement day there will therefore be larger net outstanding claims against individual banks or investment funds in respect of their financial derivatives contracts. With margin trading, much larger gross financial inflows into financial futures markets will be required to settle maturing contracts. Some proportion of this will require the sale of securities in other markets. But if liquidity in integrated cash markets for securities is reduced by declining net inflows into pension funds, a failure to meet settlement obligations in futures markets is the alternative to forced liquidation of other assets. In this way futures markets will become more fragile.

Moreover, because of the hazardous nature of financial futures, high returns for an individual firm are difficult to sustain. Disappointment is more likely to be followed by the transfer of funds to management in some other peripheral market that shows a temporary high profit. While this should not affect capacity utilization in the futures market, because of intra-market trade, it is likely to cause much more volatile trading, and an increase in the pace at which new instruments are introduced (to attract investors) and fall into disuse. Pension funds whose returns fall below those required to meet future liabilities because of such instability would normally be required to obtain additional contributions from employers and employees. The resulting drain on the liquidity of the companies affected would cause a reduction in their fixed capital investment. This would be a plausible mechanism for transmitting fragility in the financial system into full-scale decline in the real economy.

The proliferation of financial futures markets has only had been marginally successful in substituting futures contracts for Keynesian liquidity preference as a means of accommodating uncertainty. A closer look at the agents in those markets and their market mechanisms indicates that the price system in them is flawed and trading hazardous risks in them adds to uncertainty rather than reducing it. The hedging of financial futures contracts in other financial markets means that the resulting forced liquidations elsewhere in the financial system are a real source of financial instability that is likely to worsen as slower growth in stock markets makes speculative financial investments appear more attractive. Capital-adequacy regulations are unlikely to reduce such instability, and may even increase it by increasing the capital committed to trading in financial futures. Such regulations can also create an atmosphere of financial security around these markets that may increase unstable speculative flows of liquidity into the markets. For the economy as a whole, the real problems are posed by the involvement of non-financial companies in financial futures markets. With the exception of a few spectacular scandals, non-financial companies have been wary of using financial futures, and it is important that they should continue to limit their interest in financial futures markets. Industrial and commercial companies, which generate their own liquidity through trade and production and hence have more limited financial assets to realize in order to meet financial futures liabilities in times of distress, are more vulnerable to unexpected outflows of liquidity in proportion to their increased exposure to financial markets. The liquidity which they need to set aside to meet such unexpected liabilities inevitably means a reduced commitment to investment in fixed capital and new technology.

# Credit Risk Portfolio. Note Quote.

The recent development in credit markets is characterized by a flood of innovative credit risky structures. State-of-the-art portfolios contain derivative instruments ranging from simple, nearly commoditized contracts such as credit default swap (CDS), to first- generation portfolio derivatives such as first-to-default (FTD) baskets and collateralized debt obligation (CDO) tranches, up to complex structures involving spread options and different asset classes (hybrids). These new structures allow portfolio managers to implement multidimensional investment strategies, which seamlessly conform to their market view. Moreover, the exploding liquidity in credit markets makes tactical (short-term) overlay management very cost efficient. While the outperformance potential of an active portfolio management will put old-school investment strategies (such as buy-and-hold) under enormous pressure, managing a highly complex credit portfolio requires the introduction of new optimization technologies.

New derivatives allow the decoupling of business processes in the risk management industry (in banking, as well as in asset management), since credit treasury units are now able to manage specific parts of credit risk actively and independently. The traditional feedback loop between risk management and sales, which was needed to structure the desired portfolio characteristics only by selective business acquisition, is now outdated. Strategic cross asset management will gain in importance, as a cost-efficient overlay management can now be implemented by combining liquid instruments from the credit universe.

In any case, all these developments force portfolio managers to adopt an integrated approach. All involved risk factors (spread term structures including curve effects, spread correlations, implied default correlations, and implied spread volatilities) have to be captured and integrated into appropriate risk figures. We have a look on constant proportion debt obligations (CPDOs) as a leveraged exposure on credit indices, constant proportion portfolio insurance (CPPI) as a capital guaranteed instrument, CDO tranches to tap the correlation market, and equity futures to include exposure to stock markets in the portfolio.

For an integrated credit portfolio management approach, it is of central importance to aggregate risks over various instruments with different payoff characteristics. In this chapter, we will see that a state-of-the-art credit portfolio contains not only linear risks (CDS and CDS index contracts) but also nonlinear risks (such as FTD baskets, CDO tranches, or credit default swaptions). From a practitioner’s point of view there is a simple solution for this risk aggregation problem, namely delta-gamma management. In such a framework, one approximates the risks of all instruments in a portfolio by its first- and second-order sensitivities and aggregates these sensitivities to the portfolio level. Apparently, for a proper aggregation of risk factors, one has to take the correlation of these risk factors into account. However, for credit risky portfolios, a simplistic sensitivity approach will be inappropriate, as can be seen by the characteristics of credit portfolio risks shows:

• Credit risky portfolios usually involve a larger number of reference entities. Hence, one has to take a large number of sensitivities into account. However, this is a phenomenon that is already well known from the management of stock portfolios. The solution is to split the risk for each constituent into a systematic risk (e.g., a beta with a portfolio hedging tool) and an alpha component which reflects the idiosyncratic part of the risk.

• However, in contrast to equities, credit risk is not one dimensional (i.e., one risky security per issuer) but at least two dimensional (i.e., a set of instruments with different maturities). This is reflected in the fact that there is a whole term structure of credit spreads. Moreover, taking also different subordination levels (with different average recovery rates) into account, credit risk becomes a multidimensional object for each reference entity.
• While most market risks can be satisfactorily approximated by diffusion processes, for credit risk the consideration of events (i.e., jumps) is imperative. The most apparent reason for this is that the dominating element of credit risk is event risk. However, in a market perspective, there are more events than the ultimate default event that have to be captured. Since one of the main drivers of credit spreads is the structure of the underlying balance sheet, a change (or the risk of a change) in this structure usually triggers a large movement in credit spreads. The best-known example for such an event is a leveraged buyout (LBO).

These show that a simple sensitivity approach – e.g., calculate and tabulate all deltas and gammas and let a portfolio manager play with – is not appropriate. Further risk aggregation (e.g., beta management) and risk factors that capture the event risk are needed. For the latter, a quick solution is the so-called instantaneous default loss (IDL). The IDL expresses the loss incurred in a credit risk instrument in case of a credit event. For single-name CDS, this is simply the loss given default (LGD). However, for a portfolio derivative such as a mezzanine tranche, this figure does not directly refer to the LGD of the defaulted item, but to the changed subordination of the tranche because of the default. Hence, this figure allows one to aggregate various instruments with respect to credit events.

# 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.

# Hedging. Part 1.

Hedging a zero coupon bond denoted P(t,T) using other zero coupon bonds is accomplished by minimizing the residual variance of the hedged portfolio. The hedged portfolio Π(t) is represented as

Π(t) = P (t, T ) + ∑i=1NΔiP(t, Ti)

where ∆i denotes the amount of the ith bond P(t, Ti) included in the hedged portfolio. Notethe bonds P (t, T) and P (t, Ti) are determined by observing their market values at time t. It is the instantaneous change in the portfolio value that is stochastic. Therefore, the volatility of this change is computed to ascertain the efficacy of the hedge portfolio.

For starters, consider the variance of an individual bond in the field theory model. The definition P (t, T) = exp(-∫tT dxf(t, x)) for zero coupon bond prices implies that

dP(t, T)/P(t, T) = f(t, t)dt – ∫tTdxdf(t, x) = (r(t) – ∫tTdxα(t, x) – ∫tTdxσ(t, x)A(t, x))dt

and E[dP(t, T)/P(t, T) = r(t) – ∫tTdxα(t, x)dt since, E[A(t, x)] = 0. Therefore

dP(t, T)/P(t, T) – E[dP(t, T)/P(t, T) = – ∫tTdxσ(t, x)A(t, x))dt —– (1)

Squaring this expression and invoking the result that E[A(t, x)A(t, x′)] = δ(0)D(x, x′; t, TFR) = D(x, x′; t, TFR) /dt results in the instantaneous bond price variance

Var [dP(t, T)] = dt P2(t, T)∫tTdx ∫tT dx’σ(t, x) D(x, x′; t, TFR) σ(t, x’) —– (2)

As an intermediate step, the instantaneous variance of a bond portfolio is considered. For a portfolio of bonds, ∏ = ∑i=1NΔiP(t, Ti), the following results follow directly

d∏(t) – E[d∏(t)] = -dt ∑i=1NΔiP(t, Ti) ∫tTi dxσ(t, x)A(t, x) —– (3)

and

Var [d∏(t)] = dt ∑i=1Nj=1NΔiΔjP(t, Ti)P(t, Tj) ∫tTdx ∫tTj dx σ(t, x) D(x, x′; t, TFR) σ(t, x’) —– (4)

The (residual) variance of the hedged portfolio

Π(t) = P (t, T ) + ∑i=1NΔiP(t, Ti) ∫tTdx ∫tTdx’

may now be computed in a straightforward manner. For notational simplicity, the bonds P(t,Ti) (being used to hedge the original bond) and P(t,T) are denoted Pi and P respectively. Equation (4) implies the hedged portfolio’s variance equals the final result shown below

P2tTdx∫tT dx’ σ(t, x) σ(t, x’) D(x, x′; t, TFR) +2P ∑i=1NΔiPitTdx ∫tTdx’ + ∑i=1Nj=1NΔiΔjPiPjtTitTjdx’ σ(t, x) σ(t, x’) D(x, x′; t, TFR) —– (5)

Observe that the residual variance depends on the correlation between forward rates described by the propagator. Ultimately, the effectiveness of the hedge portfolio is an empirical question since perfect hedging is not possible without shorting the original bond. Minimizing the residual variance in equation (5) with respect to the hedge parameters Δi is an application of standard calculus.

# Quantum Field Theory and Evolution of Forward Rates in Quantitative Finance. Note Quote.

Applications of physics to finance are well known, and the application of quantum mechanics to the theory of option pricing is well known. Hence it is natural to utilize the formalism of quantum field theory to study the evolution of forward rates. Quantum field theory models of the term structure originated with Baaquie. The intuition behind quantum field theory models of the term structure stems from allowing each forward rate maturity to both evolve randomly and be imperfectly correlated with every other maturity. This may also be accomplished by increasing the number of random factors in the original HJM towards infinity. However, the infinite number of factors in a field theory model are linked via a single function that governs the correlation between forward rate maturities. Thus, instead of estimating additional volatility functions in a multifactor HJM framework, one additional parameter is sufficient for a field theory model to instill imperfect correlation between every forward rate maturity. As the correlation between forward rate maturities approaches unity, field theory models reduce to the standard one1 factor HJM model. Therefore, the fundamental difference between finite factor HJM and field theory models is the minimal structure the latter requires to instill imperfect correlation between forward rates. The Heath-Jarrow-Morton framework refers to a class of models that are derived by directly modeling the dynamics of instantaneous forward-rates. The central insight of this framework is to recognize that there is an explicit relationship between the drift and volatility parameters of the forward-rate dynamics in a no-arbitrage world. The familiar short-rate models can be derived in the HJM framework but in general, however, HJM models are non-Markovian. As a result, it is not possible to use the PDE-based computational approach for pricing derivatives. Instead, discrete-time HJM models and Monte-Carlo methods are often used in practice. Monte Carlo methods (or Monte Carlo experiments) are a broad class of computational algorithms that rely on repeated random sampling to obtain numerical results. Their essential idea is using randomness to solve problems that might be deterministic in principle.

A Lagrangian is introduced to describe the field. The Lagrangian has the advantage over Brownian motion of being able to control fluctuations in the field, hence forward rates, with respect to maturity through the addition of a maturity dependent gradient as detailed in the definition below. The action of the field integrates the Lagrangian over time and when exponentiated and normalized serves as the probability distribution for forward rate curves. The propagator measures the correlation in the field and captures the effect the field at time t and maturity x has on maturity x′ at time t′. In the one factor HJM model, the propagator equals one which allows the quick recovery of one factor HJM results. Previous research has begun with the propagator or “correlation” function for the field instead of deriving this quantity from the Lagrangian. More importantly, the Lagrangian and its associated action generate a path integral that facilitates the solution of contingent claims and hedge parameters. However, previous term structure models have not defined the Lagrangian and are therefore unable to utilize the path integral in their applications. The Feynman path integral, path integral in short, is a fundamental quantity that provides a generating function for forward rate curves. Although crucial for pricing and hedging, the path integral has not appeared in previous term structure models with generalized continuous random processes.

Notation

Let t0 denote the current time and T the set of forward rate maturities with t0 ≤ T . The upper bound on the forward rate maturities is the constant TFR which constrains the forward rate maturities T to lie within the interval [t0, t0 + TFR].

To illustrate the field theory approach, the original finite factor HJM model is derived using field theory principles in appendix A. In the case of a one factor model, the derivation does not involve the propagator as the propagator is identically one when forward rates are perfectly correlated. However, the propagator is non trivial for field theory models as it governs the imperfect correlation between forward rate maturities. Let A(t,x) be a two dimensional field driving the evolution of forward rates f (t, x) through time. Following Baaquie, the Lagrangian of the field is defined as

Definition:

The Lagrangian of the field equals

L[A] = -1/2TFR  {A2(t, x) + 1/μ2(∂A(t,x)∂x)2} —– (1)

Definition is not unique, other Lagrangians exist and would imply different propagators. However, the Lagrangian in the definition is sufficient to explain the contribution of field theory ∂A(t,x)∂x  that controls field fluctuations in the direction of the forward rate maturity. The constant μ measures the strength of the fluctuations in the maturity direction. The Lagrangian in the definition implies the field is continuous, Gaussian, and Markovian. Forward rates involving the field are expressed below where the drift and volatility functions satisfy the usual regularity conditions.

∂f(t,x)/∂t = α (t, x) + σ (t, x)A(t, x) —– (2)

The forward rate process in equation (2) incorporates existing term structure research on Brown- ian sheets, stochastic strings, etc that have been used in previous continuous term structure models. Note that equation (2) is easily generalized to the K factor case by introducing K independent and identical fields Ai(t, x). Forward rates could then be defined as

∂f(t, x)/∂t = α (t, x) + ∑i=1K σi(t, x)Ai(t, x) —– (3)

However, a multifactor HJM model can be reproduced without introducing multiple fields. In fact, under specific correlation functions, the field theory model reduces to a multifactor HJM model without any additional fields to proxy for additional Brownian motions.

Proposition:

Lagrangian of Multifactor HJM

The Lagrangian describing the random process of a K-factor HJM model is given by

L[A] = −1/2 A(t, x)G−1(t, x, x′)A(t, x′) —– (4)

where

∂f(t, x)/∂t = α(t, x) + A(t, x)

and G−1(t, x, x′)A(t, x′) denotes the inverse of the function.

G(t, x, x′) = ∑i=1K σi(t, x) σi(t, x’) —– (5)

The above proposition is an interesting academic exercise to illustrate the parallel between field theory and traditional multifactor HJM models. However, multifactor HJM models have the disadvantages associated with a finite dimensional basis. Therefore, this approach is not pursued in later empirical work. In addition, it is possible for forward rates to be perfectly correlated within a segment of the forward rate curve but imperfectly correlated with forward rates in other segments. For example, one could designate short, medium, and long maturities of the forward rate curve. This situation is not identical to the multifactor HJM model but justifies certain market practices that distinguish between short, medium, and long term durations when hedging. However, more complicated correlation functions would be required; compromising model parsimony and reintroducing the same conceptual problems of finite factor models. Furthermore, there is little economic intuition to justify why the correlation between forward rates should be discontinuous.

# The Politics of War on Coal. Drunken Risibility.

Coal is deemed to phase out, but the transition is going to be a slow process – an evolution/devolution simultaneously, and would be dependent largely on market conditions, for its the latter that could act the slug to phasing out. War on Coal is a political line that needs to be tread carefully for it lies on a liminal threat to slip either side, viz. war on coal as a source of energy, or war on coal as a policy to be implemented calling out for phasing out. This political line ceases to trudge the moment markets start dictating priorities as is evident in the case of the largest Sovereign Fund (Norway), or even in the US where phasing out, clauses repairment to economic-employment-geologic depression, the costs of doing which are astronomical, and thus revoking any such decrees is a trap onto eating a little bit of crow.

Incisively how the public money is channeled from source to destination in the journey of coal needs to be looked at in-depth, for mere hedging such a source would be an economic disaster rippling into sociological/ecological stalemate. Coal is cheap and dirty without doubt, but it becomes burdensome due to a host of factors, the chief among which is financialisation of it. By this is meant capital taking on garbs, which we honestly are not too equipped to understand, but equally adept at underestimating, for every ill is a result of economic liberalisation or neoliberalism (right?, pun intended!), the latter of which I personally detest using, since economies have long transcended the notion.
Please find attached the Fund’s annual report and coal criterion.

# What’s an Asset Reconstruction Company, and why does it even matter to NPAs?

As is suggestive of the name, ARCs are required to repackage assets to make them more saleable. But, in the context of bad loans, or non-performing assets, such companies often falter to garner enough firepower to root out the surging menace of NPAs. The Indian context is caused primarily by a systemic rot involving faulty practices of project finance and subsequent difficulties in recoveries on loans. ARCs are constituted to precisely address such hurdles. With their status as centralised agencies, these are programmed to buy up stressed/distressed and non-performing assets and repackage them to sell them to prospective promoters/buyers. ARCs are programmed to buy NPAs at a discounted price, which in turn help the banks and lenders to clean up their sticky balance sheets. ARCs can either be public, private or jointly owned, and are also armed to float bonds to recover dues from borrowers. Even if on paper the concept of an ARC looks robust with scalability to concoct a bad asset with a performing one in order to increase its saleability, in reality, ARCs are prone to failures for lack of buyers for their packages and limited by capital concerns. But, there are challenges galore for ARCs viz. debt aggregation is a far cry, and unless this is done, resolution will always be expeditious; hunt for fresh financial support; and discrepancy between banks and ARCs in pricing of assets, unless reached a commonality would continue to remain a contentious issue.

The whole concept of Asset Reconstruction Companies (ARCs) is closely modelled on the US model of Asset Management Companies (AMCs), and is thus a large industry in itself as far as buying and selling off of debts are concerned. It resembles a time to strike as the ducks are now lined, and the opportunities galore in private equities as ARCs get a chance to own the entire capital structure and reinstall management echelons, all thanks to Budgetary recommendations with a 100% FDI welcomed. But, it still is going a bit too far, and hence let us examine the evolution first.

The real trigger for ARCs to flourish came with Raghuram Rajan’s exhortation to banks to clean up its mess. A switch of trajectory happened sometime in 2013, when ever greening bad loans was put on to back burner and let ARCs face up. This is where banks were obligated to turnaround loans until then considered unredeemable. Adding to that trigger point is another one banking on Bankruptcy Code, which promises discount to buying off bad debts or distressed assets at a discount pitting them more profitable a venture compared with greenfield projects of similar magnitude.

ARCs are born out of SARFAESI Act, 2002, (The Securitisation and Reconstruction of Financial Assets and Enforcement of Security Interest Act, 2002) and enable the banks to acquire the securities which had been pledged. all of this was achievable without any interference from the jurisdiction of the civil courts thus lending authoritarian power to banks to cope up with NPAs. But regulatory directives prevented the smooth functioning of transactions involving bad debts, thanks to the decrepit system for enforcing securities. Sale of loans to ARCs is however the last resort banks undertake as statutory hurdles and deposed promoters speed break.

Basically, road to recovery is a step by step process involving, Bank selling a bad loan to ARC; ARCs paying 15% upfront, and issuing the remaining 85% as securities in the form of Security Receipts (SRs), which primarily deals with 5% upfront payment as was the case a year back; ARCs initiating the turnaround and in the process earning 1.5% management fee; recovery proceeds thus accrued get shared by the banks and the ARCs; and if the ARCs fail to recover the bad loans for eight years, the investments get written-off. Now, this can be seen as a clear shift from ever greening, or even liberal funding by the government year-on-year, and aptly reminds internal recapitalization as indispensable. Budgetary infusion or capital infusion or recapitalisation is putting capital for the purpose of helping the ailing public sector banks. The very understanding of PSBs would imply such an infusion required from time to time, but the problem lies in its ritualistic nature. On the contrary, if banks were to internally raise funds for recapitalisation, it would indicate a healthy practice. One reason the government does infusion is to meet Basel III norms, as reliance upon internal fund recapitalisation would not let that be accomplished. But, there is a caveat here to be always kept in mind. Written-off doesn’t necessarily mean that defaults and defaulters are not chased. They are undoubtedly sought after, with the only difference being the clean-up of balance sheets for the year such defaults happen for the banks. And, if they don’t write-offs, or alternatively called charge-offs effectuate helping banks not only erase the mess off their balance sheet, but saving them enormous tax liabilities. The real tussle is between charging-off loans and becoming industrious in selling them off. The winner takes it all, which in this case is hedging equivalent to selling off bad debts, and which is promulgated by the RBI and the government in jettisoning this messy baggage.