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

hedge2

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 The self-financing portfolio is set up with zero initial net investment cost and no additional funds are added or withdrawn afterwards.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

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

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

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

If we formally write the stochastic dynamics of V as

dV/V = μV dt + σV dZ —– (3)

then μV and σV are given by

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

and

σV = (σS ∂V/∂S)/V —– (5)

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

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

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

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

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

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

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

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

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

QS(t)/QV(t) = -∂V/∂S —– (9)

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

(8) becomes

0 = [(μ − r)MS(t) + (μV − r)MV(t)]dt

substituting the ratio factor in the above equation, we get

(μ − r)S ∂V/∂S = (μV − r)V —– (10)

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

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

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

Long Term Capital Management. Note Quote.

Long Term Capital Management, or LTCM, was a hedge fund founded in 1994 by John Meriwether, the former head of Salomon Brothers’s domestic fixed-income arbitrage group. Meriwether had grown the arbitrage group to become Salomon’s most profitable group by 1991, when it was revealed that one of the traders under his purview had astonishingly submitted a false bid in a U.S. Treasury bond auction. Despite reporting the trade immediately to CEO John Gutfreund, the outcry from the scandal forced Meriwether to resign.

Meriwether revived his career several years later with the founding of LTCM. Amidst the beginning of one of the greatest bull markets the global markets had ever seen, Meriwether assembled a team of some of the world’s most respected economic theorists to join other refugees from the arbitrage group at Salomon. The board of directors included Myron Scholes, a coauthor of the famous Black-Scholes formula used to price option contracts, and MIT Sloan professor Robert Merton, both of whom would later share the 1997 Nobel Prize for Economics. The firm’s impressive brain trust, collectively considered geniuses by most of the financial world, set out to raise a $1 billion fund by explaining to investors that their profoundly complex computer models allowed them to price securities according to risk more accurately than the rest of the market, in effect “vacuuming up nickels that others couldn’t see.”

One typical LTCM trade concerned the divergence in price between long-term U.S. Treasury bonds. Despite offering fundamentally the same (minimal) default risk, those issued more recently – known as “on-the-run” securities – traded more heavily than those “off-the-run” securities issued just months previously. Heavier trading meant greater liquidity, which in turn resulted in ever-so-slightly higher prices. As “on-the-run” securities become “off-the-run” upon the issuance of a new tranche of Treasury bonds, the price discrepancy generally disappears with time. LTCM sought to exploit that price convergence by shorting the more expensive “on-the-run” bond while purchasing the “off- the-run” security.

By early 1998 the intellectual firepower of its board members and the aggressive trading practices that had made the arbitrage group at Salomon so successful had allowed LTCM to flourish, growing its initial $1 billion of investor equity to $4.72 billion. However, the miniscule spreads earned on arbitrage trades could not provide the type of returns sought by hedge fund investors. In order to make transactions such as these worth their while, LTCM had to employ massive leverage in order to magnify its returns. Ultimately, the fund’s equity component sat atop more than $124.5 billion in borrowings for total assets of more than $129 billion. These borrowings were merely the tip of the ice-berg; LTCM also held off-balance-sheet derivative positions with a notional value of more than $1.25 trillion.

Untitled

The fund’s success began to pose its own problems. The market lacked sufficient capacity to absorb LTCM’s bloated size, as trades that had been profitable initially became impossible to conduct on a massive scale. Moreover, a flood of arbitrage imitators tightened the spreads on LTCM’s “bread-and-butter” trades even further. The pressure to continue delivering returns forced LTCM to find new arbitrage opportunities, and the fund diversified into areas where it could not pair its theoretical insights with trading experience. Soon LTCM had made large bets in Russia and in other emerging markets, on S&P futures, and in yield curve, junk bond, merger, and dual-listed securities arbitrage.

Combined with its style drift, the fund’s more than 26 leverage put LTCM in an increasingly precarious bubble, which was eventually burst by a combination of factors that forced the fund into a liquidity crisis. In contrast to Scholes’s comments about plucking invisible, riskless nickels from the sky, financial theorist Nassim Taleb later compared the fund’s aggressive risk taking to “picking up pennies in front of a steamroller,” a steamroller that finally came in the form of 1998’s market panic. The departure of frequent LTCM counterparty Salomon Brothers from the arbitrage market that summer put downward pressure on many of the fund’s positions, and Russia’s default on its government-issued bonds threw international credit markets into a downward spiral. Panicked investors around the globe demonstrated a “flight to quality,” selling the risky securities in which LTCM traded and purchasing U.S. Treasury securities, further driving up their price and preventing a price convergence upon which the fund had bet so heavily.

None of LTCM’s sophisticated theoretical models had contemplated such an internationally correlated credit market collapse, and the fund began hemorrhaging money, losing nearly 20% of its equity in May and June alone. Day after day, every market in which LTCM traded turned against it. Its powerless brain trust watched in horror as its equity shrank to $600 million in early September without any reduction in borrowing, resulting in an unfathomable 200 leverage ratio. Sensing the fund’s liquidity crunch, Bear Stearns refused to continue acting as a clearinghouse for the fund’s trades, throwing LTCM into a panic. Without the short-term credit that enabled its entire trading operations, the fund could not continue and its longer-term securities grew more illiquid by the day.

Obstinate in their refusal to unwind what they still considered profitable trades hammered by short-term market irrationality, LTCM’s partners refused a buyout offer of $250 million by Goldman Sachs, ING Barings, and Warren Buffet’s Berkshire Hathaway. However, LTCM’s role as a counterparty in thousands of derivatives trades that touched investment firms around the world threatened to provoke a wider collapse in international securities markets if the fund went under, so the U.S. Federal Reserve stepped in to maintain order. Wishing to avoid the precedent of a government bailout of a hedge fund and the moral hazard it could subsequently encourage, the Fed invited every major investment bank on Wall Street to an emergency meeting in New York and dictated the terms of the $3.625 billion bailout that would preserve market liquidity. The Fed convinced Bankers Trust, Barclays, Chase, Credit Suisse First Boston, Deutsche Bank, Goldman Sachs, Merrill Lynch, J.P. Morgan, Morgan Stanley, Salomon Smith Barney, and UBS – many of whom were investors in the fund – to contribute $300 million apiece, with $125 million coming from Société Générale and $100 million from Lehman Brothers and Paribas. Eventually the market crisis passed, and each bank managed to liquidate its position at a slight profit. Only one bank contacted by the Fed refused to join the syndicate and share the burden in the name of preserving market integrity.

That bank was Bear Stearns.

Bear’s dominant trading position in bonds and derivatives had won it the profitable business of acting as a settlement house for nearly all of LTCM’s trading in those markets. On September 22, 1998, just days before the Fed-organized bailout, Bear put the final nail in the LTCM coffin by calling in a short-term debt in the amount of $500 million in an attempt to limit its own exposure to the failing hedge fund, rendering it insolvent in the process. Ever the maverick in investment banking circles, Bear stubbornly refused to contribute to the eventual buyout, even in the face of a potentially apocalyptic market crash and despite the millions in profits it had earned as LTCM’s prime broker. In typical Bear fashion, James Cayne ignored the howls from other banks that failure to preserve confidence in the markets through a bailout would bring them all down in flames, famously growling through a chewed cigar as the Fed solicited contributions for the emergency financing, “Don’t go alphabetically if you want this to work.”

Market analysts were nearly unanimous in describing the lessons learned from LTCM’s implosion; in effect, the fund’s profound leverage had placed it in such a precarious position that it could not wait for its positions to turn profitable. While its trades were sound in principal, LTCM’s predicted price convergence was not realized until long after its equity had been wiped out completely. A less leveraged firm, they explained, might have realized lower profits than the 40% annual return LTCM had offered investors up until the 1998 crisis, but could have weathered the storm once the market turned against it. In the words of economist John Maynard Keynes, the market had remained irrational longer than LTCM could remain solvent. The crisis further illustrated the importance not merely of liquidity but of perception in the less regulated derivatives markets. Once LTCM’s ability to meet its obligations was called into question, its demise became inevitable, as it could no longer find counterparties with whom to trade and from whom it could borrow to continue operating.

The thornier question of the Fed’s role in bailing out an overly aggressive investment fund in the name of market stability remained unresolved, despite the Fed’s insistence on private funding for the actual buyout. Though impossible to foresee at the time, the issue would be revisited anew less than ten years later, and it would haunt Bear Stearns. With negative publicity from Bear’s $38.5 million settlement with the SEC regarding charges that it had ignored fraudulent behavior by a client for whom it cleared trades and LTCM’s collapse behind it, Bear Stearns continued to grow under Cayne’s leadership, with its stock price appreciating some 600% from his assumption of control in 1993 until 2008. However, a rapid-fire sequence of negative events began to unfurl in the summer of 2007 that would push Bear into a liquidity crunch eerily similar to the one that felled LTCM.

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

continuum

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.