Sunday, July 6, 2008

Revisiting Continuous-time Finance (Part 1 of 2)

If classic is the description of a work that stands the test of time, Robert Merton’s 700-page Continuous-time Finance is an apt example of unclassic. A mere 18 years after its publication, the book has fallen by the wayside, a colossal Ozymandias statue of ideas that lies in the ruins.

At the time of its publication in 1990, the book was meant as a celebration of the rise of “modern finance”. In a laudatory spirit, Merton elaborates, reworks and chronicles the ideas whose sum total defined the new discipline. The book’s size and heavily mathematical content is intended to show, if only unconsciously, that the foundation of modern finance was built on solid, scientific grounds.

All that is evident in Paul Samuelson’s Foreword to the book:

A great economist of an earlier generation said that, useful though economic theory is for understanding the world, no one would go to an economic theorist for advice on how to run a brewery or produce a mousetrap. Today, that sage would have to change his tune: economic principles really do apply and woe to the accountant or marketer who runs counter to economic law. Paradoxically, one of our most elegant and complex sectors of economic analysis – the modern theory of finance -- is confirmed daily by millions of statistical observations. When today's associate professor of security analysis is asked, “Young man, if you’re so smart, why ain’t you rich?”, he replies by laughing all the way to the bank or to his appointment as a high-paid consultant to Wall Street.
Yet, look closely, by which I mean actually read the book, and you would be struck by the shallowness of its content. What comes through is a Beckettian clerk who labors to formalize what has taken place around him without understanding the forces that drive the events. But Merton is no mere scribe. Like the general discipline he helped create, his formalization gives theoretical cover to the pragmatically crude action of traders. He validates and legitimizes them, making them acceptable and reputable. That is why Continuous-time Finance merits a revisit. An analysis of the current rot in the market cannot be complete without a look at the ideological cheerleaders who paved the way for it.

(This subjugation of theory to practice, the intellectual standing in awe of the ill-educated trader, is in plain view in Samuelson's foreword. Note his ideal of the relevance of modern finance: going to a “high-paid” consulting job on Wall Street, say, to Bear Stearns or Lehman. From Adam Smith to David Ricardo to Karl Marx to Samuelson. What a falling off was there.)

The pride of the place in Continuous-time Finance, as in “modern finance” itself, belongs to option valuation theory, or the contingent claims argument (CCA) in general, as Merton calls it. What is CCA? Simply this, that a riskless position must earn the riskless rate of return. In describing how he and his colleagues came up with the option valuation formula, Fischer Black wrote:

As the hedged position will be close to riskless, it should return an amount equal to the short-term interest rate on close-to-riskless securities. This one principle gives us the option formula.
The idea and the definition of hedging comes from the general accounting relation:

Assets = Liabilities + Owners' Equity


A = L + OE

If you have $100 in your pocket, either all of it is yours (OE = $100), or all of its is borrowed (L = $100), or a combination, say $40 borrowed and $60 your own so that $100 = $40 + $60. $100 in your pocket cannot have any other source.

The objective of hedging is to protect OE, the individual’s “net worth” so that its value would not change, no matter what happens in the markets. In mathematical terms, that means that the change in the value of OE must be zero:

change (OE) = 0
Since OE = A – L, for that condition to hold, we must have:
change(A) – change (L) = 0

change(A) = change(L)
The above equation is the necessary and sufficient condition for the hedge. It says that the change in the value of assets must be equal to the change in the value of liabilities. Students of finance know this condition under various names and guises, such as matching assets and liabilities or creating a riskless portfolio. The different names merely express the particular viewpoint of the person or entity engaged in the act; treasurers would speak of asset-liability matching, traders of creating a riskless portfolio.

The most consequential development here is the transformation of hedging to arbitrage. No better or more convincing example of a dialectical movement exists anywhere.

The purpose of hedging is preserving the owners’ equity. The hedger begins with an existing asset (liability) and seeks to find a liability (asset) which will offset its adverse price changes.

The purpose of arbitrage, by contrast, is profit. The arbitrageur has neither an asset nor a liability. He uses the hedge relation above to search for any two positions which will enable him to “lock in” a spread.

The difference between hedging and arbitrage, therefore, boils down to the purpose behind the trades, which translates itself to the sequence of execution of trades. When done sequentially, the act is defensive hedging. When done simultaneously, it is aggressive arbitrage. On the after-the-fact basis, an observer will only see a long and a short position.

Hedge fund managers and prop traders took the idea and ran with it. Speculative capital, capital engaged in arbitrage, was thus born.

One practical problem remained: how to create a “close to riskless” position? No one understood the significance of this question in theory or the implications of its execution in practice. It was left to hedge fund managers and prop traders and their quantitative underlings to find the answer.

They did.

I will return tomorrow with the second and final part.

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