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Credit Hedging and PnL Volatility

In 1959, Gerard Debreu wrote the classic treatise The Theory of Value: An Axiomatic Analysis of Economic Equilibrium. Debreu approached economics in a way that substituted the classical Newtonian calculus with differential geometry with its notion of topology and convexity. This provided a rigorous tool for analyzing markets that do not depend on rigid standards of measurement and quantification.

This led to financial engineering and what is referred to as the arbitrage pricing revolution. The financial economic theory underlying financial engineering is that every asset can be replicated by a suitably chosen portfolio of options written on some index of assets. See Fred D. Arditti & John Kose, Spanning the State Space with Options, 15 J. FIN. & QUANTITATIVE ANALYSIS (1980) (proves this technical result). Forwards and options are the basic building blocks of not only other derivatives, but also all assets because any asset—not just a derivative—can be synthesized by constructing a portfolio of the appropriate forwards and options.

By their very definition, derivatives are financial instruments whose payoffs depend on prices of underlying commodities or cash instruments. Thus, by trading in derivatives, economic agents can manage the risk of underlying market price volatility by shifting all or part of that risk to counter-parties who are more willing to assume or bear that risk. Of course, economic agents may also trade in derivatives to speculate on the risk of underlying market price volatility by shifting all or part of that risk from counter-parties who are less willing to assume or bear that risk.

Cash assets such as bonds and stocks help to allocate capital. The primary markets for primitive assets, such as bonds and stock, allow households to save or invest and permit corporations to raise capital. Secondary market trading is the result of changes in beliefs concerning the future states of the world, tastes for risk or investor patience, or shocks to investor balance sheets.

Derivatives do not allocate capital. They allocate risk associated with volatility in cash asset prices. Derivatives permit individuals and organizations to insure against fluctuations in the prices of primitive assets in exchange for paid premium. This exchange makes derivatives is a zero-sum game between the counter-parties. Derivatives permit capital market participants to hedge volatility in those underlying financial markets and facilitate speculative views on cash asset prices. Derivatives benefit the entire financial system by creating more ‘complete’ markets—offering markets to investors and traders risk and return patterns that previously were either unavailable or too costly.

Asset markets are complete if trading on asset markets can generate any conceivable pattern of payoffs over time and under alternative future scenarios. Households can insure against all financial risks by trading on complete asset markets. Indeed, under certain technical conditions, sufficiently frequent trading of a few judiciously chosen assets can substitute for assets that do not actually exist in the marketplace. See Darrell Duffie & Chi-Fu Huang, Implementing Arrow-Debreu Equilibria by Continuous Trading of Few Long-Lived Securities, 53 ECONOMETRICA 1337 (1985) (proving this technical result).

It is clear that firms and households face asset markets that are very incomplete. There is little an average family can do to insure against reversals in their employers’ fortunes, real estate prices, national income levels, employer provided health insurance coverages, and inflation. The same is true of nonfinancial businesses that remain uninsured with respect to macroeconomic and sector-specific shocks, logistics problems, or a firm supplier bottlenecks. Because of market incompleteness, the observed volume and price volatility of cash and derivative asset trading is much greater than what can be explained by differences in preferences, income, or capabilities.

Even financial firms cannot completely insure their risks. Credit cannot be hedged in the same way that a call option on a stock can be hedged. Option hedges provide identical cashflows to underlying equity in every state of the world. Credit swaps do not and cannot. The default time of the bond introduces cashflow volatility for the swap. If the counterparty defaults the instant after contracts are finalized, the hedger gains big upside because it has paid almost no premium. If the counterparty does not default, the hedger posts a loss. Between these two edge cases, PnL varies depending on precisely when the counterparty defaults. This is the way all insurance-type contracts work.

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