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Thursday, June 13, 2013

Fungibility and Option Trading

Training of trading clerks in a derivative house starts with a class about fungibility. “Fungible: being of such a nature that one part or quantity may be replaced by another equal part or quantity in the satisfaction of an obligation.” (Webster Dictionary). Thus futures contracts are fungible with corresponding cash products, and when options are involved, multitudes of synthetic products can be created that are fungible with underling cash: synthetic long and synthetic short, reversals and conversions, boxes, and jelly rolls.  All these are derived from the pretty rigorous and amazing relationship of call-put parity.

Trading is a natural process to realize the fungibility. Through trading rather than government mandate, gold prices in London and in New York will naturally converge at any instance, because gold in London is fungible with gold in New York; ETF prices will converge with the theoretical prices of the basket, because the ETF is such defined. Such rigorously fungible cases are less interesting. In derivative world, things are “kind of” fungible, but not rigorously so (except for the call-put parity).  Such less rigorous fungibility, which I coin the term "quasi-fungibility", offers opportunities as well as perils in trading.

Delta of one options are "kind of" fungible to the stocks, due to the fact that there are other important factors – especially the volatility -- that shape the value of an option.  Delta hedging an option thus is similar to pair-trading between the deltas of an option with its quasi-fungible underlying stocks.  Vega between different strikes and expiries with options of the same underlying can be aggregated (or trade against each other), and consolidated into a Vega book, under the assumption that Vega are fungible.

Fungibility originates from a linear relationship -- and quasi-fungibility from an approximated linear relationship, typically the result of a Taylor expansion of a more complex function.  The call-put parity is a perfect linear relationship, thus conversion/reversals can be priced quite tightly in the market.  For options, no matter what kind of pricing models are used, the pricing function is nonlinear.  Such a function, when linearized along the stock price and the volatility changes, forms the foundation of volatility trading.  Delta-hedging, trading vertical and calendar spreads, building a complex delta-neutral Vega book, are all taking advantage of the quasi-fungibility of the delta and Vega dimensions in options and equities.

Now let's look at the perils of quasi-fungibility.  The nonlinear shapes of the option value and volatility surface cause all kind of troubles for fungibility that is based on approximated linearity.  As I discussed in the previous Newsletter, Gamma changes rapidly when an option approaches expiration, and profitability of delta hedging is sensitively path-dependent.  Such characteristics cause the delta to be less accurate, and offsetting Vega of different expiries to be less exact.  This is a systematic risk that needs to be paid close attention to.  On the other hand, Gamma gives us the convexity that can be a lot of leverage and opportunities, beyond the simple fungible delta and Vega trading.

Then there is the peril of statistics.  Armed with a few data points and a linear regression function in Excel, anyone can create a linear relationship (or correlation), and thus some kind of "quasi-fungibility" to trade on.  Thus there are many pair-trading strategies -- looking at historical data of INTC vs AMD, HPQ vs DELL, gold vs equity market, etc., probably one can come up with some “strong” statistics, then many would pile on the similar trades on such statistical correlation.  Unfortunately, everyone can see the history, nobody can see the future.  Most of statistics that are based on historical data are tenuous at best.  Similarly, quasi-fungible idea based on "beta" carry the similar peril, since “beta” is a statistical measurement based on historical data.

It is important to distinguish 2 kinds of relationships:  constructive relationship and statistical relationship.  The first kind, the relationship is defined by some construction.  It can be simple (like the futures vs spot, ETF vs basket or benchmark), or intuitive, like the call-put parity, or complex, like option pricings.  Vol trading, based on the fungible delta and Vega from the constructive relationship of option pricing, faces the nonlinearity perils.  The second kind, statistics relationship, is too easy to be "discovered", even the ones built upon strong historical statistics can be easily broken by any fundamental factors.

As a practitioner, either as a trader trading on strategies that are based on certain quasi-fungibility, or a risk manager watching a complex book with many dimensions of fungible assumptions, first of all, one should be always very skeptical of relationships or correlations that are based on the statistics (remember Mark Twain’s famous saying: "There are 3 kinds of lies: lies, damned lies, and statistics.")  Secondly, for strong constructive relationship, we should be vigilant of any linear approximations that are being made, and keep a watchful eye on the pitfalls of nonlinearity.

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