### 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.