Degeneracy in the Financial Markets
31st August 2014
Degeneracy is a concept in mathematical physics. In quantum mechanics degeneracy refers to the fact that more than one quantum state has the same energy level. Mathematically speaking, degeneracy implies identical roots of a polynomial equation. For example, if we find the roots of the equation we will find that it is 2. In the simplest possible sense, degeneracy arises when the two possible values of a variable, , are equal.
There is an alternative conceptual meaning of degeneracy in pure mathematics, and one that is perhaps intuitively more close to the notion of degeneracy in quantitative finance. In math, degeneracy arises when a key measurable variable associated with an object approaches zero such that the object changes are form or class. For example, if we take a circle and make its radius go to zero then it will become a point. Hence, a point is a degenerate case of a circle. Similarly, if we make the eccentricity of an ellipse approach zero, then the ellipse will collapse into a circle. Hence a circle is a degenerate case of an ellipse.
When do we say that a financial asset is degenerate? When the volatility of an asset is zero, i.e. the asset has no volatility, then that asset is said to be degenerate. Of course, in real life every financial asset, be it a stock, a bond, an interest rate futures contract or a foreign exchange pair, has some volatility, even if it is very small. If today's price of a stock is exactly the same íV up to the second decimal place, i.e. cents íV as yesterday's price and if tomorrow's price turns out to be exactly the same as today's price then that stock is degenerate. However, that almost never ever happens. Even if the change is in cents, or in the tiniest permissible dollar or unit variation, there's always going to be a price difference between two period's prices.
However, at times markets can really become degenerate. And, in such cases, theoretical rules of options market usually break down.
Consider the Eurodollar futures market (the dollar interest rate futures). These futures contracts are capped at 100 and cannot go (trade) any higher. Say, the interest rates are at an all-time low (a situation somewhat similar to what currently exists in the G7 world) and so much so that they have hit rock bottom, i.e. almost zero percent. This means that the Eurodollar futures contract would trade almost at 100. Now, what happens when the futures contract hit 100? They cannot go any higher. Thus, the price of a 100 strike Eurodollar futures call option will be zero.
However, given the put-call parity rules in options theory, how do we estimate the price of a 100 strike Eurodollar futures put option when the futures contract is trading at 100. A deceptively easy íV and yet, both mathematically wrong and practically inconsistent íV answer would be zero.
Just input 100 for the asset and the strike price as well as zero for the value of the call (with discount rates making no impact) in the put-call parity equation and get a zero value for the put option. But this isn't the correct answer. The put-call parity relationship actually breaks down at 100 (at the boundary).
In fact, near or at 100, the Eurodollar call option itself will not trade at zero (otherwise, traders can partake in a free lunch). Both mathematically as well as from an options trading point of view the values of 100 strike Eurodollar calls and puts at (or very nearly at) the 100 level will be different from zero. For those interested in a more market related explanation of degeneracy we would like to suggest Nassim Taleb's Dynamic Hedging. Even setting mathematics aside, from a pure market based argument, one can show that a Eurodollar futures market trading at 100 will become degenerate and cause put-call parity to break down.
Taleb, Nassim, Dynamic Hedging, John Wiley & Sons, 1997
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