* continued from the previous article.... *

Just as in life, so it is in the financial markets; the "zero-space" is a fascinating concept within the context of financial markets as well. Extremely close to, or exactly at zero, we observe "degenerate" assets. It is difficult to delve deeper in the analysis of degeneracy within the context of financial assets without resorting to some very tedious math, but suffice it to say, that we can define a state of degeneracy where the volatility of an asset is zero. When that happens, a lot of mathematical relationships, such as the well known put-call relationship of financial options or the equilibrium Capital Asset Pricing Model (CAPM), break down. In fact, degeneracy causes equilibrium to break down.

Mathematically speaking, a degenerate case is a limiting case where an object changes its class. Take a circle of radius ^{r} ( ^{r} can have any positive value, say, 10 cm or 10 km) and keep collapsing it, i.e. keep reducing the value of the radius. As the radius, ^{r} shrinks in value the circle becomes smaller and smaller until finally at zero value, i.e ^{r=0} the circle disappears and what is left is just a point. A **point**, therefore, is a degenerate case of a **circle** and at "zero space" one changes to the other.

Value of financial instruments cannot be less than zero. The stock price of a company can only fall up to zero but not beyond that. However, can you really have the value of a stock (share) exactly equal to zero, i.e. a perfect zero? It is highly unlikely. For most cases, the stock price can be extremely close to zero (for worthless companies), but not exactly zero. Even bankrupt companies, in liquidation process, have some residual values – howsoever, infinitesimal. Within the context of finance, an asset value of zero can correspond to a singular event.

One exception is the short term interest rates. The nominal value of short term interest rates of a country which is set by the Central Bank of that country can indeed be exactly equal to zero. This is totally in the hands of the Central Bank of a country. However, even the nominal short term interest rates cannot go below zero. Here again, just like other assets like stocks and bonds, zero is the limiting case on the downside. Nominal short term rates cannot be negative.

And around the zero value, or very close to it, financial assets, of all kinds, start to display the property of degeneracy. Mathematically speaking, a degenerate asset has zero volatility. Even interest rates display this property. And degeneracy can give rise to some interesting issues in the options market. Let’s look at short term interest rates as an example (it’s worthwhile to do so now because the short term rates in the US and Japan are almost, but not exactly equal to, zero).

Zero interest rates also correspond to 100 (one hundred) on the interest rate futures. These instruments, the interest rate futures, are bounded from above and below, meaning that their value is bounded on both sides. The most popular and heavily traded interest rate futures is the 3 month Eurodollar futures which roughly correspond to the 3 month short term USD interest rates as given in the LIBOR market.

The value of Eurodollar (ED) futures, on the upside, cannot cross 100. If the Eurodollar futures are at 100, then it means that the corresponding 3 month US Dollar interest rate (yield) – equivalently, the 3 month USD LIBOR – stands at zero (a reading of 96 on the ED futures will correspond to 4% on the short term yield and so on). With a certain adjustment, called the convexity adjustment, this yield will reflect the forward rate in the market. Since the interest rates cannot go below zero the Eurodollar futures cannot go above 100.

What happens to a 100 strike call on Eurodollar (ED) futures? What would be the price of such a call option on Eurodollars? If the strike is 100 and if the asset is at 100 (which is the upside limit) and it cannot trade any higher then the price of a 100 strike call on ED should be zero. And indeed it is. But then what should be the price of a corresponding 100 strike put option on ED futures?

Put-call parity states that a call option minus a put option is equal to the (underlying) spot minus the discounted strike. Then, by this construction, the put should trade at zero as well. Is this correct? Actually, no. Why? This is because a situation like this, where an asset which is bounded at 100 and cannot trade any higher will, in reality, never happen. The asset will approach zero and will be very close to zero but it will not be at zero exactly. One can think of such a situation – though not in a strictly mathematical sense – as an equivalent of “financial singularity”.

An asset which trades at and on the boundary, such as ED futures being at 100 (with 100 being the upside limit) is said to be degenerate. Simply speaking, degeneracy implies zero volatility. However, in or around the boundary, i.e. 100 in this case, the volatility would become so low that it would take the asset forever to reach the limiting point. In other words, the asset would never touch the boundary, but would forever hover very close to it. In a state of degeneracy the asset would begin to paste around the boundary but would never actually touch it.

In option parlance, around the value of zero – the limiting value of 100 on ED futures – the put-call parity of financial options breaks down. It’s a bit like laws of physics breaking down at “singularity”, i.e. at the creation of the universe (where “time” is bounded and is a singularity in time) or event horizon inside a black hole (where the “space” is bounded and is a singularity in space). And financial singularity and degeneracy are very closely related, one leads to the other.

Any comments and queries can
be sent through our
web-based form.

__
More on Quantitative Finance >>__

back to top