These days when volatilities are exploding on the upside, it may seem odd to talk about zero volatilities and negative volatility. Zero volatility is theoretically possible, but never actually observed in the financial markets.

In a recent training session with a group of quants someone asked us what would happen to option prices if the vols became zero? If the volatility is exactly zero then option prices cannot be calculated as becomes infinity (with vol as zero) and in the Black-Scholes formula, explodes. Option prices aside, what does zero volatility actually signify?

If volatility is taken to mean dispersion of asset prices (around a mean) then volatility, theoretically speaking, is bounded by zero on the downside. Zero vols imply, zero standard deviation, or no dispersion in the asset prices around the drift. Zero volatility means as asset is degenerate. Degeneracy is mathematical concept and not easy to relate in market terms.

Ordinarily, zero volatility would mean that a Brownian motion is no longer stochastic and will move in the direction of its drift. However, zero volatility has interesting (theoretical) consequences for bounded assets such as short rates or interest rate futures. This is where the concept of degeneracy comes in.

Let’s look at example of a degenerate asset*. Eurodollar futures are capped at 100.00. ED futures cannot go above 100.00, and hence that is the upper bound. ED futures trading at 100.00 means that short term US Dollar interest rates are zero. Take a hypothetical scenario. Let’s say that the current sub-prime and the credit crisis worsens to such an extent that its doomsday all around and the Fed has lowered rates practically to zero. Thus the Eurodollar futures are trading at 100.00 which imply short term dollar rates are zero. Now, at 100.00 what is the price of a Eurodollar futures at the money (ATM) call option, struck at 100.00 (i.e. the spot is 100.00, the strike is also 100.00 and the spot cannot go beyond 100.00 on the upside)? Your answer will immediately be zero. Since the markets cannot go over 100.00 the ATM calls on ED futures struck at 100.00 should be zero, using the call payoff function. Now what is the value of the ATM put option struck at 100.00 for the ED futures? This is not an easy question to answer.

Applying the put-call parity we can say that:

**Call – Put = Spot (underlying) + Discounted Strike**

Therefore, since by construction the rates are zero (or near zero) the value of the put is also zero. Therefore, at 100.00 both the ATM calls and puts on ED futures will be zero.

This argument, though mathematically true and feasible, cannot hold in practice. In fact, a situation such as this will not arise in real life. A market stuck at limits (like ED futures at 100.00) is degenerate because the volatility is zero; in fact, a market will never reach limits, like in this case (or in the case when a stock price becomes zero) because volatility near the limit will be so low that it will take forever for the market to reach that level (thus rates will approach zero but will never be zero, or ED futures will approach 100.00 but will never be 100.00). There is also another argument, more from a trading point of view, which will show why such a situation cannot happen.

This is an interesting theoretical result, which shows that if the asset is bounded then at limits the put-call parity breaks down for degenerate markets (i.e. zero vol). However, for assets in general a zero vol implies that if we assume a model of geometric brownian motion for assets, then an asset will grow forever at its expected rate of return; all forward rates (values) will be realized and the markets will be a one way bet. This can be seen from the equation:

Again, such a situation cannot happen in real life, and hence the notion of zero vol is at best a theoretical concept with some interesting results.

* * See the excellent explanation of this problem in Taleb’s Dynamic Hedging. *

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