Absurd Results in Quantitative Finance
August 13, 2006
There are many paradoxes and absurdities in quantitative finance, some truly puzzling some arising out of some mathematical incongruity with the practical world. Below we list two such absurd results which many of us may have come across either through reading or in actual practice.
Absurdity # 1
uration of Inverse Floating Rate Notes may exceed their Term to Maturity!
Inverse floaters (Inverse FRNs) are contracts where the contract rate (coupon rate) moves inversely to the reference rate, generally the LIBOR rate. As LIBOR increases the coupon rate decreases and vice versa.
Vanilla floating rate notes (FRNs), by the obvious nature of their construction, have extremely small duration. However, inverse FRNs, though they are floating rate notes have a duration that in fact exceeds by far their term to maturity. This absurd result was first observed by the market professionals more than a decade and half ago when inverse floaters became a part of the family of collateralized mortgage obligations (CMOs). And apparently the market participants were completely thrown off guard by this result.
The above result can be mathematically proven using the simple duration formula and the formula to construct an inverse FRN. We shall talk about it in greater detail in these columns later.
Zero Value of both Calls and Puts for a bounded Interest Rate Futures market!
This example, discussed by many veteran options trader during the mid nineties, and quoted succinctly and brilliantly (through a slightly different example in Nassim Taleb's Dynamic Hedging), is a rare example of what we call degenerate volatility. In practical traders' lingo degenerate volatility signifies a market that does not move, i.e. volatility of the underlying is zero (some quantitative traders call the underlying asset as degenerate rather than implying that the vol is zero). If the vol is zero then the asset will go nowhere and be stuck at a point. Or if the asset is bounded from the top, some sort of a band or a permanent upper bound, and if the asset is trading very, very close to that bound then the volatility could be such that market may take an eternity to hit that bound.
Consider the Euroyen futures market (the Japanese Yen interest rate futures). These futures contracts are bounded from the above at 100. This means a Euroyen (or Eurodollar for that matter) futures contract cannot go beyond 100, that is, the limit. A value of 100 on the Euroyen futures means that the underlying short term interest rate is zero. In the mid to late nineties the Yen interest rates were hovering very close to zero. And the yen interest rate futures were trading very close to 100.
If the market is capped at 100 and cannot go any further then a call option struck at 100 will have a zero value. And by the mathematical relationship of Put-Call parity the value of a put option struck at 100 should also be zero.
Actually, the above is a nonsensical result and does not hold in practice. Mathematically the problem gets resolved by the concept of degenerate volatility. Even in practice the action of traders who would be selling calls very near the bound of 100 will make the call trade at a value above zero. One can intuitive understand this result as well without resorting to any math.
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