A flexi-cap gives the holder the right to cap some but not all the period rates at a specified strike level over a given time span. For instance, a flexi-cap may give its holder the right to cap at 4% (at most) three out of the seven consecutive quarterly LIBOR fixings over the next two years. Flexi-caps come in two varieties; auto-flex, where all caplets with in-the-money fixings are automatically exercised until there are no more caplets to exercise and the contract has become worthless; and chooser-flex (or 'u-choose'), where the holder can choose on any fixing day whether to cap the corresponding rate until there are no more caplets to exercise and the contract has become worthless.
Since flexible caps offer less protection than ordinary caps they are less expensive. For this reason they will be preferred by users who do not expect to need to cap all the period rates covered by the contract. Flexi-caps may also be used to "soften the blow" from a rise in interest rates by delaying the investor's exposure to higher rates for a certain period of time after rates have gone up.
The optimal number of caplets to include in the flexi-cap as well as the choice between auto- and chooser-flex caps will depend on the investor's risk preferences and view on future interest rates.
If the number of caplets in the contract equals the number of periods in the time span covered then both the auto- and chooser-flex cap are equivalent to the ordinary cap covering the same time span. Also, because the holder of a chooser-flex cap is able to reproduce the corresponding auto-flex cap (but is not restricted to this choice of exercise strategy) it follows that a chooser-flex cap must always be worth at least as much as the corresponding auto- ex cap. By the same line of reasoning it follows that the value of a chooser-flex cap with N caplets is always greater than or equal to the sum of the values of the N most expensive caplets. Chooser-flex caps are actually American style securities and this, as usual, necessitates an optimization over exercise strategies for their pricing. However, unlike usual American style securities there are several interdependent exercise choices to be made so the choice of the exercise strategy is more involved. Still, chooser-flex (and auto-flex) caps may be priced using an extension of the well-known lattice or (recombining) tree methodology (see Pedersen and Sidenius, Journal of Derivatives 1998). The pricing method applies equally well to all of the stochastic interest rate models which can be approximated by lattices in the short rate. Alternatively, pricing can be done using Monte-Carlo simulation. Market observations on August 6 a 10 year auto-flex cap with a strike of 5% on 3 months FRF PIBOR traded at 597 bp. On this auto-flex the first 20 in-the-money caplets (out of the total of 39 caplets) are exercised automatically while remaining caplets are forfeited. On September 5 an 8 year auto-flex cap with a strike of 6.5% on 6 months DEM LIBOR traded at 215 bp. On this auto-flex the first 7 in-the-money caplets (out of the total of 15 caplets) are exercised while remaining caplets are forfeited.
To price these two trades we calibrated lattices for the Black-Karasinski (BK) model and the Black-Derman-Toy (BDT) model to market cap prices. The BDT model produced flexi-cap prices agreeing with the trade prices to within 1% while the BK model prices were a few percent o. For comparison we calibrated LIBOR Market Models (BGM) with 1, 2, 3, 5, 7, and 10 factors to the FRF market. Figure 1 contains our results. It is interesting to note that the prices computed using LIBOR market models converge to some level as more factors are added. This can be explained by noting that models with few factors exaggerate short-term correlation between rates. When such a model is fitted to (normal) market prices of caps and swaptions it leads to a rather rigid term structure evolution where scenarios with greatly varying rate fixings are heavily suppressed. These scenarios are precisely those for which the flexi-cap generates the greatest pay compared to a standard cap. As the suppression is lifted when more factors are added, the flexi-cap price increases towards its "true" level in the complete model with a maximal number of factors.
If the flexible dynamics embodied by many-factor models is more realistic than the rigid few-factor models, then the market misprices flexi-caps and there is an arbitrage opportunity.
This week's Learning Curve was written by Morten Bjerregaard Pedersen, from SimCorpand Jakob Sidenius, from Skandinaviska Enskilda Banken.
