9.2.2 Example: Holdings Remapping of Interest Rate Caps
Cárdenas et al. (1999) dramatically expanded the scope of holdings remappings, showing how techniques applicable to cash flows could be extended to a variety of instruments, including interest rate caps, floors, and swaptions. Consider caps. We assume a 1-dayvalue-at-risk horizon. Today is March 23, 2000, and 3-month Euribor is 3.767%. A portfolio holds EUR interest rate caps. All are linked to 3-month Euribor, settle quarterly and make payments in arrears. Contracts are detailed in Exhibit 9.7.
Assets represent individual caplets with a notional amount of 1MM EUR.1 Each asset is uniquely specified by a rate-determination date (settlement occurs 3 months later) and its strike rate. We determine holdings ω and define
Holdings are summarized graphically in Exhibit 9.8. Each caplet is represented with a dot positioned according to its rate-determination date and strike rate. Sizes of dots correspond to notional amounts. As indicated, caplets are aggregated into rectangular buckets according to both rate-determination date and strike rate.
We apply a holdings remapping as follows. Caplets with essentially no market value—those that are far out-of-the-money and whose rate-determination date is imminent—are discarded. Within each rectangular bucket, the total market value, delta, and vega of remaining caplets are calculated. The remapping represents those caplets with a single proxy caplet that has the same market value, delta, and vega. The process is repeated for each bucket.
We don’t attempt to match the net gamma of caplets within a bucket. This is because we have only three degrees of freedom to work with—rate-determination date, strike rate, and notional amount. Because vega is closely related to gamma, this is not a significant problem.
Determining a single caplet with a desired market value, delta, and vega is a nonlinear problem. We solve it for each bucket using Newton’s method with line searches. Because there is no guarantee that this will converge, it is advisable to start with seed values for the rate-determination date, strike rate, and notional amount as close to the solution as possible. For this purpose, we use as seed values:
- the weighted average rate-determination date,
- the weighted average strike rate, and
- the sum notional amount
of all caplets in a bucket. Caplet market values are used as weights in the two averages.
Based upon the above analysis, remapped holdings are determined as indicated in Exhibit 9.9. With this holdings remapping, we have replaced 80 caplets with just 13 proxy caplets. If our portfolio held more caps, economies would be greater.
Let’s assess how accurately our remapped portfolio value approximates 1P. To facilitate the analysis, we first specify key factors. We define 1S = φ(1R), where 1R represents forward rates and implied volatilities. A joint distribution for 1R is constructed from historical data using techniques from Chapter 7. Schematically, we have
Note that both 1P and depend upon the same key vector 1R. We generate 1,000 pseudorandom realizations 1r[k] for 1R. Corresponding realizations 1p[k] and are calculated. Pairs (1p[k],) are plotted to form the scatter diagram in Exhibit 9.10. All points fall near a line passing at a 45° angle through the center of the graph. This indicates that is an excellent approximation for 1P.