9.2.2 Example: Holdings Remapping of Interest Rate Caps

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.

Exhibit 9.7: Contracts comprising the portfolio in the example. All contracts are interest-rate caps on 3-month Euribor. They settle quarterly in arrears.

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

[9.6]

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.

Exhibit 9.8: Portfolio holdings are illustrated graphically. Individual caplets are represented with dots positioned according to the number of years to the caplet’s rate-determination date and its strike rate. The size of each dot corresponds to notional amount. Caplets are aggregated into rectangular buckets, as demarked by horizontal and vertical lines.

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.

Exhibit 9.9: Remapped portfolio holdings are illustrated graphically. Proxy caplets for each bucket are represented with dots positioned according to the number of years to the caplet’s rate-determination date and its strike rate. The size of each dot corresponds to notional amount. The upper left bucket has no proxy caplet. All caplets within that bucket were discarded for having essentially no market value.

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

[9.7]

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.

Exhibit 9.10: A scatter diagram illustrates that the particular holdings remapping of our example provides an excellent approximation for ¹P.