###### 9.2.3 Designing Holdings Remappings

Our last example hints at the tremendous potential of holdings remappings. They:

- are applicable to a wide variety of instruments;
- can replace a portfolio mapping function with another that requires far less computations to evaluate; and
- offer an excellent approximation for a portfolio’s value
^{1}*P*.

We have much flexibility in how we design a holdings remapping, and it is worth exploring several options to find which works best in any given situation. Decisions include:

- how to bucket assets,
- what assets to employ as proxy assets, and
- what characteristics of bucketed assets to match with proxy assets.

Bucketing can be performed in several dimensions, depending upon the assets. In our fixed cash-flow example, we bucketed according to a single dimension: maturity. In our cap example, we bucketed according to two: rate-determination date and strike. For a vanilla swaption portfolio, three dimensions would be appropriate: expiration, strike, and tenor of the underlying swap.

You can also vary the size of buckets. With options, gamma tends to be most extreme for contracts that are close to expiration and at-the-money. For this reason, it may be advantageous to use smaller buckets for options close to expiration and for those that are near-the-money. In our cap example, we varied bucket size by expiration, but not by strike. If options will expire during the value-at-risk horizon, it may be appropriate to not bucket them. Simply include such holdings, unaltered in .

Consider ignoring assets that contribute essentially nothing to risk or market value. In our cap example, we discarded from the analysis those caplets that had essentially no market value. In many cases, this will leave some buckets empty, which will further reduce the active holdings of .

As our fixed cash-flow example illustrates, we can employ proxy assets in various ways. In the second approach of that example, we used cash flows at either end of each bucket as proxy assets. We could vary the size of each, allowing us two degrees of freedom to match both the market value and duration of bucketed assets. In the third approach of that example, we employed a single cash flow as a proxy asset for each bucket. We could select both its maturity and its size. This also afforded two degrees of freedom, allowing us again to match both total market value and duration.

Various characteristics of bucketed assets can be matched with proxy assets. It is desirable that market value be matched. Various sensitivities may also be matched: duration, convexity, delta, gamma, vega, etc. Since the purpose of a remapping is to approximate ^{1}*P*, it makes sense to match characteristics reflective of the portfolio at time 1. Consider time-1 sensitivities defined by the first and second partial derivatives2

[9.8]

which are analogous to time-0 sensitivities delta and gamma. Time-1 market value, evaluated perhaps at ^{0}*E*(^{1}** R**), might also be matched in lieu of time-0 market value.

In general, determining from **ω** entails solving a system of equations for each bucket. These may be linear or non linear. In either case, it is possible that a system will have no solution, but this is rarely an issue. If a system is nonlinear, it can be solved using Newton’s method with line searches. To ensure convergence to a reasonable solution, it is important to identify seed values as close to the solution as possible. This is illustrated by our choice of seed values in our cap example.

###### Exercises

In our discussion of holdings remappings, we illustrated three approaches for fixed cash-flows and one for interest-rate caps. Of the three fixed cash-flow approaches, which is most analogous to the cap approach?

Solution

Consider a 1-dayvalue-at-risk horizon with 2^{nd}-day valuation. A Canadian firm holds a portfolio of call USD put CAD options. The options’ prices, deltas, and vegas can be determined with the Garman and Kohlhagen (1983) option-pricing model as:

[9.9]

[9.10]

[9.11]

where:

[9.12]

[9.13]

and

*n*= notional amount (USD);*s*= USD/CAD exchange rate;*x*= strike rate;*y*= time to expiration in years;*r*= continuously compounded CAD interest rate for maturity_{CAD}*y*;*r*= continuously compounded USD interest rate for maturity_{USD}*y*;*v*= implied volatility for strike*x*and maturity*y*;- Φ = CDF of the standard normal distribution;
- ϕ = PDF of the standard normal distribution.

Define

[9.14]

where assets represent individual options with notional amount USD 1MM. A holdings remapping

[9.15]

is proposed. Options with essentially 0 market value are discarded from the analysis. Remaining options are aggregated into buckets for maturities 0 - 1 month, 1 - 3 months, 3 - 6 months and 6 - 12 months and strikes spaced at .025 increments.

For each bucket, the total value, delta and vega are determined. The options are represented with a single option having the same value, delta, and vega. The process is repeated for each bucket.

Consider the bucket for maturities 3 - 6 months and strikes .6250 - .6500. It contains the contracts indicated in Exhibit 9.11. In this exercise, you will represent the options of this bucket with a single option that matches their total value, delta, and vega.

- Current values for CAD interest rates
*r*at maturities_{CAD}*y*= .25 and*y*= .50 are .0393 and .0385. Use linear interpolation to obtain an expression for*r*of the form_{CAD}[9.16]

for any maturity

*y*between .25 and .50. - Current values for USD interest rates
*r*at maturities_{USD}*y*= .25 and*y*= .50 are .0352 and .0354. Use linear interpolation to obtain an expression for*r*of the form_{USD}[9.17]

for any maturities

*y*between .25 and .50. - Implied volatility
*v*varies with both strike*x*and expiration*y*. Current implied volatilities are indicated in Exhibit 9.12.Exhibit 9.12: Current implied volatilities*v*by strike*x*and expiration*y*.Use quadratic interpolation to obtain an expression for

*v*of the form[9.18]

for values

*x*between .625 and .650 and values*y*between .25 and .50. - The current exchange rate
*s*is .6414 USD/CAD. Substitute this and your results for items (a), (b), and (c) into [9.9] through [9.11] to obtain formulas for an option’s value, delta, and vega that are applicable across a range of strikes*x*and maturities*y*. - Use your results from item (d) to determine the prices, deltas, and vegas of the options in Exhibit 9.11. Sum your results to obtain the total price, total delta, and total vega.
- Calculate the weighted average strike, weighted average expiration, and total notional amount of the options in Exhibit 9.11. Use the options’ values as weights in the first two items.
- Employ Newton’s method with line searches and your results from item (d) to find the strike, maturity, and notional amount of a single option whose value, delta, and vega match the total value, total delta, and total vega of the options of Exhibit 9.11. In your search routine, use your results from item (f) as seed values.