Pricing a Callable CMS Spread Note with FINCAD XL v10
Overview
A constant maturity swap (CMS) spread note is a derivative with a payoff based on the difference of two swap rates of specific maturities. For example, a CMS spread note might pay quarterly coupons based on the difference between quarterly fixings of the 10-year and 5-year semi-annual swap rates. The coupons of such a structure depend on the slope of the yield curve; the Note would therefore be traded by parties who wish to take a view on future relative changes in different parts of the yield curve. The steeper the yield curve, the greater the coupon - giving rise to the term "steepener" for certain CMS spread instruments.
Since the beginning of 2006, the yield curve in both the US and Europe has flattened dramatically and has thus led to a rapidly shrinking CMS spread. This has caused the classic steepener to under-perform and investors who had positions in CMS spread instruments experienced large losses due to a significantly lower coupon. For these reasons, such products have become less attractive, and so their level of liquidity is thus also decreasing. However, mark-to-market valuations still need to be performed for the many outstanding products on investors' books.
A typical CMS spread product may have coupons depending on CMS rates with different maturities and multiplicative factors. Rate margins, caps and floors are also common, as are embedded Bermudan-style call and/or put options. FINCAD has the capacity to price CMS spread notes with all these features. The CMS functions in FINCAD XL v10 can be used to price instruments with the following generalized coupon structure (j is an index which runs over the coupons):
By appropriately choosing the specific parameters of this generalized coupon, FINCAD XL v10 can thus be used to value the following instruments:
| Instrument | Typical CMS Coupon |
| (Callable) CMS note | max(CMSx + m, 0) |
| (Callable) CMS spread note | max(CMSx - bCMSy + m, 0) |
| (Callable) CMS spread cap | max(CMSx - bCMSy - K, 0) |
| (Callable) CMS spread floor | max(K - CMSx + bCMSy, 0) |
| (Callable) CMS inverse floater | max(m - aCMSx , G) |
The FINCAD approach to pricing CMS derivatives is to use a Monte Carlo simulation of the Libor Market Model (LMM), with extensions to allow for non-lognormal evolution of the underlying Libor forward rates. The LMM model must be properly calibrated and FINCAD provides a general framework by which to do this, allowing the user to calibrate to the smile of caplets and/or to at-the-money swaption prices.
» To download the latest trial version of FINCAD Analytics, contact a FINCAD Representative.
Valuation
FINCAD XL v10 contains a workbook to price a CMS Spread Note. The workbook is comprised of 14 worksheets and we will work it through using the following example of a callable CMS spread note:
| Nominal Amount | EUR 100,000,000.00 |
| Trade Date | 31st October 2006 |
| Start Date | 13th Rebruary 2006 |
| Maturity Date | 14th Rebruary 2011 |
| Payments | Quarterly payments Year 1 : 7.60% Year 2 : 100% of CMS10 - CMS2 Year 3-5 : 80% of CMS 10 - CMS 2 Additional agreements:
|
| Interest Payment Dates | 13th May, August, November, February |
| Business Days | Target, Zurich, modified following, adjusted |
| Fixing Dates | 2 Business days prior to start date of interest period |
Step 1: Update the Curve and Holidays worksheets with data corresponding to the specific jurisdiction/currency.
Step 2: On the Main worksheet, input the general details of the CMS spread note. The main choice to make is which local volatility model to use to value the instrument. Here, we will the displaced diffusion model (that is, we assume the underlying forward rates to follow a displaced diffusion process). Since the Note is callable, we must also choose regression variables and basis functions - the standard choice is to take IR curve moments for our regression variables and polynomial basis functions (see the Math Reference document for details of this). We assume that both the 10- and 2-year swap rates are for semi-annual swaps, and that both the accrual method for coupon periods and that for rate terms is 30/360 (ISDA). Finally, we choose an exponential interpolation method for the discount factor curve (calculated in Step 1). The resulting inputs are as follows (the value for the local volatility parameter is found through the calibration procedure described in Step 4 below):
Step 3: In the Coupons worksheet, input the coupon details and specify the terms of the coupon rate in the coupon periods table. The coupon details are as follows:
Note that the terminating date in the above screenshot is unadjusted.
The CMS rates can be set in-advance or in-arrears. For the case at hand, they are set in-advance, the rate effective/reset dates being 2 business days prior to the coupon effective dates. To generate the coupon periods, the workbook uses the FINCAD function aaDateGen, cycling off the Maturity Date. The rate effective/reset dates are then found by adjusting the coupon effective dates backward by 2 business days, using the FINCAD function aaDateAdjust. The following coupon periods table shows the results of this set-up procedure (white cells are the user defined fields of the coupon table, corresponding to the terms of the above example):
The coupon details match the above example: the first two coupons (those left in Year 1) have a fixed payment of 7.60%. The coupons of Year 2 are equal to CMS10-CMS2+0.50%, and the remaining coupons are equal to 0.8*(CMS10-CMS2)+0.50%. There is a floor of 0% and a cap of 6%, and the instrument is callable at par on the February coupon terminating dates.
Step 4: A recommended calibration procedure using the FINCAD functions is to (1) calibrate the volatility parameters a, b, c, and d to caplet prices; then (2) to hold these parameters fixed while calibrating the instantaneous correlation parameters β1 and β2 to swaption prices. Various sheets are provided in the workbook to perform this calibration. Adanced users may prefer to fix the correlation parameters β1 and β2 from historical data and/or experience, and to calibrate the volatility parameters to swaption prices.
The results of a typical calibration are shown below.
Step 5: Once all data has been entered and calibration is performed, the user can calculate the fair value and other statistics for this CMS spread note by clicking the Calculate All macro button. The results are as follows:
The 95% confidence interval for the clean price of the note is thus $2,785,486 ± 208,003. Note that this simulation was performed with only 1000 trials - a much greater number of trials is recommended once the other inputs have been established. The note is certain (probability = 1.000), the expected time of exercise being 13-Feb-2008 (which is approximately 1.41 years from today, 31-Oct-2006).
Tips:
- In order to generate an accurate valuation, using Monte Carlo simulation, the number of trials should be set in the range 100,000 - 1,000,000. It is recommended to start from 1000 trials and consecutively increase the number until the "accuracy", given in the last column of the results table, is acceptable. As a rule of thumb, the execution time will be proportional the number of trials, while the "accuracy" will be inversely proportional to the square root of the number of trials. For example, if the number of trials is quadrupled, the execution time will also be approximately quadrupled, and the "accuracy" (i.e., the half-width of the 95% confidence interval) will be approximately halved.
- Users should be careful with the discount factor curve. It should start on the valuation date and extend to at least as far as the note maturity date plus the longer maturity of the two CMS rates. For example, if the note matures in 2025 and the longer CMS rate is 20 years, the curve needs to extend to 2045.
Disclaimer
Your use of the information in this article is at your own risk. The information in this article is provided on an "as is" basis and without any representation, obligation, or warranty from FINCAD of any kind, whether express or implied. We hope that such information will assist you, but it should not be used or relied upon as a substitute for your own independent research.
» For more information or a customized demonstration of the FINCAD analytics, contact a FINCAD Representative.