Pricing of Automatic Redemption Instruments with FINCAD Introduction
Introduction
In recent years, instruments with automatic redemption features have become increasingly popular. Typically, these instruments offer high fixed coupons initially and then pay like an inverse floater, i.e. pay a fixed rate minus the forward (Libor) rate. Redemption is governed by the cumulative coupon (the sum of all coupons paid up to a particular date), relative to a "target" or "lifetime cap". If the cumulative coupon on a particular date exceeds the lifetime cap, the instrument redeems, i.e. is bought back by the issuer at a cost of the remaining principal payments. Instruments with automatic redemption (AR) features offer the holder a method of taking a position on the future structure of interest rates, and offer potentially very high yields if redemption occurs early. The risk to the investor occurs if future interest rates do not behave as expected and redemption does not occur early, leaving the principal tied up for possibly a significant period of time.
Details
Instruments subject to AR often also have a "lifetime floor" in addition to the lifetime cap. The lifetime floor guarantees the holder will receive some minimum coupon over the lifetime of the instrument. This feature was particularly popular with Asian investors when interest rates were low in the early 2000's. Often the lifetime cap and floor are the same, stipulating that the holder receives no more, and no less, than the common value of the lifetime cap/floor. In situations where they are different, the AR feature does not come into play until either boundary is hit. The simplest instrument that contains an AR feature is a Target Redemption Note, or TARN. This is essentially a vanilla floating rate note with an AR feature.
The AR feature of a TARN can be applied to many other financial instruments, for example CMS and CMS Spread notes and Snowballs. A Snowball is a structured instrument in which each coupon is dependent on the previous coupon based on a predefined formula. A Snowblade is a Snowball with an AR feature added to it. Additionally, it can also apply to swaps where one leg has an AR feature.
In FINCAD version 11, functions were introduced to price TARNs, Snowblades, and CMS and CMS Spread notes with AR. FINCAD version 11 also offers functions to price swap versions of the above instruments. For example, a Snowblade Swap is a swap where the structured leg pays snowblade style coupons and the funding leg pays either fixed or floating rate.
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This article will focus on the pricing of a Snowblade, which will illustrate some of the features common to all instruments subject to automatic redemption.
Valuation of Snowblade with FINCAD
The nature of the coupons paid by a Snowblade, namely that the current coupon can depend in some way on the previous one, as well as the AR feature, introduce a strong path-dependency into the note. The AR feature effectively means that a Snowblade embeds a knock out barrier option depending on the cumulative coupon. If the sum of all coupons paid is equal to or greater than the lifetime cap, the note is terminated early with the outstanding principal paid back plus the final coupon scaled down such that the cap is hit exactly. Otherwise, the note will mature on the terminating date of the contract with the return of principal and the last coupon, which will be scaled up if the cumulative coupon does not reach the lifetime floor.
The Snowblade also has volatility smile dependence because of the varying knock out barrier determined by the coupon left to date to reach the target cap. The strong path-dependency is best handled using Monte Carlo simulation. For these reasons we value the Snowblade using the LIBOR market model (LMM) in conjunction with local volatility (other AR instruments are also priced similarly). See the LIBOR Market Model and Option Pricing using a Local Volatility Model FINCAD Math Reference's for details.
The example below will outline how one can price a Snowblade note using the FINCAD pre-built workbook.
Sample deal terms
| 1. | Nominal amount: | JPY 100,000,000. |
| 2. | Issue date: | 24 March 2005 |
| 3. | Maturity date: | 24 March 2020, subject to Modified Following Business Day Convention |
| 4. | Interest basis: | Fixed to Floating Rate (further particulars specified below) |
| 5. | Redemption/payment basis: | Redemption at par |
| 6. | Fixed Rate Note Provisions: | Applicable for the period from and including the Issue Date to but excluding 24 March 2006 |
| (i) Rate of interest: | 2.20% per annum payable semi-annually in arrear | |
| (ii) Interest payment date(s): | 24 September 2005 and 24 March 2006, subject to Modified Following Business Day Convention | |
| (iii) Fixed rate day count: | 30/360 (unadjusted) | |
| 7. | Floating Rate Note Provisions: | Applicable for the period from and including 24 March 2006 to but excluding the maturity Date |
| (i) Business day convention: | Modified Following Business Day Convention||
| (ii) Rates to be determined: | From 24 March 2006 to 24 March 2008 Previous Coupon + 0.25% - 2 x 6 month JPY-LIBOR-BBA | |
| From 24 March 2008 to 24 March 2010 Previous Coupon + 0.35% - 2 x 6 month JPY-LIBOR-BBA | ||
| From 24 March 2010 to 24 March 2012 Previous Coupon + 0.50% - 2 x 6 month JPY-LIBOR-BBA | ||
| From 24 March 2012 to 24 March 2014 Previous Coupon + 1.00% - 2 x 6 month JPY-LIBOR-BBA | ||
| From 24 March 2014 24 March 2016 Previous Coupon + 1.50% - 2 x 6 month JPY-LIBOR-BBA | ||
| From 24 March 2016 to 24 March 2018 Previous Coupon + 2.50% - 2 x 6 month JPY-LIBOR-BBA | ||
| From 24 March 2018 to the Maturity Date Previous Coupon + 3.50% - 2 x 6 month JPY-LIBOR-BBA | ||
| (iii) Reference rate determination: | 6 month JPY-LIBOR-BBA, determined 10 London business days prior to each relevant specified interest payment date | |
| (iv) Day count fraction: | 30/360 (unadjusted) | |
| 8. | Other terms or special conditions: | The note will automatically be redeemed if the total coupon level is greater than or equal to 4.40%. |
Step 1:
In the Payment worksheet, provide payment details, such as effective and terminating dates, cash flow and rate frequencies, and known rate resets. Notional amounts and coupon payoff formula are specified for each coupon period in the white area. A previous coupon scale factor can also be given, determining the degree to which the previous coupon will influence the current coupon.
Step 2:
For the purpose of calibration, first go to the Swaption Data worksheet, and follow the "how to" instruction to generate the swaption data table shown below.
Step 3:
In the Calibration to Swaption sheet, select the terms and methodology for calibration. For the boundaries of model parameters, one could start with the default values. Calibration results will appear on the same page to the right.
A thorough test on the goodness of calibration is performed on the Calibration Check worksheet. If the overall calibration result indicates "Inadequate", one should investigate the reasons why the calibration was not successful and take appropriate action. This could mean investigating the reliability of questionable data points, changing the parameter ranges, or selecting a different minimization method, to name a few. Suggestions are provided on the Calibration Check worksheet.
Step 4:
The user needs to fill in the deposit and swap rates in the standardized Curve sheet as in most FINCAD workbooks.
Step 5:
In the Main sheet, the local volatility method and lifetime cap/floor are specified. The pricing of the deal and other statistics are returned based on the information provided in other worksheets. For example, we can see the target has a 100% chance of being hit, resulting in automatic redemption after only one year and giving a fairly large value to the TARN features.
Note: given the complexity of the instrument itself, the pricing accuracy comes at the cost of calculation time. It is recommended to start with a small number of random trials (for example 1000) and increase it gradually until the pricing stabilizes. Accuracy values next to the pricings provide an estimated range that prices can deviate within, up or down given a 95% confidence interval. Therefore, the smaller the accuracy values, the more accurate the pricings are. To reduce the accuracy value, one can increase the number of Monte Carlo paths.
Conclusion
With FINCAD version 11, users now can value a wide range of instruments that have AR features using LMM MC with local volatility. In the process of pricing, FINCAD also now offers a more comprehensive check on the quality of calibration, making the procedure more transparent.
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.
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