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How to Value Snowballs Using FINCAD®XL Version 9

FINCAD now offers support for pricing snowballs in Version 9. To view our callable snowball workbook, download the latest trial version of FINCAD Analytics.

Overview

A family of complex debt instruments, referred to as "callable LIBOR exotics" [1], are available to investors who wish to enhance the yield on their portfolios. As discussed in the June 2005 FINCAD newsletter, one such instrument is a "callable range accrual note". This instrument is priced in the FINCAD XL v9 product, along with other exotic instruments such as callable range accrual swaps, callable capped floaters (both notes and swaps), callable inverse floaters (both notes and swaps), and callable snowballs. Each of these instruments has a complicated coupon that depends on floating interest rates, with multiplicative factors, additive spreads, and caps and floors. Investors in these instruments are therefore exposed to interest rate risk. In addition, these instruments contain embedded Bermudan-style call and/or put options. The instrument issuer (e.g., a bank) is long the call option, which exposes the investor to re-investment risk. To compensate the investor for these risks, the issuer offers a high initial coupon and sets an initial lock-out period during which the call option cannot be exercised. If the investor predicts movements in interest rates correctly, then the high coupon will persist after the initial period and the investor will obtain an above-market yield, at least until such time as the issuer calls back the note. If the investor does not predict movements in interest rates correctly, then the investor may get stuck with a long-dated instrument that pays little or no coupon over its life.

Snowballs

Snowballs (also referred to as callable snowballs or callable inverse snowballs) are somewhat different from range accruals, capped floaters, and inverse floaters because snowballs are "path-dependent" instruments. The current coupon is given by the previous coupon plus a spread minus a floating interest rate, floored at 0%. In this way, the coupon builds upon previous coupons, much like a snowball that accumulates snow and grows as it rolls along the ground. For example, the first 2 coupons in a 10-year semi-annual snowball might be fixed at an above-market rate of 6.5%. The remaining coupons, i = 3 to 20, are given by Ci = Ci-1 + 2% - 6-month Libor setting in arrears. If 6-month LIBOR consistently fixes at a small value (e.g., < 2%), then the snowball coupons grow over time. If 6-month LIBOR consistently fixes at a larger value (e.g., > 2%), then the snowball coupons decrease over time. For a very large fixing of 6-month LIBOR (e.g., 9%), the snowball can immediately melt all the way down to the floor of 0%.

FINCAD Valuation

Methods for the valuation of interest rate derivatives include closed-form solutions, lattice methods (e.g., trees), and Monte Carlo simulations. Closed-form solutions are typically only available for the simplest of interest rate derivatives (e.g., European exercise and a simple coupon). The features of snowballs that impact the choice of valuation method include: (1) the coupon is path-dependent; (2) rates can set in advance or in arrears; and (3) early exercise is possible (i.e., there are multiple call/put dates). The path-dependent feature means that one must know about past coupons in order to determine the current coupon. This suggests that valuation is best accomplished using a (forward-looking) Monte Carlo method, rather than a (backward-looking) tree-based method. Rates setting on arbitrary dates (in advance or in arrears) are also more easily handled by a Monte Carlo method than a tree-based method. However, the early exercise feature means that one must know about the future in order to determine the value of holding the option as opposed to exercising early. The Bermudan exercise feature is more easily handled by a (backward-looking) tree-based method than a (forward-looking) Monte Carlo method.

The choice of an interest rate model for valuing an interest rate derivative is driven by the instrument features and the valuation method. The LIBOR Market Model (LMM) is a popular model because the modeled quantities are the market-observed forward rates, and the LMM is consistent with the market-standard approach for valuing caps using Black's formula. The LMM is well-suited to Monte Carlo methods, and Bermudan exercise can be handled in a Monte Carlo method using the Least Squares Monte Carlo (LSMC) algorithm proposed by Longstaff and Schwartz [2]. The FINCAD snowball valuation functions use the LMM and LSMC.

Calibration of the LMM

The first step in the snowball valuation is to calibrate the LMM. The liquid calibration instruments in the interest rate market are caps and European swaptions. The implied volatilities of caps contain information on the volatilities of forward rates. The implied volatilities of swaptions contain information on both the volatilities and correlations of forward rates. Snowballs are sensitive to both the volatilities and correlations of forward rates. Therefore, one could choose to calibrate the LMM to both caps and swaptions or to swaptions alone. For example, the LMM could be calibrated to liquid European swaption data, with swaption expiries corresponding to snowball rate fixing dates, and swap floating leg rate terms corresponding to snowball rate terms. Alternatively, the LMM could be calibrated to liquid cap data alone, with caplet expiries corresponding to snowball rate fixing dates, and caplet rate terms corresponding to snowball rate terms. This calibration would capture the volatilities of the forward rates that underlie the snowball, while the correlations between forward rates could be set based on historical data. Calibration is a topic in the September 2005 FINCAD newsletter. Workbooks are available in the FINCAD XL v9 product for calibration of various interest rate models including the LMM.

Monte Carlo Generation of Forward Rate Paths

The second step is to define a path of dates for the Monte Carlo simulation. These dates define a set of periods over which forward rates are evolved according to the LMM. Path dates include important instrument dates, such as rate fixing dates, coupon payment dates, and exercise dates. Given the paths of forward rates and a set of possible exercise dates, the LSMC algorithm is used to estimate the optimal exercise date for each path. Given the optimal exercise dates, the callable snowball is priced on each path. The price of the callable snowball is the average price over all paths.

Early Exercise via Longstaff & Schwartz

The LSMC algorithm estimates the optimal exercise strategy by estimating an option's "continuation" and "exercise" values on exercise dates. The continuation value is the value of the embedded option assuming the option is not exercised immediately. The exercise value is the value of the option resulting from immediate exercise. For snowballs, computations of the continuation and exercise values require nested Monte Carlo simulations, which are computationally expensive. The LSMC algorithm is a "dimension-reducing" algorithm. Instead of running nested Monte Carlo simulations of forward rates on each path, the LSMC algorithm regresses the path-specific continuation values and exercise values over all paths against a small number of functions of simple variables that are observable on the exercise date (e.g., a swap price and a forward rate). The exercise strategy found by the LSMC algorithm is typically a sub-optimal exercise strategy (at best, the optimal exercise strategy), so a lower bound on the value of the embedded option is obtained. Therefore, the LSMC algorithm gives an upper bound for the price of the callable snowball because the investor is short the call option.

Selection of Regression Variable

The choice of regression variables affects how closely the LSMC algorithm approximates the optimal exercise strategy [1, 2]. The FINCAD snowball pricing functions provide a choice between two sets of regression variables: either the 0th and 1st moments of the interest rate curve (i.e., the level and slope of the interest rate curve, represented by a swap price per unit notional and a forward rate) or the sums of the random factors used to evolve each forward rate. In the case of snowballs, the swap price and forward rate are expected to be the best choice for regression variables because they are financially meaningful and significantly change in value when continuation is optimal vs. when immediate exercise is optimal.

Example

Suppose we have a deal sheet for a Medium Term Note Snowball, with the following properties:

  • The snowball has a maturity of 5 years and a notional of 100 denominated in USD.
  • The product is callable by the issuer at par starting from the first interest payment date and on every interest payment date thereafter with a notice of 5 business days.
  • The note has a Trade Date of 14-Feb-2000 and a Redemption Date of 14-Mar-2005. The note is effective on 14-Mar-2000.
  • Coupons are semi-annual, such that Coupon(i) refers to Interest Period(i), and i goes from 1 to 10.
  • For Interest Periods 1 and 2, Coupon(i) = 6.5% per annum. For Interest Periods 3 through 10, Coupon(i) = Coupon(i-1) + Spread(i) - Reference Index(i).
  • For Interest Periods 3 and 4, Spread(i) = 2%; for Interest Periods 5 and 6, Spread(i) = 3%; for Interest Periods 7 and 8, Spread(i) = 4%, and for Interest Periods 9 and 10, Spread(i) = 5%.
  • The Reference Index is the fixing of the 6-month USD LIBOR 7 business days before the end of the semi-annual interest period, published on Reuters LIBOR01 at 11:00 AM London time.
  • The Business Day Convention is Modified Following Adjusted, and the Daycount is 30/360.
  • For payment purposes, Business Days are NY and LDN Business Days. For the purpose of the delivery of the notice, Business Days are NY, LDN, and TARGET Business days. For the purpose of the Reference Index, Business Days are LDN Business Days.

First we calibrate the LIBOR Market Model using co-terminal semi-annual European swaptions that expire on 5-Sep-2000, 5-Mar-2001, 5-Sep-2001, …, 3-Mar-2005 such that the forward rates underlying the swaptions are the same forward rates that underlie the snowball (i.e., 6-month LIBOR forward rates that fix 7 business days before the end of coupon periods, which are on the 14th day of September and March). We use the FINCAD function aaCalibrateSwaption2_LMM to obtain calibrated values of the LMM volatility and correlation parameters.

Second, we generate the Monte Carlo path dates using the FINCAD function aaCallSnowball_LMM_fs_tbl. The generated dates and the LMM parameters are input into the FINCAD function aaCovarMatGen2_LMM to generate a forward rate-forward rate covariance matrix.

Finally, the covariance matrix and the snowball details are input into the FINCAD function aaCallSnowball_LMM_fs to get the price of the snowball. Using 10000 Monte Carlo paths, the 95% confidence interval for the price of the callable snowball is [88.26, 88.38] per 100 notional. An example spreadsheet showing this last step is given in the Figure below. Recall that this price range is an upper bound on the price. The upper bound approaches the true price as the exercise boundary estimated by the LSMC algorithm approaches the optimal exercise boundary.

Below is a screenshot of FINCAD XL v9's snowball valuation function.

References

[1] Piterbarg, V. V. (2003) 'A Practioner's Guide To Pricing And Hedging Callable Libor Exotics In Forward Libor Models', SSRN Working Paper.
[2] Longstaff, F. A. and Schwartz, E. S. (1998) 'Valuing American Options by Simulation: A Simple Least Squares Approach', Working Paper, The Anderson School, UCLA.

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 software, contact a FINCAD Representative.