A snowball is a complex debt instrument known as a "callable LIBOR exotic". Many snowballs are "callable inverse snowballs" in that the snowball coupon depends on a previous coupon rate plus spread minus a (scaled) floating rate, and the snowball can be called on coupon payment dates by the issuer. Snowball coupons are typically floored at 0%, and rates can be set in-arrears.
The "snowball" feature, whereby the previous coupon is added into the current coupon, means that snowballs are "path-dependent instruments". That is, 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, snowballs are typically Bermudan callable, which 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 makes valuation using Monte Carlo somewhat difficult.
The LIBOR Market Model (LMM) is a flexible interest rate model that evolves a set of market-observable forward rates. Each forward rate evolves according to a stochastic log-normal process, written in terms of the market-observed volatilities and correlations of the forward rates. The LMM is popular with practitioners and is well-suited to Monte Carlo methods. Bermudan exercise in Monte Carlo can be modeled using the Least Squares Monte Carlo (LSMC) algorithm proposed by Longstaff and Schwartz.
The FINCAD snowball valuation functions use the LMM and LSMC. Local volatility extensions to the LMM are available which allow volatility smile effects to be modeled. To evaluate the lastest version of FINCAD Analytics Suite to value a snowball, contact a FINCAD Representative
Formulas & Technical Details
Suppose that a snowball pays coupons over its life. The coupon (per unit notional) for period is given by:
is the accrual factor for the period to .
The coupon rate is given by:
is the floating rate set in-advance or in-arrears,
is the rate scale factor,
is the spread,
is the previous coupon rate, and
is another scale factor that multiplies the previous coupon rate.
the floating rates are contiguous forward rates (i.e., the terminating date of one rate term is the effective date of the next rate term) with one forward rate setting in each coupon period.
is the floor for the coupon rate.
The LMM is used to evolve forward rates along a Monte Carlo "path" of dates. Path dates include 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 date, the callable snowball is priced on each path. The price of the callable snowball is the average price over all paths.
The LSMC algorithm for Bermudan callable snowballs is identical to that for Bermudan swaptions (see LIBOR Market Model), except that both the "exercise values" and "continuation values" must be regressed. For a Bermudan-callable instrument, the continuation value on an exercise date is the value of the embedded option assuming the option is not exercised immediately. This value is obtained via regression (see below). The exercise value is the value of the option resulting from immediate exercise. For Bermudan swaptions, the exercise value is the value of a plain-vanilla swap, which can be priced exactly "off the curve" on a given date and path. In contrast, the exercise value of a callable snowball on a given date and path requires a nested Monte Carlo simulation (in order to value the underlying snowball), which is computationally expensive. Therefore, for snowballs, regression is used to estimate both the exercise and continuation values.
The payoff of a call option on a snowball (from the snowball issuer’s perspective) on coupon date is given by the difference between the expected present value of future coupons and the call price:
is given by Equation 1 and Equation 2, and
the forward rates in are evolved (simulated) in the measure that is forward risk neutral with respect to the numeraire asset
is the price of the numeraire asset on coupon date .
The LSMC algorithm for snowballs loops backwards from the last exercise date to previous exercise dates as follows:
- Calculate the value of on the exercise date for each path.
- Regress the values from each path in Step 1 against a linear combination of "basis functions" evaluated at "regression variables" whose values are known on the exercise date for each path.
- Use the regression parameters, regression variables, and basis functions from Step 2 to estimate the expected snowball price(the expectation in Equation 3) and therefore the payoff on the exercise date for each path. The payoff is the exercise value.
- If this is the last exercise date, set both the continuation value and the discounted continuation value to 0 for each path and go to Step 8.
- For each path, set the continuation value on the exercise date equal to the discounted continuation value from the later exercise date.
- Regress the continuation values from each path in Step 5 against a linear combination of basis functions evaluated at the regression variables.
- Use the regression parameters, regression variables, and basis functions from Step 6 to estimate the continuation value on the exercise date for each path.
- For each path, if the exercise value continuation value, then set the exercise flag for this date = TRUE and set continuation value = exercise value. Otherwise, set the exercise flag = FALSE and set continuation value = discounted continuation value.
- Move to the previous exercise date.
- Repeat steps 1-9 until there are no more exercise dates.
The earliest date on each path for which exercise flag = TRUE is the optimal exercise date for that path. Let this exercise date be the -th coupon payment date. The price of the callable snowball for the path is:
the summation over is a sum over coupon payment dates,
and the index 0 refers to the value (settlement) date.
If there are no dates for which exercise flag = TRUE, then set and replace in Equation 4 with the principal repayment at maturity. 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. An embedded call option makes the note less expensive to the buyer (who is short the option), so the price of a callable snowball in Equation 4 is an upper bound on the price. Conversely, an embedded put option makes the note more expensive to the buyer (who is long the option), so the calculated price of a puttable snowball is a lower bound. The price of a snowball that is both callable and puttable is obtained assuming that the issuer has the right to call before the holder has the right to put, and that the call price is not less than the put price.
The choice of regression variables and choice of basis functions affect how closely the LSMC algorithm approximates the optimal exercise strategy, and depend on the specific instrument being priced. The FINCAD snowball pricing functions provide a choice between two sets of regression variables: either the 0-th 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. A choice between two different sets of basis functions is also provided: either 2nd-order polynomials or 2nd-order Laguerre polynomials. The price of a snowball is not expected to be particularly sensitive to the basis functions. Instead, one should focus on selecting regression variables that are indicative of continuation values relative to exercise values. The sums of random factors are general variables that can be used for any derivative. In the case of snowballs, the 0-th and 1st moments of the interest rate curve are expected to be the best choice for regression variables.
The LSMC algorithm is described further in the LIBOR Market Model and Bermudan and American Style Basket Options FINCAD Math Reference documents.
Regardless of whether or not rates are set in advance, in arrears, or somewhere in the middle of a coupon period, the numeraire asset in Equation 3 and Equation 4 is chosen to be a zero coupon bond paying 1.0 on the terminating date of the last forward rate. For a snowball where rates are set in advance, the maturity date of the zero coupon bond is the same as the maturity date of the snowball. In the Monte Carlo simulation, forward rates must be evolved to both rate fixing dates and coupon payment dates so that the correct coupon payment can be calculated (using the rate on the rate fixing date) and discounted through the numeraire (using the numeraire prices on the coupon payment date and the value date for a particular path). If rates set somewhere in the middle of a coupon period, then a "stub" rate is needed (in addition to the family of contiguous forward rates that determine the coupon rates) in order to perform the discounting. The stub rate is not simulated; rather, it is approximated via an interpolation of evolved and already-fixed forward rates on the coupon payment date in question.
The FINCAD snowball pricing functions output price, expected time to exercise, probability of exercise, and other quantities. To find out more information about FINCAD products and services, contact a FINCAD Representative