American Swaption Model

A model is presented for pricing single-currency, American style fixed-for-floating interest rate swaptions

Pricing American Swaption

A model is presented for pricing single-currency, American style fixed-for-floating interest rate swaptions

If an American swaption is exercised at a point that is not a reset date, in practice, the effective Libor rate at the point of exercise is a blended rate, which is linearly interpolated from a pair of Libor rates with respective accrual periods that bracket the remaining time interval to the next reset date. The effective Libor rate at the exercise point is taken to be the simple interest rate implied from the zero-coupon bond price to the next reset date. This treatment represents a compromise between accuracy and computational efficiency, since it avoids having to determine bracketing Libor rate values.

We consider a single currency swap specified as follows,

§ reset point, , for , where ,

§ floating-leg payment, , at settlement time, , for , where

o is a notional amount,

o ,

o is the simple interest rate[1] applicable at for the accrual period, ,

§ fixed-leg payment, , at settlement time, , for , where is a simple, annualized rate (ref https://finpricing.com/lib/IrCurveIntroduction.html).

An American style swaption allows the holder to choose the entry point, into the tail of the swap, from a list of possible exercise times (e.g., a window of successive business days). Specifically let and , for , where ,

,

and

,

denote a respective notification time and exercise time. Next consider a particular such pair of times, and , and assume that

for some . If notification is given at time , then the respective floating rate and fixed rate payments,

and

,

must be made at , for . In addition a blended Libor rate, , is determined at , and the respective floating rate and fixed rate payments,

and

,

must be made at time .

We consider the “BK” method for valuing single-currency, fixed-for-floating interest rate, American style swaptions with features of the type described in Section 2. The BK method is an implementation of a “disconnected” tree discretization of a one factor Black-Karazinski (BK) risk-neutral short-rate process of the form below.

Let denote the short-interest rate. We consider a short-interest rate process such that satisfies a risk-neutral SDE of the form,

, (3.1)

where

· is a piecewise constant mean reversion rate,

· is a piecewise constant volatility function,

· is chosen to fit the initial term structure of discount factors,

· is a standard Brownian motion.

A disconnected tree discretization of the short-rate process above is non-recombinant by design, but employs an interpolation scheme to approximate short-rate values at tree nodes along a time slice.

Calibration is accomplished by matching, in a least squares sense, the model price against the market price for each respective European style payer swaption in a cache of calibration securities. The volatility break points are related to the forward start times of the respective swaptions in the calibration portfolio (see Section 4.2 for a typical specification). Given an American swaption,

Consider the American swaption. Let denote the short-interest rate at time . We assume that the process,, satisfies a risk-neutral SDE of the Black-Karazinski form, (3.1).

We construct a trinomial tree to approximate the short-rate process,, based on the algorithm. Let

,

where , be a partition of the interval, , from valuation time to the last settlement point. We assume that the respective volatility and mean reversion functions, and , are constant over each interval, , for . Moreover, we ensure that the set of time slices,, includes the following events,

· the reset point, , for

· the respective exercise point and notification point, and , for

· the respective sets of volatility time and mean reversion time break points.

Consider a particular exercise point, , and assume that

where ; furthermore, let be the corresponding notification time. Consider an option to enter, at , into the tail of the swap above with right to pay fixed and receive float. Based on Section 2 specifications and the treatment in Optex of the effective Libor rate value at exercise time, this swap has price at notification time, ,

where denotes the risk-neutral probability measure. The American swaption value is then given by

where denotes a stopping time.

Consider the Bermudan swaption specified in Section 2. Let

,

where , be a subset of the set of reset points, . We consider a portfolio of European style payer swaptions. In particular, the () swaption is specified as follows,

· forward start time, ,

· resets, ,

· payoff at , , where is a strike level.

For each option in the portfolio above, we obtain from WM a corresponding implied Black’s volatility. We then price the payer swaption above based on Black’s analytical formula. Let denote the price for the () swaption calculated using Black’s model as described above.

We assume that the volatility function,, is constant over the respective periods, , . Let , for , be the constant volatility value, which corresponds to the respective intervals , . Furthermore let , for , denote the benchmark option price for the swaption where . We seek to solve in a least squares sense,

,

for the unknown, ; that is, we seek to minimize

with respect to . A necessary condition for a minimum is that

. (4.2.1)

We solve (4.2.1) for the unknown, , using Newton’s method based on our short-rate trinomial tree.

---

[1] Here denotes the price at of a zero-coupon bond with maturity, , and unit face value.