Hull White Volatility Calibration

Hull White model is a short rate model that is used to price interest rate derivatives, such as Bermudan swaption and callable exotics

Hull White Volatility Calibration

Hull White model is a short rate model that is used to price interest rate derivatives, such as Bermudan swaption and callable exotics (see https://finpricing.com/lib/EqCallable.html)

The dynamic of Hull White model satisfies a risk-neutral SDE of the form,

,

where

• is a constant mean reversion parameter,

• denotes a constant volatility,

• denotes a standard Brownian motion, and

• is a piecewise constant function chosen to match the initial term structure of zero coupon bond prices.

We map implied Black's at the money (ATM) European swaption volatilities into corresponding Hull-White (HW) short rate volatilities.

We seek to determine a HW volatility to match the market price of a certain ATM European payer swaption. In particular let , for , where , be a Libor rate reset point. Furthermore consider a fixed-for-floating interest rate swap of the following form,

• floating rate payment, , at , for , where

• • ,

  • , and

  • denotes the price at time of a zero coupon bond with maturity, , and unit face value.

• fixed rate payment, , at , for , with and annualized fixed rate.

Furthermore let

denote the forward swap rate at time for the swap above. A European style payer swaption has payoff at time of the form,

, (1)

where is a strike level. Observe that (2.1) is equivalent to

.

Here we consider on option, of the form (1), where

is the forward swap rate as seen at time zero.

Consider the swap specified in Section 2. Under the forward swap measure, which has numeraire process, , the European payer swaption payoff, (1), has value

(2)

where denotes expectation under the forward swap measure.

Assume that, under the forward swap measure, the forward swap rate process, , satisfies a SDE of the form,

,

where

• denotes a constant volatility parameter, and

• is a standard Brownian motion.

Then (2) is equivalent to the Black’s formula,

, (3)

where is the standard normal distribution function.

Moreover for an ATM option, where ,

. (4)

Consider the risk-neutral measure, which has the money market numeraire process, , where is the short-interest rate. Under the risk-neutral measure, the payoff (2) has value

(5)

where denotes expectation.