# Hull White Volatility Calibration

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>)

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

&#x20;                       ,                                                       &#x20;

where

* &#x20;is a constant mean reversion parameter,
* &#x20;denotes a constant volatility,
* &#x20;denotes a standard Brownian motion, and
* &#x20;is a piecewise constant function chosen to match the initial term structure of zero coupon bond prices.

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

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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,

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* floating rate payment, , at , for , where
* * &#x20;,
  * , and
  * &#x20;denotes the price at time  of a zero coupon bond with maturity, , and unit face value.

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

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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

.

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

&#x20;is the forward swap rate as seen at time zero.

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

&#x20;                                                                            (2)

where  denotes expectation under the forward swap measure.

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

&#x20;                                                           ,

where

* &#x20;denotes a constant volatility parameter, and
* &#x20;is a standard Brownian motion.

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

,        (3)

where  is the standard normal distribution function.

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Moreover for an ATM option, where ,

.                    (4)

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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

&#x20;                                       (5)

where  denotes expectation.

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