# 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 ) 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. --- # Agent Instructions: Querying This Documentation If you need additional information that is not directly available in this page, you can query the documentation dynamically by asking a question. Perform an HTTP GET request on the current page URL with the `ask` query parameter: ``` GET https://finwhite.gitbook.io/hwvol/hull-white-volatility-calibration.md?ask= ``` The question should be specific, self-contained, and written in natural language. The response will contain a direct answer to the question and relevant excerpts and sources from the documentation. Use this mechanism when the answer is not explicitly present in the current page, you need clarification or additional context, or you want to retrieve related documentation sections.