## Calibrating time-dependent volatility to swaption prices

We have seen in a previous post how to fit initial discount curves to swap rates in a model-independent way. What if we want to control the volatility parameter to match vanilla rates derivatives as well? Just as we found for vanilla calls and puts, we will need to chose a model, for example the Hull-White extended Vasicek (HWeV) model that we’ve seen before (there are a few reasons why this isn’t a great choice, discussed later).

Our next choice is which vanilla rates options we want to use for the calibration. A common choice is the interest rate swaption, which is the right to enter a swap at some future time with fixed payment dates and a strike . These are fairly liquid contracts so present a good choice for our calibration. A ‘payer swaption’ is one in which we pay fixed and recieve floating, and a ‘receiver swaption’ is the opposite. For simplicity, for the rest of this post we will assume all payments are annual, so year fractions are ignored.

A reciever swaption can be seen as a call option on a coupon-paying bond with fixed payments equal to at the same payment dates as the swap. To see this, consider the price of a swap discussed before: