是是ATM swaption strike = \frac{P(0,T_\alpha) - P(0,T_\beta)}{\sum_{i=\alpha+1}^{\beta} \tau_i P(0,T_i)}
P(0, T_i): pure discount bond price with expiration T_i
T_{\alpha}: expiration of swaption
T_{\alpha+1}, ... , T_{\beta}: payment dates of swap
\tau_{\alpha+1}, ... \tau_{\beta}: year fractionsJamshidian or trinomial tree

You also need to know the initial term structure to fit theta(t).
Swaption price has closed form formula under Hull-White using Jamshidian's decomposition. So it's just a multidimension minimization problem.
It's dangerous to let sigma be time-dependent. This may lead to nonsense future volatility structure. The way to fit initial volatility structure exactly is to let a be time dependent.
The paper "The General Hull-White Model and Supercalibration " is available for free at Hull's website. It describes every step of how to build a trinomial tree with piecewise linear a and sigma and calibrate to swaptions.

谢谢你的回复。我看过你说的那篇文章，可能因为理解力比较差。。。其实还是比较模糊的概念。我不知道你提到的theta(t)是否可以通过估值日当天的收益率曲线求解出来？我目前也是已知估值日当天的收益率曲线的。
我个人对波动率校正的理解是：1.根据互换期权的波动率报价，对其中的某一期T到期的互换期权，按照公式LA[S0*N(d1)-Sk*N(d2)]（其中L本金可以假设为1，A=(1/m)*sum(DF(0,Ti))，公式中用的波动率为对应的波动率报价）计算市场价格。不知道是否是这样？还有其中的S0=Sk=远期互换利率，这个怎么求取呢？
2.假设波动率值，计算模型价值使不断逼近通过“1”里的公式计算出的市场价格。但是这里的“模型价值”是用什么公式或者如何计算的呢？
不好意思因为之前学得比较浅，所以可能大家觉得理所当然一笔带过的东西，我理解起来还是有点问题。您是否能说得详细点呢？非常感谢
2# irvingy