When you mention the difference between Black and Terminal Swap Rate (TSR) for swaptions, I guess you mean for cash-settled swaptions (EUR style).

The cash-settled swaptions usually require some type of approximation to be priced. The Terminal Swap Rate model (which is not a model but an approximation technique) can be useful for that. I did an analysis on different pricing model for cash-settled swaptions. The presentation, titled “Cash-settled swaptions: How wrong are we?”, is available on our documentation site: http://docs.opengamma.com/display/DOC/Presentations It is based on the working paper with the same title http://ssrn.com/abstract=1703846

To come back to your remark stating a 40 bps difference, I guess it is for in-the-money options. The TSR are approximation techniques. In the presentation I take the example of the Linear TSR. The approximation is exact only in t=0 and ATM. In the money, the approximation errors can become quite large. In the table on page 21 of the presentation there is a table were the difference between the two approaches is above 100 bps (for a swaption 4% ITM). In this case, this is mainly due to approximation errors.

In general the different pricing techniques will give different prices, even if the difference is not so large when there is no approximation error.

Marc