Question regarding "swaption under Black & Sholes framework and swaption under TSR framework"

The following question was posted in another thread:


First, great job for OG. By any chance, have you ever analyzed the price differences between swaption under Black & Sholes framework and swaption under TSR framework?

I developed the same framework of TSR pricing, however I found a wide difference between both pricing framework. (40 Bps). I am wondering if it will make sense for you ?
I am not sure if it’s the right place to post my comment…
Many Thanks for your help


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: It is based on the working paper with the same title

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.


Hi Marc,

Your answer really helps, thanks for your time. I forgot to tell you that I am pricing the cash settled swaption with muti curves approach (eonia curve for discount + Eurib6M for forecast).

What about taking into account 2nd order approximation ( alpha0X^2+alpha1X+alpha0) instead of pitterbarg approximation ? maybe it will help to catch the convexity ?


I have not looked at other approximations beyond the linear one. I would be careful with a second order. It is probably locally better, but the problem that is causing a large discrepancy comes from large ratess. For those strikes, a quadratic function may cause worst explosions. Adding a quadratic term may cause more harm than good. A swap-yield TSR model is probably a better idea, but I have not implemented it.