So here are some sample calculations again, for example, so

if you look at t equals to 3, this is the pay off of, This swaption.

So in fact all of these values here from t equals 4 and 5 are simply

the values that we see that we calculated earlier for the underlying swap.

So we see 0.0512 here, well that's the same 0.0512 here.

Down here however, is the value of the swaption at maturity.

And now this is equal to the maximum of 0 and S3.

So if you recall, the only change is actually down here.

Because in the underlying swap value,

we had a negative value minus 0.0085 at this node.

But now, the holder of the swaption would not choose to exercise at this node.

Why would they buy something for 0 when it has a negative value?

They won't do that.

So they will exercise however at all of these three nodes,

because these three nodes the swap has a positive value.

So certainly the whole of the swaption will exercise at these three nodes and

receive these three payoffs.

So what we have is here are the values,

the payoff values of the swaption at maturity, at t = 3.

It's equal to the maximum of 0 and S3.

And now we actually just compute the fair value of the swaption

by computing the value of this payoff backwards in the lattice.

So here what we're using in our risk-neutral pricing again,

if you like I won't use s here, I'll say Z.

So Zt = the expected value of Zt

+ 1 divided by 1 + over t.

This is our risk neutral pricing for the swaption.

Note, I don't have any intermediate coupon or cash flow in here.

And that's because the swaption doesn't pay any intermediate coupons or

cash flows between t = 0 and t = 3.

So I just iterate this backwards to get the value of 0.0620 as

being the value of the swaption at t = 0.

So here's a sample calculation, 0.0908.

Well 0.0908 is just 1 over 1 plus, or

1 over 1 is going to be 7.5% of this node times the expected value of the swaptions

one period head while the risk mutual probabilities are half in a half.

These are the values of the swaption one period ahead 0.1286 and

0.0665 and so that's how I get 0.0908.