Forward Rates: Notation and Binomial Tree

[mikey10011]

In the screencast you mentioned that you did not use Tuchman's notation for forward rates. Is the notation on slide [Acrobat p. 17] right? Note that on the next and last year's slides you used f(1.5,2.0).

On page 106 on your study notes, an example is presented on calculating f(1.5,2.0). Are there typos?

Finally, I am having trouble understanding the "binomial assumption" regarding the forward rate f(0.5,1.0) where I am assuming the notation to mean the forward rate from 0.5 years to 1.0 years. [You used f(0.5,0.5); is this a typo?] If I understand Chapter 2 right, f(0.5,1.0) can be calculated if we know the spot rates s(0.5)=5.00% and s(1.0)=5.15% using

[1+s(1.0)/2]^2 = [1+s(0.5)/2] x [1+f(0.5,1.0)/2] = [1+5.15%/2]^2 = [1+5%/2] x [1+f(0.5,1.0)/2]

Solving, I get f(0.5,1.0)=5.30%.

Now to Chapter 9 where Tuchman (p. 172) says to "assume that six months from now the six-month rate will be either 4.50% or 5.50% with equal probability." To me that suggests f(0.5,1.0)=1/2 x 4.5% + 1/2 x 5.5% = 5.0%.

Question: I seem to be getting two different numbers for f(0.5,1.0). How do I reconcile the two?

 

[David Harper CFA FRM]

I don't use Tuckman notation because he uses r and r-carrot and it's hard (for me) to tell forward and spot rates apart in his text. P. 17,18 and the notes look fine to me. I probably should use just one alternative method, but they each reflect the arrow:

f(0.5,1.0) = 1 f 0.5 = 6 mos forward 6 mos from now
f(1.5,2.0) = 1 f 1.5 = 6 mos forward 18 months from now

There are at least two additional ways to annotate these....but the math looks good to me...

In regard to the binomial, yes, I should not have f(.5,.5), for consistency it should say f(0.5,1.0)...that's a typo...or at least inconsistent with previous notation

regarding your two different solutions,

I agree with your first since, (1+5%/2)*(1+5.3%/2) = (1+5.15%/2)^2.
So, at time zero that is the implied forward rate.

The reconciliation is profound and is sort of the whole point of risk neutral pricing. See slide 53 or Tuckman 177. The risk nuetral probabilities are 80% and 20%, so that

80% * 5.5% + 20% * 4.5 = about 5.3%!

The reconciliation is, these spot rates and forward rate imply the up/down probabilities are not 50/50...

David

 

[David Harper CFA FRM]

actually, correction:

bottom of slide 17, the yellow box is wrong: it should say 1 f 3 to indicate 6 months forward starting in 18 months (where each period = 6 months). But i regret trying to show alternative notation, i should have stuck with p 18: f(1.5, 2.0) as

1 f 3 = f (1.5,2.0) = 6 month forward starting in 18 months (i.e., 3 six month periods)

David