Binomial Interest Rate Tree at 10% Interest Rate Volatility

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helloatlas

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Here’s the bionomial interest rate tree (exhibit 11 of reading 45)
at year 0, it has a rate of 2.5
at year 1, it has a rate of Ru = 3.8695 and Rd = 3.1681
at year 2, it has a rate of Ruu - 5.5258, Rud / Rdu = 4.5242 and Rdd = 3.7041
it says that Ru = Rd x e^(2s.d. * sqrt of t where t = time)
What I don’t understand is that at year 2 this relationship doesn’t seem to hold -
the gap between the middle rate that is 4.5242 (Rud/Rdu) and the corresponding higher rate 5.5258 (Ru) is not e^(2*0.1*sqrt of 2).
same goes for Rud/Rdu to Rdd.
if anyone care to explain?
thanks!
 
helloatlas wrote:Here’s the bionomial interest rate tree (exhibit 11 of reading 45)
at year 0, it has a rate of 2.5
at year 1, it has a rate of Ru = 3.8695 and Rd = 3.1681
at year 2, it has a rate of Ruu - 5.5258, Rud / Rdu = 4.5242 and Rdd = 3.7041
it says that Ru = Rd x e^(2s.d. * sqrt of t where t = time between slices)
What I don’t understand is that at year 2 this relationship doesn’t seem to hold -
the gap between the middle rate that is 4.5242 (Rud/Rdu) and the corresponding higher rate 5.5258 (Ru) is not e^(2*0.1*sqrt of 2).
same goes for Rud/Rdu to Rdd.
if anyone care to explain?
thanks!
Click to expand...
You missed a couple of words; I added them above.
The time in year 2 is not 2; it’s 1: one year periods between nodes.
 
Saviour! Thanks very much
May I ask one more question? (i know many threads say we are not required to know the calculation of how these interest rates are derived) but just out of curiousity
How is the rate Rud/Rdu getting calculated as 4.5242?
From exhibit 1 the forward rate is given as 4.564
I would reckon it’ll go through several iteration by a software to arrive at 4.5242. But how?
Thanks in advance
 
helloatlas wrote:Saviour! Thanks very much
May I ask one more question? (i know many threads say we are not required to know the calculation of how these interest rates are derived) but just out of curiousity
How is the rate Rud/Rdu getting calculated as 4.5242?
From exhibit 1 the forward rate is given as 4.564
I would reckon it’ll go through several iteration by a software to arrive at 4.5242. But how?
Thanks in advance
Click to expand...
It’s done by pricing par bonds at each maturity.
You price a 1-year risk-free par bond, and adjust Rd until the price comes out at par. This could be done analytically (solving a quadratic equation), but is most often done using numerical methods, such as Goal Seek or Solver in Excel.
Then, you price a 2-year risk-free par bond, and adjust Rdd until the price comes out at par (leaving Rd fixed). This could be done analytically (solving a cubic equation), but is most often done using numerical methods.
Then, you price a 3-year risk-free par bond, and adjust Rddd until the price comes out at par (leaving Rd and Rdd fixed). This could be done analytically (solving a quartic equation), but is most often done using numerical methods.
Then, you price a 4-year risk-free par bond, and adjust Rdddd until the price comes out at par (leaving Rd, Rdd, and Rddd fixed). This cannot be done analytically (general 5th order (quintic) and higher polynomial equations do not have analytical solutions); it must be done using numerical methods.
And so on.
 
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