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Vasicek model recombining tree

SalinaMiao

New Member
Hi,

Can anyone demonstrate an example of how the rates in the Vasicek model tree is calculated? I am looking at Tuckman's chapter 9 study notes on page 57 (see attached). For example, how is the rate at (1,1) of 5.5060% calculated?

Thank you
 

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SalinaMiao

New Member
also im not sure if its a typo? in the picture the formula says (k*theta-r) but at the top of chapter it says k*(theta-r). Which one is correct?
 

David Harper CFA FRM

David Harper CFA FRM
Staff member
Subscriber
HI @SalinaMiao The formula featured in the study note text is correct: dr = k (θ -r)dt + σdw. The displayed blue formula (that is part of the XLS screen) is incorrect (although the calculations are fine), apologies, and should match the text and consequently should appear as follows:


In regard to the calculations, they are in the XLS and explained by Tuckman in Chapter 7 (see below). However, the problem is that the node does not recombine naturally, see difference in values at node[2,1] inside red box:



Which he resolves as follows, I hope this explains(!):

 

nivethasridhar3

New Member
HI @SalinaMiao The formula featured in the study note text is correct: dr = k (θ -r)dt + σdw. The displayed blue formula (that is part of the XLS screen) is incorrect (although the calculations are fine), apologies, and should match the text and consequently should appear as follows:


In regard to the calculations, they are in the XLS and explained by Tuckman in Chapter 7 (see below). However, the problem is that the node does not recombine naturally, see difference in values at node[2,1] inside red box:



Which he resolves as follows, I hope this explains(!):

Hi, can you explain the sentence " The expected perturbation due to volatility over each time step is zero, the drift alone determines the expected value of the process after each time step". Why is that the case?
 

David Harper CFA FRM

David Harper CFA FRM
Staff member
Subscriber
Hi @nivethasridhar3 It is a good question because it relates to a key feature of these interest rate trees: they are not simulations themselves, nor are they they term structures, these binomial interest rate trees are "maps" that illustrate one sigma (standard deviation) jumps up and down. In the case of the Vasicek model, given by dr = k (θ -r)dt + σdw, this is a model of the movement in the short rate and it has two components: the deterministic drift, k(θ -r)dt, and the random shock (aka, stochastic) component, σdw. If we remove the stochastic component, then the tree would be a single (bending) line that bends from the current rate to the long-run mean, much like in GARCH(1,1), the prediction is a volatility that bends toward the long-run mean.

But the model contains a random shock, σdw. The binomial rate tree is a map of up/down jumps where σ = +/- 1.0. But the expected value (aka, mean) of this random normal variable is zero. The tree is showing how the rate looks if it jump up one or down one sigma, but the expected value is zero. That is the meaning of "Since the expected perturbation due to volatility over each time step is zero, the drift alone determines the expected value of the process after each time step". Good question, IMO, because it forces us to understand what this trees represent; over the years, I have noticed that many confuse them with a single simulation trial, or even the implied term structures (plots of the rate against maturity that are implied by the model). I hope that's helpful!
 
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