Square root rule in short rate models

[afterworkguinness]

Hi ,
In the practice questions for Tuckman chapter 9, the random normal in the short rate simulation models is scaled by sqrt(dt). In Tuckman's equation 9.2 (source page 252), the volatility is annual with a monthly time step, but he doesn't scale dw at all. Should we scale dw when the time step doesn't match the volatility?

 

[afterworkguinness]

I may have posted too soon. Reading the text and the practice question more closely, Tuckman's dw has a standard deviation of sqrt(1/12) already and the practice question text says dw is a standard normal implying a standard deviation of 1 and not 1/12

....great practice questions btw.

 

[David Harper CFA FRM]

Thanks @afterworkguinness. Yes, you noticed the difference. I explained in more detail here at https://forum.bionicturtle.com/thre...rt-term-interest-rate-models.6688/#post-37884 i.e.,

Hi @LaCrema--Good observation, but these are consistent (I awarded you a star because I really appreciate your eye for detail: these nuances matter!). Please note that you are correct when you write "it appears that the standard deviation of √1/12 is already implied in the value of random variable dw according to GARP." In the reading, consequently, Tuckman expresses Model 1 by the following: dr = σ*dw. However, please note that my question does not say that 1.240 is dw, rather the assumption given is "the random uniform variable is 0.8925 such that, via inverse transformation, the associated random standard normal value is 1.240." Consequently, we must multiply 1.240, which has a standard deviation of 1.0 because it is a standard random normal, by the square root of the time step; i.e., sqrt(1/12). Contrast this with the reading which, by providing 0.15 is providing dw directly ; i.e., "random variable dw, with its zero mean and its standard deviation of sqrt(1/12) or 0.2887, happens to take on a value of 0.15."

Model 1 is given by:

dr = σ*dw; or because dw = z*sqrt(Δt) when z = random standard normal,
dr = σ*z*sqrt(Δt)

Candidly, my parsing out the standard random normal from the time step, such that the sqrt(1/12) multiplier is necessary, was deliberate. It's minor, but I believe that just providing the dw helps people too easily "forget" that time is being scaled (like in the binomial trees that are displayed in the reading); it's easy to forget in the binomial tree. I hope that helps, great observation