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?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 @QuantFFM dw is how Tuckman specifies the models; he scales the random standard normal rather than scaling the annual basis point volatility input (which would be more intuitive to me, too!). But I'm not sure it matters because the essential random shock is the product of three variables:
[random normal Z = N^(-1)(random p)] * σ[annual basis point volatility] * sqrt(Δt/12_months); i.e.,
It's just the case that models are specified by (random normal Z) * σ[annual basis point volatility] * sqrt(Δt) = [(random normal Z) *sqrt(Δt)] * σ[annual basis point volatility] = dw * σ[annual basis point volatility], so that dw is not random standard normal but instead a random normal, with standard deviation of sqrt(Δt), that scales (i.e., is a multiplier on) the annual basis point volatility. To me it's not different than itemizing all three with Z*σ*sqrt(Δt), and if he gains an advantage by using (dw) I don't really know what it is?! To your point, I don't know why his is better than: (random normal Z) * σ[annual basis point volatility] * sqrt(Δt) = (random normal Z) *[sqrt(Δt) * σ(annual basis point volatility)] = Z * σ[t-period basis point volatility]
- Random normal Z: a random standard normal, by definition µ = 0, σ = 1.0
- The annual basis point volatility; e.g., 1.60% or 160 basis points per annum. As usual, inputs should be in per annum terms
- Scaling factor per the usual square root rule (SRR) that assumes i.i.d. Notice I elaborated the full SRR to sqrt(Δt/12_months) because the denominator is whatever are the time dimension of the volatility input, in the case and as usual, per annum = 12 months. So to your second point, of course the 1-year volatility input can be scaled to monthly with 1.60% * SQRT(1 month/12 months) when our tree step is one month (i.e., numerator) and our input volatility is 12 months (i.e., denominator). Because we can assume this re-scaled monthly volatility is normal, it randomized by multiplying by a random normal Z.
I hope that's helpful, have a good week yourself!