Company X’s models populate a binomial interest rate tree assuming interest rates follow a lognormal random walk.
Assumption 1: the lognormal model ensures that interest rates are nonnegative
Assumption 2: the lognormal model ensures a constant volatility of interest rates at all levels of interest rates.
Solution: A1 is correct, A2 is not.
Anyone understands why??
A1 -> Log is always positive. So numbers can never go negative.
A2 -> LogNormal when plotted would be a straight line. So the slope of the line is constant. That does not mean that the volatility is constant but that the change in volatility is constant.
I hope I am interpreting A2 correctly. S2000 or TickerSu should be able to validate my response.
Sorry for necro but am hoping someone can provide more color on this question.
The source of my confusion is the fact that when we build out a binomial interest rate tree in excel (i.e., using goal-seek at each time step), we use e2*sigma as the distance between adjacent nodes at all time steps. In other words, I thought that the fact that sigma does not change for any portion of the tree implies that volatility is constant throughout the tree?
Am hoping someone can help me understand why volatility is higher at higher interest rates (as per the solution to the problem).
Update: It looks like volatility in this situation is NOT referring to the sigma term but rather to the variance of interest rates at a given point in time, which is indeed higher for higher rates.