# LO 13.1: Construct and describe the effectiveness o f a short term interest rate tree

LO 13.1: Construct and describe the effectiveness o f a short term interest rate tree assum ing normally distributed rates, both with and without drift.
T e r m S t r u c t u r e M o d e l w i t h N o D r i f t (M o d e l I)
This topic begins with the simplest model for predicting the evolution of short rates (Model 1), which is used in cases where there is no drift and interest rates are normally distributed. The continuously compounded instantaneous rate, denoted rt, will change (over time) according to the following relationship:
dr = crdw
where: dr = change in interest rates over small time interval, dt dt = small time interval (measured in years) (e.g., one month = 1/12) a = annual basis-point volatility of rate changes dw = normally distributed random variable with mean 0 and standard deviation Vdt
Given this definition, we can build an interest rate tree using a binomial model. The probability of up and down movements will be the same from period to period (30% up and 30% down) and the tree will be recombining. Since the tree is recombining, the up- down path ends up at the same place as the down-up path in the second time period.
For example, consider the evolution of interest rates on a monthly basis. Assume the current short-term interest rate is 6% and annual volatility is 120bps. Using the above notation, rQ = 6%, a = 1.20%, and dt = 1/12. Therefore, dw has a mean of 0 and standard deviation of Vl
112 = 0.2887.
After one month passes, assume the random variable dw takes on a value of 0.2 (drawn from a normal distribution with mean = 0 and standard deviation = 0.2887). Therefore, the
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Topic 13 Cross Reference to GARP Assigned Reading – Tuckman, Chapter 9
change in interest rates over one month is calculated as: dr = 1.20% x 0.2 = 0.24% = 24 basis points. Since the initial rate was 6% and interest rates changed by 0.24%, the new spot rate in one month will be: 6% + 0.24% = 6.24%.