TextView?type=daily_treasury_yield_curve&field_tdr_date_value=all

2. Interest Rate Data: Daily Treasury Par Yield Curve Rates

All interest rate models are special cases of the general form of the short-term rate:

𝑑𝑓(𝑟𝑡 ) = (𝜃𝑡 + 𝜌𝑡𝑔(𝑟𝑡))𝑑𝑡 + 𝜎(𝑟𝑡 , 𝑡)𝑑𝑍

where f and g are suitably chosen functions of the short-term rate and are the same for most models.

θ is the drift of the short-term rate, and ρ is the mean reversion term. The term σ is the local volatility

of the short-term rate, and Z is a normally distributed Wiener process that captures the randomness

of future changes in the short-term rate.

The Kalotay-Williams-Fabozzi model assumes that changes in the short-term rate can be modeled by

using the above equation and setting f(r) = ln (r) (where ln is the natural logarithm) and ρ = 0.

𝑑 𝑙𝑛(𝑟) = 𝜃𝑡𝑑𝑡 + 𝜎𝑡𝑑𝑍

The explicit solution to 𝑟𝑡 can be found through Ito’s Lemma:

𝐸[𝑟𝑡+Δ𝑡] = 𝑟𝑡 𝑒(𝜃𝑡 )Δ𝑡+ 𝜎𝑍√Δ𝑡

where Z is a normally distributed random variable with mean = 0 and variance = 1

Estimate θ and σ using historical data. In the context of this model, θ is the expected logarithmic change

in yield and σ is expected standard deviation. Simulate 100 interest rate paths for the next month for

the 3-year & 5-year treasury rates using the Kalotay-Williams-Fabozzi Model and display the simulated

paths for each maturity in a separate graph.

𝜃 = 𝐸 [𝑙𝑛 [

𝑟𝑡+Δ𝑡

𝑟𝑡

]]

DATA SET

https://home.treasury.gov/resource-center/data-chart-center/interest-rates/TextView?type=daily_treasury_yield_curve&field_tdr_date_value=all