The purpose of this document is to demonstrate methodology to estimate the parameters of Black Karasinski (BK) interest rate model. The methodology is linear regression based. The parameters are estimated, assuming that model will be used only for scenario generation under real world measure.
It is assumed that the reader of this document is well versed with the model and its advantages, hence only a brief description of the model will be given.
In financial mathematics, the Black–Karasinski model is a mathematical model of the term structure of interest rates. It is a one-factor model as it describes interest rate movements as driven by a single source of randomness. It belongs to the class of no-arbitrage models, i.e. it can fit today’s zero coupon bond prices, and in its most general form, today’s prices for a set of caps, floors or European swaptions. The model was introduced by Fischer Black and Piotr Karasinski in 1991.
Please find further details about this model in
Model parameter process
Assume we have an Ornstein-Uhlenbeck process, of the form (in continuous time)
dq(t) = κ×(μ – q(t)) ×dt + σ×dW(t) ———–(1)
q = is the process under consideration
κ = is the mean reversion speed of the process
μ = long term mean of the process
σ = volatility of the process
dt = time step between q(t+1) and q(t)
W(t) = Weiner process
The simulation model samples the discrete form equivalent of this process as
Δq = q(t+1) – q(t) = κ×(μ – q(t)) ×dt + σ × ε × sqrt(dt) ———–(2)
Here ε is standard normal distribution N(0,1).
Now the equation becomes
q(t+1) = q(t) + κ×(μ – q(t)) ×dt + σ × ε × sqrt(dt) ———–(3)
q(t+1) = q(t)×(1- κ×dt) + κ×dt×μ + σ×ε×sqrt(dt) ———–(4)
From this last equation, we can see that the expected level of future value of the process is a weighted average of the most recent value and a mean reversion level of the process.
Hence we can estimate the parameters of the process with the time series equation
q(t+1) = q(t) × α + β + εt ———–(5)
Now α = (1- κ × dt), this implies κ = (1- α)/dt
β = κ×dt×μ = ((1- α)/dt) ×dt ×μ or μ = β/(1- α)
Now if we replace q = log(r), where r is the short rate then the model becomes Black Karasinski interest rate model.
In this paper we wanted to demonstrate the methodology of estimating parameters of Black Karasinski interest rate model, conditioned that the scenarios will be generated under real world measure.
For simplicity we have assumed that the model parameters are constant, but under the same assumptions and methodology the parameters can be made time varying.
Modeling Financial Scenarios – A Framework for the Actuarial Profession – Research sponsored by Casualty Actuarial Society and Society of Actuaries.