Black-Karasinski Model

Overview

In financial mathematics, the Black-Karasinski model is a mathematical model of the term structure of interest rates; see short rate model. 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 (from Wiki).

Implementation

This section mainly introduces the implementation process of short-rate and discount, which is core part of option pricing, and applied in Tree Engine.

As a critical part of Tree Engine, the class \(BKModel\) implements the single-factor Black-Karasinski model to calculate short-rate and discount by using continuous compounding. The implementation process is introduced as follows:

    1. The short-rate is calculated at time point \(t\) with the duration \(dt\) from 0 to N point-by-point by functions treeShortRate, initRate and iterRate. As the core part of the treeShortRate, the outer loop_rateModel is used to ensure the results under pre-specified tolerance. For the internal functions, the functionality of initRate and iterRate is similar with each other, but initRate can produce 3 intermediate results while the iterRate gives only one per iteration. In order to achieve intiation interval (II)=1, the array values16 is added to store the intermediate results. Then an addtion tree is performed subsequently for the whole process.
    2. For implementing the generic Tree framework, the \(state\_price\) calculating process is moved from Tree Lattice to this Model.
  2. The discount is calculated at time point \(t\) with the duration \(dt\) based on the short-rate.