Black-Karasinski Short Rate Tree Algorithm

The Black-Karasinski model is a short rate model that assumes the short-term interest rates to be log-normally distributed. We implement the one factor Black-Karasinski model as a tree.

Black-Karasinski Short Rate Tree Analytics

The Black-Karasinski model is a short rate model that assumes the short-term interest rates to be log-normally distributed. We implement the one factor Black-Karasinski model as a binomial or trinomial tree.

Assume that short term interest rate process, , satisfies, under the risk neutral probability measure, a SDE of Black-Karasinski form,

where

· denotes standard Brownian motion,

· is the volatility,

· , with , is the mean reversion,

· is chosen to match the initial term structure of zero coupon bond prices.

Our approach towards building a tree for the short-term interest rate process, , is based on the single-factor tree construction technique. Specifically let

,

where the process satisfies the SDE

(2.1)

Then

where . We first build a tree for the process as described below.

Let . From Ito’s Lemma,

,

Then

where .

Next let

,

where and , be a partition of the interval ; furthermore, let be an additional time slice. We can view our tree for the process as a directed graph, which is defined by a set of vertices and directed edges. Let and , for , respectively denote a tree node at time slice and the associated value for ; here , for , denotes the total number of nodes on the time slice. Furthermore let

denote the random variable

where . Then

.

Since

Then

We build a tree for , based on Myint’s equity tree construction technique, using the expressions above for and . Here we employ a partition with spacing of

at time slice , for , where

Let , for and , denote a child, at the time slice, of the tree node ; here denotes the number of children emanating from the parent node, (e.g., for a trinomial tree). Let , for and , denote the price at time zero of an Arrow-Debreu security at the node , that is, a security that pays 1 currency unit if the node is reached at time and zero otherwise.

Black-Karasinski short rate tree approach can be used to price convertible bond. Convertible bond is not only a coupon paying bond but also can be converted at the discretion of the holder within the periods of time specified by the conversion schedule. Typically, the issuer has the option to buy the bond back at a predetermined strike price(s) during the callable period(s). Also, there are provisions that allow the holder to return the bond to the issuer in exchange for a predetermined cash price during certain period(s) (see https://finpricing.com/lib/FiBond.html)

Let denote the price at time zero of a zero coupon bond with maturity of and face value of 1 currency unit. We determine at each time slice by matching the initial term structure of zero coupon bond prices. We first solve

for , that is, . We then set the Arrow-Debreu security values at the time slice to

for , where denotes the tree root node.

For , let

Sequentially, for , we then numerically solve

(1)

for the unknown . Here we employ the Newton iteration scheme

,

for . Observe that

which we denote by . An initial guess to the Newton iteration scheme above, , is then obtained by solving

for the unknown ; that is,

.