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.
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,
.
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