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

Tim Xiao

Humanities Commons

Black-Karasinski Short Rate Tree Model  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,  0 trt , satisfies, under the risk neutral probability measure, a SDE of Black-Karasinski form,  ( ) ,0      ,log)(log +−= tdWdtratrd rtrtt    where    •  0 tW rt  denotes standard Brownian motion, • r  is the volatility,  • a , with 0a ,  is the mean reversion, • ( )t  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,  ],0[ Ttrt  ,  is based on the single-factor tree construction technique.   Specifically let   ( ) tttt rrrr~~loglog +−= ,   where the process  ],0[ ~ Ttrt   satisfies the SDE  =+−=                     .0~,~~0rdWrard rtrtt      (2.1)  Then   ( )=−=                  ,log      ,00 rdtad ttt  where ttt rr~log −= .  We first build a tree for the process  ],0[ ~ Ttrt   as described below.     Let tatt rex~= .  From Ito’s Lemma,  rtatrt dWedx = , .~00 rx =   Then  ( )( ) ,~~,~~,++−−+++++++=+=+=tttrsasttarttatttttrsasrtatttttatttrsasrtttdWeererdWereredWexx  where 0t .    Next let   Niit0== ,  where 0N  and Ttt N == 00 , be a partition of the interval ],0[ T ; furthermore, let ( )t t t tN N N N+ −= + −1 1  be an additional time slice. We can view our tree for the process  ],0[ ~ Ttrt   as a directed graph, which is defined by a set of vertices and directed edges.  Let i j  and jir~ , for j ni= 1,..., , respectively denote a tree node at time slice it  and the associated value for jtr~ ; here ni , for i N= 0,..., , denotes the total  number of nodes on the ith  time slice.  Furthermore let   ( )ji itrr ~ ~1+  denote the random variable    ,~11 ++−− +iiijittrsasatritadWeere    where  iii ttt −= +1 .   Then   ( )( )jiji itait rerrE~ ~ ~ 1−=+.  Since ,1111++++−−=iiiiiiiittrsasrtatrtatttrsas dsWeaWeWedWe      ( ( )( )( )( )( ),              2 22, ),min(           2 22,222                ,111111111111111111111 11111112212221222222dsduesduueeadsetaedsseaetetetedsduuseadsetaedsseaetetetedsWeWaedsWeWaeWWedsduWWeaWeWeEdsWeaWeWeVardWeVariiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiitusuausutuauastststtasiatttasatittaiatiattutuusatststtasiatttasatittaiatiatttrsasrtatttrsasrtatrtrtttatststuturursusartatrtatttrsasrtatrtatttrsas+++−−+=++−−+=+−−++=−−= +++++++++++++++++++++ +++++++======++==+==+++====+  Then  ( )( ) ( )( ).12                       ,2 ~ ~ 222222111iiiijitaratatatriteaeeearrVar−−−=−= +++  We build a tree for  ],0[ ~ Ttrt  , based on Myint’s equity tree construction technique, using the expressions above for ( )( ) ~ ~ 1 ji itrrE+ and ( )( ) ~ ~ 1 ji itrrVar+.  Here we employ a partition with spacing of   ( )( ) ~ ~ 1 ji itrrrVar+   at time slice 1+it , for 1,...,0 −= Ni , where .95.1=r   Let ( )e ik j , for k mi j= 1,...,  and i N= 0,..., , denote a child, at the ( )i + 1st time slice, of the tree node i j ; here mi j  denotes the number of children emanating from the parent node, i j  (e.g., 3=jim  for a trinomial tree).   Let Ai j , for i N= 0,...,  and j ni= 1,..., , denote the price at time zero of an Arrow-Debreu security at the node i j , that is, a security that pays 1 currency unit if the node i j  is reached at time ti  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). Find convertible bond valuation details at https://finpricing.com/lib/EqConvertible.html  Let ),0( tP  denote the price at time zero of a zero coupon bond with maturity of t  and face value of 1 currency unit.  We determine ttt rr~log −=  at each time slice by matching the initial term structure of zero coupon bond prices.  We first solve   ),0( 110tPete=−  for 0 , that is, −=110),0(loglogttP .  We then set the Arrow-Debreu security values at the time slice t1  to  ( ) ( )( ),0,0 110 101 ktee eprobeA k−=   for k n= 1 01,..., , where 01  denotes the tree root node.  For i N= 1,..., ,  let  ( ) ).,0( 11~+=−−=+injteiii tPeAfiiijirj   Sequentially, for i N= 1,..., , we then numerically solve  ( ) 0=iif       (1)  for the unknown i .   Here we employ the Newton iteration scheme  ( ) ( )( )( ) ( )( )jiiijiijiji fdfd11   −+−=  ,  for 1j .  Observe that  ( )( )),,0(          ),,0(11~111~1+=−+−+= ++−−=−injtritinjtriiitPeAetPeAfiijijiiiiijij  which we denote by ( )iig  .   An initial guess to the Newton iteration scheme above, ( )0i , is then obtained by solving  ( )( ) 00 =iig    for the unknown ( )0i ; that is,   ( )injtriiiteAtPiijij−−==−+1~10),0(log1 .    

Black-Karasinski Short Rate Tree Model

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Black-Karasinski Short Rate Tree Model

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