Financial Derivatives Toolbox

Using BDT Trees in MATLAB

When working with the BDT model, the Financial Derivatives Toolbox uses trees to represent interest rates, prices, etc. At the highest level, these trees contain several MATLAB structures. The structures encapsulate information needed to interpret completely the information contained in a tree.

Because BDT trees are essentially MATLAB structures, you can examine their contents manually, just as you can for HJM trees. Consider this example, which uses the data in the MAT-file deriv.mat included in the toolbox.

Load the data into the MATLAB workspace.

• load deriv.mat
  

Display the list of the variables loaded from the MAT-file.

• whos
  
  Name               Size         Bytes  Class
  
  BDTInstSet         1x1          22708 struct array
  BDTTree            1x1           5522  struct array
  HJMInstSet         1x1          22700  struct array
  HJMTree            1x1           6318  struct array
  ZeroInstSet        1x1          14442  struct array
  ZeroRateSpec       1x1           1580  struct array
  

Structure of a BDT Tree

You can now examine in some detail the contents of the BDTTree structure.

• BDTTree 
  
  BDTTree = 
  
        FinObj: 'BDTFwdTree'
       VolSpec: [1x1 struct]
      TimeSpec: [1x1 struct]
      RateSpec: [1x1 struct]
          tObs: [0 1 2 3]
          TFwd: {[4x1 double]  [3x1 double]  [2x1 double]  [3]}
        CFlowT: {[4x1 double]  [3x1 double]  [2x1 double]  [4]}
       FwdTree: {1x4 cell}
  

FwdTree contains the actual rate tree. It is represented in MATLAB as a cell array with each cell array element containing a tree level.

The other fields contain other information relevant to interpreting the values in FwdTree. The most important of these are VolSpec, TimeSpec, and RateSpec, which contain the volatility, rate structure, and time structure information respectively.

Look at the RateSpec structure used in generating this tree to see where these values originate. Arrange the values in a single array.

• [BDTTree.RateSpec.StartTimes BDTTree.RateSpec.EndTimes... 
  BDTTree.RateSpec.Rates]
  
  ans =
  
           0    1.0000    0.1000
           0    2.0000    0.1100
           0    3.0000    0.1200
           0    4.0000    0.1250
  

Note    The Financial Derivatives Toolbox uses inverse discount notation for forward rates in the tree. An inverse discount represents a factor by which the present value of an asset is multiplied to find its future value. In general, these forward factors are reciprocals of the discount factors.

Look at the rates in FwdTree. The first node represents the valuation date, tObs = 0. The second node represents tObs = 1. Examine the rates at the second, third and fourth nodes.

• BDTTree.FwdTree{2}  
  
  ans =
  
        1.0979    1.1432
  

The second node represents the first observation time, tObs = 1. This node contains a total of two states, one representing the branch going up (1.0979) and the other representing the branch going down (1.1432).

Note    The convention in this document is to display prices going up on the upper branch. Consequently, when displaying rates, rates are falling on the upper branch and increasing on the lower.

• BDTTree.FwdTree{3}
  
  ans =
  
      1.0976    1.1377    1.1942
  

The third node represents the second observation time, tObs = 2. This node contains a total of three states, one representing the branch going up (1.0976), one representing the branch in the middle (1.1377) and the other representing the branch going down (1.1942).

• BDTTree.FwdTree{4}
  
  ans =
  
      1.0872    1.1183    1.1606    1.2179
  

The fourth node represents the third observation time, tObs = 3. This node contains a total of four states, one representing the branch going up (1.0872), two representing the branches in the middle (1.1183 and 1.1606) and the other representing the branch going down (1.2179).

Verifying Results with treepath

The function treepath isolates a specific node by specifying the path to the node as a vector of branches taken to reach that node. As an example, consider the node reached by starting from the root node, taking the branch up, then the branch down, and finally another branch down. Given that the tree has only two branches per node, branches going up correspond to a 1, and branches going down correspond to a 2. The path up-down-down becomes the vector [1 2 2].

• FRates = treepath(BDTTree.FwdTree, [1 2 2]) 
  
  FRates =
  
      1.1000
      1.0979
      1.1377
      1.1606
  

treepath returns the short rates for all the nodes touched by the path specified in the input argument, the first one corresponding to the root node, and the last one corresponding to the target node.

Graphical View of Interest Rate Tree

The function treeviewer provides a graphical view of the path of interest rates specified in BDTTree. For example, load the file deriv.mat. Here is a treeviewer representation of the rates along several branches of BDTTree.

• treeviewer(BDTTree)
  
  
  

Note    When using treeviewer with BDT trees, you must click on each node in succession from the beginning to the end. Because BDT trees can recombine, treeviewer is unable to compute the path automatically.

A previous example used treepath to find the path of interest rates taking the first branch up and then two branches down the rate tree.

• FRates = treepath(BDTTree.FwdTree, [1 2 2])
  
  FRates =
  
      1.1000
      1.0979
      1.1377
      1.1606
  

The treeviewer function displays the same information obtained by clicking along the sequence of nodes, as shown next.

Black-Derman-Toy Model (BDT)

Pricing and Sensitivity from BDT