Financial Derivatives Toolbox

Black-Derman-Toy Model (BDT)

The Black-Derman-Toy (BDT) model is an analytical model used for pricing interest rate derivatives. The model considers a given initial zero rate term structure of interest rates and a specification of the yield volatilities of long rates to build a tree representing the evolution of the interest rates. For further explanation, see the paper "A One Factor Model of Interest Rates and its Application to Treasury Bond Options" by Fischer Black, Emanuel Derman, and William Toy.

Building a BDT Interest Rate Tree

The BDT interest rate tree represents the evolution of interest rates in a given period of time. This section explains how to create the BDT interest rate tree using the Financial Derivatives Toolbox.

The MATLAB function that creates the BDT interest rate tree is bdttree. This function takes three structures as input arguments:

• The volatility model VolSpec. (See Specifying the Volatility (VolSpec).)
• The interest rate term structure RateSpec. (See Specifying the Interest Rate Term Structure (RateSpec).)
• The tree time layout TimeSpec. (See Specifying the Time Structure (TimeSpec).)

Creating the BDT Interest Rate Tree (bdttree)

Calling the function bdttree creates the structure, BDTTree, containing time and interest rate information of the recombining tree.

This structure is a self-contained unit that includes the BDT tree of rates (found in the FwdTree field), and the volatility, rate, and time specifications used in building this tree.

The calling syntax for bdttree is

• BDTTree = bdttree(VolSpec, RateSpec, TimeSpec)
  

where:

• VolSpec is a structure that specifies the interest rate volatility process. VolSpec is created using the function bdtvolspec.
• RateSpec is the interest rate specification of the initial rate curve. This structure is created with the function intenvset. (See Interest Rate Term Structure.)
• TimeSpec is the tree time layout specification. This variable is created with the function bdttimespec. It represents the mapping between level times and level dates for rate quoting. This structure determines indirectly the number of levels of the tree generated in the call to bdttree.

Specifying the Volatility (VolSpec)

The function bdtvolspec generates the structure VolSpec, which specifies the volatility process. The function requires three input arguments:

• The valuation date ValuationDate
• The yield volatility end dates VolDates
• The yield volatility values VolCurve

An optional fourth argument InterpMethod, specifying the interpolation method, can be included.

The syntax used for calling bdtvolspec is

• VolSpec = bdtvolspec(ValuationDate, VolDates, VolCurve,... 
  InterpMethod)
  

where:

• ValuationDate is the first observation date in the tree.
• VolDates is a vector of dates representing yield volatility end dates.
• VolCurve is a vector of yield volatility values.
• InterpMethod is the method of interpolation to use. The default is 'linear'.

Consider the example

• ValuationDate = datenum('01-01-2000');
  EndDates = datenum(['01-01-2001'; '01-01-2002'; '01-01-2003'; 
  '01-01-2004'; '01-01-2005']);
  Volatility = [.2; .19; .18; .17; .16];
  

Use bdtvolspec to create a volatility specification. Because no interpolation method is explicitly specified, the function uses the 'linear' default.

• BDTVolSpec = bdtvolspec(ValuationDate, EndDates, Volatility)
  
  BDTVolSpec = 
               FinObj: 'BDTVolSpec'
        ValuationDate: 730486
             VolDates: [5x1 double]
             VolCurve: [5x1 double]
      VolInterpMethod: 'linear'
  

Specifying the Interest Rate Term Structure (RateSpec)

The structure RateSpec is an interest term structure that defines the initial interest rate specification from which the tree rates are derived. The section Interest Rate Term Structure explains how to create these structures using the function intenvset, given the interest rates, the starting and ending dates for each rate, and the compounding value.

Consider the example

• Compounding = 1;
  Rates = [0.02; 0.02; 0.02; 0.02];
  StartDates = ['01-Jan-2000';   
                '01-Jan-2001';  
                '01-Jan-2002';  
                '01-Jan-2003'];
  EndDates =   ['01-Jan-2001'; 
                '01-Jan-2002';  
                '01-Jan-2003'; 
                '01-Jan-2004'];
  ValuationDate = '01-Jan-2000';
  
  RateSpec = intenvset('Compounding',1,'Rates', Rates,... 
  'StartDates', StartDates, 'EndDates', EndDates,...
  'ValuationDate', ValuationDate)
  
  RateSpec = 
  
          FinObj: 'RateSpec'
     Compounding: 1
            Disc: [4x1 double]
           Rates: [4x1 double]
        EndTimes: [4x1 double]
      StartTimes: [4x1 double]
        EndDates: [4x1 double]
      StartDates: [4x1 double]
   ValuationDate: 730486
           Basis: 0
    EndMonthRule: 1
  

Use the function datedisp to examine the dates defined in the variable RateSpec. For example,

• datedisp(RateSpec.ValuationDate)
  01-Jan-2000
  

Specifying the Time Structure (TimeSpec)

The structure TimeSpec specifies the time structure for a BDT tree. This structure defines the mapping between the observation times at each level of the tree and the corresponding dates.

TimeSpec is built using the function bdttimespec. The bdttimespec function requires three input arguments:

• The valuation date ValuationDate
• The maturity date Maturity
• The compounding rate Compounding

The syntax used for calling bdttimespec is

• TimeSpec = bdttimespec(ValuationDate, Maturity, Compounding)
  

where:

• ValuationDate is the first observation date in the tree.
• Maturity is a vector of dates representing the cash flow dates of the tree. Any instrument cash flows with these maturities fall on tree nodes.
• Compounding is the frequency at which the rates are compounded when annualized.

Calling bdttimespec with the same data used to create the interest rate term structure, RateSpec builds the structure that specifies the time layout for the tree.

• Maturity = EndDates;
  TimeSpec = bdttimespec(ValuationDate, Maturity, Compounding)
  
  TimeSpec = 
  
         FinObj: 'BDTTimeSpec'
  ValuationDate: 730486
       Maturity: [4x1 double]
    Compounding: 1
          Basis: 0
   EndMonthRule: 1
  

Note that the maturities specified when building TimeSpec do not have to coincide with the EndDates of the rate intervals in RateSpec. Since TimeSpec defines the time-date mapping of the BDT tree, the rates in RateSpec are interpolated to obtain the initial rates with maturities equal to those found in TimeSpec.

Example: Creating a BDT Tree

Use the previously computed values for VolSpec, RateSpec, and TimeSpec as input to the function bdttree to create a BDT Tree.

• BDTTree = bdttree(BDTVolSpec, RateSpec, TimeSpec)
  
  BDTTree = 
  
    FinObj: 'BDTFwdTree'
   VolSpec: [1x1 struct]
  TimeSpec: [1x1 struct]
  RateSpec: [1x1 struct]
      tObs: [0 1.00 2.00 3.00]
      TFwd: {[4x1 double]  [3x1 double]  [2x1 double]  [3.00]}
    CFlowT: {[4x1 double]  [3x1 double]  [2x1 double]  [4.00]}
   FwdTree: {[1.02] [1.02 1.02] [1.01 1.02 1.03] [1.01 1.02 1.02 1.03]}
  

Calculating Prices and Sensitivities

Using BDT Trees in MATLAB