Financial Derivatives Toolbox

Pricing and Sensitivity from HJM

This section explains how to use the Financial Derivatives Toolbox to compute prices and sensitivities of several financial instruments using the Heath-Jarrow-Morton (HJM) model. For information, see:

• Pricing and the Price Tree for a discussion of using the hjmprice function to compute prices for a portfolio of instruments.
• Calculating Prices and Sensitivities for a discussion of using the hjmsens function to compute delta, gamma, and vega portfolio sensitivities.

Pricing and the Price Tree

For the HJM model, the function hjmprice calculates the price of any set of supported instruments, based on an interest rate tree. The function is capable of pricing these instrument types:

• Bonds
• Bond options
• Arbitrary cash flows
• Fixed-rate notes
• Floating-rate notes
• Caps
• Floors
• Swaps

The syntax used for calling hjmprice is

• [Price, PriceTree] = hjmprice(HJMTree, InstSet, Options)
  

This function requires two input arguments: the interest rate tree, HJMTree, and the set of instruments, InstSet. An optional argument Options further controls the pricing and the output displayed.

HJMTree is the Heath-Jarrow-Morton tree sampling of a forward rate process, created using hjmtree. See Building an HJM Forward Rate Tree to learn how to create this structure.

InstSet is the set of instruments to be priced. This structure represents the set of instruments to be priced independently using the HJM model. The section Creating and Managing Instrument Portfolios explains how to create this variable.

Options is an options structure created with the function derivset. This structure defines how the HJM tree is used to find the price of instruments in the portfolio, and how much additional information is displayed in the command window when calling the pricing function. If this input argument is not specified in the call to hjmprice, a default Options structure is used.

hjmprice classifies the instruments and calls the appropriate pricing function for each one of the instrument types. The pricing functions are bondbyhjm, cfbyhjm, fixedbyhjm, floatbyhjm, optbndbyhjm, and swapbyhjm. You can also use these functions directly to calculate the price of sets of instruments of the same type. See the documentation for these individual functions for further information.

Example: HJM Pricing

Consider the following example, which uses the portfolio and interest rate data in the MAT-file deriv.mat included in the toolbox. Load the data into the MATLAB workspace.

• load deriv.mat
  

Use the MATLAB whos command to display a 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
  

HJMTree and HJMInstSet are the input arguments needed to call the function hjmprice.

Use the function instdisp to examine the set of instruments contained in the variable HJMInstSet.

• instdisp(HJMInstSet)
  
  
  

  Index Type CouponRate Settle       Maturity     Period Basis ......Name      Quantity 1     Bond 0.04       01-Jan-2000  01-Jan-2003  1      NaN.........4% bond   100 2  Bond 0.04       01-Jan-2000  01-Jan-2004  2      NaN.........4% bond  50 Index Type    UnderInd OptSpec Strike ExerciseDates AmericanOpt Name        Quantity 3 OptBond 2 call 101 01-Jan-2003 NaN Option 101  -50 Index Type  CouponRate Settle      Maturity    FixedReset Basis Principal Name     Quantity 4     Fixed 0.04       01-Jan-2000 01-Jan-2003 1          NaN NaN 4% Fixed 80 Index Type Spread Settle      Maturity    FloatReset  Basis Principal Name       Quantity 5 Float 20     01-Jan-2000 01-Jan-2003 1           NaN NaN       20BP Float 8 Index Type Strike Settle Maturity CapReset Basis Principal Name Quantity 6 Cap 0.03 01-Jan-2000 01-Jan-2004 1 NaN NaN 3% Cap 30 Index Type Strike Settle      Maturity       FloorReset Basis Principal Name     Quantity 7 Floor 0.03   01-Jan-2000 01-Jan-2004 1          NaN NaN 3% Floor 40 Index Type LegRate   Settle      Maturity     LegReset Basis Principal LegType  Name         Quantity 8 Swap [0.06 20] 01-Jan-2000 01-Jan-2003  [1 1]   NaN NaN      [NaN]    6%/20BP Swap  10

Note that there are eight instruments in this portfolio set: two bonds, one bond option, one fixed rate note, one floating rate note, one cap, one floor, and one swap. Each instrument has a corresponding index that identifies the instrument prices in the price vector returned by hjmprice.

Now use hjmprice to calculate the price of each instrument in the instrument set.

• Price = hjmprice(HJMTree, HJMInstSet)
  Warning: Not all cash flows are aligned with the tree. Result will 
  be approximated.
  
  Price =
  
     98.7159
     97.5280
      0.0486
     98.7159
    100.5529
      6.2831
      0.0486
      3.6923
  

Note    The warning shown above appears because some of the cash flows for the second bond do not fall exactly on a tree node. This situation is discussed further in HJM Pricing Options Structure.

Price Vector

The prices in the output vector Price correspond to the prices at observation time zero (tObs = 0), which is defined as the valuation date of the interest rate tree. The instrument indexing within Price is the same as the indexing within InstSet. In this example, the prices in the Price vector correspond to the instruments in the following order.

• InstNames = instget(HJMInstSet, 'FieldName','Name')
  
  InstNames =
  
  4% bond     
  4% bond     
  Option 101  
  4% Fixed    
  20BP Float  
  3% Cap      
  3% Floor    
  6%/20BP Swap
  

Consequently, in the Price vector, the fourth element, 98.7159, represents the price of the fourth instrument (4% fixed-rate note); the sixth element, 6.2831, represents the price of the sixth instrument (3% cap).

Price Tree Structure

If you call the hjmprice function with two output arguments, e.g.,

• [Price, PriceTree] = hjmprice(HJMTree, HJMInstSet)
  

you generate a price tree along with the price information.

The optional output price tree structure PriceTree holds all the pricing information. The first field of this structure, FinObj, indicates that this structure represents a price tree. The second field, PBush is the tree holding the price of the instruments in each node of the tree. The third field, AIBush is the tree holding the accrued interest of the instruments in each node of the tree. Finally, the fourth field, tObs, represents the observation time of each level of PBush and AIBush, with units in terms of compounding periods.

In this example the price tree looks like

• PriceTree = 
  
  FinObj: 'HJMPriceTree'
   PBush: {[8x1 double]  [8x1x2 double]  ...[8x8 double]}
  AIBush: {[8x1 double]  [8x1x2 double] ... [8x8 double]}
    tObs: [0 1 2 3 4]
  

Both PBush and AIBush are actually 1-by-5 cell arrays, consistent with the five observation times of tObs. The data display has been shortened here to fit on a single line.

Using the command line interface, you can directly examine PriceTree.PBush, the field within the PriceTree structure that contains the price tree with the price vectors at every state. The first node represents tObs = 0, corresponding to the valuation date.

• PriceTree.PBush{1}
  
  ans =
  
     98.7159
     97.5280
      0.0486
     98.7159
    100.5529
      6.2831
      0.0486
      3.6923
  

With this interface you can observe the prices for all instruments in the portfolio at a specific time.

Using HJM Trees in MATLAB

Using treeviewer to View Instrument Prices Through Time