Financial Derivatives Toolbox

Calculating Prices and Sensitivities

The function hjmsens computes the delta, gamma, and vega sensitivities of instruments using an interest rate tree created with hjmtree. It also optionally returns the calculated price for each instrument. hjmsens requires the same two input arguments used by hjmprice, namely HJMTree and InstSet.

hjmsens calculates the dollar value of delta and gamma by shifting the observed forward yield curve by 100 basis points in each direction, and the dollar value of vega by shifting the volatility process by 1%. To obtain the per-dollar value of the sensitivities, divide the dollar sensitivity by the price of the corresponding instrument.

The calling syntax for the function is

[Delta, Gamma, Vega, Price] = hjmsens(HJMTree, HJMInstSet)

Use the previous example data to calculate the price of instruments.

load deriv.mat
[Delta, Gamma, Vega, Price] = hjmsens(HJMTree, HJMInstSet);
Warning: Not all cash flows are aligned with the tree. Result will 
be approximated.

Note    The warning appears because some of the cash flows for the second bond do not fall exactly on a tree node. This situation was discussed in "HJM Pricing Options Structure" on page 1-56.

The sensitivities and the prices can be examined conveniently by arranging them into a single matrix.

All = [Delta, Gamma, Vega, Price]

All =

  1.0e+003 *

   -0.2997    1.1603   -0.0000    0.1058
   -0.3958    1.9012   -0.0003    0.1077
   -0.0818    0.5566    0.0610    0.0073
   -0.2997    1.1603   -0.0000    0.1058
   -0.0011    0.0037         0    0.1006
    0.1085   -0.6975    0.1665    0.0154
   -0.1741    0.9569    0.1615    0.0154
   -0.2986    1.1567   -0.0000    0.0052

As with the prices, each row of the sensitivity vectors corresponds to the similarly indexed instrument in HJMInstSet. To view the per-dollar sensitivities, divide each dollar sensitivity by the corresponding instrument price.

All = [Delta ./ Price, Gamma ./ Price, Vega ./ Price, Price]

All =

   -2.8337   10.9706   -0.0000  105.7678
   -3.6759   17.6566   -0.0026  107.6773
  -11.1729   76.0234    8.3312    7.3217
   -2.8337   10.9706   -0.0000  105.7678
   -0.0112    0.0364         0  100.5768
    7.0273  -45.1870   10.7873   15.4367
  -11.3117   62.1593   10.4911   15.3938
  -57.5215  222.8243   -0.0000    5.1910

Pricing and Sensitivity from HJM

Hedging