# Documentation

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# bdtsens

Instrument prices and sensitivities from Black-Derman-Toy interest-rate tree

## Syntax

[Delta,Gamma,Vega,Price] = bdtsens(BDTTree,InstSet,Options)

## Arguments

 BDTTree Interest-rate tree structure created by bdttree. InstSet Variable containing a collection of NINST instruments. Instruments are categorized by type. Each type can have different data fields. The stored data field is a row vector or character vector for each instrument. Options (Optional) Derivatives pricing options structure created with derivset.

## Description

[Delta,Gamma,Vega,Price] = bdtsens(BDTTree,InstSet,Options) computes instrument sensitivities and prices for instruments using an interest-rate tree created with the bdttree function. NINST instruments from a financial instrument variable, InstSet, are priced. bdtsens handles instrument types: 'Bond', 'CashFlow', 'OptBond', 'OptEmBond', 'OptEmBond', 'OptFloat', 'OptEmFloat', 'Fixed', 'Float', 'Cap', 'Floor', 'RangeFloat', 'Swap'. See instadd for information on instrument types.

Delta is an NINST-by-1 vector of deltas, representing the rate of change of instrument prices with respect to changes in the interest rate. Delta is computed by finite differences in calls to bdttree. See bdttree for information on the observed yield curve.

Gamma is an NINST-by-1 vector of gammas, representing the rate of change of instrument deltas with respect to the changes in the interest rate. Gamma is computed by finite differences in calls to bdttree.

Vega is an NINST-by-1 vector of vegas, representing the rate of change of instrument prices with respect to the changes in the volatility $\sigma \left(t,T\right)$. Vega is computed by finite differences in calls to bdttree. See bdtvolspec for information on the volatility process.

 Note   All sensitivities are returned as dollar sensitivities. To find the per-dollar sensitivities, divide by the respective instrument price.

Price is an NINST-by-1 vector of prices of each instrument. The prices are computed by backward dynamic programming on the interest-rate tree. If an instrument cannot be priced, NaN is returned.

Delta and Gamma are calculated based on yield shifts of 100 basis points. Vega is calculated based on a 1% shift in the volatility process.

## Examples

collapse all

Load the tree and instruments from the deriv.mat data file.

BDTSubSet = instselect(BDTInstSet,'Type', {'Bond', 'Cap'});

instdisp(BDTSubSet)
Index Type CouponRate Settle         Maturity       Period Basis EndMonthRule IssueDate FirstCouponDate LastCouponDate StartDate Face Name     Quantity
1     Bond 0.1        01-Jan-2000    01-Jan-2003    1      NaN   NaN          NaN       NaN             NaN            NaN       NaN  10% Bond 100
2     Bond 0.1        01-Jan-2000    01-Jan-2004    2      NaN   NaN          NaN       NaN             NaN            NaN       NaN  10% Bond  50

Index Type Strike Settle         Maturity       CapReset Basis Principal Name    Quantity
3     Cap  0.15   01-Jan-2000    01-Jan-2004    1        NaN   NaN       15% Cap 30

Compute Delta and Gamma for the cap and bond instruments contained in the instrument set.

[Delta, Gamma] = bdtsens(BDTTree, BDTSubSet)
Warning: Not all cash flows are aligned with the tree. Result will be approximated.
Delta =

-232.6681
-281.0517
63.8102

Gamma =

1.0e+03 *

0.8037
1.1819
1.8535