Pricing Using Interest-Rate Term Structure

Introduction

The instruments can be presented to the functions as a portfolio of different types of instruments or as groups of instruments of the same type. The current version of the toolbox can compute price and sensitivities for five instrument types of using interest-rate curves:

• Bonds

• Fixed-rate notes

• Floating-rate notes

• Swaps

• OAS for callable and puttable bonds

• Agency OAS

In addition to these instruments, the toolbox also supports the calculation of price and sensitivities of arbitrary sets of cash flows.

Options and interest-rate floors and caps are absent from the above list of supported instruments. These instruments are not supported because their pricing and sensitivity function require a stochastic model for the evolution of interest rates. The interest-rate term structure used for pricing is treated as deterministic, and as such is not adequate for pricing these instruments.

Financial Instruments Toolbox™ also contains functions that use the Heath-Jarrow-Morton (HJM) and Black-Derman-Toy (BDT) models to compute prices and sensitivities for financial instruments. These models support computations involving options and interest-rate floors and caps. See Pricing Using Interest-Rate Tree Models for information on computing price and sensitivities of financial instruments using the HJM and BDT models.

Computing Instrument Prices

The main function used for pricing portfolios of instruments is intenvprice. This function works with the family of functions that calculate the prices of individual types of instruments. When called, intenvprice classifies the portfolio contained in InstSet by instrument type, and calls the appropriate pricing functions. The map between instrument types and the pricing function intenvprice calls is

bondbyzero:	Price a bond by a set of zero curves
fixedbyzero:	Price a fixed-rate note by a set of zero curves
floatbyzero:	Price a floating-rate note by a set of zero curves
swapbyzero:	Price a swap by a set of zero curves

You can use each of these functions individually to price an instrument. Consult the reference pages for specific information on using these functions.

intenvprice takes as input an interest-rate term structure created with intenvset, and a portfolio of interest-rate contingent derivatives instruments created with instadd.

The syntax for using intenvprice to price an entire portfolio is

Price = intenvprice(RateSpec, InstSet)

where:

• RateSpec is the interest-rate term structure.

• InstSet is the name of the portfolio.

Example: Pricing a Portfolio of Instruments

Consider this example of using the intenvprice function to price a portfolio of instruments supplied with Financial Instruments Toolbox software.

The provided MAT-file deriv.mat stores a portfolio as an instrument set variable ZeroInstSet. The MAT-file also contains the interest-rate term structure ZeroRateSpec. You can display the instruments with the function instdisp.

load deriv.mat;
instdisp(ZeroInstSet)

Index Type CouponRate Settle         Maturity       Period Basis...
1     Bond 0.04       01-Jan-2000    01-Jan-2003    1      NaN... 
2     Bond 0.04       01-Jan-2000    01-Jan-2004    2      NaN... 
 
Index Type  CouponRate Settle      Maturity    FixedReset Basis...
3     Fixed 0.04       01-Jan-2000 01-Jan-2003 1          NaN... 
 
Index Type  Spread Settle         Maturity       FloatReset Basis...
4     Float 20     01-Jan-2000    01-Jan-2003    1          NaN... 
 
Index Type LegRate    Settle         Maturity       LegReset Basis...
5     Swap [0.06 20]  01-Jan-2000    01-Jan-2003    [1  1]   NaN...

Use intenvprice to calculate the prices for the instruments contained in the portfolio ZeroInstSet.

format bank
Prices = intenvprice(ZeroRateSpec, ZeroInstSet)

Prices =

         98.72
         97.53
         98.72
        100.55
          3.69

The output Prices is a vector containing the prices of all the instruments in the portfolio in the order indicated by the Index column displayed by instdisp. So, the first two elements in Prices correspond to the first two bonds; the third element corresponds to the fixed-rate note; the fourth to the floating-rate note; and the fifth element corresponds to the price of the swap.

Computing Instrument Sensitivities

In general, you can compute sensitivities either as dollar price changes or as percentage price changes. The toolbox reports all sensitivities as dollar sensitivities.

Using the interest-rate term structure, you can calculate two types of derivative price sensitivities, delta and gamma. Delta represents the dollar sensitivity of prices to shifts in the observed forward yield curve. Gamma represents the dollar sensitivity of delta to shifts in the observed forward yield curve.

The intenvsens function computes instrument sensitivities and instrument prices. If you need both the prices and sensitivity measures, use intenvsens. A separate call to intenvprice is not required.

Here is the syntax

[Delta, Gamma, Price] = intenvsens(RateSpec, InstSet)

where, as before:

• RateSpec is the interest-rate term structure.

• InstSet is the name of the portfolio.

Example: Sensitivities and Prices

Here is an example that uses intenvsens to calculate both sensitivities and prices.

format bank
load deriv.mat;
[Delta, Gamma, Price] = intenvsens(ZeroRateSpec, ZeroInstSet);

Display the results in a single matrix in bank format.

All = [Delta Gamma Price]

All =

       -272.64       1029.84       98.72
       -347.44       1622.65       97.53
       -272.64       1029.84       98.72
         -1.04          3.31      100.55
       -282.04       1059.62        3.69

To view the per-dollar sensitivity, divide the first two columns by the last one.

[Delta./Price, Gamma./Price, Price]

ans =

         -2.76         10.43         98.72
         -3.56         16.64         97.53
         -2.76         10.43         98.72
         -0.01          0.03        100.55
        -76.39        286.98          3.69

OAS for Callable and Puttable Bonds

Option Adjusted Spread (OAS) is a useful way to value and compare securities with embedded options, like callable or puttable bonds. Basically, when the constant or flat spread is added to the interest-rate curve/rates in the tree, the pricing model value equals the market price. Financial Instruments Toolbox supports pricing American, European, and Bermuda callable and puttable bonds using different interest rate models. The pricing for a bond with embedded options is:

• For a callable bond, where the holder has bought a bond and sold a call option to the issuer:

  Price callable bond = Price Option free bond − Price call option

• For a puttable bond, where the holder has bought a bond and a put option:

  Price puttable bond = Price Option free bond + Price put option

There are two additional sensitivities related to OAS for bonds with embedded options: Option Adjusted Duration and Option Adjusted Convexity. These are similar to the concepts of modified duration and convexity for option-free bonds. The measure Duration is a general term that describes how sensitive a bond’s price is to a parallel shift in the yield curve. Modified Duration and Modified Convexity assume that the bond’s cash flows do not change when the yield curve shifts. This is not true for OA Duration or OA Convexity because the cash flows may change due to the option risk component of the bond.

Function	Purpose
oasbybdt	Compute OAS using a BDT model.
oasbybk	Compute OAS using a BK model.
oasbyhjm	Compute OAS using an HJM model.
oasbyhw	Compute OAS using an HW model.

Agency OAS

Often bonds are issued with embedded options, which then makes standard price/yield or spread measures irrelevant. For example, a municipality concerned about the chance that interest rates may fall in the future might issue bonds with a provision that allows the bond to be repaid before the bond’s maturity. This is a call option on the bond and must be incorporated into the valuation of the bond. Option-adjusted spread (OAS), which adjusts a bond spread for the value of the option, is the standard measure for valuing bonds with embedded options. Financial Instruments Toolbox supports computing option-adjusted spreads for bonds with single embedded options using the agency model.

The Securities Industry and Financial Markets Association (SIFMA) has a simplified approach to compute OAS for agency issues (Government Sponsored Entities like Fannie Mae and Freddie Mac) termed “Agency OAS.” In this approach, the bond has only one call date (European call) and uses Black’s model (see The BMA European Callable Securities Formula at https://www.sifma.org) to value the bond option. The price of the bond is computed as follows:

Price_Callable = Price_NonCallable – Price_Option

where

Price_Callable is the price of the callable bond.

Price_NonCallable is the price of the noncallable bond, that is, price of the bond using bndspread.

Price_Option is the price of the option, that is, price of the option using Black’s model.

The Agency OAS is the spread, when used in the previous formula, yields the market price. Financial Instruments Toolbox supports these functions:

Agency OAS

Agency OAS Functions	Purpose
agencyoas	Compute the OAS of the callable bond using the Agency OAS model.
agencyprice	Price the callable bond OAS using the Agency OAS model.

For more information on agency OAS, see Agency Option-Adjusted Spreads.