Learn more about Financial Derivatives Toolbox

Introduction

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Financial Instruments Interest-Rate Modeling Rate and Price Trees Viewing Rate or Price Movement with This Toolbox Hedging

Financial Instruments

Financial Derivatives Toolbox software extends the capabilities of Financial Toolbox™ software in the areas of fixed-income derivatives and of securities contingent on interest rates. The toolbox provides components for analyzing individual financial derivative instruments and portfolios. Specifically, it provides the necessary functions for calculating prices and sensitivities, for hedging, and for visualizing results. The toolbox provides a set of functions that perform computations on portfolios containing up to nine types of interest-rate based financial instruments.

Bond.   A long-term debt security with preset interest rate and maturity, by which the principal and interest must be paid.

Bond Option.   Puts and calls on portfolios of bonds. The toolbox supports three types of put and call options on bonds:

• American option: An option that can be exercised any time until its expiration date.

• European option: An option that can be exercised only on its expiration date.

• Bermuda option: A Bermuda option is somewhat like a hybrid of American and European options. It can be exercised on predetermined dates only, usually once a month.

Bond with Embedded Options.   Use bondbyhw to price bonds for a HW tree as well as optembndbyhw to price bonds with embedded options for a Hull-White interest-rate tree.

Fixed-Rate Note.   A long-term debt security with preset interest rate and maturity, by which the interest must be paid. The principal may or may not be paid at maturity. In this version of Financial Derivatives Toolbox software, the principal is always paid at maturity.

Floating-Rate Note.   A security similar to a bond, but in which the note's interest rate is reset periodically, relative to a reference index rate, to reflect fluctuations in market interest rates.

Cap.   A contract that includes a guarantee that sets the maximum interest rate to be paid by the holder, based on an otherwise floating interest rate.

Floor.   A contract that includes a guarantee setting the minimum interest rate to be received by the holder, based on an otherwise floating interest rate.

Swap.   A contract between two parties obligating the parties to exchange future cash flows. This version of Financial Derivatives Toolbox software handles only the vanilla swap, which is composed of a floating-rate leg and a fixed-rate leg.

Additionally, the toolbox provides functions for the creation and pricing of arbitrary cash flow instruments based on zero-coupon bonds or on any of the various interest rate models that the toolbox supports. (See Interest-Rate Modeling.)

Swaption.   An option to enter into an interest rate swap contract. A call swaption allows the option buyer to enter into an interest rate swap in which the buyer of the option pays the fixed rate and receives the floating rate. A put swaption allows the option buyer to enter into an interest rate swap in which the buyer of the option receives the fixed rate and pays the floating rate.

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Interest-Rate Modeling

Financial Derivatives Toolbox software computes prices and sensitivities of interest-rate contingent claims based on several methods of modeling changes in interest rates over time:

• The interest-rate term structure

  This model uses sets of zero-coupon bonds to predict changes in interest rates.

• Heath-Jarrow-Morton (HJM) model

  The HJM model considers a given initial term structure of interest rates and a specification of the volatility of forward rates to build a tree representing the evolution of the interest rates, based on a statistical process.

• Black-Derman-Toy (BDT) model

  In the BDT model, all security prices and rates depend on the short rate (annualized 1-period interest rate). The model uses long rates and their volatilities to construct a tree of possible future short rates. The resulting tree can then be used to determine the value of interest-rate sensitive securities from this tree.

• Hull-White (HW) model

  The Hull-White model incorporates the initial term structure of interest rates and the volatility term structure to build a trinomial recombining tree of short rates. The resulting tree is used to value interest-rate dependent securities. The implementation of the HW model in Financial Derivatives Toolbox software is limited to one factor.

• Black-Karasinski (BK) model

  The BK model is a single-factor, log-normal version of the HW model.

For detailed information about interest-rate models, see:

• Computing Prices and Sensitivities Using the Interest-Rate Term Structure for a discussion of price and sensitivity based on portfolios of zero-coupon bonds

• Computing Prices and Sensitivities Using Interest-Rate Models for a discussion of price and sensitivity based on the HJM and BDT interest-rate models

  Note   Historically, the initial version of Financial Derivatives Toolbox software provided only the HJM interest-rate model. A later version added the BDT model. The current version adds both the HW and BK models. This chapter provides extensive examples of using the HJM and BDT models to compute prices and sensitivities of interest-rate based financial derivatives. The HW and BK tree structures are similar to the BDT tree structure. To avoid needless repetition throughout this chapter, documentation is provided only where significant deviations from the BDT structure exist. Specifically, HW and BK Tree Structures explains the few noteworthy differences among the various formats. If you need more detailed information about functions that use the HW and BK tree structures, see Function Reference, which provides extensive reference information for all functions that compose this toolbox.

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Rate and Price Trees

The interest-rate or price trees supported in this toolbox can be either binomial (two branches per node) or trinomial (3 branches per node). Typically, binomial trees assume that underlying interest rates or prices can only either increase or decrease at each node. Trinomial trees allow for a more complex movement of rates or prices. With trinomial trees the movement of rates or prices at each node is unrestricted (for example, up-up-up or unchanged-down-down).

Types of Trees

Financial Derivatives Toolbox trees can be classified as bushy or recombining. A bushy tree is a tree in which the number of branches increases exponentially relative to observation times; branches never recombine. In this context, a recombining tree is the opposite of a bushy tree. A recombining tree has branches that recombine over time. From any given node, the node reached by taking the path up-down is the same node reached by taking the path down-up. A bushy tree and a recombining binomial tree are illustrated next.

In this toolbox the Heath-Jarrow-Morton model works with bushy trees. The Black-Derman-Toy model, on the other hand, works with recombining binomial trees.

The other two interest rate models supported in this toolbox, Hull-White and Black-Karasinski, work with recombining trinomial trees.

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Viewing Rate or Price Movement with This Toolbox

This toolbox provides the data file deriv.mat that contains four interest-rate based trees:

• HJMTree — A bushy binomial tree

• BDTTree — A recombining binomial tree

• HWTree and BKTree — Recombining trinomial trees

The toolbox also provides the treeviewer function, which graphically displays the shape and data of price, interest rate, and cash flow trees. Viewed with treeviewer, the bushy shape of an HJM tree and the recombining shape of a BDT tree are apparent.

With treeviewer, you can also see the recombining shape of HW and BK trinomial trees.

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Hedging

Financial Derivatives Toolbox software also includes hedging functionality, allowing the rebalancing of portfolios to reach target costs or target sensitivities, which you can set to zero for a neutral-sensitivity portfolio. Optionally, the rebalancing process can be self-financing or directed by a set of user-supplied constraints. For information, see:

• Hedging for a discussion of the hedging process

• hedgeopt for a description of the function that allocates an optimal hedge

• hedgeslf for a description of the function that allocates a self-financing hedge

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Interest-Rate Derivatives

Understanding the Interest-Rate Term Structure

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