MathWorks. Financial Instruments Toolbox™ User's Guide

4124 Pages·2022·17.195 MB·English

by  The MathWorks, Inc.

Financial Instruments Toolbox™
User's Guide
R2022b
How to Contact MathWorks
Latest news: www.mathworks.com
Sales and services: www.mathworks.com/sales_and_services
User community: www.mathworks.com/matlabcentral
Technical support: www.mathworks.com/support/contact_us
Phone: [phone]
The MathWorks, Inc.
1 Apple Hill Drive
Natick, MA 01760-2098
Financial Instruments Toolbox™ User's Guide
© COPYRIGHT 2012–2022 by The MathWorks, Inc.
The software described in this document is furnished under a license agreement. The software may be used or copied
only under the terms of the license agreement. No part of this manual may be photocopied or reproduced in any form
without prior written consent from The MathWorks, Inc.
FEDERAL ACQUISITION: This provision applies to all acquisitions of the Program and Documentation by, for, or through
the federal government of the United States. By accepting delivery of the Program or Documentation, the government
hereby agrees that this software or documentation qualifies as commercial computer software or commercial computer
software documentation as such terms are used or defined in FAR 12.212, DFARS Part 227.72, and DFARS 252.227-7014.
Accordingly, the terms and conditions of this Agreement and only those rights specified in this Agreement, shall pertain
to and govern the use, modification, reproduction, release, performance, display, and disclosure of the Program and
Documentation by the federal government (or other entity acquiring for or through the federal government) and shall
supersede any conflicting contractual terms or conditions. If this License fails to meet the government's needs or is
inconsistent in any respect with federal procurement law, the government agrees to return the Program and
Documentation, unused, to The MathWorks, Inc.
Trademarks
MATLAB and Simulink are registered trademarks of The MathWorks, Inc. See
www.mathworks.com/trademarks for a list of additional trademarks. Other product or brand names may be
trademarks or registered trademarks of their respective holders.
Patents
MathWorks products are protected by one or more U.S. patents. Please see www.mathworks.com/patents for
more information.
Revision History
September 2012 Online only Version 1.0 (Release 2012b)
March 2013 Online only Version 1.1 (Release 2013a)
September 2013 Online only Version 1.2 (Release 2013b)
March 2014 Online only Version 1.3 (Release 2014a)
October 2014 Online only Version 2.0 (Release 2014b)
March 2015 Online only Version 2.1 (Release 2015a)
September 2015 Online only Version 2.2 (Release 2015b)
March 2016 Online only Version 2.3 (Release 2016a)
September 2016 Online only Version 2.4 (Release 2016b)
March 2017 Online only Version 2.5 (Release 2017a)
September 2017 Online only Version 2.6 (Release 2017b)
March 2018 Online only Version 2.7 (Release 2018a)
September 2018 Online only Version 2.8 (Release 2018b)
March 2019 Online only Version 2.9 (Release 2019a)
September 2019 Online only Version 2.10 (Release 2019b)
March 2020 Online only Version 3.0 (Release 2020a)
September 2020 Online only Version 3.1 (Release 2020b)
March 2021 Online only Version 3.2 (Release 2021a)
September 2021 Online only Version 3.3 (Release 2021b)
March 2022 Online only Version 3.4 (Release 2022a)
September 2022 Online only Version 3.5 (Release 2022b)
Contents
Getting Started
1
Financial Instruments Toolbox Product Description . . . . . . . . . . . . . . . . . 1-2
Interest-Rate-Based Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-3
Equity-Based Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-4
Expected Users . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-5
Portfolio Creation Using Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-6
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-6
Interest-Rate-Based Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-6
Equity Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-7
Adding Instruments to an Existing Portfolio Using Functions . . . . . . . . . 1-8
Pricing a Portfolio Using the Black-Derman-Toy Model . . . . . . . . . . . . . . 1-10
Instrument Construction and Portfolio Management Using Functions . 1-15
Instrument Constructors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-15
Creating Instruments or Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-16
Searching or Subsetting a Portfolio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-17
Get Started with Workflows Using Object-Based Framework for Pricing
Financial Instruments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-22
Workflow to Price an Interest-Rate Instrument . . . . . . . . . . . . . . . . . . . . 1-25
Price Vanilla Fixed Bond Instrument Using ratecurve and Discount Pricer
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-25
Workflow to Price an Inflation Instrument . . . . . . . . . . . . . . . . . . . . . . . . 1-28
Analyze Inflation-Indexed Instruments . . . . . . . . . . . . . . . . . . . . . . . . . . 1-28
Workflow to Price an Equity, Commodity, or FX Instrument . . . . . . . . . . 1-37
Price Vanilla Instrument Using Black-Scholes Model and Black-Scholes
Pricer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-37
Workflow to Price a Credit Derivative Instrument . . . . . . . . . . . . . . . . . . 1-40
Price CDS Instrument Using Default Probability Curve and Credit Pricer
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-40
Workflow to Create and Price a Portfolio of Instruments . . . . . . . . . . . . 1-42
Create and Price Portfolio of Instruments . . . . . . . . . . . . . . . . . . . . . . . 1-42
iii
Workflow for Creating and Analyzing a ratecurve and parametercurve
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-46
Convert RateSpec to a ratecurve Object . . . . . . . . . . . . . . . . . . . . . . . . . 1-49
Workflow for Creating and Analyzing a defprobcurve . . . . . . . . . . . . . . . 1-51
Choose Instruments, Models, and Pricers . . . . . . . . . . . . . . . . . . . . . . . . . 1-53
Interest-Rate Instruments with Associated Models and Pricers . . . . . . . . 1-53
Equity, Commodity, FX, and Energy Instruments with Associated Models and
Pricers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-56
Inflation Instruments with Associated Models and Pricers . . . . . . . . . . . . 1-60
Credit Derivative Instruments with Associated Models and Pricers . . . . . 1-61
Supported Exercise Styles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-62
Mapping Financial Instruments Toolbox Functions to Object-Based
Framework for Instruments, Models, and Pricers . . . . . . . . . . . . . . . . 1-70
Mapping Financial Instruments Toolbox Functions for Interest-Rate
Instrument Objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-73
Mapping Financial Instruments Toolbox Functions for Equity, Commodity,
FX Instrument Objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-84
Mapping Financial Instruments Toolbox Functions for Credit Derivative
Instrument Objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-94
Mapping Financial Instruments Toolbox Curve Functions to Object-Based
Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-95
Price European Vanilla Call Options Using Black-Scholes Model and
Different Equity Pricers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-97
Price Weather Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-103
Interest-Rate Derivatives
2
Supported Interest-Rate Instrument Functions . . . . . . . . . . . . . . . . . . . . . 2-3
Bond . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-3
Convertible Bond . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-4
Stepped Coupon Bonds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-5
Sinking Fund Bonds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-5
Bonds with an Amortization Schedule . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-6
Bond Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-6
Bond with Embedded Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-7
Stepped Coupon Bonds with Calls and Puts . . . . . . . . . . . . . . . . . . . . . . . 2-8
Sinking Fund Bonds with an Embedded Option . . . . . . . . . . . . . . . . . . . . . 2-8
Amortizing Callable or Puttable Bond . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-9
Fixed-Rate Note . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-9
Floating-Rate Note . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-10
Floating-Rate Note with an Amortization Schedule . . . . . . . . . . . . . . . . . 2-10
iv Contents
Floating-Rate Note with Caps, Collars, and Floors . . . . . . . . . . . . . . . . . 2-10
Floating-Rate Note Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-11
Floating-Rate Note with Embedded Options . . . . . . . . . . . . . . . . . . . . . . 2-11
Cap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-12
Floor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-12
Range Note . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-13
Swap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-13
Swap with an Amortization Schedule . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-14
Forward Swap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-14
Swaption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-14
Bond Futures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-15
Work with Negative Interest Rates Using Functions . . . . . . . . . . . . . . . . 2-18
Interest-Rate Modeling Options for Negative Rates . . . . . . . . . . . . . . . . 2-18
Modeling Negative Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-18
Work with Negative Interest Rates Using Objects . . . . . . . . . . . . . . . . . . 2-22
Interest-Rate Modeling Options for Negative Rates . . . . . . . . . . . . . . . . 2-22
Modeling Negative Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-22
Price Swaptions with Negative Strikes Using the Shifted SABR Model
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-26
Calibrate the SABR Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-33
Load Market Implied Black Volatility Data . . . . . . . . . . . . . . . . . . . . . . . 2-33
Method 1: Calibrate Alpha, Rho, and Nu Directly . . . . . . . . . . . . . . . . . . 2-33
Method 2: Calibrate Rho and Nu by Implying Alpha from At-The-Money
Volatility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-34
Use the Calibrated Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-35
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-36
Price a Swaption Using the SABR Model . . . . . . . . . . . . . . . . . . . . . . . . . . 2-38
Overview of Interest-Rate Tree Models . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-44
Interest-Rate Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-44
Rate and Price Trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-45
Viewing Rate or Price Movement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-45
Understanding the Interest-Rate Term Structure . . . . . . . . . . . . . . . . . . 2-48
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-48
Interest Rates Versus Discount Factors . . . . . . . . . . . . . . . . . . . . . . . . . . 2-48
Interest-Rate Term Conversions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-53
Spot Curve to Forward Curve Conversion . . . . . . . . . . . . . . . . . . . . . . . . 2-53
Alternative Syntax (ratetimes) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-54
Modeling the Interest-Rate Term Structure . . . . . . . . . . . . . . . . . . . . . . . 2-57
Creating or Modifying (intenvset) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-57
Obtaining Specific Properties (intenvget) . . . . . . . . . . . . . . . . . . . . . . . . 2-58
Pricing Using Interest-Rate Term Structure . . . . . . . . . . . . . . . . . . . . . . . 2-61
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-61
Computing Instrument Prices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-61
Computing Instrument Sensitivities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-63
OAS for Callable and Puttable Bonds . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-64
v
Agency OAS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-64
Understanding Interest-Rate Tree Models . . . . . . . . . . . . . . . . . . . . . . . . 2-66
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-66
Building a Tree of Forward Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-66
Specifying the Volatility Model (VolSpec) . . . . . . . . . . . . . . . . . . . . . . . . 2-68
Specifying the Interest-Rate Term Structure (RateSpec) . . . . . . . . . . . . . 2-70
Specifying the Time Structure (TimeSpec) . . . . . . . . . . . . . . . . . . . . . . . 2-70
Creating Trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-72
Examining Trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-72
Pricing Using Interest-Rate Tree Models . . . . . . . . . . . . . . . . . . . . . . . . . 2-81
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-81
Computing Instrument Prices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-81
Computing Instrument Sensitivities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-89
HJM Sensitivities Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-89
BDT Sensitivities Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-89
Calibrating Hull-White Model Using Market Data . . . . . . . . . . . . . . . . . . 2-92
Hull-White Model Calibration Example . . . . . . . . . . . . . . . . . . . . . . . . . . 2-92
Interest-Rate Derivatives Using Closed-Form Solutions . . . . . . . . . . . . . 2-99
Pricing Caps and Floors Using the Black Option Model . . . . . . . . . . . . . . 2-99
Price Swaptions with Interest-Rate Models Using Simulation . . . . . . . 2-100
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-100
Construct a Zero Curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-100
Define Swaption Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-102
Compute the Black Model and the Swaption Volatility Matrix . . . . . . . . 2-102
Select Calibration Instruments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-102
Compute Swaption Prices Using Black's Model . . . . . . . . . . . . . . . . . . . 2-102
Define Simulation Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-103
Simulate Interest-Rate Paths Using the Hull-White One-Factor Model . . 2-103
Simulate Interest-Rate Paths Using the Linear Gaussian Two-Factor Model
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-106
Simulate Interest-Rate Paths Using the LIBOR Market Model . . . . . . . . 2-108
Compare Interest-Rate Modeling Results . . . . . . . . . . . . . . . . . . . . . . . 2-112
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-113
Pricing Bermudan Swaptions with Monte Carlo Simulation . . . . . . . . . 2-114
Managing Interest-Rate Risk with Bond Futures . . . . . . . . . . . . . . . . . . 2-125
Analyze Inflation-Indexed Instruments . . . . . . . . . . . . . . . . . . . . . . . . . . 2-132
Bootstrapping a Swap Curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-141
Fitting Interest-Rate Curve Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-144
Fitting the Diebold Li Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-150
Calibrating Caplets Using the Normal (Bachelier) Model . . . . . . . . . . . 2-155
Calibrating Floorlets Using the Normal (Bachelier) Model . . . . . . . . . . 2-159
vi Contents
Calibrate the SABR Model Using Normal (Bachelier) Volatilities with
Negative Strikes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-163
Calibrate Shifted SABR Model Parameters for Swaption Instrument 2-167
Price Portfolio of Bond and Bond Option Instruments . . . . . . . . . . . . . 2-172
Calibrate SABR Model Using Normal (Bachelier) Volatilities with Analytic
Pricer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-177
Calibrate SABR Model Using Analytic Pricer . . . . . . . . . . . . . . . . . . . . . 2-181
Price a Swaption Using SABR Model and Analytic Pricer . . . . . . . . . . . 2-185
Compute LIBOR Fallback . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-192
Use treeviewer to Examine HWTree and PriceTree When Pricing
European Callable Bond . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-194
Select Cheapest-to-Deliver Bond Using BondFuture Instrument . . . . . 2-212
Graphical Representation of Trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-219
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-219
Observing Interest Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-219
Observing Instrument Prices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-222
Basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-228
Equity Derivatives
3
Understanding Equity Trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-2
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-2
Building Equity Binary Trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-3
Building Implied Trinomial Trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-6
Building Standard Trinomial Trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-11
Examining Equity Trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-14
Differences Between CRR and EQP Tree Structures . . . . . . . . . . . . . . . . 3-17
Supported Equity Derivative Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-19
Asian Option . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-19
Barrier Option . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-20
Double Barrier Option . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-21
Basket Option . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-22
Chooser Option . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-23
Compound Option . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-23
Convertible Bond . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-24
Lookback Option . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-25
Digital Option . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-26
Rainbow Option . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-27
Vanilla Option . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-27
Spread Option . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-30
vii
One-Touch and Double One-Touch Options . . . . . . . . . . . . . . . . . . . . . . . 3-30
Forwards Option . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-31
Futures Option . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-32
Supported Energy Derivative Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 3-34
Asian Option . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-34
Barrier Option . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-35
Double Barrier Option . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-36
Vanilla Option . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-37
Spread Option . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-38
Lookback Option . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-39
Forwards Option . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-40
Futures Option . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-41
Pricing Swing Options Using the Longstaff-Schwartz Method . . . . . . . . 3-43
Simulating Electricity Prices with Mean-Reversion and Jump-Diffusion
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-53
Pricing Equity Derivatives Using Trees . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-64
Computing Instrument Prices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-64
Computing Prices Using CRR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-65
Computing Prices Using EQP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-66
Computing Prices Using ITT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-68
Computing Prices Using STT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-69
Examining Output from the Pricing Functions . . . . . . . . . . . . . . . . . . . . . 3-70
Graphical Representation of Equity Derivative Trees . . . . . . . . . . . . . . . 3-73
Computing Equity Instrument Sensitivities . . . . . . . . . . . . . . . . . . . . . . . 3-75
CRR Sensitivities Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-75
ITT Sensitivities Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-76
Equity Derivatives Using Closed-Form Solutions . . . . . . . . . . . . . . . . . . . 3-79
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-79
Black-Scholes Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-79
Black Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-80
Roll-Geske-Whaley Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-80
Bjerksund-Stensland 2002 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-81
Barone-Adesi-Whaley Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-81
Pricing Using the Black-Scholes Model . . . . . . . . . . . . . . . . . . . . . . . . . . 3-82
Pricing Using the Black Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-83
Pricing Using the Roll-Geske-Whaley Model . . . . . . . . . . . . . . . . . . . . . . 3-84
Pricing Using the Bjerksund-Stensland Model . . . . . . . . . . . . . . . . . . . . . 3-84
Compute American Option Prices Using the Barone-Adesi and Whaley
Option Pricing Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-86
Pricing European Call Options Using Different Equity Models . . . . . . . . 3-88
Compute the Option Price on a Future . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-95
Pricing European and American Spread Options . . . . . . . . . . . . . . . . . . . 3-97
Pricing Asian Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-110
viii Contents
Price Spread Instrument for a Commodity Using Black-Scholes Model and
Analytic Pricers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-123
Price Vanilla Instrument Using Heston Model and Multiple Different
Pricers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-125
Create and Price Portfolio of Instruments . . . . . . . . . . . . . . . . . . . . . . . 3-131
Use Black-Scholes Model to Price Asian Options with Several Equity
Pricers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-135
Calibrate Option Pricing Model Using Heston Model . . . . . . . . . . . . . . 3-143
Use Deep Learning to Approximate Barrier Option Prices with Heston
Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-148
Hedging Portfolios
4
Hedging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-2
Hedging Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-3
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-3
Hedging with hedgeopt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-4
Self-Financing Hedges with hedgeslf . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-9
Pricing and Hedging a Portfolio Using the Black-Karasinski Model . . . 4-13
Specifying Constraints with ConSet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-24
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-24
Setting Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-24
Portfolio Rebalancing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-26
Hedging with Constrained Portfolios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-28
Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-28
Example: Fully Hedged Portfolio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-28
Example: Minimize Portfolio Sensitivities . . . . . . . . . . . . . . . . . . . . . . . . 4-30
Example: Under-Determined System . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-30
Example: Portfolio Constraints with hedgeslf . . . . . . . . . . . . . . . . . . . . . 4-31
Hedging Strategies Using Spread Options . . . . . . . . . . . . . . . . . . . . . . . . 4-35
Mortgage-Backed Securities
5
What Are Mortgage-Backed Securities? . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-2
Fixed-Rate Mortgage Pool . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-3
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-3
ix
Inputs to Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-3
Generating Prepayment Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-4
Mortgage Prepayments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-5
Risk Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-6
Mortgage Pool Valuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-7
Computing Option-Adjusted Spread . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-9
Prepayments with Fewer Than 360 Months Remaining . . . . . . . . . . . . . . 5-12
Pools with Different Numbers of Coupons Remaining . . . . . . . . . . . . . . . 5-14
Summary of Prepayment Data Vector Representation . . . . . . . . . . . . . . . 5-14
Prepayment Modeling with a Two Factor Hull White Model and a LIBOR
Market Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-16
Pricing Mortgage Backed Securities Using the Black-Derman-Toy Model
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-34
Using Collateralized Mortgage Obligations (CMOs) . . . . . . . . . . . . . . . . 5-40
What Are CMOs? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-40
Prepayment Risk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-41
Sequential Tranches Without a Z-Bond . . . . . . . . . . . . . . . . . . . . . . . . . . 5-41
Sequential Tranches with a Z-Bond . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-42
PAC Tranches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-42
TAC Tranches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-44
CMO Workflow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-47
Calculate Underlying Mortgage Cash Flows . . . . . . . . . . . . . . . . . . . . . . 5-47
Define CMO Tranches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-47
If Using a PAC or TAC CMO, Calculate Principal Schedule . . . . . . . . . . . 5-47
Calculate Cash Flows for Each Tranche . . . . . . . . . . . . . . . . . . . . . . . . . 5-47
Analyze CMO by Computing Price, Yield, and Spread of CMO Cash Flows
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-48
Create PAC and Sequential CMO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-49
Debt Instruments
6
Agency Option-Adjusted Spreads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-2
Computing the Agency OAS for Bonds . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-2
Using Zero-Coupon Bonds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-5
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-5
Measuring Zero-Coupon Bond Function Quality . . . . . . . . . . . . . . . . . . . . 6-5
Pricing Treasury Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-5
Pricing Corporate Bonds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-7
Stepped-Coupon Bonds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-8
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-8
x Contents