The Complete Guide to Option Pricing Formulas

by Espen Gaarder Haug

Overview

Long-established as a definitive resource by Wall Street professionals, The Complete Guide to Option Pricing Formulas has been revised and updated to reflect the realities of today's options markets. The Second Edition contains a complete listing of virtually every pricing formula_all presented in an easy-to-use dictionary format, with expert author commentary and ready-to-use programming code.

The Second Edition of this classic guide now includes more than 60 new option models and formulas…extensive tables providing an overview of all formulas…new examples and applications…and an updated CD containing all pricing formulas, with VBA code and ready-to-use Excel spreadsheets.

The volume also features several new chapters covering such things as: option sensitivities, discrete dividend, commodity options, and two chapters on numerical methods covering trees, finite difference and Monte Carlo Simulation.

The new edition of The Complete Guide to Option Pricing Formulas offers quick access to:

• Options Pricing Overview
• Black-Scholes-Merton
• Black-Scholes-Merton Greeks
• Analytical Formulas for American Options
• Exotic Options Single Asset
• Exotic Options on Two Assets
• Black-Scholes-Merton Adjustments and Alternatives
• Trees and Finite Difference Methods
• Monte Carlo Simulation
• Options on Stocks that Pay Discrete Dividends
• Commodity and Energy Options
• Interest Rate Derivatives
• Volatility and Correlation
• Distributions
• Some Useful Formulas: Interpolation, Interest Rates, and Risk-Reward Measures

This all-in-one options pricing guide contains a numerical example or a table with values for each option pricing formula. The book also includes a helpful glossary of notations, as well as an extensive bibliography of related books and articles.

Product Details

ISBN-13:	9780071389976
Publisher:	McGraw Hill LLC
Publication date:	12/20/2006
Edition description:	REV
Pages:	492
Product dimensions:	7.70(w) x 9.50(h) x 1.60(d)

About the Author

Espen Gaarder Haug, has more than 15 years of experience in derivatives trading and research. He has worked as a proprietary option trader at J.P. Morgan Chase in New York, and as an option trader for the hedge funds Amaranth Advisors and Paloma Partners. Dr. Haug has published extensively in journals such as Quantitative Finance, International Journal of Theoretical and Applied Finance, and Wilmott Magazine. He is also a popular lecturer on option pricing, hedging, and risk management and an Adjunct Associate Professor at Norwegian University of Science and Technology.

Table of Contents

Introduction     xvii
Acknowledgments     xix
What Is New in the Second Edition?     xxi
Option Pricing Formulas Overview     xxiii
Glossary of Notations     xxxv
Black-Scholes-Merton     1
Black-Scholes-Merton     2
The Black-Scholes Option Pricing Formula     2
Options on Stock Indexes     4
Options on Futures     4
Margined Options on Futures     5
Currency Options     6
The Generalized Black-Scholes-Merton Option Pricing Formula     7
Parities and Symmetries     9
Put-Call Parity for European Options     9
At-the-Money Forward Value Symmetry     10
Put-Call Symmetry     10
Put-Call Supersymmetry     11
Black-Scholes-Merton on Variance Form     11
Before Black-Scholes-Merton     12
The Bachelier Model     12
The Sprenkle Model     13
The Boness Model     14
The Samuelson Model     14
Appendix A: The Black-Scholes-Merton PDE     15
Ito's Lemma     15
Dynamic Hedging     16
Black-Scholes-Merton Greeks     21
DeltaGreeks     21
Delta     21
Delta Mirror Strikes and Assets     29
Strike from Delta     30
Futures Delta from Spot Delta     31
DdeltaDvol and DvegaDspot     32
DvannaDvol     34
DdeltaDtime, Charm     35
Elasticity     36
Gamma Greeks     38
Gamma     38
Maximal Gamma and the Illusions of Risk     39
GammaP     42
Gamma Symmetry     45
DgammaDvol, Zomma     45
DgammaDspot, Speed     47
DgammaDtime, Color     49
Vega Greeks     50
Vega     50
Vega Symmetry     55
Vega-Gamma Relationship     55
Vega from Delta     56
VegaP     56
Vega Leverage, Vega Elasticity     57
DvegaDvol, Vomma     57
DvommaDvol, Ultima     60
DvegaDtime     61
Variance Greeks     62
Variance Vega     62
DdeltaDvar     63
Variance Vomma     63
Variance Ultima     63
Volatility-Time Greeks     64
Theta Greeks      64
Theta     64
Theta Symmetry     68
Rho Greeks     68
Rho     68
Phi/Rho-2     71
Carry Rho     73
Probability Greeks     75
In-the-Money Probability     76
DzetaDvol     79
DzetaDtime     80
Risk-Neutral Probability Density     80
From in-the-Money Probability to Density     80
Probability of Ever Getting in-the-Money     80
Greeks Aggregations     81
Net Weighted Vega Exposure     82
At-the-Money Forward Approximations     84
Approximation of the Black-Scholes-Merton Formula     84
Delta     84
Gamma     84
Vega     84
Theta     84
Rho     85
Cost-of-Carry     85
Numerical Greeks     85
First-Order Greeks     85
Second-Order Greeks     86
Third-Order Greeks     86
Mixed Greeks     87
Third-Order Mixed Greeks     87
Greeks from Closed-Form Approximations     89
Appendix B: Taking Partial Derivatives     90
Analytical Formulas for American Options     97
The Barone-Adesi and Whaley Approximation     97
The Bjerksund and Stensland (1993) Approximation     101
The Bjerksund and Stensland (2002) Approximation     104
Put-Call Transformation American Options     109
American Perpetual Options     109
Exotic Options-Single Asset     111
Variable Purchase Options     111
Executive Stock Options     114
Moneyness Options     114
Power Contracts and Power Options     115
Power Contracts     115
Standard Power Option     116
Capped Power Option     117
Powered Option     118
Log Contracts     119
Log(S) Contract     120
Log Option     121
Forward Start Options     121
Fade-in Option     122
Ratchet Options     124
Reset Strike Options-Type 1     124
Reset Strike Options-Type 2     125
Time-Switch Options     127
Chooser Options     128
Simple Chooser Options     128
Complex Chooser Options     129
Options on Options     132
Put-Call Parity Compound Options     135
Compound Option Approximation     136
Options with Extendible Maturities     138
Options That Can Be Extended by the Holder     138
Writer-Extendible Options     140
Lookback Options     141
Floating-Strike Lookback Options     141
Fixed-Strike Lookback Options     143
Partial-Time Floating-Strike Lookback Options     144
Partial-Time Fixed-Strike Lookback Options     147
Extreme-Spread Options     148
Mirror Options     150
Barrier Options     152
Standard Barrier Options     152
Standard American Barrier Options     154
Double-Barrier Options     156
Partial-Time Single-Asset Barrier Options     160
Look-Barrier Options     163
Discrete-Barrier Options     164
Soft-Barrier Options     165
Use of Put-Call Symmetry for Barrier Options     167
Barrier Option Symmetries     168
First-Then-Barrier Options     169
Double-Barrier Option Using Barrier Symmetry     171
Dual Double-Barrier Options     172
Binary Options     174
Gap Options      174
Cash-or-Nothing Options     174
Asset-or-Nothing Options     175
Supershare Options     176
Binary Barrier Options     176
Double-Barrier Binary Options     180
Double-Barrier Binary Asymmetrical     181
Asian Options     182
Geometric Average-Rate Options     182
Arithmetic Average-Rate Options     186
Discrete Arithmetic Average-Rate Options     192
Equivalence of Floating-Strike and Fixed-Strike Asian Options     199
Asian Options with Volatility Term-Structure     199
Exotic Options on Two Assets     203
Relative Outperformance Options     203
Product Options     205
Two-Asset Correlation Options     205
Exchange-One-Asset-for-Another Options     206
American Exchange-One-Asset-for-Another Option     208
Exchange Options on Exchange Options     209
Options on the Maximum or the Minimum of Two Risky Assets     211
Spread-Option Approximation     213
Two-Asset Barrier Options     215
Partial-Time Two-Asset Barrier Options     217
Margrabe Barrier Options     219
Discrete-Barrier Options      221
Two-Asset Cash-or-Nothing Options     221
Best or Worst Cash-or-Nothing Options     223
Options on the Minimum or Maximum of Two Averages     224
Currency-Translated Options     226
Foreign Equity Options Struck in Domestic Currency     226
Fixed Exchange Rate Foreign Equity Options     228
Equity Linked Foreign Exchange Options     230
Takeover Foreign Exchange Options     232
Greeks for Two-Asset Options     232
Black-Scholes-Merton Adjustments and Alternatives     233
The Black-Scholes-Merton Model with Delayed Settlement     234
The Black-Scholes-Merton Model Adjusted for Trading Day Volatility     235
Discrete Hedging     236
Hedging Error     236
Discrete-Time Option Valuation and Delta Hedging     237
Discrete-Time Hedging with Transaction Cost     238
Option Pricing in Trending Markets     240
Alternative Stochastic Processes     242
Constant Elasticity of Variance     242
Skewness-Kurtosis Models     244
Definition of Skewness and Kurtosis     244
The Skewness and Kurtosis for a Lognormal Distribution     245
Jarrow and Rudd Skewness and Kurtosis Model      246
The Corrado and Su Skewness and Kurtosis Model     247
Modified Corrado-Su Skewness-Kurtosis Model     250
Skewness-Kurtosis Put-Call Supersymmetry     252
Skewness-Kurtosis Equivalent Black-Scholes-Merton Volatility     252
Gram Charlier Density     252
Skewness-Kurtosis Trees     253
Pascal Distribution and Option Pricing     253
Jump-Diffusion Models     253
The Merton Jump-Diffusion Model     253
Bates Generalized Jump-Diffusion Model     255
Stochastic Volatility Models     258
Hull-White Uncorrelated Stochastic Volatility Model     259
Hull-White Correlated Stochastic Volatility Model     261
The SABR Model     265
Variance and Volatility Swaps     271
Variance Swaps     271
Volatility Swaps     274
More Information     278
Trees and Finite Difference Methods     279
Binomial Option Pricing     279
Cox-Ross-Rubinstein American Binomial Tree     284
Greeks in CRR Binomial Tree     287
Rendleman Bartter Binomial Tree     289
Leisen-Reimer Binomial Tree     290
Convertible Bonds in Binomial Trees      292
Binomial Model with Skewness and Kurtosis     297
Trinomial Trees     299
Exotic Options in Tree Models     303
Options on Options     303
Barrier Options Using Brownian Bridge Probabilities     305
American Barrier Options in CRR Binomial Tree     307
European Reset Options Binomial     308
American Asian Options in a Tree     314
Three-Dimensional Binomial Trees     315
Implied Tree Models     321
Implied Binomial Trees     321
Implied Trinomial Trees     327
Finite Difference Methods     334
Explicit Finite Difference     335
Implicit Finite Difference     338
Finite Difference in ln(S)     340
The Crank-Nicolson Method     342
Monte Carlo Simulation     345
Standard Monte Carlo Simulation     345
Greeks in Monte Carlo     347
Monte Carlo for Callable Options     349
Two Assets     350
Three Assets     352
N Assets, Cholesky Decomposition     353
Monte Carlo of Mean Reversion     355
Generating Pseudo-Random Numbers     356
Variance Reduction Techniques      358
Antithetic Variance Reduction     358
IQ-MC/Importance Sampling     359
IQ-MC Two Correlated Assets     361
Quasi-Random Monte Carlo     362
American Option Monte Carlo     364
Options on Stocks That Pay Discrete Dividends     367
European Options on Stock with Discrete Cash Dividend     368
The Escrowed Dividend Model     368
Simple Volatility Adjustment     369
Haug-Haug Volatility Adjustment     369
Bos-Gairat-Shepeleva Volatility Adjustment     370
Bos-Vandermark     371
Non-Recombining Tree     372
Black's Method for Calls on Stocks with Known Dividends     375
The Roll, Geske, and Whaley Model     375
Benchmark Model for Discrete Cash Dividend     378
A Single Dividend     378
Multiple Dividends     382
Applications     382
Options on Stocks with Discrete Dividend Yield     390
European with Discrete Dividend Yield     390
Closed-Form American Call     390
Recombining Tree Model     393
Commodity and Energy Options     397
Energy Swaps/Forwards     397
Energy Options      400
Options on Forwards, Black-76F     400
Energy Swaptions     401
Hybrid Payoff Energy Swaptions     405
The Miltersen-Schwartz Model     406
Mean Reversion Model     410
Seasonality     411
Interest Rate Derivatives     413
FRAs and Money Market Instruments     413
FRAs From Cash Deposits     413
The Relationship between FRAs and Currency Forwards     414
Convexity Adjustment Money Market Futures     415
Simple Bond Mathematics     417
Dirty and Clean Bond Price     417
Current Yield     417
Modified Duration and BPV     417
Bond Price and Yield Relationship     418
Price and Yield Relationship for a Bond     418
From Bond Price to Yield     419
Pricing Interest Rate Options Using Black-76     419
Options on Money Market Futures     420
Price and Yield Volatility in Money Market Futures     421
Caps and Floors     421
Swaptions     422
Swaption Volatilities from Caps or FRA Volatilities     424
Swaptions with Stochastic Volatility     425
Convexity Adjustments      425
European Short-Term Bond Options     427
From Price to Yield Volatility in Bonds     428
The Schaefer and Schwartz Model     428
One-Factor Term Structure Models     429
The Rendleman and Bartter Model     429
The Vasicek Model     430
The Ho and Lee Model     432
The Hull and White Model     433
The Black-Derman-Toy Model     434
Volatility and Correlation     445
Historical Volatility     445
Historical Volatility from Close Prices     445
High-Low Volatility     447
High-Low-Close Volatility     448
Exponential Weighted Historical Volatility     449
From Annual Volatility to Daily Volatility     450
Confidence Intervals for the Volatility Estimate     451
Volatility Cones     452
Implied Volatility     453
The Newton-Raphson Method     453
The Bisection Method     455
Implied Volatility Approximations     456
Implied Forward Volatility     458
From Implied Volatility Surface to Local Volatility Surface     458
Confidence Interval for the Asset Price     459
Basket Volatility      460
Historical Correlation     460
Distribution of Historical Correlation Coefficient     461
Implied Correlations     462
Implied Correlation from Currency Options     462
Average Implied Index Correlation     462
Various Formulas     463
Probability of High or Low, the Arctangent Rule     463
Siegel's Paradox and Volatility Ratio Effect     464
Distributions     465
The Cumulative Normal Distribution Function     465
The Hart Algorithm     465
Polynomial Approximations     467
The Inverse Cumulative Normal Distribution Function     469
The Bivariate Normal Density Function     470
The Cumulative Bivariate Normal Distribution Function     470
The Trivariate Cumulative Normal Distribution Function     482
Some Useful Formulas     487
Interpolation     487
Linear Interpolation     487
Log-Linear Interpolation     487
Exponential Interpolation     487
Cubic Interpolation: Lagrange's Formula     488
Cubic-Spline Interpolation     488
Two-Dimensional Interpolation     490
Interest Rates     491
Future Value of Annuity     491
Net Present Value of Annuity     491
Continuous Compounding     491
Compounding Frequency     491
Zero-Coupon Rates from Par Bonds/Par Swaps     492
Risk-Reward Measures     493
Treynor's Measure     493
Sharpe Ratio     494
Confidence Ratio     494
Sortino Ratio     495
Burke Ratio     495
Return on VaR     495
Jensen's Measure     496
Appendix C: Basic Useful Information     496
The Option Pricing Software     497
Bibliography     499
Index     521