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Overview
This class discusses the frontiers of the option pricing literature. After a review of the Black-Merton-Scholes (BSM) model and stylized evidence on the common violations of BMS assumptions, we will go through old and new option pricing models from the perspective of modeling security returns with time-changed Lévy processes. It is a new framework that can encompass pretty much all existing models. It also provides an intuitive way to designing new models. Once we obtain a general understanding of the theoretical framework, we will shift our focus on designing and estimating models that match specific features of the options data.
Prerequisite Readings
Hull, Options, Futures, & Other Derivatives.
Required Readings
Modeling Financial Security Returns Using Lévy Processes. This handbook chapter explains the underlying ideas and reviews the relevant literature on option pricing with time changed Lévy processes.
Class Notes
Related Articles
Violations of BMS assumptions
Backus, Foresi, and Wu,
1997, Accounting
for Biases in Black-Scholes, wp. Carr and Wu,
2003, Finite
Moment Log Stable Process and Option Pricing, Journal of Finance, 58(2), 753--777. Wu, 2006,
Dampened Power Law: Reconciling the Tail Behavior of Financial
Security Returns, Journal of Business, 79(3), 1445--1474. Wu, 2005,
Crash-O-Phobia: A Domestic Fear or a Worldwide Concern?, Journal of Derivatives, 13(2), 8--21. Bakshi, Kapadia, and
Madan, 2003, Stock Return Characteristics, Skew Laws,
and Differential Pricing of Individual Equity Options, Review of Financial Studies, 16(1), 101--143. Option pricing using transform methods
Duffie, Pan, Singleton,
2000, Transform Analysis and Asset Pricing for Affine Jump Diffusions,
Econometrica, 68(6), 1343--1376. Carr and Madan,
1999, Option Valuation Using the Fast Fourier Transform,
Journal of Computational Finance, 2(4), 61--73. Lewis,
2000, A Simple Option Formula for General Jump-Diffusion and Other Exponential
Levy Processes, wp. Leippold and Wu,
2002, Asset pricing under the Quadratic Class, JFQA,
37(2), 271--295. Lee, 2004, Option pricing by transform methods: extensions, unification and error control, Journal of Computational Finance, 7(3), 51--86. Chourdakis, 2005, Option pricing using
fractional FFT, Journal of Computational Finance, 8(2), 1--18. Fang and Oosterlee, 2008, A novel pricing method for European options based
on Fourier-cosine series expansions, wp. Lévy (jump-diffusion) processes
Bertoin,
1996, Lévy Processes, Cambridge Sato,
1999, Lévy Processes and Infinitely Divisible Distributions,
Cambridge Merton,
1976, Option Pricing When Underlying Stock Returns Are Discontinuous,
Journal of Financial Economics, 3(1),125-144. Carr, Geman, Madan, Yor,
2002, The Fine Structure of Asset Returns: An Empirical Investigation,
Journal of Business, 75(2), 305--332. Madan, Carr, Chang,
1998, The Variance Gamma Process and Option Pricing,
European Finance Review, 2(1), 79--105.> Wu, 2006,
Dampened Power Law: Reconciling the Tail Behavior of Financial
Security Returns, Journal of Business, 79(3), 1445--1474.
Stochastic volatility models
Heston,
1993, A Closed-Form Solution for Options with Stochastic Volatility, with Application to Bond and Currency Options,
Review of Financial Studies, 6(2), 327--343. Hull and White,
1987, The Pricing of Options on Assets with Stochastic Volatilities,
Journal of Finance, 42(2), 281--300. Time-Changed Lévy processes--Combining jumps with stochastic
volatility (and higher moments)
Carr and Wu,
2004, Time-Changed
Lévy Processes and Option Pricing, Journal of Financial
Economics, 17(1), 113--141. Huang and Wu,
2004, Specification
Analysis of Option Pricing Models Based on Time-Changed Lévy Processes,
Journal of Finance, 59(3), 1405--1439. Research Area I: Specification analysis
Pan,
2000, The Jump-Risk Premia Implicit in Options: Evidence from an Integrated Time-Series Study,
Journal of Financial Economics, 63(1), 3--50. Eraker,
2004, Do Stock Prices and Volatility Jump? Reconciling Evidence from Spot and Option Prices,
Journal of Finance, 59(3), 1367--1404. Carr and Wu, 2007,
Stochastic Skew in Currency Options, Journal of Financial Economics, 86(1), 213--247.
Wu, 2005, Variance Dynamics: Joint Evidence from Options and High Frequency Data, wp.
Egloff, Leippold, Wu, 2007, Variance Risk Dynamics, Variance Risk Premia, and Optimal Variance Swap Investments, wp. Research Area II: Inferring
market price of risks from options data
Ait-Sahalia
and Lo, 1998, Nonparametric Estimation of State-Price Densities Implicit in Financial Asset Prices,
Journal of Finance, 53(2), 499--547. Jackwerth,
2000, Recovering Risk Aversion from Option Prices and Realized Returns, Review
of Financial Studies, 13(2), 433--451. Engle and
Rosenberg, 2002, Empirical Pricing Kernels, Journal of Financial
Economics, 64(3), 341--372. Bakshi
and Kapadia, 2003, Delta-Hedged Gains and the Negative Market Volatility
Risk Premium, Review of Financial Studies, 16(2), 527--566. Jackwerth
and Rubinstein, 2004, Recovering Probabilities and Risk Aversion from Option Prices and Realized
Returns, in: Essays in Honor of Fisher Black, editor: Bruce Lehmann,
Oxford University Press, Oxford. Bliss and
Panigirtzoglou, 2004, Option-Implied Risk Aversion Estimates, Journal
of Finance, 59(1), 407--446.
Carr and
Wu, forthcoming, Variance Risk Premia,
Review of Financial Studies.
Bakshi, Carr, and Wu, forthcoming, Stochastic
Risk Premiums, Stochastic Skewness in Currency Options, and Stochastic Discount Factors in International Economies,
Journal of Financial Economics. Mo and Wu, 2007, International Capital Asset Pricing:
Evidence from Options,
Journal of Empirical Finance, 14(4), 465--498. Bakshi and Wu, 2007, Investor Irrationality and the Nasdaq Bubble,wp. Research Area III: Cross-market linkages
Carr and Wu, 2007, Theory and Evidence on the Dynamic
Interactions Between Sovereign Credit Default Swaps and Currency Options,
Journal of Banking and Finance, 31(8), 2383--2403. Carr and Wu, Stock Options and Credit Default Swaps: A Joint Framework for Valuation and Estimation, wp. Cremers, Driessen, and Maenhout, Explaining the Level of Credit Spreads: Option-implied Jump Risk Premia in a Firm Value Model, Review of Financial Studies, forthcoming. (my discussion of the paper at 2006 EFA) Carr and Wu, 2007, Simple Robust Linkages Between CDS and Equity Options, wp. Research
Area IV: New specifications Carr and Wu, 2009, Leverage Effect,
Volatility Feedback, and Self-Exciting Market Disruptions:
Disentangling the Multi-Dimensional Variations in S&P 500
Index Options, separate modeling of asset value dynamics
and leverage variation. Multifractal volatility
dynamics, high-dimensional specification with very few
parameters.
Some Useful References
Jean Jacod and Albert N. Shiryaev, 1987, Limit Theorems for Stochastic Processes, Springer-Verlag.
Uwe Kuchler and Michael Sorensen, 1997, Exponential Families of Stochastic Processes, Springer.
Philip Protter, 1990, Stochastic Integration and Differential Equations: A New Approach, Springer.
Stuart Alan and J. Keith Ord, 1987, Kendall's Advanced Theory of Statistics, 5th edition, Oxford University Press.
Dan Simon, 2006, Optimal State Estimation: Kalman, H Infinity, and Nonlinear Approaches, John Wiley.
Data
One year's worth (1996) of options data on S&P 500 index. The data format: date, maturity in actual days, call(1) or put (0), strike, mid option price.
Libor and swap rates from 1996 on US dollar (downloaded from Bloomberg).