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Fall 2007, FIN 890 Mondays: 5:00-7:00pm, VC10-215 |
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Overview
This class discusses the frontiers of the option pricing literature. After a brief review of the options market, including market conventions and stylized facts, I'll 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.
Prerequisite Readings
Hull, Options, Futures, & Other Derivatives, 6th Ed, Chapters 8-16
Required Readings
Modeling Financial Security Returns Using Lévy Processes. I will go through this handbook chapter in detail.
Homework Assignments (I prefer that you do them in matlab, but you can
also write the code in C or C++.)
[Due 10/15/2007] Write a numerical function that returns a portfolio of call options (including forward, spot, or call at zero strike) that replicate an arbitrary piece-wise linear payoff function.
(optional) Write a numerical function that uses finite number of call options to approximate an arbitrary continuous, twice-differentiable payoff function.
price American options using a binomial tree. Test the accuracy of the program using a call option (compare with the European analytical value).
Compare the relative effectiveness of using (1) averaging and (2) replacing the last step with the BS formula to reduce pricing errors.
Write matlab functions that price American options using a binomial tree. Test the accuracy of the program using a call option (compare with the European analytical value).
Compare the relative effectiveness of using (1) averaging and (2) replacing the last step with the BS formula to reduce pricing errors.
Write matlab functions that
compute the characteristic function of the security return under (1) the Black-Scholes model, (2) the Merton (1976) Jump-Diffusion Model,
(3) the Heston (1993) stochastic volatility model, and (4) the Lévy pure jump process described by the exponentially
dampened power law. For the last case, the function should accommodate all special cases. The functions should be two dimensional,
with each row corresponding to one characteristic coefficient u and each column corresponding to one return maturity. Allow
interest rates and dividend yields to vary across maturities.
Write a matlab program that computes the density function of the
security return based on the above characteristic functions. Use rounded off numbers from the literature on the parameter values to generate
density functions on the return. Plot the density function.
Use no less than ten-years' worth of daily return data on some equity indexes or exchange rates
to estimate the models (1), (2), (3) (the Lévy models) using maximum likelihood method. Report the model parameters and their standard errors. Think of ways to pre-process the return data to make the estimation
more stable. Compare the likelihood estimates from different models.
(Optional) Propose and implement a method to
estimate the Heston model using time-series returns.
Write matlab functions that compute option values based on the
characteristic functions of the security return. Consider different Fourier inversion methods: inversion of a cumulative
distribution, FFT, Fractional FFT.
Calibrate the above four models to one day's worth of options data on equity indexes or single name stocks. You can download/print
these data from Bloomberg.
(Optional) propose and implement a method to estimate
a jump-diffusion stochastic volatility model with dynamic consistency using both options and time-series returns.
Grading
The final grade is 50% from
homework assignments and 50% from a final exam (closed book) at the end of the semester.
You can also write an optional term paper to improve your grade. If you plan to do so, talk to me for potential topics.
Class Notes
Related Articles
Biases in the Black-Scholes model
(Empirical Evidence)
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. Lévy Jump (Diffusion) Models
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. 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 Process--Combining Jumps with Stochastic
Volatility
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. Carr and Wu,
2004, Stochastic Skew in Currency Options, Journal of Financial Economics, forthcoming. Mo and Wu, International Capital Asset Pricing:
Evidence from Options,
Journal of Empirical Finance, 14(4), 465--498. 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. Semi-Static
Hedging of Standard an Exotic Options
Breeden
and Litzenberger, 1978, Prices of State-Contingent Claims Implicit in Option Prices,
Journal of Business, 51(4), 621--651. Carr and
Wu, 2003, Static Hedging of Standard Options,
wp. Carr and Lee, 2004, Robust Replication of Volatility Derivatives,
talk slides. Carr and Madan, 2001, Optimal positioning in derivative securities, Quantitative Finance, 1(1), 19--37. Inferring
Market Price of Risk from Options Data
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. 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. 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. Engle and
Rosenberg, 2002, Empirical Pricing Kernels, Journal of Financial
Economics, 64(3), 341--372. Bliss and
Panigirtzoglou, 2004, Option-Implied Risk Aversion Estimates, Journal
of Finance, 59(1), 407--446. Bakshi
and Kapadia, 2003, Delta-Hedged Gains and the Negative Market Volatility
Risk Premium, Review of Financial Studies, 16(2), 527--566.
Bakshi, Carr, and Wu, Stochastic
Risk Premiums, Stochastic Skewness in Currency Options, and Stochastic Discount Factors in International Economies,
Journal of Financial Economics, forthcoming.
Carr and
Wu, 2004, Variance Risk Premia,
Review of Financial Studies, forthcoming.
Wu, 2005, Variance Dynamics: Joint Evidence from Options and High Frequency Data, wp. Linking options with credits
Carr and Wu, Stock Options and Credit Default Swaps: A Joint Framework for Valuation and Estimation, wp. 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. Cremers, Driessen, and Maenhout, Explaining the Level of Credit Spreads: Option-implied Jump Risk Premia in a Firm Value Model, wp. (my discussion of the paper at 2006 EFA) Options Market Microstructure
Holowczak, Simaan, and Wu,
2006, Price Discovery in the U.S. Stock and Stock Options Markets: A Portfolio Approach,
Review of Derivatives Research, 9, 37--65. Simaan and Wu,
2003, Price Discovery in the U.S. Stock Options Market,
Journal of Derivatives, forthcoming. Mayhew,
2002, Competition, Market Structure and Bid-Ask Spreads in Stock Option Markets, Journal of Finance, 57(2), 931--958. Chakravarty, Gulen and Mayhew,
2004, Informed Trading in Stock and Option Markets, Journal of Finance, 59(3), 1235--1258. Battalio, Hatch, Jennings,
2004, Toward a National Market System for U.S. Exchange-Listed Equity Options, Journal of Finance, 59(2), 933--962. Ofek, Richardson, and Whitelaw
2004, Limited Arbitrage and Short Sale Constraints: Evidence from the Options Market, Journal of Financial Economics, 74(2), 305--342. Pan and Poteshman
2006, The Information in Option Volume for Future Stock Prices, Review of Financial Studies, 19(3), 871--980. 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. 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).