Option Pricing: Theory and Practice
FIN 890, Fall 2026, Thursdays, 1-3 p.m. Location: VC 10-210.
Office Hours: Thursdays, 3:00-4:00 p.m. Or by appointment. liuren.wu@baruch.cuny.edu.
Overview
The class starts with an overview of the philosophies underlying the classic option valuation models and their contrasts with valuation on primary securities. It then examines three progressing threads in option pricing. The first is a general framework for designing and estimating bottom-up option pricing models based on time-changed Levy processes. The second is a framework on decentralized top-down option pricing, which relies more on data than on bottom-up dynamic structures. The third approach pushes further to rely even more on data (for distributional behaviors) and even less on model structures.
There can be many different objectives for developing option pricing models. I structure the class and discuss the modeling efforts from the perspective of option investments.
Projects
Grades will be based on completing 2-3 of the following 4 projects. Each one is designed for you to really digest what we have discussed in class. Each project is worth 40%. Each optional topic is worth 10%. You can pass the class with a good grade by doing a good job on 2-3 of the 4 projects.
Use the difference between implied volatility and some volatility forecast to predict the option returns.
The starting point can be Goyal and Saretto (2009). You can either replicate their results, or apply their method to other options market. If processing options on a large universe of stocks becomes too daunting for you, try to pick one underlying and examine whether the prediction works in the time series. You can look into Tian and Wu (RAPS, 2023), Wu and Xu (RAPS, 2026) (appendix) for option return calculation and discussion.
(Optional) They focus on one-month at-the-money options. Wu and Xu show why this happens to work well for the strategy. You can examine how the prediction varies at other moneyness and maturities.
(Optional) When investing in an option for volatility exposures, you can/should perform delta hedge to remove delta risk and focus on volatility risk. How to do the delta hedge is an important question. You can compute delta with the BMS model, with either the contract's implied volatility (as in Wu and Xu) or some volatility forecast. You can also make other adjustments (such as "total delta" adjusting for volatility move and volatility-return correlation --- There is a whole literature on this). Some more recent work also tries to learn delta empirically with machine learning... Examine which method is better in removing the delta risk. A simple comparison would be between a delta computed with implied volatiity and one with a one-year historical vol estimator. Try to think through the rationale behind the performance difference.
Due end of week 4.
Implement a jump-diffusion stochastic volatility model.
Use options on one name (index, individual stock, or currency).
Minimum requirement is to use nonlinear least squares to fit the model parameters (and variance rates) to the options data on a few days when different implied volatility surface shapes (e.g., upward v downward sloping term structure, flat v strong skew across strike). Show the fit in the implied volatility space.
(Optional) Do state-space estimation over a sample period, with extended (unscented) Kalman filter and maximum likelihood estimation. Example, Carr and Wu (JFE 2007)
(Optional) Do parameter sensitivity analysis to understand the effect of different model parameters on the shape of the implied volatility surface.
Due end of week 9.
Use option sensitivities as features to explain option returns and estimate option risk premiums, as in Tian and Wu (2023), Wu and Zhang (RFS 2025), Wu and Xu (RAPS, 2026)
Use options on one name (index, individual stock, or currency). Estimate one-year realized variance and covariance estimators on return and implied volatility. Perform analysis with at least one year of data.
(Optional) Experiment with different representations (say instead of computing the risk sensitivities using BS model, use the Bachelier model, the JTE model, or something else)
(Optional) Experiment ways of capturing the effects of discontinuous movements
(Optional) Try to derive finite-horizon P&L attribution under different assumptions
Due end of week 11.
Bootstrap risk premiums on options or bonds or something else.
Pick any security.
(Optional) Explore embedding a model into the bootstrapping so you can bootstrap both the model dynamics and the pricing errors
Due end of semester
Data sources: School has subscription to IvyDB for US exchange-traded stock and stock index/ETF options. The data are organized one file per day, making it particular easy for cross-stock analysis for a given date. You can also download implied volatility quotes from Bloomberg on OTC FX options and interest rate options such as caps/floors and swaptions. These are better suited for examining the time-series variation on a given underlying security.
Present the results in a well-written paper format. Describe how you did the implementation (methodology). Summarize your findings. Discuss implications of your findings. Keep your writing simple, short, and clear. Use your own words.
Feel free to search the web, ask AI, for help in coding. Feel free to discuss with your fellow students or other faculties. But I do want to see your personal analysis. I hate long-winded AI-edited sentences with fluffy words but no insights. Keep it personal, concise, and to the point. Don't cut/paste cliches.
Class Outline (target dates)
Introduction (9/3)
Grinold and Kahn: Active Portfolio Management
Penaranda and Wu, 2022, Targets, predictability, and performance, Management Science, 68(2), 1537--1555.
The Black-Merton-Scholes model
(9/10) Primary v derivative securities, binomial tree
(9/17) BMS
(9/24) Biases in BMS
Hull: Options, Futures, and Other Derivatives
Goyal and Saretto, 2009, Cross-Section of Option Returns and Volatility, Journal of Financial Economics, 94(2), 310--326.
Bottom-up centralized option pricing models
(10/1) Fourier transforms for option pricing
Zemanian: Distribution Theory and Transform Analysis
Kendall's Advanced Theory of Statistics, Volume I, chapter 4
Carr and Madan, 1999, Option valuation using the fast Fourier transform, Journal of Computational Finance, 2(4), 61--73.
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, SIAM Journal on Scientific Computing, 31 (2), 826-848.
(10/8) Levy process to model security returns
Bertoin: Levy Processes
Sato: Levy Processes and Infinitely Divisible Distributions
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.
Wu, 2006, Dampened power law: Reconciling the tail behavior of financial security returns, Journal of Business, 79(3), 1445--1474.
(10/15) Stochastic time changes to capture stochastic volatilities and skews
Jacod and Shiryaev: Limit Theorems for Stochastic Processes
Kuchler and Sorensen: Exponential Families of Stochastic Processes
Carr and Wu, 2004, Time-changed levy processes and option pricing, Journal of Financial Economics, 17(1), 113--141.
Heston, 1993, A closed-form solution for options with stochastic volatility, Review of Financial Studies, 6(2), 327--343.
Bates, 1996, Jumps and stochastic volatility: Exchange rate processes implicit in Deutsche Mark options, Review of Financial Studies, 1996, 9(1), 69--107.
Huang and Wu, 2004, Specification analysis of option pricing models based on time-changed levy processes, Journal of Finance, 59(3), 1405--1439.
Carr and Wu, 2007, Stochastic skew in currency options, Journal of Financial Economics, 86(1), 213--247.
(10/22)State-space setup and unscented Kalman filter for model estimation
Simon: Optimal State Estimation: Kalman, H Infinity, and Nonlinear Approaches
Top-down decentralized option pricing
(10/29) Risk representation and instantaneous return attribution
Tian and Wu, 2023, Limits of Arbitrage and Primary Risk Taking in Derivative Securities, Review of Asset Pricing Studies, 13(3), 405--439
(11/5) Pricing pool assuming common dynamics
Carr and Wu, 2016, Analyzing Volatility Risk and Risk Premium in Option Contracts: A New Theory, Journal of Financial Economics, 120(1), 1--20.
Carr and Wu, 2020, Option profit and loss attribution and pricing: A new framework, Journal of Finance, 75(4), 2271--2316.
(11/12) Pricing pooling assuming common pricing of similar risk
Wu and Zhang, 2025, Common pricing of decentralized risk: A linear option pricing model, Review of Financial Studies, 38(6), 1822-1867.
Wu and Xu, 2026, Cross-sectional variation of risk-targeting option portfolios, Review of Asset Pricing Studies, 16(1), 133--161.
(11/19) Long-horizon return attribution and alternative risk representation
Nawalkha and Zhuo (2022), A theory of equivalent expectation measures for contingent claim returns
Wu (2026), Jump to expiry: A new framework for pricing short-dated options
Data-driven option pricing (12/3, 12/10)
(12/3) Bootstrapping conditional option risk premium and covariance matrix
Wu, 2025: Bootstrapping option risk premiums.
(12/10) Alternative bootstrapping procedure designs and applications