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.

1. 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.

   • Due end of week 4.

2. 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 (say 2 years), with extended Kalman filter and maximum likelihood estimation. Example, Carr and Wu (JFE 2007)

   • (Optional) Compare the pricing performance of 2 specifications, or 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.

3. Use option sensitivities as features to explain option returns and estimate option risk premiums, as in 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, or something else)

   • (Optional) Try to derive finite-horizon P&L attribution under different assumptions

   • Due end of week 11.

4. 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, 9/17, 9/24)

    • Primary v derivative securities, forward pricing 

    • Optional pricing under binomial tree 

    • Black-Merton-Model

    • Beyond BMS

    • References

      • 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.

      • Meng Tian and Liuren Wu, Limits of Arbitrage and Primary Risk Taking in Derivative Securities, Review of Asset Pricing Studies, 2023, 13(3), 405--439.

  • Bottom-up centralized option pricing models 

    • Technical background: From characteristic functions and Fourier transforms to densities and option prices (10/1)

      • Zemanian: Distribution Theory and Transform Analysis

      • Kendall's Advanced Theory of Statistics, Volume I, chapter 4

      • Sample code for FFT 

      • 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.

    • Levy process to model security returns (10/8)

      • 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.

    • Stochastic time changes to capture stochastic volatilities and skews (10/15)

      • 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.

    • Model estimation and statistical arbitrage trading (10/22)

      • Simon: Optimal State Estimation: Kalman, H Infinity, and Nonlinear Approaches

  • Top-down decentralized option pricing (10/29, 11/5, 11/12)

    • Instantaneous P&L attribution based on BMS representation (10/29)

      • Tian and Wu, 2023, Limits of Arbitrage and Primary Risk Taking in Derivative Securities, Review of Asset Pricing Studies, 13(3), 405--439

    • Decentralized pricing of option contracts (11/5)

      • Carr and Wu, 2020, Option profit and loss attribution and pricing: A new framework, Journal of Finance, 75(4), 2271--2316.

    • Common pricing of decentralized risk (11/12)

      • 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.

    • P&L attribution over discrete (long) investment horizons (11/19)

      • Nawalkha and Zhuo (2022)

    • Other representations (11/19)

      • Wu (2026), Jump to expiry: A new framework for pricing short-dated options 

  • Data-driven option pricing (12/3, 12/10)

    • Wu, 2025: Bootstrapping option risk premiums.