Option Pricing: Theory and Practice

FIN 890, Fall 2026, Thursdays, 1-3 p.m.

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 first-generation 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 second-generation 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 will discuss the modeling efforts from the perspective of option investments.

• Class formats and requirements

  • Grades will be based on the completion of 3 projects. Each one 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 of the 3 projects.

1. Use the difference between implied volatility and some realized 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. This will be a big achievement. 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 prices and option returns, 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) Compare the risk premium estimate on each option contract from this approach and that from a bootstrapping approach

   • Due end of semester.

• 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.  But I still want to see your personal analysis. Don't hesitate to talk to me for clarification or general discussion.

• Class Outline

  • An overview of investment considerations

    • Grinold and Kahn: Active Portfolio Management

    • Penaranda and Wu, 2022, Targets, predictability, and performance, Management Science, 68(2), 1537--1555.

  • Introduction of option markets and valuation

    • 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.

  • Second-generation bottom-up option pricing models

    • Technical background: From characteristic functions and Fourier transforms to densities and option prices

      • 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

      • 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

      • 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

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

  • Decentralized top-down option pricing

    • 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.

    • 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.

  • Data-driven option pricing

    • Wu, 2025: Bootstrapping option risk premiums.