Fourier Method for Valuation of Options under Parameter and State Uncertainty

Abstract

Mainstream option valuation theory relies implicitly on the assumption that latent states (such as stochastic volatility) and parameters are perfectly known, an assumption that is dubious in many ways. Computing the value of options under the assumption of perfect knowledge will typically introduce bias. Correcting for the bias is straightforward but can be computationally expensive.

Fourier-based methods for computing option values are nowadays the preferred computational technique in the financial industry as a result of speed and accuracy. The author shows that the bias correction for parameter and state uncertainty for a large class of processes can be incorporated into the Fourier framework, resulting in substantial computational savings compared with Monte Carlo methods or deterministic quadrature rules previously used. In addition, the author proposes extensions, such as time varying parameters and hyperparameters, to the class of uncertainty models.

The author finds that the proposed Fourier method is retaining all the good properties that are associated with Fourier methods; it is fast, accurate, and applicable to a wide range of models. Furthermore, the empirical performance of the corrected models is almost uniformly better than that of their noncorrected counterparts when evaluated on S&P 500 option data.

TOPICS: Derivatives, options, factor-based models, analysis of individual factors/risk premia

Key Findings

• • Parameter and state uncertainty in option models is often ignored but this leads to bias.

• • The bias correction introduced in this paper can be computed through the standard Fourier methodology, being fast and accurate.

• • The methodology results in better model in-sample and out-of-sample for a wide range of models, and the best results are found for parameters where the uncertainty is substantial.