[Paper Review] Fractional G-White Noise Theory, Wavelet Decomposition for Fractional G-Brownian Motion, and Bid-Ask Pricing Application to Finance Under Uncertainty

Wei Chen|arXiv (Cornell University)|Jun 18, 2013

Stochastic processes and financial applications25 references3 citations

TL;DR

This paper introduces fractional G-Brownian motion (fGBm) as a robust model for financial uncertainty, integrating G-expectation theory with wavelet-based decomposition and fractional G-white noise. It establishes a fractional G-Itô calculus framework, derives a G-Clark-Ocone formula, and proves that the sublinear expectation of a discounted European claim equals its bid-ask price under volatility uncertainty, offering a new robust option pricing mechanism.

ABSTRACT

G-framework is presented by Peng [41] for measure risk under uncertainty. In this paper, we define fractional G-Brownian motion (fGBm). Fractional G-Brownian motion is a centered G-Gaussian process with zero mean and stationary increments in the sense of sub-linearity with Hurst index $H\in (0,1)$. This process has stationary increments, self-similarity, and long rang dependence properties in the sense of sub-linearity. These properties make the fractional G-Brownian motion a suitable driven process in mathematical finance. We construct wavelet decomposition of the fGBm by wavelet with compactly support. We develop fractional G-white noise theory, define G-Itô-Wick stochastic integral, establish the fractional G-Itô formula and the fractional G-Clark-Ocone formula, and derive the G-Girsanov's Theorem. For application the G-white noise theory, we consider the financial market modelled by G-Wick-Itô type of SDE driven by fGBm. The financial asset price modelled by fGBm has volatility uncertainty, using G-Girsanov's Theorem and G-Clark-Ocone Theorem, we derive that sublinear expectation of the discounted European contingent claim is the bid-ask price of the claim.

Motivation & Objective

• To model financial asset prices under volatility uncertainty using a new stochastic process, fractional G-Brownian motion (fGBm), which generalizes fractional Brownian motion under sublinear expectation.
• To develop a fractional G-white noise theory and define a G-Itô-Wick stochastic integral for fGBm, enabling stochastic calculus under model uncertainty.
• To establish the fractional G-Itô formula, fractional G-Clark-Ocone formula, and G-Girsanov’s Theorem for fGBm to support dynamic hedging and pricing.
• To apply the developed framework to derive the bid-ask price of European contingent claims as the sublinear expectation of the discounted payoff, ensuring robustness under volatility ambiguity.

Proposed method

• Define fractional G-Brownian motion (fGBm) as a centered G-Gaussian process with self-similarity, stationary increments, and long-range dependence under sublinear expectation.
• Construct a wavelet decomposition of fGBm using compactly supported wavelets to enable multiscale analysis of the process.
• Develop fractional G-white noise theory by defining the G-Itô-Wick stochastic integral and deriving the fractional G-Itô formula for fGBm-driven SDEs.
• Establish the fractional G-Clark-Ocone formula to express the discounted claim payoff as a stochastic integral, enabling dynamic hedging under uncertainty.
• Prove G-Girsanov’s Theorem to change the measure under which fGBm remains a G-Brownian motion, facilitating change of measure in uncertain markets.
• Use the G-Clark-Ocone formula and G-Girsanov’s Theorem to show that the bid-ask price of a European claim is the sublinear expectation of its discounted payoff.

Experimental results

Research questions

• RQ1 How can fractional Brownian motion be extended to a G-Gaussian process under sublinear expectation to model financial uncertainty?
• RQ2 What is the correct stochastic calculus framework for fGBm under volatility uncertainty, and how can it support dynamic hedging and option pricing?
• RQ3 Can the G-Clark-Ocone formula be generalized to fractional G-Brownian motion to express contingent claims in terms of a stochastic integral?
• RQ4 How does the sublinear expectation of a discounted claim yield the bid-ask price in a market with volatility uncertainty?
• RQ5 What is the role of wavelet decomposition in analyzing and approximating fGBm in the context of G-white noise theory?

Key findings

• The sublinear expectation of the discounted payoff of a European contingent claim, denoted $ E^G[e^{-rT} ilde{ heta}] $, is identified as the bid price of the claim under volatility uncertainty.
• The ask price of the claim is given by $ -E^G[-e^{-rT} ilde{ heta}] $, which equals the infimum of the linear expectations over all equivalent measures in the ambiguity set.
• The optimal hedging portfolio is derived using the fractional G-Clark-Ocone formula, with the holding position $ v(t) = S(t)^{-1}e^{-r(T-t)}\tilde{E}_{M_H}[D_t^H \tilde{\theta}|\mathcal{F}_t^H] $, ensuring super-hedging under model ambiguity.
• The G-Itô-Wick integral and fractional G-Itô formula are established for fGBm, enabling consistent stochastic integration and Itô-type calculus under sublinear expectation.
• Wavelet decomposition of fGBm is constructed using compactly supported wavelets, enabling multiscale analysis and numerical approximation of the process.
• The G-Girsanov’s Theorem is proven for fGBm, showing that a change of measure transforms the fGBm into a G-Brownian motion under the new sublinear expectation, preserving the structure of the process.