[Paper Review] Time Consistent Bid-Ask Dynamic Pricing Mechanisms for Contingent Claims and Its Numerical Simulations Under Uncertainty

Wei Chen|arXiv (Cornell University)|Nov 18, 2011

Stochastic processes and financial applications15 references3 citations

TL;DR

This paper proposes a time-consistent dynamic bid-ask pricing mechanism for European contingent claims under volatility and drift uncertainty using Peng's G-framework. It formulates the pricing as a viscosity solution to a fully nonlinear PDE derived from the G-heat equation, and implements monotone implicit finite difference schemes for numerical simulation, demonstrating convergence and stability in bid-ask price trajectories for digital and butterfly options.

ABSTRACT

We study time consistent dynamic pricing mechanisms of European contingent claims under uncertainty by using G framework introduced by Peng ([24]). We consider a financial market consisting of a riskless asset and a risky stock with price process modelled by a geometric generalized G-Brownian motion, which features the drift uncertainty and volatility uncertainty of the stock price process. Using the techniques on G-framework we show that the risk premium of the asset is uncertain and distributed with maximum distribution. A time consistent G-expectation is defined by the viscosity solution of the G-heat equation. Using the time consistent G-expectation we define the G dynamic pricing mechanism for the claim. We prove that G dynamic pricing mechanism is the bid-ask Markovian dynamic pricing mechanism. The full nonlinear PDE is derived to describe the bid (resp. ask) price process of the claim. Monotone implicit characteristic finite difference schemes for the nonlinear PDE are given, nonlinear iterative schemes are constructed, and the simulations of the bid (resp. ask) prices of contingent claims under uncertainty are implemented.

Motivation & Objective

• To develop a time-consistent dynamic pricing mechanism for European contingent claims under model uncertainty involving drift and volatility.
• To model asset price dynamics using generalized geometric G-Brownian motion to capture both drift and volatility uncertainty.
• To define a bid-ask dynamic pricing mechanism based on time-consistent G-expectation derived from the viscosity solution of the G-heat equation.
• To derive and numerically solve the full nonlinear PDE governing the bid and ask price processes of contingent claims.
• To implement monotone implicit finite difference schemes for numerical simulation and validate convergence and stability of the bid-ask price surfaces.

Proposed method

• Model the risky asset price using a generalized geometric G-Brownian motion to represent drift and volatility uncertainty within bounded intervals.
• Define a time-consistent G-expectation via the viscosity solution of the G-heat equation, ensuring dynamic consistency in pricing.
• Construct a bid-ask dynamic pricing mechanism by solving the nonlinear PDEs derived from the G-expectation framework.
• Develop monotone implicit characteristic finite difference schemes for the nonlinear PDEs to ensure stability and convergence.
• Implement nonlinear iterative schemes to solve the discrete nonlinear systems at each time step, with convergence proven via M-matrix properties.
• Simulate bid and ask price surfaces for digital and butterfly options using specified grid parameters and tolerance thresholds.

Experimental results

Research questions

• RQ1 How can a time-consistent dynamic bid-ask pricing mechanism be constructed for contingent claims under drift and volatility uncertainty?
• RQ2 What is the role of the G-expectation in ensuring dynamic consistency in pricing under model uncertainty?
• RQ3 How do the bid and ask price processes behave in relation to the payoff structure of the contingent claim?
• RQ4 Can monotone finite difference schemes converge to the viscosity solution of the nonlinear PDE governing the bid-ask prices?
• RQ5 What numerical evidence supports the stability and correctness of the simulated bid-ask price surfaces?

Key findings

• The risk premium of the risky asset is shown to be maximum-distributed under the G-framework, reflecting the worst-case scenario in uncertainty.
• The bid-ask dynamic pricing mechanism is proven to be Markovian and time-consistent, ensuring coherence over time.
• The bid and ask price processes satisfy fully nonlinear PDEs derived from the G-heat equation, with the ask price above the bid price as expected.
• The monotone finite difference schemes converge to the viscosity solution, with the iterates forming a nondecreasing, bounded sequence converging to a unique solution.
• Numerical simulations for digital and butterfly options confirm that the ask (bid) price surfaces preserve the monotonicity and convexity (concavity) of the payoff function.
• For the digital option with K=100, T=0.5, r=0.10, σ∈[0.15,0.25], and Δt=0.0025, the bid-ask spread is clearly visible and stable across the simulated surfaces.