Numerical convergence properties of option pricing PDEs with uncertain volatility. (English) Zbl 1040.91053

The present paper provides numerical convergence properties of option pricing PDEs \[ U_{\tau}=\frac{\sigma(\Gamma)^2}{2}S^2U_{SS}+rSU_{S}-rU, \] for the fair price \(U\) of a contingent claim on an asset \(S\) in the Black-Scholes model with uncertain volatility \(\sigma(\Gamma),\) which is assumed to range in the interval \[ 0<\sigma_{\min}\leq \sigma(\Gamma)\leq \sigma_{\max}, \] where \(\Gamma:=U_{SS},\) \(r>0\) is the risk-free interest rate.
The authors show that the iterative scheme developed in the paper is globally convergent. They also derive conditions which ensure that the discrete scheme is monotone and hence converges to the financially relevant solution.
As a numerical example, they show that non-monotone, but implicit, schemes can lead to incorrect solutions, or to instability.


91G60 Numerical methods (including Monte Carlo methods)
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
35L70 Second-order nonlinear hyperbolic equations
91G20 Derivative securities (option pricing, hedging, etc.)
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