Numerical convergence properties of option pricing PDEs with uncertain volatility

Abstract

The pricing equations derived from uncertain volatility models in finance are often cast in the form of nonlinear partial differential equations. Implicit timestepping leads to a set of nonlinear algebraic equations which must be solved at each timestep. To solve these equations, an iterative approach is employed. In this paper, we prove the convergence of a particular iterative scheme for one factor uncertain volatility models. We also demonstrate how non-monotone discretization schemes (such as standard Crank-Nicolson timestepping) can converge to incorrect solutions, or lead to instability. Numerical examples are provided.

An ENO-Based Method for Second-Order Equations and Application to the Control of Dike Levels

“…The convergence results of [16]. The higher order schemes show fast convergence and reach highly satisfactory approximations for N = 120, although the differences in this test are relatively small.…”

“…where the order of the truncation term is now decreased by one compared to (16), since the second-order derivative in h := d 2 H dx 2 replaces the first-order derivative in (12). 2 Changing the indices k to k + 1 and (k) to (k−1) and separating the first term in the summation gives us:…”

“…Pooley et al [16] show that a non-monotone scheme can lead to incorrect solutions. They utilise a locally first-order upwind-like finite difference discretization With n L the number of points in the stretched region, we use the transformation:…”

Numerical methods for controlled Hamilton-Jacobi-Bellman PDEs in finance

“…Examples where such nonlinear PDEs arise include transaction cost/uncertain volatility models [28,4,38], passport options [3,43], unequal borrowing/lending costs [13], large investor effects [2], risk control in reinsurance [32], pricing options and insurance in incomplete markets using an instantaneous Sharpe ratio [51,31,11], and optimal consumption [12,15]. A recent survey article on the theoretical aspects of this topic is given in [35].…”

“…For our problems we need to ensure that our discretization methods converge to the financially relevant solution, which in this case is the viscosity solution [18]. As demonstrated in [38], seemingly reasonable discretization methods can converge to non-viscosity solutions. We show that an optimal control formulation is in fact quite convenient for verifying monotonicity, l ∞ stability and consistency of our discrete schemes.…”