A nonsmooth version of Newton's method
Mathematical Programming: Series A and B
On quadratic cost criteria for option hedging
Mathematics of Operations Research
Variance-optimal hedging in discrete time
Mathematics of Operations Research
Quadratic Convergence for Valuing American Options Using a Penalty Method
SIAM Journal on Scientific Computing
A Smoothing Newton Method for Minimizing a Sum of Euclidean Norms
SIAM Journal on Optimization
Hedging Derivative Securities and Incomplete Markets: An e-Arbitrage Approach
Operations Research
Numerical methods for nonlinear equations in option pricing
Numerical methods for nonlinear equations in option pricing
A penalty method for American options with jump diffusion processes
Numerische Mathematik
A Semi-Lagrangian Approach for American Asian Options under Jump Diffusion
SIAM Journal on Scientific Computing
A Finite Difference Scheme for Option Pricing in Jump Diffusion and Exponential Lévy Models
SIAM Journal on Numerical Analysis
| Hi-index | 7.29 |
Hedging a contingent claim with an asset which is not perfectly correlated with the underlying asset results in unhedgeable residual risk. Even if the residual risk is considered diversifiable, the option writer is faced with the problem of uncertainty in the estimation of the drift rates of the underlying and the hedging instrument. If the residual risk is not considered diversifiable, then this risk can be priced using an actuarial standard deviation principle in infinitesimal time. In both cases, these models result in the same nonlinear partial differential equation (PDE). A fully implicit, monotone discretization method is developed for solution of this pricing PDE. This method is shown to converge to the viscosity solution. Certain grid conditions are required to guarantee monotonicity. An algorithm is derived which, given an initial grid, inserts a finite number of nodes in the grid to ensure that the monotonicity condition is satisfied. At each timestep, the nonlinear discretized algebraic equations are solved using an iterative algorithm, which is shown to be globally convergent. Monte Carlo hedging examples are given to illustrate the profit and loss distribution at the expiry of the option.