scholarly journals A Numerical Algorithm for a Fully Nonlinear PDE Involving the Jacobian Determinant

2014 ◽  
pp. 143-151
Author(s):  
Alexandre Caboussat ◽  
Roland Glowinski
Keyword(s):  
Numerical Algorithm ◽  
Nonlinear Pde ◽  
Jacobian Determinant ◽  
Fully Nonlinear ◽  
Fully Nonlinear Pde

scholarly journals Asian Option Pricing under an Uncertain Volatility Model

2020 ◽  
Vol 2020 ◽  
pp. 1-10
Author(s):  
Yuecai Han ◽  
Chunyang Liu
Keyword(s):  
Boundary Condition ◽  
Nonlinear Pde ◽  
Asian Option ◽  
Option Prices ◽  
Case Scenario ◽  
Worst Case ◽  
Volatility Model ◽  
Black Scholes ◽  
Uncertain Volatility ◽  
Fully Nonlinear Pde

In this paper, we study the asymptotic behavior of Asian option prices in the worst-case scenario under an uncertain volatility model. We derive a procedure to approximate Asian option prices with a small volatility interval. By imposing additional conditions on the boundary condition and splitting the obtained Black–Scholes–Barenblatt equation into two Black–Scholes-like equations, we obtain an approximation method to solve a fully nonlinear PDE.

scholarly journals A numerical algorithm for fully nonlinear HJB equations: An approach by control randomization

2014 ◽  
Vol 20 (2) ◽  
Author(s):  
Idris Kharroubi ◽  
Nicolas Langrené ◽  
Huyên Pham
Keyword(s):  
Numerical Algorithm ◽  
Detailed Comparison ◽  
Feedback Form ◽  
Hjb Equations ◽  
Alternative Scheme ◽  
Fully Nonlinear ◽  
Numerical Resolution ◽  
Partial Analysis ◽  
Hamilton Jacobi Bellman ◽  
Parametric Estimate

Abstract.We propose a probabilistic numerical algorithm to solve Backward Stochastic Differential Equations (BSDEs) with nonnegative jumps, a class of BSDEs introduced in [`Feynman–Kac representation for Hamilton–Jacobi–Bellman IPDE', Ann. Probab., to appear] for representing fully nonlinear HJB equations. This includes in particular numerical resolution for stochastic control problems with controlled volatility, possibly degenerate. Our backward scheme, based on least-squares regressions, takes advantage of high-dimensional properties of Monte Carlo methods, and also provides a parametric estimate in feedback form for the optimal control. A partial analysis of the algorithm error is presented, as well as numerical tests on the problem of option superreplication with uncertain volatilities and/or correlations, including a detailed comparison with the numerical results from the alternative scheme proposed in [J. Comput. Finance 14 (2011), 37–71].

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