Duy-Minh Dang

Better than Pre-Commitment Mean-Variance Portfolio Allocation Strategies: A Semi-Self-Financing Hamilton-Jacobi-Bellman Equation Approach

We generalize the idea of semi-self-financing strategies, originally discussed in Ehrbar (1990), and later formalized in Cui et al (2012), for the pre-commitment mean-variance (MV) optimal portfolio allocation problem. The proposed semi-self-financing strategies are built upon a numerical solution framework for Hamilton–Jacobi–Bellman equations, and can be readily employed in a very general setting, namely continuous or discrete re-balancing, jump-diffusions with finite activity, and realistic portfolio constraints. We show that if the portfolio wealth exceeds a threshold, an MV optimal strategy is to withdraw cash. These semi-self-financing strategies are generally non-unique. Numerical results confirming the superiority of the efficient frontiers produced by the strategies with positive cash withdrawals are presented. Tests based on estimation of parameters from historical time series show that the semi-self-financing strategy is robust to estimation ambiguities.

Adaptive and High-Order Methods for Valuing American Options

We develop space-time adaptive and high-order methods for valuing American options using a partial differential equation (PDE) approach. The linear complementarity problemarising due to the free boundary is handled by a penalty method. Both finite difference and finite element methods are considered for the space discretization of the PDE, while classical finite differences, such as Crank-Nicolson, are used for the time discretization. The high-order discretization in space is based on an optimal finite element collocation method, the main computational requirements of which are the solution of one tridiagonal linear system at each time step, while the resulting errors at the gridpoints and midpoints of the space partition are fourth-order. To control the space error, we use adaptive gridpoint distribution based on an error equidistribution principle. A time stepsize selector is used to further increase the efficiency of the methods. Numerical examples show that our methods converge fast and provide highly accurate options prices, Greeks, and early exercise boundaries

An efficient graphics processing unit-based parallel algorithm for pricing multi-asset American options

We develop highly efficient parallel PDE‐based pricing methods on graphics processing units (GPUs) for multi‐asset American options. Our pricing approach is built upon a combination of a discrete penalty approach for the linear complementarity problem arising because of the free boundary and a GPU‐based parallel alternating direction implicit approximate factorization technique with finite differences on uniform grids for the solution of the linear algebraic system arising from each penalty iteration. A timestep size selector implemented efficiently on GPUs is used to further increase the efficiency of the methods. We demonstrate the efficiency and accuracy of the parallel numerical methods by pricing American options written on three assets. Copyright

Quadratic spline collocation for one-dimensional linear parabolic partial differential equations

New methods for solving general linear parabolic partial differential equations (PDEs) in one space dimension are developed. The methods combine quadratic-spline collocation for the space discretization and classical finite differences, such as Crank-Nicolson, for the time discretization. The main computational requirements of the most efficient method are the solution of one tridiagonal linear system at each time step, while the resulting errors at the gridpoints and midpoints of the space partition are fourth order. The stability and convergence properties of some of the new methods are analyzed for a model problem. Numerical results demonstrate the stability and accuracy of the methods. Adaptive mesh techniques are introduced in the space dimension, and the resulting method is applied to the American put option pricing problem, giving very competitive results.

A Parallel Implementation on GPUs of ADI Finite Difference Methods for Parabolic PDEs with Applications in Finance

We study a parallel implementation on a Graphics Processing Unit (GPU) of Alternating Direction Implicit (ADI) time-discretization methods for solving time-dependent parabolic Partial Differential Equations (PDEs) in three spatial dimensions with mixed spatial derivatives in a variety of applications in computational finance. Finite differences on uniform grids are used for the spatial discretization of the PDEs. As examples, we apply the GPU-based parallel methods to price European rainbow and European basket options, each written on three assets. Numerical results showing the efficiency of the parallel methods are provided.

Better than pre-commitment optimal mean-variance portfolio allocation: a semi-self-financing Hamilton-Jacobi-Bellman approach

We generalize the idea of semi-self-financing strategies, originally discussed in Ehrbar (1990), and later formalized in Cui et al (2012), for the pre-commitment mean-variance (MV) optimal portfolio allocation problem. The proposed semi-self-financing strategies are built upon a numerical solution framework for Hamilton–Jacobi–Bellman equations, and can be readily employed in a very general setting, namely continuous or discrete re-balancing, jump-diffusions with finite activity, and realistic portfolio constraints. We show that if the portfolio wealth exceeds a threshold, an MV optimal strategy is to withdraw cash. These semi-self-financing strategies are generally non-unique. Numerical results confirming the superiority of the efficient frontiers produced by the strategies with positive cash withdrawals are presented. Tests based on estimation of parameters from historical time series show that the semi-self-financing strategy is robust to estimation ambiguities.

GPU Pricing of Exotic Cross-Currency Interest Rate Derivatives with a Foreign Exchange Volatility Skew Model

We present a Graphics Processing Unit (GPU) parallelization of the computation of the price of exotic cross-currency interest rate derivatives via a Partial Differential Equation (PDE) approach. In particular, we focus on the GPU-based parallel pricing of long-dated foreign exchange (FX) interest rate hybrids, namely Power Reverse Dual Currency (PRDC) swaps with Bermudan cancelable features. We consider a three-factor pricing model with FX volatility skew which results in a time-dependent parabolic PDE in three spatial dimensions. Finite difference methods on uniform grids are used for the spatial discretization of the PDE, and the Alternating Direction Implicit (ADI) technique is employed for the time discretization. We then exploit the parallel architectural features of GPUs together with the Compute Unified Device Architecture (CUDA) framework to design and implement an efficient parallel algorithm for pricing PRDC swaps. Over each period of the tenor structure, we divide the pricing of a Bermudan cancelable PRDC swap into two independent pricing subproblems, each of which can efficiently be solved on a GPU via a parallelization of the ADI timestepping technique. Numerical results indicate that GPUs can provide significant increase in performance over CPUs when pricing PRDC swaps. An analysis of the impact of the FX skew on such derivatives is provided.

Graphics processing unit pricing of exotic cross-currency interest rate derivatives with a foreign exchange volatility skew model

We present a graphics processing unit (GPU) parallelization of the computation of the price of exotic cross‐currency interest rate derivatives via a partial differential equation (PDE) approach. In particular, we focus on the GPU‐based parallel pricing of long‐dated foreign exchange (FX) interest rate hybrids, namely power reverse dual currency (PRDC) swaps with Bermudan cancelable features. We consider a three‐factor pricing model with FX volatility skew, which results in a time‐dependent parabolic PDE in three spatial dimensions. Finite difference methods on uniform grids are used for the spatial discretization of the PDE, and the alternating direction implicit (ADI) technique is employed for the time discretization. We then exploit the parallel architectural features of GPUs together with the Compute Unified Device Architecture framework to design and implement an efficient parallel algorithm for pricing PRDC swaps. Over each period of the tenor structure, we divide the pricing of a Bermudan cancelable PRDC swap into two independent pricing subproblems, each of which can efficiently be solved on a GPU via a parallelization of the ADI timestepping technique. Numerical results indicate that GPUs can provide significant increase in performance over CPUs when pricing PRDC swaps. An analysis of the impact of the FX skew on such derivatives is provided. Copyright

Continuous time mean-variance optimal portfolio allocation under jump diffusion: an numerical impulse control approach

We present efficient partial differential equation (PDE) methods for continuous time mean-variance portfolio allocation problems when the underlying risky asset follows a jump-diffusion. The standard formulation of mean-variance optimal portfolio allocation problems, where the total wealth is the underlying stochastic process, gives rise to a one-dimensional (1-D) non-linear Hamilton-Jacobi- Bellman (HJB) partial integro-differential equation (PIDE) with the control present in the integrand of the jump term, and thus is difficult to solve efficiently. In order to preserve the efficient handling of the jump term, we formulate the asset allocation problem as a 2-D impulse control problem, one dimension for each asset in the portfolio, namely the bond and the stock. We then develop a numerical scheme based on a semi-Lagrangian timestepping method, which we show to be monotone, consistent, and stable. Hence, assuming a strong comparison property holds, the numerical solution is guaranteed to converge to the unique viscosity solution of the corresponding HJB PIDE. The correctness of the proposed numerical framework is verified by numerical examples. We also discuss the effects on the efficient frontier of realistic financial modeling, such as different borrowing and lending interest rates, transaction costs and constraints on the portfolio, such as maximum limits on borrowing and solvency.

Pricing of cross-currency interest rate derivatives on Graphics Processing Units

We present a Graphics Processing Unit (GPU) parallelization of the computation of the price of cross-currency interest rate derivatives via a Partial Differential Equation (PDE) approach. In particular, we focus on the GPU-based parallel computation of the price of long-dated foreign exchange interest rate hybrids, namely Power Reverse Dual Currency (PRDC) swaps with Bermudan cancelable features. We consider a three-factor pricing model with foreign exchange skew which results in a time-dependent parabolic PDE in three spatial dimensions. Finite difference methods on uniform grids are used for the spatial discretization of the PDE, and the Alternating Direction Implicit (ADI) technique is employed for the time discretization. We then exploit the parallel architectural features of GPUs together with the Compute Unified Device Architecture (CUDA) framework to design and implement an efficient parallel algorithm for pricing PRDC swaps. Over each period of the tenor structure, we divide the pricing of a Bermudan cancelable PRDC swap into two independent pricing subproblems, each of which can efficiently be solved on a GPU via a parallelization of the ADI scheme at each timestep. Using this approach on two NVIDIA Tesla C870 GPUs of an NVIDIA 4-GPU Tesla S870 to price a Bermudan cancelable PRDC swap having a 30 year maturity and annual exchange of fund flows, we have achieved an asymptotic speedup by a factor of 44 relative to a single thread on a 2.0GHz Xeon processor.