A PDE pricing framework for cross-currency interest rate derivatives

Dang, Duy Minh; Christara, Christina C.; Jackson, Kenneth R.; Lakhany, Asif

doi:10.1016/j.procs.2010.04.267

Cited by 18 publications

(17 citation statements)

References 6 publications

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“…Table 1. It is evident that second-order convergence is achieved, as expected from the numerical scheme (7) and the interpolation scheme (4). The prices are expressed as a percentage of the notional N d .…”

Section: Numerical Resultsmentioning

confidence: 65%

“…For the discretization of the PDE, we extend the second-order ADI FD approach developed in [4]. We truncate the unbounded domain into a finite-sized computational one {(s, .…”

Section: Methodsmentioning

confidence: 99%

“…We truncate the unbounded domain into a finite-sized computational one {(s, . We use the following ADI splitting scheme [4], which, starting from the vector of initial conditions u 0;n , generates an approximation u M+1;n by Let u m;n denote the vector of approximate values u(i∆s, j∆r d , k∆r f , m∆τ; n∆a), i = 0, .…”

Section: Methodsmentioning

confidence: 99%

“…More specifically, apart from dates {T α } β α=1 , for each fixed value of a, the function u satisfies the PDE[4] Denote by u ≡ u(s, r d , r f ,t; a) the domestic value function of a FX-TARN PRDC swap.…”

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confidence: 99%

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A PDE Pricing Framework for Cross-Currency Interest Rate Derivatives with Target Redemption Features

Christara¹,

Dang²,

Jackson³

et al. 2010

AIP Conference Proceedings

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We propose a general framework for efficient pricing via a partial differential equation (PDE) approach for exotic cross-currency interest rate (IR) derivatives, with strong emphasis on long-dated foreign exchange (FX) IR hybrids, namely Power Reverse Dual Currency (PRDC) swaps with a FX Target Redemption (FX-TARN) provision. The FX-TARN provision provides a cap on the FX-linked PRDC coupon amounts, and once the accumulated coupon amount reaches this cap, the underlying PRDC swap terminates. Our PDE pricing framework is based on an auxiliary state variable to keep track of the total accumulated PRDC coupon amount. Finite differences on uniform grids and the Alternating Direction Implicit (ADI) method are used for the spatial and time discretizations, respectively, of the model-dependent PDE corresponding to each discretized value of the auxiliary variable. Numerical examples illustrating the convergence properties of the numerical methods are provided.

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Section: Numerical Resultsmentioning

confidence: 65%

“…For the discretization of the PDE, we extend the second-order ADI FD approach developed in [4]. We truncate the unbounded domain into a finite-sized computational one {(s, .…”

Section: Methodsmentioning

confidence: 99%

Section: Methodsmentioning

confidence: 99%

mentioning

confidence: 99%

See 2 more Smart Citations

A PDE Pricing Framework for Cross-Currency Interest Rate Derivatives with Target Redemption Features

Christara¹,

Dang²,

Jackson³

et al. 2010

AIP Conference Proceedings

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“…A discussion of the calibration of the aforementioned pricing model can be found in the literature, for example, [3,9]. A discussion of the calibration of the aforementioned pricing model can be found in the literature, for example, [3,9].…”

Section: Introductionmentioning

confidence: 99%

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

Dang

Christara

Jackson

2012

Concurrency and Computation

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We present a graphics processing unit (GPU) parallelization of the computation of the price of exotic crosscurrency 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. PRDC swaps. The use of a local volatility model provides better modeling for the skewness of the FX rate and at the same time avoids introducing more stochastic factors into the model.The ever growing interest in cross-currency interest rate derivatives in general, and PRDC swaps in particular, has created a need for efficient pricing and hedging strategies for them. The popular choice for pricing PRDC swaps is Monte-Carlo (MC) simulation, but this approach has several major disadvantages, including slow convergence, and the limitation that the price is obtained at a single point only in the domain as opposed to the global character of the partial differential equation (PDE) approach. In addition, accurate hedging parameters, such as delta and gamma, are generally harder to compute via an MC approach than via a PDE approach. On the other hand, when pricing PRDC swaps by the PDE approach, each stochastic factor in the pricing model gives rise to a spatial variable in the PDE. Because of the 'curse of dimensionality' associated with highdimensional PDEs-not to mention additional complexity due to multiple cash flows and exotic features, such as Bermudan cancelability-the pricing of such derivatives via the PDE approach is not only mathematically challenging but also very computationally intensive.Over the last few years, the rapid evolution of graphics processing units (GPUs) into powerful, cost efficient, programmable computing architectures for general purpose computations has provided application potential beyond the primary purpose of graphics processing. In the area of computational finance, although there has been great interest in utilizing GPUs in developing eff...

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A semi‐Lagrangian ε$$ \varepsilon $$‐monotone Fourier method for continuous withdrawal GMWBs under jump‐diffusion with stochastic interest rate

Lu,

Dang

2023

Numerical Methods Partial

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We develop an efficient pricing approach for guaranteed minimum withdrawal benefits (GMWBs) with continuous withdrawals under a realistic modeling setting with jump‐diffusions and stochastic interest rate. Utilizing an impulse stochastic control framework, we formulate the no‐arbitrage GMWB pricing problem as a time‐dependent Hamilton‐Jacobi‐Bellman (HJB) Quasi‐Variational Inequality (QVI) having three spatial dimensions with cross derivative terms. Through a novel numerical approach built upon a combination of a semi‐Lagrangian method and the Green's function of an associated linear partial integro‐differential equation, we develop an ‐monotone Fourier pricing method, where is a monotonicity tolerance. Together with a provable strong comparison result for the HJB‐QVI, we mathematically demonstrate convergence of the proposed scheme to the viscosity solution of the HJB‐QVI as . We present a comprehensive study of the impact of simultaneously considering jumps in the subaccount process and stochastic interest rate on the no‐arbitrage prices and fair insurance fees of GMWBs, as well as on the holder's optimal withdrawal behaviors.

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A PDE pricing framework for cross-currency interest rate derivatives

Cited by 18 publications

References 6 publications

A PDE Pricing Framework for Cross-Currency Interest Rate Derivatives with Target Redemption Features

A PDE Pricing Framework for Cross-Currency Interest Rate Derivatives with Target Redemption Features

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

A semi‐Lagrangian ε$$ \varepsilon $$‐monotone Fourier method for continuous withdrawal GMWBs under jump‐diffusion with stochastic interest rate

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