Stochastic Grid Bundling Method Research Articles

Efficient parallel Monte-Carlo techniques for pricing American options including counterparty credit risk

In this article we mainly propose a numerical scheme, based on the novel Stochastic Grid Bundling Method (SGBM), to price American options in the presence of counterparty credit risk. More precisely, we consider the regression techniques (regress later) employed in the SGBM method and take advantage of the bundling structure to develop an efficient parallel strategy that is implemented on a GPU architecture. Also, a novel interpolation-based technique is efficiently applied in the XVA computation. Besides the advantages obtained in the sequential version, when compared with the more classical Least Squares Method, we show the relevant speedup of the parallel GPU-based version with respect to the sequential CPU-based one.

An SGBM-XVA demonstrator: a scalable Python tool for pricing XVA

In this work, we developed a Python demonstrator for pricing total valuation adjustment (XVA) based on the stochastic grid bundling method (SGBM). XVA is an advanced risk management concept which became relevant after the recent financial crisis. This work is a follow-up work on Chau and Oosterlee in (Int J Comput Math 96(11):2272–2301, 2019), in which we extended SGBM to numerically solving backward stochastic differential equations (BSDEs). The motivation for this work is basically two-fold. On the application side, by focusing on a particular financial application of BSDEs, we can show the potential of using SGBM on a real-world risk management problem. On the implementation side, we explore the potential of developing a simple yet highly efficient code with SGBM by incorporating CUDA Python into our program.

Stochastic grid bundling method for backward stochastic differential equations

ABSTRACTIn this work, we apply the Stochastic Grid Bundling Method (SGBM) to numerically solve backward stochastic differential equations (BSDEs). The SGBM algorithm is based on conditional expectations approximation by means of bundling of Monte Carlo sample paths and a local regress-later regression within each bundle. The basic algorithm for solving the backward stochastic differential equations will be introduced and an upper error bound is established for the local regression. A full error analysis is also conducted for the explicit version of our algorithm and numerical experiments are performed to demonstrate various properties of our algorithm.

Rolling Adjoints: Fast Greeks along Monte Carlo scenarios for early-exercise options

In this paper we extend the Stochastic Grid Bundling Method (SGBM), a regress-later Monte Carlo scheme for pricing early-exercise options, with an adjoint method to compute in a highly efficient manner the option sensitivities (the “Greeks”) along the Monte Carlo paths, with reasonable accuracy. The path-wise SGBM Greeks computation is based on the conventional path-wise sensitivity analysis, however, for a regress-later technique. The resulting sensitivities at the end of the monitoring period are implicitly rolled over into the sensitivities of the regression coefficients of the previous monitoring date. For this reason, we name the method Rolling Adjoints, which facilitates Smoking Adjoints [M. Giles, P. Glasserman, Smoking adjoints: fast Monte Carlo Greeks, Risk 19 (1) (2006) 88–92] to compute conditional sensitivities along the paths for options with early-exercise features.

KVA, Mind Your P's and Q's!

There is an increasing consensus within banks, on the need to recognize the impact of rising capital requirements on their derivative business in the form of capital valuation adjustment (KVA). However, because of varied reasons, there are still concerns over how exactly KVA should be computed, charged, and managed. The focus of our analysis here is on the numerical aspects of costs arising due to holding of counterparty credit risk (CCR) capital. CCR capital for a portfolio today under the internal model method (IMM) approach is based on its future exposure profile, usually computed by Monte Carlo methods. For the corresponding KVA, we need (outer) Monte Carlo simulation of the future capital, which ideally would then involve (inner) Monte Carlo simulation. Additionally, the measures under which the outer and inner scenarios should be simulated can be different, ideally real-world (P) in risk-neutral (Q). In this work we, first, propose the use of stochastic grid bundling method (SGBM), an American Monte Carlo based computational technique to circumvent the problem of nested simulation working under hybrid measures to efficiently solve the P-in-Q problem. Second, we demonstrate that hybrid measures can substantially affect the KVA values. Third, through a simple hedging example we demonstrate the implication of hedging KVA under various approximations and hybrid measures.

Rolling Adjoints : Fast Greeks Along Monte Carlo Scenarios for Early-Exercise Options

In this paper we extend the stochastic grid bundling method (SGBM), a regress-later based Monte Carlo scheme for pricing early-exercise options, with an adjoint method to compute in a highly efficient manner sensitivities along the paths, with reasonable accuracy. With the ISDA standard initial margin model being adopted by the financial markets, computing sensitivities along scenarios is required to compute quantities like the margin valuation adjustment.

Wrong Way Risk Modeling and Computation in Credit Valuation Adjustment for European and Bermudan Options

We study the impact of wrong-way-risk (WWR) on credit valuation adjustment (CVA) for European and Bermudan options, based on an intensity model. WWR is modeled by a dependency between the underlying asset and the intensity of the counterparty’s default. We consider three different models. We take the difference between the default-free value and the default-adjusted value for the purpose of the CVA calculations. Two numerical algorithms, the COS method and the Stochastic Grid Bundling Method (SGBM) are generalized and employed for the computations. By varying correlation coefficients, we perform a CVA stress test for European options, show differences in the optimal exercise boundaries, in corresponding Bermudan option values and compute the CVA shortfall.

Counterparty Credit Exposures for Interest Rate Derivatives using the Stochastic Grid Bundling Method

ABSTRACTThe regulatory credit value adjustment (CVA) for an outstanding over-the-counter (OTC) derivative portfolio is computed based on the portfolio exposure over its lifetime. Usually, the future portfolio exposure is approximated using the Monte Carlo simulation, as the portfolio value can be driven by several market risk-factors. For derivatives, such as Bermudan swaptions, that do not have an analytical approximation for their Mark-to-Market (MtM) value, the standard market practice is to use the regression functions from the least squares Monte Carlo method to approximate their MtM along simulated scenarios. However, such approximations have significant bias and noise, resulting in inaccurate CVA charge. In this paper, we extend the Stochastic Grid Bundling Method (SGBM) for the one-factor Gaussian short rate model, to efficiently and accurately compute Expected Exposure, Potential Future exposure and CVA for Bermudan swaptions. A novel contribution of the paper is that it demonstrates how different measures, for instance spot and terminal measure, can simultaneously be employed in the SGBM framework, to significantly reduce the variance and bias of the solution.

Counterparty Credit Exposures for Interest Rate Derivatives Using the Stochastic Grid Bundling Method

The regulatory credit value adjustment (CVA) for an outstanding over-the-counter (OTC) derivative portfolio, is computed based on the portfolio exposure over its lifetime. Usually the future portfolio exposure is approximated using Monte Carlo simulation, as the portfolio value can be driven by several market risk-factors. For derivatives, such as Bermudan swaptions, that do not have an analytical approximation for their Mark-to-Market (MtM) value, the standard market practice is to use the regression functions from least squares Monte Carlo method to approximate their MtM along simulated scenarios. However, such approximations have significant bias and noise, resulting in inaccurate CVA charge. In this paper we extend the Stochastic Grid Bundling Method (SGBM) for the one-factor Gaussian short rate model, to efficiently and accurately compute Expected Exposure, Potential Future exposure and CVA for Bermudan swaptions. A novel contribution of the paper is that it demonstrates, how different measures, for instance spot and terminal measure, can simultaneously be employed in the SGBM framework, to significantly reduce the variance and bias of the solution.

Accurate and Robust Numerical Methods for the Dynamic Portfolio Management Problem

This paper enhances a well-known dynamic portfolio management algorithm, the BGSS algorithm, proposed by Brandt et al. (Review of Financial Studies, 18(3):831–873, 2005). We equip this algorithm with the components from a recently developed method, the Stochastic Grid Bundling Method (SGBM), for calculating conditional expectations. When solving the first-order conditions for a portfolio optimum, we implement a Taylor series expansion based on a nonlinear decomposition to approximate the utility functions. In the numerical tests, we show that our algorithm is accurate and robust in approximating the optimal investment strategies, which are generated by a new benchmark approach based on the COS method (Fang and Oosterlee, in SIAM Journal of Scientific Computing, 31(2):826–848, 2008).

Fast and accurate exercise policies for Bermudan swaptions in the LIBOR market model

This paper describes an American Monte Carlo approach for obtaining fast and accurate exercise policies for pricing of callable LIBOR Exotics (e.g., Bermudan swaptions) in the LIBOR market model using the Stochastic Grid Bundling Method (SGBM). SGBM is a bundling and regression based Monte Carlo method where the continuation value is projected onto a space where the distribution is known. We also demonstrate an algorithm to obtain accurate and tight lower–upper bound values without the need for nested Monte Carlo simulations.

Pricing Bermudan options under Merton jump-diffusion asset dynamics

In this paper, a recently developed regression-based option pricing method, the Stochastic Grid Bundling Method (SGBM), is considered for pricing multidimensional Bermudan options. We compare SGBM with a traditional regression-based pricing approach and present detailed insight in the application of SGBM, including how to configure it and how to reduce the uncertainty of its estimates by control variates. We consider the Merton jump-diffusion model, which performs better than the geometric Brownian motion in modelling the heavy-tailed features of asset price distributions. Our numerical tests show that SGBM with appropriate set-up works highly satisfactorily for pricing multidimensional options under jump-diffusion asset dynamics.

The Stochastic Grid Bundling Method: Efficient pricing of Bermudan options and their Greeks

This paper describes a practical simulation-based algorithm, which we call the Stochastic Grid Bundling Method (SGBM) for pricing multi-dimensional Bermudan (i.e. discretely exercisable) options. The method generates a direct estimator of the option price, an optimal early-exercise policy as well as a lower bound value for the option price. An advantage of SGBM is that the method can be used for fast approximation of the Greeks (i.e., derivatives with respect to the underlying spot prices, such as delta, gamma, etc.) for Bermudan-style options. Computational results for various multi-dimensional Bermudan options demonstrate the simplicity and efficiency of the algorithm proposed.

ALGORITHMIC COUNTERPARTY CREDIT EXPOSURE FOR MULTI-ASSET BERMUDAN OPTIONS

For an efficient computation of the counterparty credit exposure profiles of the multi-asset options, a simulation-based method, named the Stochastic Grid Bundling Method (SGBM), is applied. The method is based on a 'regression later' technique used for the conditional expectation approximation and a bundling (or 'binning') technique used for state space partitioning. In the case of high-dimensional underlying asset processes, by using the bundling technique, the accuracy of exposure profiles is improved significantly, and the computation speed is reasonably fast. A detailed analysis for the bundling technique and regression approximation technique used in SGBM is given via various numerical examples. We provide an efficiency comparison of SGBM, the Standard Regression Method (SRM), and the Standard Regression Bundling Method (SRBM). We also show that for discontinuous payoffs, such as digital options, by using the bundling technique appropriately, SGBM can get accurate and stable results of option prices and exposure profiles. Compared with the benchmark results of one-dimensional European and Bermudan options, the SGBM has high accuracy in the computation of exposure profiles. The efficient calculation of the expected exposure (EE) by using SGBM forms the basis of the credit value adjustment (CVA) for multi-asset portfolios.

Pricing Bermudan Options Under Merton Jump-Diffusion Asset Dynamics

In this paper, a recently developed regression-based option pricing method, the Stochastic Grid Bundling Method (SGBM), is implemented for pricing multidimensional Bermudan options. We compare SGBM to a traditional regression-based pricing approach and present detailed insight in the application of SGBM, including how to configure it and how to reduce the uncertainty of its estimates by control variates. We consider the Merton jump-diffusion model, which performs better than the geometric Brownian motion in modeling the heavy-tail feature of the asset price distributions. Our numerical tests show that SGBM with appropriate setup works satisfactorily on pricing multidimensional options under jump-diffusion asset dynamics.

GPU Acceleration of the Stochastic Grid Bundling Method for Early-Exercise Options

In this work, a parallel GPU version of the Monte Carlo Stochastic Grid Bundling Method (SGBM) for pricing multi-dimensional early-exercise options is presented. To extend the method's applicability, the problem dimensions and the number of bundles will be increased drastically. This makes SGBM very expensive in terms of computational costs on conventional hardware systems based on CPUs. A parallelization strategy of the method is developed and the GPGPU paradigm is used to reduce the execution time. An improved technique for bundling asset paths, which is more efficient on parallel hardware is introduced. Thanks to the performance of the GPU version of SGBM, a general approach for computing the early-exercise policy is proposed. Comparisons between sequential and GPU parallel versions are presented.

Efficient Computation of Exposure Profiles for Counterparty Credit Risk

Three computational techniques for approximation of counterparty exposure for financial derivatives are presented. The exposure can be used to quantify so-called Credit Valuation Adjustment (CVA) and Potential Future Exposure (PFE), which are of utmost importance for modern risk management in the financial industry, especially since the recent credit crisis. The three techniques all involve a Monte Carlo path discretization and simulation of the underlying entities. Along the generated paths, the corresponding values and distributions are computed during the entire lifetime of the option. Option values are computed by either the finite difference method for the corresponding partial differential equations, or the simulation based Stochastic Grid Bundling Method (SGBM), or by the COS method, based on Fourier-cosine expansions. In this research, numerical results are presented for early-exercise options. The underlying asset dynamics are given by either the Black-Scholes or the Heston stochastic volatility model.

The Stochastic Grid Bundling Method: Efficient Pricing of Multidimensional Bermudan Options and the Greeks

This paper describes a practical simulation-based algorithm, which we call the Stochastic Grid Bundling Method (SGBM) for pricing multi-dimensional Bermudan (i.e. discretely exercisable) options. The method generates a direct estimator of the option price, an optimal early-exercise policy as well as a lower bound value for the option price. An advantage of SGBM is that the method can be used for fast approximation of the Greeks (i.e., derivatives with respect to the underlying spot prices, such as delta, gamma, etc) for Bermudan-style options. Computational results for various multi-dimensional Bermudan options demonstrate the simplicity and efficiency of the algorithm proposed.

Fast and Accurate Exercise Policies for Bermudan Swaptions in Libor Market Model

This paper describes an American Monte Carlo approach for obtaining fast and accurate exercise policies for pricing of callable LIBOR Exotics (e.g., Bermudan swaptions) in the LIBOR market model using the Stochastic Grid Bundling Method (SGBM). SGBM is a bundling and regression based Monte Carlo method where the continuation value is projected onto a space where the distribution is known. We also demonstrate an algorithm to obtain accurate and tight lower-upper bound values without the need for nested Monte Carlo simulations.