Least Squares Monte Carlo Research Articles

Investment Decision for Long-Term Battery Energy Storage System Using Least Squares Monte Carlo

The use of renewable energy sources to achieve carbon neutrality is increasing. However, the uncertainty and volatility of renewable resources are causing problems in power systems. Flexible and low-carbon resources such as Energy Storage Systems (ESSs) are essential for solving the problems of power systems and achieving greenhouse gas reduction goals. However, ESSs are not being installed because of Korea’s fuel-based electricity market. To address this issue, this paper presents a method for determining the optimal investment timing of Battery Energy Storage Systems (BESSs) using the Least Squares Monte Carlo (LSMC) method. A case study is conducted considering the System Marginal Price (SMP) and Capacity Payment (CP), which are electricity rates in Korea. Revenue is calculated through the arbitrage of a 10 MW/40 MWh lithium-ion BESS, and linear programming optimization is performed for ESS scheduling to maximize revenue. The ESS revenue with uncertainty is modeled as a stochastic process using Geometric Brownian Motion (GBM), and the optimal time to invest in an ESS is determined using an LSMC simulation considering investment costs. The proposed method can be used as a decision-making tool for ESS investors to provide information on facility investments in arbitrage situations.

Distributed Least-Squares Monte Carlo for American Option Pricing

Option pricing is an important research field in financial markets, and the American option is a common financial derivative. Fast and accurate pricing solutions are critical to the stability and development of the market. Computational techniques, especially the least squares Monte Carlo (LSMC) method, have been broadly used in optimizing the pricing algorithm. This paper discusses the application of distributed computing technology to enhance the LSMC in American option pricing. Although parallel computing has been used to improve the LSMC method, this paper is the first to explore distributed computing technology for LSMC enhancement. Compared with parallel computing, distributed computing has several advantages, including reducing the computational complexity by the “divide and conquer” method, avoiding the complicated matrix transformation, and improving data privacy as well as security. Moreover, LSMC is suitable for distributed computing because the price paths can be simulated and regressed separately. This research aims to show how distributed computing, particularly the divide and conquer approach implemented by Apache Spark, can be used to improve the efficiency and accuracy of LSMC in American option pricing. This paper provides an innovative solution to the financial market and could contribute to the advancement of American option pricing research.

Foreign Exchange Options on Heston-CIR Model Under Lévy Process Framework

In this paper, we consider the Heston-CIR model with Lévy process for pricing in the foreign exchange (FX) market by providing a new formula that better fits the distribution of prices. To do that, first, we study the existence and uniqueness of the solution to this model. Second, we examine the strong convergence of the Lévy process with stochastic domestic short interest rates, foreign short interest rates and stochastic volatility. Then, we apply Least Squares Monte Carlo (LSMC) method for pricing American options under our model with stochastic volatility and stochastic interest rate. Finally, by considering real-world market data, we illustrate numerical results for the four-factor Heston-CIR Lévy model.

A machine learning approach to portfolio pricing and risk management for high‐dimensional problems

AbstractWe present a general framework for portfolio risk management in discrete time, based on a replicating martingale. This martingale is learned from a finite sample in a supervised setting. Our method learns the features necessary for an effective low‐dimensional representation, overcoming the curse of dimensionality common to function approximation in high‐dimensional spaces, and applies for a wide range of model distributions. We show numerical results based on polynomial and neural network bases applied to high‐dimensional Gaussian models. In these examples, both bases offer superior results to naive Monte Carlo methods and regress‐now least‐squares Monte Carlo (LSMC).

Neural networks meet least squares Monte Carlo at internal model data

In August 2020 we published “Comprehensive Internal Model Data for Three Portfolios” as an outcome of our work for the committee “Actuarial Data Science” of the German Actuarial Association. The data sets include realistic cash-flow models outputs used for proxy modelling of life and health insurers. Using these data, we implement the hitherto most promising model in proxy modeling consisting of ensembles of feed-forward neural networks and compare the results with the least squares Monte Carlo (LSMC) polynomial regression. To date, the latter represents—to our best knowledge—the most accurate proxy function productively in use by insurance companies. An additional goal of this publication is a more precise description of “Comprehensive Internal Model Data for Three Portfolios” for other researchers, practitioners and regulators interested in developing solvency capital requirement (SCR) proxy models.

JDOI variance reduction method and the pricing of American-style options

This article revisits the Diffusion Operator Integral (DOI) variance reduction technique originally proposed in Heath and Platen (2002) and extends its theoretical concept to the pricing of American-style options under (time-homogeneous) Lévy stochastic differential equations. The resulting Jump Diffusion Operator Integral (JDOI) method can be combined with numerous Monte Carlo-based stopping-time algorithms, including the ubiquitous least-squares Monte Carlo (LSMC) algorithm of Longstaff and Schwartz (cf. Carriere (1996) and Longstaff and Schwartz (2001)). We exemplify the usefulness of our theoretical derivations under a concrete, though very general jump-diffusion stochastic volatility dynamics and test the resulting LSMC-based version of the JDOI method. The results provide evidence of a strong variance reduction when compared with a simple application of the LSMC algorithm and proves that applying our technique on top of Monte Carlo-based pricing schemes provides a powerful way to speed-up these methods.

The Acceleration of Least Squares Monte Carlo in Risk Management

The Least Squares Monte Carlo (LSMC) method was first proposed by Longstaff and Schwartz [1] to price the American option, since then it has been applied in different industries from banking [2] to energy sector [3]. In the last decade, there is an increasing demand for sophisticated risk modeling [4]. To overcome the computational complexity of those models, the proxy techniques have gain popularity in both risk management practice and research over the last decade [5].

A Multilevel Monte Carlo Method for the Valuation of Swing Options

In this study, we propose a novel approach for the valuation of swing options. Swing options are a kind of American options with multiple exercise rights traded in energy markets. Longstaff and Schwartz have suggested a regression-based Monte Carlo method known as the least-squares Monte Carlo (LSMC) method to value American options. In this work, first we introduce the LSMC method for the pricing of swing options. Then, to achieve a desired accuracy for the price estimation, we combine the idea of LSMC with multilevel Monte Carlo (MLMC) method. Finally, to illustrate the proper behavior of this combination, we conduct numerical results based on the Black–Scholes model. Numerical results illustrate the efficiency of the proposed approach.

JDOI Variance Reduction Method and the Pricing of American-Style Options

The present article revisits the Diffusion Operator Integral (DOI) variance reduction technique originally proposed in [HP02] and extends its theoretical concept to the pricing of American-style options under (time-homogeneous) Levy stochastic differential equations. The resulting Jump Diffusion Operator Integral (JDOI) method can be combined with numerous Monte Carlo based stopping-time algorithms, including the ubiquitous least-squares Monte Carlo (LSMC) algorithm of Longstaff and Schwartz (cf. [Car96], [LS01]). We exemplify the usefulness of our theoretical derivations under a concrete, though very general jump-diffusion stochastic volatility dynamics and test the resulting LSMC based version of the JDOI method. The results provide evidence of a strong variance reduction when compared with a simple application of the LSMC algorithm and proves that applying our technique on top of Monte Carlo based pricing schemes provides a powerful way to speed-up these methods.

A new lattice-based scheme for swing option pricing under mean-reverting regime-switching jump–diffusion processes

Swing options are complex path-dependent contracts, granting their holders a prefixed number of transaction rights to buy/sell a variable amount of the underlying asset (e.g. energy commodities) subject to daily or periodic constraints. Stating the swing option price as the solution of a stochastic optimal control problem, we employ a dynamic programming formulation in which the underlying asset price is modeled by a mean-reverting regime-switching jump–diffusion process. We explore a newly devised lattice-based pricing framework to find the premium of swing options in a cost-effective and easily implementable manner. We compare the performance of the proposed tree building procedure with a simulation-based Least-Squares Monte-Carlo (LSM) approach.

Least-squares Monte-Carlo methods for optimal stopping investment under CEV models

The optimal stopping investment is a kind of mixed expected utility maximization problems with optimal stopping time. The aim of this paper is to develop the least-squares Monte-Carlo methods to solve the optimal stopping investment under the constant elasticity of variance (CEV) model. Such a problem has no closed-form solutions for the value functions, optimal strategies and optimal exercise boundaries due to the early exercised feature. The dual optimal stopping problem is first derived and then the strong duality between the dual and prime problems is established. The least-squares Monte-Carlo methods based on the dual control theory are developed and numerical simulations are provided. Both the power and non-HARA utilities are studied.

Machine Learning in Least-Squares Monte Carlo Proxy Modeling of Life Insurance Companies

Under the Solvency II regime, life insurance companies are asked to derive their solvency capital requirements from the full loss distributions over the coming year. Since the industry is currently far from being endowed with sufficient computational capacities to fully simulate these distributions, the insurers have to rely on suitable approximation techniques such as the least-squares Monte Carlo (LSMC) method. The key idea of LSMC is to run only a few wisely selected simulations and to process their output further to obtain a risk-dependent proxy function of the loss. In this paper, we present and analyze various adaptive machine learning approaches that can take over the proxy modeling task. The studied approaches range from ordinary and generalized least-squares regression variants over generalized linear model (GLM) and generalized additive model (GAM) methods to multivariate adaptive regression splines (MARS) and kernel regression routines. We justify the combinability of their regression ingredients in a theoretical discourse. Further, we illustrate the approaches in slightly disguised real-world experiments and perform comprehensive out-of-sample tests.

Pricing bounds and bang-bang analysis of the Polaris variable annuities

This paper studies the no-arbitrage pricing of the ‘Polaris Income Plus Daily’ structured in the ‘Polaris Choice IV’ variable annuities recently issued by the American International Group. Distinct from the withdrawal benefits studied in the literature, Polaris allows the income base to ‘lock in’ the high water mark of the investment account over a certain monitoring period which is related to the timing of the policyholder's first withdrawal. By prudently introducing certain auxiliary state and decision variables, we manage to formulate the pricing model under a Markovian stochastic optimal control framework. By a slight modification of the fee structure, we show the existence of a bang-bang solution to the stochastic control problem: the optimal withdrawal strategy is among a few explicit choices. We consequently design a novel Least Squares Monte Carlo (LSMC) algorithm to approach the optimal solution. Convergence results are established for the algorithm by applying the theory of nonparametric sieve estimation. Compared with existing LSMCs, our algorithm possesses a number of advantages such as memory reduction, preservation of convexity and monotonicity of the continuation value, reducing computational cost of the tuning parameter selection, and evading extrapolation of the value function estimate. Finally, we prove that the obtained pricing result works as an upper bound of the no-arbitrage price of Polaris with the real fee structure. Numerical experiments show that this upper bound is fairly tight.

Quasi-Regression Monte-Carlo Scheme for Semi-Linear PDEs and BSDEs with Large Scale Parallelization on GPUs

In this paper, we design a novel algorithm based on Least-Squares Monte Carlo (LSMC) in order to approximate the solution of discrete time Backward Stochastic Differential Equations (BSDEs). Our algorithm allows massive parallelization of the computations on multicore devices such as graphics processing units (GPUs). Our approach consists of a novel method of stratification which appears to be crucial for large scale parallelization.

LASSO-Based Simulation for High-Dimensional Multi-Period Portfolio Optimization

This paper proposes a regression-based simulation algorithm for multi-period mean-variance portfolio (MVP) optimization problems with constraints under a high-dimensional setting. We show that the least-squares Monte Carlo (LSMC) algorithm for portfolio optimization is either unstable or degenerates in a high-dimensional portfolio. The problem stems from the estimation error of the ordinary least-squares (OLS) in the regression steps. By introducing l1-regularization technique, we greatly extend the LSMC algorithm to a high-dimensional setting. Specifically, we propose to replace the OLS by the least absolute shrinkage and selection operator (LASSO). We prove the asymptotic convergence of the novel LASSO-based simulation under such a recursive regression setting. Numerical experiments suggest our algorithm achieves great stability under both low- and higher-dimensional cases.

Skewed target range strategy for multiperiod portfolio optimization using a two-stage least squares Monte Carlo method

In this paper, we propose a novel investment strategy for portfolio optimization problems. The proposed strategy maximizes the expected portfolio value bounded within a targeted range, composed of a conservative lower target representing a need for capital protection and a desired upper target representing an investment goal. This strategy favorably shapes the entire probability distribution of returns, as it simultaneously seeks a desired expected return, cuts off downside risk and implicitly caps volatility and higher moments. To illustrate the effectiveness of this investment strategy, we study a multiperiod portfolio optimization problem with transaction costs and develop a two-stage regression approach that improves the classical least squares Monte Carlo (LSMC) algorithm when dealing with difficult payoffs, such as highly concave, abruptly changing or discontinuous functions. Our numerical results show substantial improvements over the classical LSMC algorithm for both the constant relative risk-aversion (CRRA) utility approach and the proposed skewed target range strategy (STRS). Our numerical results illustrate the ability of the STRS to contain the portfolio value within the targeted range. When compared with the CRRA utility approach, the STRS achieves a similar mean-variance efficient frontier while delivering a better downside risk-return trade-off.

A Backward Simulation Method for Stochastic Optimal Control Problems

A number of optimal decision problems with uncertainty can be formulated into a stochastic optimal control framework. The Least-Squares Monte Carlo (LSMC) algorithm is a popular numerical method to approach solutions of such stochastic control problems as analytical solutions are not tractable in general. This paper generalizes the LSMC algorithm proposed in Shen and Weng (2017) to solve a wide class of stochastic optimal control models. Our algorithm has three pillars: a construction of auxiliary stochastic control model, an artificial simulation of the post-action value of state process, and a shape-preserving sieve estimation method which equip the algorithm with a number of merits including bypassing forward simulation and control randomization, evading extrapolating the value function, and alleviating computational burden of the tuning parameter selection. The efficacy of the algorithm is corroborated by an application to pricing equity-linked insurance products.

Sharp Target Range Strategies with Application to Dynamic Portfolio Selection

In this paper, we propose a novel investment strategy for portfolio optimization problems. The proposed strategy maximizes the expected portfolio value bounded within a targeted range, composed of a conservative lower target representing a need for capital protection and a desired upper target representing an investment goal. This strategy favorably shapes the entire probability distribution of returns, as it simultaneously seeks a desired expected return, cuts off downside risk and implicitly caps volatility and higher moments. To illustrate the effectiveness of this investment strategy, we study a multiperiod portfolio optimization problem with transaction costs and develop a two-stage regression approach that improves the classical least squares Monte Carlo (LSMC) algorithm when dealing with difficult payoffs, such as highly concave, abruptly changing or discontinuous functions. Our numerical results show substantial improvements over the classical LSMC algorithm for both the constant relative risk-aversion (CRRA) utility approach and the proposed skewed target range strategy (STRS). Our numerical results illustrate the ability of the STRS to contain the portfolio value within the targeted range. When compared with the CRRA utility approach, the STRS achieves a similar mean–variance efficient frontier while delivering a better downside risk–return trade-off.

Pricing multi-asset American option under Heston stochastic volatility model

In this paper, we employ the Least-Squares Monte-Carlo (LSM) algorithm regarding three discretization schemes, namely, the Euler–Maruyama discretization scheme, the Milstein scheme and the Quadratic Exponential (QE) scheme to price the multiple assets American put option under the Heston stochastic volatility model. Some numerical results are presented to demonstrate the effectiveness of the proposed methods.

A Least-Squares Monte Carlo Framework in Proxy Modeling of Life Insurance Companies

The Solvency II directive asks insurance companies to derive their solvency capital requirement from the full loss distribution over the coming year. While this is in general computationally infeasible in the life insurance business, an application of the Least-Squares Monte Carlo (LSMC) method offers a possibility to overcome this computational challenge. We outline in detail the challenges a life insurer faces, the theoretical basis of the LSMC method and the necessary steps on the way to a reliable proxy modeling in the life insurance business. Further, we illustrate the advantages of the LSMC approach via presenting (slightly disguised) real-world applications.