Least-squares Monte Carlo Algorithm Research Articles

Bias Correction in the Least-Squares Monte Carlo Algorithm

Bias Correction in the Least-Squares Monte Carlo Algorithm

Lower bounds for American option prices with control variates

This article discusses the application of optimally sampled control variates in the context of the Least-Squares Monte Carlo algorithm for pricing American options. We demonstrate theoretically that optimal sampling introduces bias when estimated exercise times are not stopping times. Numerical results show that this bias is an accurate proxy for the positive foresight bias of American option price estimates, effectively yielding in-sample lower-bound estimates.

Valuing photovoltaic power plants by compound real options

Meeting electricity demand with renewable generation requires spending trillions of dollars in investments exposed to financial risk. Renewable projects have multiple strategic real options to manage investment risk, but they are usually priced separately in isolation. This paper proposes a framework for valuing multistage decisions in a grid-scale photovoltaic (PV) power plant under uncertainty. The project is modelled as a compound portfolio of American sequential and mutually exclusive options to defer, expand and relocate. The compound options are jointly priced by a hierarchically nested least-squares Monte Carlo algorithm. The framework incorporates uncertainties related to decreasing costs in solar technology and the emergence of more profitable renewable tariffs elsewhere. The results show that the value of the compound option portfolio significantly exceeds the value of the single deferral option commonly considered in practice. Ignoring expansion and relocation flexibilities may lead to underinvestment by mispricing the value of PV projects. Incorporating these options encourages early commitment of the PV investment. Moreover, summation of individual option prices may not be a good approximation of the true portfolio value, as interactions among options exist. These findings may improve allocation efficiency of renewable investments, accelerate energy transition and benefit investors, funding agencies and regulators.

Bias Correction in the Least-Squares Monte Carlo Algorithm

Bias Correction in the Least-Squares Monte Carlo Algorithm

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.

“Regression anytime” with brute-force SVD truncation

We propose a new least-squares Monte Carlo algorithm for the approximation of conditional expectations in the presence of stochastic derivative weights. The algorithm can serve as a building block for solving dynamic programming equations, which arise, for example, in nonlinear option pricing problems or in probabilistic discretization schemes for fully nonlinear parabolic partial differential equations. Our algorithm can be generically applied when the underlying dynamics stem from an Euler approximation to a stochastic differential equation. A built-in variance reduction ensures that the convergence in the number of samples to the true regression function takes place at an arbitrarily fast polynomial rate, if the problem under consideration is smooth enough.

Real options approach for a staged field development with optional wells

With the decreasing average size of new discoveries in mature production areas, the uncertainties in the base of oil field investment decisions are continually increasing. Fewer appraisal wells, which allow to decrease the amount of subsurface uncertainty, are typically drilled before the development of a small field compared to large fields. In this context, novel solutions must be established to commercialize small discoveries under technical and market uncertainties. In such conditions, managerial flexibilities, which enable to change the course of the project in the event of new information acquisition, must be critically considered in the investment valuation process. Combining the real options approach and decision analysis, we establish a novel model to identify the additional value created by a sequential drilling strategy for field development under oil price and resource uncertainty. In particular, we capture the sequence of the key investment and operating decisions pertaining to a marginal field development in cooperation with an oil industry partner, which corresponds to a synthetic yet realistic project case. By considering the flexibility in dividing the production well drilling into two stages, we adopt the least-squares Monte Carlo algorithm to evaluate the option to wait to expand the production by drilling additional wells. Furthermore, we identify the conditions in which the staged (phased) development is preferable against standard development. We propose a decision rule to determine the optimal expansion timing based on the acquisition of new information on the reservoir and oil price uncertainty. Our results suggest that staged development carries large upside potential for the marginal field development under extensive reservoir uncertainty. In addition, partial hedging against the downside risks in the staged development can enhance the project's economy in a sufficiently significant manner to justify investment.

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.

Efficient Variance Reduction with Least-Squares Monte Carlo Pricing

This paper examines the efficiency of standard variance reduction techniques across option characteristics when pricing American-style call and put options with the Least-Squares Monte Carlo algorithm of Longstaff & Schwartz (2001). Our numerical experiments evaluate the efficiency of antithetic sampling, control variates, importance sampling, and combinations thereof. Whereas most of the American option pricing literature has focused on either put or call options individually, we employ the symmetry relation of McDonald & Schroder (1998) to compare performance for pairs of call and put options whose solution coincide. Our results first show that variance reduction is generally more efficient for put than call options and that control variates is the most efficient stand-alone method. We also find that marginal gains in efficiency are typically achieved by combining variance reduction techniques, though some techniques may interact conflictingly. Finally, since valuation of American-style call options can be improved by pricing symmetric put options instead (Stentoft 2019), we demonstrate that drastic reductions in the standard deviation of the call is obtained by combining all three variance reduction techniques in a symmetric pricing approach, which reduces the standard deviation by a factor of over 20 for long maturity call options on highly volatile assets.

Real Options Approach for a Staged Field Development With Optional Wells

The decreasing average size of new discoveries in mature production areas makes the base of oil field investment decisions more uncertain than ever before. Fewer appraisal wells, which allow to decrease the amount of subsurface uncertainty, are typically drilled before the development of a small field compared to large fields. Therefore, new solutions are required to make small discoveries commercial given both technical and market uncertainties. Therefore, accounting for managerial flexibilities that enable to change the course of the project due to new information, becomes even more important for investment valuation. Combining the real options approach and decision analysis we present a novel model that allows to identify additional value created by a sequential drilling strategy for field development under oil price and resource uncertainty. We capture the sequence of key investment and operating decisions of a marginal field development in cooperation with an oil industry partner building a synthetic (yet realistic) project case. Addressing the flexibility to divide production wells drilling into two stages, we evaluate the option to wait to expand the production by drilling additional wells after the revelation of reservoir information based on a least-squares Monte Carlo algorithm. We identify the conditions under which the staged (phased) development is preferred compared to standard development. Furthermore, we propose a decision rule determining the optimal expansion timing based on new information on the reservoir and the oil price uncertainty. Our results suggest that staged development carries large upside potential for marginal field development under extensive reservoir uncertainty. We also illustrate that partial hedging against the downside risks within a staged development can improve project's economy significant enough to justify investment.

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.

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.

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.

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.

Bias-Corrected Least-Squares Monte Carlo for Utility Based Optimal Stochastic Control Problems

The Least-Squares Monte Carlo (LSMC) method has gained popularity in recent years due to its ability to handle multi-dimensional stochastic control problems, including problems with state variables affected by control. However, when applied to the stochastic control problems in the multi-period expected utility models, the regression fit tends to contain errors which accumulate over time and typically blow up the numerical solution. In this paper we propose to transform the value function of the problems to improve the regression fit, and then using either the smearing estimate or smearing estimate with controlled heteroskedasticity to avoid the re-transformation bias in the estimates of the conditional expectations calculated in the LSMC algorithm. We also present and utilise recent improvements in the LSMC algorithms such as control randomisation with policy iteration to avoid accumulation of regression errors over time. Presented numerical examples demonstrate that transformation method leads to an accurate solution. In addition, in the forward simulation stage of the control randomisation algorithm, we propose a re-sampling of the state and control variables in their full domain at each time t and then simulating corresponding state variable at t+1, to improve the exploration of the state space that also appears to be critical to obtain a stable and accurate solution for the expected utility models.

LSM Algorithm for Pricing American Option Under Heston–Hull–White’s Stochastic Volatility Model

In this paper, we present American option pricing under Heston---Hull---White's stochastic volatility and stochastic interest rate model. To do this, we first discretize the stochastic processes with Euler discretization scheme. Then, we price American option by using least-squares Monte Carlo algorithm. We also compare the numerical results of our model with the Heston-CIR model. Finally, numerical results show the efficiency of the proposed algorithm for pricing American option under the Heston---Hull---White model.

Dynamic Portfolio Optimisation with Intermediate Costs: A Least-Squares Monte Carlo Simulation Approach

We propose a dynamic portfolio optimisation strategy that takes liquidity cost into account. Our liquidity cost model is built upon the so-called Marginal Supply-Demand Curve which describes the asset price as a function of the trading volume. We extend the least-squares Monte Carlo algorithm to a stochastic control problem with switching costs and endogenous state variables. This approach is simulation-based, with great flexibility in the choice of investor objective, asset dynamics, portfolio constraints, intermediate consumption and incorporation of different sources of costs, making it easy to implement in practice. We study a portfolio investing in four major sectors in the U.S. market. We benchmark our dynamic strategy against several alternative portfolio strategies in terms of the out-of-sample distribution of terminal wealth and the real market performance. Overall, our dynamic strategy outperforms other standard asset allocation strategies.

Optimal LNG (liquefied natural gas) regasification scheduling for import terminals with storage

We describe a stochastic dynamic programming model for maximising the revenue generated by regasification of LNG (liquefied natural gas) from storage tanks at importation terminals in relation to a natural gas spot market. We present three numerical resolution strategies: a posterior optimal strategy, a rolling intrinsic strategy and a full option strategy based on a least-squares Monte Carlo algorithm. We then compare model simulation results to the observed behaviour of three LNG importation terminals in the UK for the period April 2011 to April 2012, and find that there was low correlation between the observed regasification decisions of the operators and those suggested by the three simulated strategies. However, the actions suggested by the model simulations would have generated significantly higher revenues, suggesting that the facilities might have been operated sub-optimally. A further numerical experiment shows that increasing the storage and regasification capacities of a facility can significantly increase the achievable revenue, even without altering the amount of LNG received, by allowing operators more flexibility to defer regasification.

ESTIMATING RESIDUAL HEDGING RISK WITH LEAST-SQUARES MONTE CARLO

Frequently, dynamic hedging strategies minimizing risk exposure are not given in closed form, but need to be approximated numerically. This makes it difficult to estimate residual hedging risk, also called basis risk, when only imperfect hedging instruments are at hand. We propose an easy to implement and computationally efficient least-squares Monte Carlo algorithm to estimate residual hedging risk. The algorithm approximates the variance minimal hedging strategy within general diffusion models. Moreover, the algorithm produces both high-biased and low-biased estimators for the residual hedging error variance, thus providing an intrinsic criterion for the quality of the approximation. In a number of examples we show that the algorithm delivers accurate hedging error characteristics within seconds.

How to Value a Gas Storage Facility

We show how to value a storage facility using Least Squares Monte Carlo (LSMC). We present a toy model to understand how to employ the LSMC algorithm and then show how to incorporate realistic constraints in the valuation including: the maximum capacity of the storage, injection and withdrawal rates and costs, and market constraints such as bid-ask spread in the spot market and transaction costs.