Least-squares Monte Carlo Method Research Articles

A static replication approach for callable interest rate derivatives: mathematical foundations and efficient estimation of SIMM–MVA

The computation of credit risk measures such as exposure and Credit Value Adjustments (CVA) requires the simulation of future portfolio prices. Recent metrics, such as dynamic Initial Margin (IM) and Margin Value Adjustments (MVA) additionally require the simulation of future conditional sensitivities. For portfolios with non-linear instruments that do not admit closed-form valuation formulas, this poses a significant computational challenge. This problem is addressed by proposing a static replication algorithm for interest rate options with early-exercise features under an affine term-structure model. Under the appropriate conditions, we can find an equivalent portfolio of vanilla options that replicate these products. Specifically, we decompose the product into a portfolio of European swaptions. The weights and strikes of the portfolio are obtained by regressing the target option value with interpretable, feed-forward neural networks. Once an equivalent portfolio of European swaptions is determined, we can leverage on closed-form expressions to obtain the conditional prices and sensitivities, which serve as an input to exposure and SIMM-driven MVA quantification. For a consistent forward sensitivity estimation, this involves the differentiation of the portfolio-weights. The accuracy and convergence of the method is demonstrated through several representative numerical examples, benchmarked against the established least-square Monte Carlo method.

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.

Optimal Annuitisation, Housing Decisions and Means-Tested Public Pension in Retirement

In this paper we develop a retirement model under the expected utility stochastic control framework to find optimal decisions with respect to the consumption, risky asset allocation, access to annuities, reverse mortgage and the option to scale housing. The model is solved numerically using Least-Squares Monte Carlo method adapted to handle optimal stochastic control problems in the expected utility models. To demonstrate the applicability of the framework, the model is applied in the context of the Australian retirement system. Few retirees in Australia utilise financial products in retirement, such as annuities or reverse mortgages. Since the government-provided means-tested Age Pension in Australia is an indirect annuity stream which is typically higher than the average consumption floor, it is argued that this is the reason why Australians do not annuitise. In addition, in Australia where assets allocated to the family home are not included in the means tests of Age Pension, the incentive to over allocate wealth into housing assets is high. This raises the question whether a retiree is really better off over allocating into the family home, while accessing home equity later on either via downsizing housing or by taking out a reverse mortgage. Our findings confirm that means-tested pension crowds out voluntary annuitisation in retirement, and that annuitisation is optimal sooner rather than later once retired. We find that it is never optimal to downscale housing with the means-tested Age Pension when a reverse mortgage is available; only when there is no other way to access equity then downsizing is the only option.

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.

Valuation of callable accreting interest rate swaps: Least squares Monte-Carlo method under Hull-White interest rate model

Using the Hull-White interest rate model, this paper proposes a valuation method of callable accreting interest rate swap (CAIRS) and how it can be used for managing the risk of zero callable bonds (ZCBs). Firstly, CAIRS can be decomposed into accreting payer interest rate swaps and Bermudan options. Considering the financial valuation of both components, the former can be valued directly while the latter has no close-form due to its early exercise characteristics. Using the Least Squares Monte-Carlo method (LSM) proposed by Longstaff and Schwartz (2001), we find that the two options embedded in ZCB and CAIRS have the same exercise strategy since the terms of the swaps will include the bonds in practice. However, the cash flow of risk management in swaps and bonds can differ when considering the time value. Hence, CAIRS is not the best financial instrument for managing risks of ZCB under the current design.

Least-Squares Monte Carlo for Proxy Modeling in Life Insurance: Neural Networks

The least-squares Monte Carlo method has proved to be a suitable approximation technique for the calculation of a life insurer’s solvency capital requirements. We suggest to enhance it by the use of a neural network based approach to construct the proxy function that models the insurer’s loss with respect to the risk factors the insurance business is exposed to. After giving a mathematical introduction to feed forward neural networks and describing the involved hyperparameters, we apply this popular form of neural networks to a slightly disguised data set from a German life insurer. Thereby, we demonstrate all practical aspects, such as the hyperparameter choice, to obtain our candidate neural networks by bruteforce, the calibration (“training”) and validation (“testing”) of the neural networks and judging their approximation performance. Compared to adaptive OLS, GLM, GAM and FGLS regression approaches, an ensemble built of the 10 best derived neural networks shows an excellent performance. Through a comparison with the results obtained by every single neural network, we point out the significance of the ensemble-based approach. Lastly, we comment on the interpretability of neural networks compared to polynomials for sensitivity analyses.

Neural Network Pricing of American Put Options

In this study, we use Neural Networks (NNs) to price American put options. We propose two NN models—a simple one and a more complex one—and we discuss the performance of two NN models with the Least-Squares Monte Carlo (LSM) method. This study relies on American put option market prices, for four large U.S. companies—Procter and Gamble Company (PG), Coca-Cola Company (KO), General Motors (GM), and Bank of America Corp (BAC). Our dataset is composed of all options traded within the period December 2018 until March 2019. Although on average, both NN models perform better than LSM, the simpler model (NN Model 1) performs quite close to LSM. Moreover, the second NN model substantially outperforms the other models, having an RMSE ca. 40% lower than the presented by LSM. The lower RMSE is consistent across all companies, strike levels, and maturities. In summary, all methods present a good accuracy; however, after calibration, NNs produce better results in terms of both execution time and Root Mean Squared Error (RMSE).

Optimal decision policy for real options under general Markovian dynamics

The Least-Squares Monte Carlo Method (LSM) has become the standard tool to solve real options modeled as an optimal switching problem. The method has been shown to deliver accurate valuation results under complex and high dimensional stochastic processes; however, the accuracy of the underlying decision policy is not guaranteed. For instance, an inappropriate choice of regression functions can lead to noisy estimates of the optimal switching boundaries or even continuation/switching regions that are not clearly separated. As an alternative to estimate these boundaries, we formulate a simulation-based method that starts from an initial guess of them and then iterates until reaching optimality. The algorithm is applied to a classical mine under a wide variety of underlying dynamics for the commodity price process. The method is first validated under a one-dimensional geometric Brownian motion and then extended to general Markovian processes. We consider two general specifications: a two-factor model with stochastic variance and a rich jump structure, and a four-factor model with stochastic cost-of-carry and stochastic volatility. The method is shown to be robust, stable, and easy-to-implement, converging to a more profitable strategy than the one obtained with LSM.

Re-evaluating natural resource investments under uncertainty: An alternative to limited traditional approaches

The need to evaluate natural resource investments under uncertainty has given rise to the development of real options valuation; however, the analysis of such investments has been restricted by the capabilities of existing valuation approaches. We re-visit the well-known example of a copper mine project under a one-factor and two multi-factor models using the influence diagram simulation-and-regression (IDSR) approach. The one-factor setting was originally proposed by Brennan and Schwartz (J Bus 58(2):135–157, 1985), who used partial differential equations (PDEs) and finite differences to approximately solve the valuation problem; extensions to two and three factors were later analysed by Tsekrekos et al. (Eur Financ Manag 18(4):543–575, 2012) using the least-squares Monte Carlo method. We apply the IDSR approach to perform a detailed portfolio analysis of the one-factor benchmark investment and find issues in both the definitions and values of the fixed-output-rate mine and closure option at the portfolio decomposition stage in Brennan and Schwartz (1985). We then apply the IDSR approach to re-evaluate two multi-factor extensions of Tsekrekos et al. (2012) and detect issues in their sensitivity analyses that impact on the reliability of some of their findings. To confirm this and validate the values we obtained, we integrate PDE-based analytical expressions that describe the volatilities implied by the multi-factor models into our IDSR-based analysis. Using the investment-uncertainty relationship, we are able to correctly analyse the impact of the complex multi-factor model parameters on investment value. We conclude that the limitations of PDE-based finite difference approaches may invalidate their use in portfolio situations, but analytical expressions obtained from PDE-based modelling may be profitably integrated into a simulation-based, numerical analysis to validate results and gain new insights.

Primal–dual quasi-Monte Carlo simulation with dimension reduction for pricing American options

The pricing of American options is one of the most challenging problems in financial engineering due to the involved optimal stopping time problem, which can be solved by using dynamic programming (DP). But applying DP is not always practical, especially when the state space is high dimensional. However, the curse of dimensionality can be overcome by Monte Carlo (MC) simulation. We can get lower and upper bounds by MC to ensure that the true price falls into a valid confidence interval. During the recent decades, progress has been made in using MC simulation to obtain both the lower bound by least-squares Monte Carlo method (LSM) and the upper bound by duality approach. However, there are few works on pricing American options using quasi-Monte Carlo (QMC) methods, especially to compute the upper bound. For comparing the sample variances and standard errors in the numerical experiments, randomized QMC (RQMC) methods are usually used. In this paper, we propose to use RQMC to replace MC simulation to compute both the lower bound (by the LSM) and the upper bound (by the duality approach). Moreover, we propose to use dimension reduction techniques, such as the Brownian bridge, principal component analysis, linear transformation and the gradients based principle component analysis. We perform numerical experiments on American–Asian options and American max-call options under the Black–Scholes model and the variance gamma model, in which the options have the path-dependent feature or are written on multiple underlying assets. We find that RQMC in combination with dimension reduction techniques can significantly increase the efficiency in computing both the lower and upper bounds, resulting in better estimates and tighter confidence intervals of the true price than pure MC simulation.

Testing the Least-Squares Monte Carlo Method for the Evaluation of Capital Requirements in Life Insurance

In this paper, we test the efficiency of least-squares Monte Carlo method to estimate capital requirements in life insurance. We choose a simplified Gaussian evaluation framework where closed-form formulas are available and allow us to obtain solid benchmarks. Extensive numerical experiments were conducted by considering different combinations of simulation runs and basis functions, and the corresponding results are illustrated.

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.

Efficient Numerical Pricing of American Call Options Using Symmetry Arguments

This paper demonstrates that it is possible to improve significantly on the estimated call prices obtained with the regression and simulation-based least-squares Monte Carlo method by using put-call symmetry. The results show that, for a large sample of options with characteristics of relevance in real-life applications, the symmetric method performs much better on average than the regular pricing method, is the best method for most of the options, never performs poorly and, as a result, is extremely efficient compared to the optimal, but unfeasible method that picks the method with the smallest Root Mean Squared Error (RMSE). A simple classification method is proposed that, by optimally selecting among estimates from the symmetric method with a reasonably small order used in the polynomial approximation, achieves a relative efficiency of more than 98 % . The relative importance of using the symmetric method increases with option maturity and with asset volatility. Using the symmetric method to price, for example, real options, many of which are call options with long maturities on volatile assets, for example energy, could therefore improve the estimates significantly by decreasing their bias and RMSE by orders of magnitude.

A New Approach for American Option Pricing: The Dynamic Chebyshev Method

We introduce a new method to price American options based on Chebyshev interpolation. In each step of a dynamic programming time-stepping we approximate the value function with Chebyshev polynomials. The key advantage of this approach is that it allows us to shift the model-dependent computations into an offline phase prior to the time-stepping. In the offline part a family of generalized (conditional) moments is computed by an appropriate numerical technique such as a Monte Carlo, PDE, or Fourier transform based method. Thanks to this methodological flexibility the approach applies to a large variety of models. Online, the backward induction is solved on a discrete Chebyshev grid, and no (conditional) expectations need to be computed. For each time step the method delivers a closed form approximation of the price function along with the options' delta and gamma. Moreover, the same family of (conditional) moments yield multiple outputs including the option prices for different strikes, maturities, and different payoff profiles. We provide a theoretical error analysis and find conditions that imply explicit error bounds for a variety of stock price models. Numerical experiments confirm the fast convergence of prices and sensitivities. An empirical investigation of accuracy and runtime also shows an efficiency gain compared with the least-squares Monte Carlo method introduced by Longstaff and Schwartz [Rev. Financ. Stud., 14 (2001), pp. 113--147]. Moreover, we show that the proposed algorithm is flexible enough to price barrier and multivariate barrier options.

VALUATION OF CONTINGENT GUARANTEES USING LEAST-SQUARES MONTE CARLO

AbstractWe consider the problem of pricing modern guarantee concepts in unit-linked life insurance, where the guaranteed amount grows contingent on the performance of an investment fund that acts simultaneously as the underlying security and the replicating portfolio. Using the Martingale Method, this nonstandard pricing problem can be transformed into a fixed-point problem, whose solution requires the evaluation of conditional expectations of highly path-dependent payoffs. By adapting the least-squares Monte Carlo method for American option pricing problems, we develop a new numerical approach to approximate the value of contingent guarantees and prove its convergence. Our valuation procedure can be applied to large-scale pricing problems, for which existing methods are infeasible, and leads to significant improvements in performance.

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.

Optimal Annuitisation, Housing Decisions and Means-Tested Public Pension in Retirement Under Expected Utility Stochastic Control Framework

In this paper, we develop a retirement model under the expected utility stochastic control framework to find optimal decisions with respect to consumption, risky asset allocation, access to annuities, reverse mortgage and the option to scale housing. The model is solved numerically using Least-Squares Monte Carlo method adapted to handle expected utility stochastic control problem in high dimensions. To demonstrate the applicability of the framework, the model is applied in the context of the Australian retirement system. Few retirees in Australia utilise financial products in retirement, such as annuities or reverse mortgages. Since the government-provided means-tested Age Pension in Australia is an indirect annuity stream which typically is higher than the average consumption floor, it is argued that this is the reason why Australians do not annuitise. In addition, in Australia where assets allocated to the family home are not included in the means tests of Age Pension, the incentive to over allocate wealth into housing assets is high. This raises the question whether a retiree is really better off over allocating into the family home, while accessing home equity later on either via downsizing housing or by taking out a reverse mortgage. Our findings confirm that means-tested pension crowds out voluntary annuitisation in retirement, and that annuitisation is optimal sooner rather than later once retired. We find that it is never optimal to downscale housing with the means-tested Age Pension when a reverse mortgage is available, only when there is no other way to access equity then downsizing is the only option.

Pricing American put option on zero-coupon bond under fractional CIR model with transaction cost

ABSTRACTThe motivation of this study is to evaluate the American put option on zero-coupon bond, when the interest rate model is governed by a fractional CIR (FCIR) interest rate model. Since the existence of fractional Brownian motion, leading to create the arbitrage, we employ the transaction cost for eliminating the arbitrage. We first of all apply the Leland's hedging strategy for a self-financing portfolio that contains an American option and zero-coupon bond and derive a formula for the transaction cost. We perform the least-square Monte Carlo (LSM) method for pricing American option under the proposed interest rate model.

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.

ON THE CALCULATION OF RISK MEASURES USING LEAST-SQUARES MONTE CARLO

The paper establishes a convergence result for the Least-Squares Monte Carlo method, applied to the estimation of risk measures (such as VaR) depending on the distribution of the future value of some derivative contract (or portfolio). We extend previous results in the literature by focusing on the specific case where the distribution of the underlying factors is known. In particular, we are able to remove the requirement of bounded conditional variance which is usually not satisfied by common financial models and payoffs at the cost of using a weaker notion of convergence. Our main result shows the convergence of the empirical distribution function when the number of simulations [Formula: see text] and of basis variables [Formula: see text] are jointly sent to infinity under certain conditions on the payoff and the basis. With a suitable choice of basis functions, we finally prove that convergence can be ensured when [Formula: see text].