Peter Forsyth

Peter Forsyth
University of Waterloo | UWaterloo · David R. Cheriton School of Computer Science

About

213
Publications
25,239
Reads
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7,716
Citations
Additional affiliations
November 2016 - January 2017
University of Waterloo
Position
  • Professor Emeritus
June 1987 - November 2016
University of Waterloo
Position
  • Professor
September 1979 - May 1985
Computer Modelling Group
Position
  • Researcher

Publications

Publications (213)
Preprint
Full-text available
We analyze dynamic investment strategies for benchmark outperformance using two widely-used objectives of practical interest to investors: (i) maximizing the information ratio (IR), and (ii) obtaining a favorable tracking difference (cumulative outperformance) relative to the benchmark. In the case of the tracking difference, we propose a simple an...
Preprint
Full-text available
We present a data-driven neural network approach to find optimal dynamic (multi-period) factor investing strategies in the presence of transaction costs. The factor investing problem is formulated as a stochastic optimal control problem, which we solve and analyze using two objectives, namely a (i) one-sided quadratic target objective (closely rela...
Article
This paper examines the strategic interactions of two large regions making choices about greenhouse gas emissions in the face of rising global temperatures. Three central features are highlighted: uncertainty, the incentive for free riding, and asymmetric characteristics of decision makers. Optimal decisions are modelled in a fully dynamic, feedbac...
Article
Full-text available
We consider the practical investment consequences of implementing the two most popular formulations of the scalarization (or risk-aversion) parameter in the time-consistent dynamic mean–variance (MV) portfolio optimization problem. Specifically, we compare results using a scalarization parameter assumed to be (i) constant and (ii) inversely proport...
Article
We extend the Annually Recalculated Virtual Annuity (ARVA) spending rule for retirement savings decumulation (Waring and Siegel (2015) Financial Analysts Journal , 71 (1), 91–107) to include a cap and a floor on withdrawals. With a minimum withdrawal constraint, the ARVA strategy runs the risk of depleting the investment portfolio. We determine the...
Article
Optimal stochastic control methods are used to examine decumulation strategies for a defined contribution (DC) plan retiree. An initial investment horizon of 15 years is considered, since the retiree will attain this age with high probability. The objective function reward measure is the expected sum of the withdrawals. The objective function tail...
Preprint
Full-text available
We extend the Annually Recalculated Virtual Annuity (ARVA) spending rule for retirement savings decumulation to include a cap and a floor on withdrawals. With a minimum withdrawal constraint, the ARVA strategy runs the risk of depleting the investment portfolio. We determine the dynamic asset allocation strategy which maximizes a weighted combinati...
Preprint
Full-text available
1 Optimal stochastic control methods are used to examine decumulation strategies for a defined 2 contribution (DC) plan retiree. An initial investment horizon of fifteen years is considered, since 3 the retiree will attain this age with high probability. The objective function reward measure is 4 the expected sum of the withdrawals. The objective f...
Preprint
Full-text available
We pose the decumulation strategy for a Defined Contribution (DC) pension plan as a problem in optimal stochastic control. The controls are the withdrawal amounts and the asset allocation strategy. We impose maximum and minimum constraints on the withdrawal amounts, and impose no-shorting no-leverage constraints on the asset allocation strategy. Ou...
Article
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In single-period portfolio optimization settings, Mean-Variance (MV) optimization can result in notoriously unstable asset allocations due to small changes in the underlying asset parameters. This has resulted in the widespread questioning of whether and how MV optimization should be implemented in practice, and has also resulted in a number of alt...
Preprint
Full-text available
We propose a data-driven Neural Network (NN) optimization framework to determine the optimal multi-period dynamic asset allocation strategy for outperforming a general stochastic target. We formulate the problem as an optimal stochastic control with an asymmetric, distribution shaping, objective function. The proposed framework is illustrated with...
Article
We determine the optimal asset allocation to bonds and stocks using an annually recalculated virtual annuity (ARVA) spending rule for DC pension plan decumulation. Our objective function minimizes downside withdrawal variability for a given fixed value of total expected withdrawals. The optimal asset allocation is found using optimal stochastic con...
Article
Members of defined contribution (DC) pension plans must take on additional responsibilities for their investments, compared to participants in defined benefit (DB) pension plans. The transition from DB to DC plans means that more employees are faced with these responsibilities. We explore the extent to which DC plan members can follow financial str...
Article
Full-text available
The trend towards eliminating defined benefit (DB) pension plans in favour of defined contribution (DC) plans implies that increasing numbers of pension plan participants will bear the risk that final realized portfolio values may be insufficient to fund desired retirement cash flows. We compare the outcomes of various asset allocation strategies f...
Article
We consider optimal asset allocation for an investor saving for retirement. The portfolio contains a bond index and a stock index. We use multi-period criteria and explore two types of strategies: deterministic strategies are based only on the time remaining until the anticipated retirement date, while adaptive strategies also consider the investor...
Article
Full-text available
We investigate the time-consistent mean–variance (MV) portfolio optimization problem, popular in investment–reinsurance and investment-only applications, under a realistic context that involves the simultaneous application of different types of investment constraints and modelling assumptions, for which a closed-form solution is not known to exist....
Article
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Stochastic control problems in finance having complex controls inevitably give rise to low order accuracy, usually at most second order. Fourier methods are efficient at advancing the solution between control monitoring dates, but are not monotone. This gives rise to possible violations of arbitrage inequalities. We devise a preprocessing step for...
Article
Although high performance computers and advanced numerical methods have made the application of fully-integrated surface and subsurface flow and transport models such as HydroGeoSphere common place, run times for large complex basin models can still be on the order of days to weeks, thus, limiting the usefulness of traditional workhorse algorithms...
Article
We consider a portfolio consisting of a risk-free bond and an equity index which follows a jump diffusion process. Parameters for the inflation-adjusted return of the stock index and the risk-free bond are determined by examining 89 years of data. The optimal dynamic asset allocation strategy for a long-term pre-commitment mean variance (MV) invest...
Article
Full-text available
Investors in Target Date Funds are automatically switched from high risk to low risk assets as their retirements approach. Such funds have become very popular, but our analysis brings into question the rationale for them. Based on both a model with parameters fitted to historical returns and on bootstrap resampling, we find that adaptive investment...
Article
This paper explores dynamic mean-variance (MV) asset allocation over long horizons. This is equivalent to target-based investing with a quadratic loss penalty for deviations from the target level of terminal wealth. We provide a number of illustrative examples in a setting with a risky stock index and a risk-free asset. Our underlying model is very...
Article
We consider optimal asset allocation for an investor saving for retirement. The portfolio contains a bond index and a stock index. We use multi-period criteria and explore two types of strategies: deterministic strategies are based only on the time remaining until the anticipated retirement date, while adaptive strategies also consider the investor...
Article
We present efficient partial differential equation (PDE) methods for continuous-time mean-variance portfolio allocation problems when the underlying risky asset follows a stochastic volatility process. The standard formulation for mean-variance optimal portfolio allocation problems gives rise to a two-dimensional nonlinear Hamilton-Jacobi-Bellman (...
Article
In contrast to single-period mean-variance (MV) portfolio allocation, multi-period MV optimal portfolio allocation can be modified slightly to be effectively a down-side risk measure. With this in mind, we consider multi-period MV optimal portfolio allocation in the presence of periodic withdrawals. The investment portfolio can be allocated between...
Article
Under the assumption that two asset prices follow an uncertain volatility model, the maximal and minimal solution values of an option contract are given by a two-dimensional Hamilton–Jacobi–Bellman Partial Differential Equation (PDE). A fully implicit, unconditionally monotone finite difference numerical scheme is developed in this article. Consequ...
Article
This work is motivated by numerical solutions to Hamilton-Jacobi-Bellman quasivariational inequalities (HJBQVIs) associated with combined stochastic and impulse control problems. In particular, we consider (i) direct control, (ii) penalized, and (iii) semi-Lagrangian discretization schemes applied to the HJBQVI problem. Scheme (i) takes the form of...
Article
Full-text available
This work is motivated by numerical solutions to Hamilton-Jacobi-Bellman quasi-variational inequalities (HJBQVIs) associated with combined stochastic and impulse control problems. In particular, we consider (i) direct control, (ii) penalized, and (iii) explicit control schemes applied to the HJBQVI problem. Scheme (i) takes the form of a Bellman pr...
Article
A numerical technique based on the embedding technique proposed in (Math Finan 10:387–406, 2000), (Appl Math Optim 42:19–33, 2000) for dynamic mean-variance (MV) optimization problems may yield spurious points, i.e. points which are not on the efficient frontier. In (SIAM J Control Optim 52:1527–1546 2014), it is shown that spurious points can be e...
Article
An advantageous feature of piecewise constant policy timestepping for Hamilton-Jacobi-Bellman (HJB) equations is that different linear approximation schemes, and indeed different meshes, can be used for the resulting linear equations for different control parameters. Standard convergence analysis suggests that monotone (i.e., linear) interpolation...
Article
Full-text available
A large collection of financial contracts offering guaranteed minimum benefits are often posed as control problems, in which at any point in the solution domain, a control is able to take any one of an uncountable number of values from the admissible set. Often, such contracts specify that the holder exert control at a finite number of deterministi...
Article
In contrast to single-period mean-variance portfolio allocation, optimal multi-period mean-variance allocation can be modified slightly to be effectively a down-side risk measure. With this in mind, we consider optimal multi-period mean-variance portfolio allocation in the presence of periodic withdrawals. The investment portfolio can be allocated...
Article
Hydrologic modeling requires the handling of a wide range of highly nonlinear processes from the scale of a hill slope to the continental scale, and thus the computational efficiency of the model becomes a critical issue for water resource management. This work is aimed at implementing and evaluating a flexible parallel computing framework for hydr...
Article
An implicit partial differential equation (PDE) method is used to determine the cost of hedging for a Guaranteed Lifelong Withdrawal Benefit (GLWB) variable annuity contract. In the basic setting, the underlying risky asset is assumed to evolve according to geometric Brownian motion, but this is generalized to the case of a Markov regime switching...
Article
A continuous time mean variance (MV) problem optimizes the biobjective criteria (V, ε), representing variance V and expected value ε, respectively, of a random variable at the end of a time horizon T. This problem is computationally challenging since the dynamic programming principle cannot be directly applied to the variance criterion. An embeddin...
Chapter
A general methodology is described in which policyholder behaviour is decoupled from the pricing of a variable annuity based on the cost of hedging it, yielding two weakly coupled systems of partial differential equations (PDEs): the pricing and utility systems. The utility system is used to generate policyholder withdrawal behaviour, which is in t...
Article
The application of adaptive implicit methods to a dead-oil steam simulator is discussed. Various test results are presented. The adaptive implicit method works well on thermal problems. demonstrating a 30 to 50% computing time reduction compared with a fully implicit solution technique, while giving essentially the same results. Introduction Recen...
Article
We present efficient partial differential equation (PDE) methods for continuous time mean-variance portfolio allocation problems when the underlying risky asset follows a jump-diffusion. The standard formulation of mean-variance optimal portfolio allocation problems, where the total wealth is the underlying stochastic process, gives rise to a one-d...
Article
The embedding technique proposed in [13, 22] for mean-variance (MV) optimization problems may yield spurious points. These are points in the MV objective set, derived from the embedding technique, but are not MV scalarization optimal points (SOPs) with respect to this set. In [17], it is shown that a spurious point is the point at which a supportin...
Article
We generalize the idea of semi-self-financing strategies, originally discussed in Ehrbar, Journal of Economic Theory (1990), and later formalized in em Cui et al, Mathematical Finance 22 (2012), for the pre-commitment mean-variance (MV) optimal portfolio allocation problem. The proposed semi-self-financing strategies are built upon a numerical solu...
Chapter
Full-text available
Several examples of nonlinear Hamilton Jacobi Bellman (HJB) partial differential equations are given which arise in financial applications. The concept of a visocisity solution is introduced. Sufficient conditions which ensure that a numerical scheme converges to the viscosity solution are discussed. Numerical examples based on an uncertain volatil...
Article
Full-text available
We determine the optimal dynamic investment policy for a mean quadratic variation objective function by numerical solution of a nonlinear Hamilton-Jacobi-Bellman (HJB) partial differential equation (PDE). We compare the efficient frontiers and optimal investment policies for three mean variance like strategies: pre-commitment mean variance, time-co...
Article
Full-text available
A theoretical analysis tool, iterated optimal stopping, has been used as the basis of a numerical algorithm for American options under regime switching [19]. Similar methods have also been proposed for American options under jump diffusion [3] and Asian options under jump diffusion [4]. We show that a re-arrangement of the numerical algorithm in th...
Article
Full-text available
Implicit methods for Hamilton Jacobi Bellman (HJB) partial differential equations give rise to highly nonlinear discretized algebraic equations. The classic policy iteration approach may not be efficient in many circumstances. In this article, we derive sufficient conditions to ensure convergence of a combined fixed point-policy iteration scheme fo...
Article
The no arbitrage pricing of Guaranteed Minimum Withdrawal Benefits (GMWB) contracts results in a singular stochastic control problem which can be formulated as a Hamilton Jacobi Bellman (HJB) Variational Inequality (VI). Recently, a penalty method has been suggested for solution of this HJB variational inequality (Dai et al., 2008). This method is...
Article
Implicit methods for Hamilton-Jacobi-Bellman (HJB) partial differential equations give rise to highly nonlinear discretized algebraic equations. The classic policy iteration approach may not be efficient in many circumstances. In this article, we derive sufficient conditions to ensure convergence of a combined fixed point policy iteration scheme fo...
Article
Full-text available
We compare optimal liquidation policies in continuous time in the presence of trading impact using numerical solutions of Hamilton Jacobi Bellman (HJB) partial differential equations (PDE). In particu-lar, we compare the path dependent, time-consistent mean-quadratic-variation strategy with the path-independent, time-inconsistent (pre-commitment) m...
Article
In order to ensure convergence to the viscosity solution, it is common to use a positive coeffi-cient discretization for Hamilton Jacobi Bellman (HJB) Partial Integro Differential Equations (PIDEs) in finance. In this article, we focus on a specific HJB PIDE, namely American options under jump diffusion. A positive coefficient discretization then i...
Article
Full-text available
Discretized singular control problems in finance result in highly nonlinear algebraic equations which must be solved at each timestep. We consider a singular stochastic control problem arising in pricing a Guaranteed Minimum Withdrawal Benefit (GMWB), where the underlying asset is assumed to follow a jump diffusion process. We use a scaled direct c...
Article
We develop a numerical scheme for determining the optimal asset allocation strategy for time-consistent, continuous time, mean variance optimization. Any type of constraint can be applied to the investment policy. The optimal policies for time-consistent and pre-commitment strategies are compared. When realistic constraints are applied, the efficie...
Article
The optimal trade execution problem is formulated in terms of a mean-variance tradeoff, as seen at the initial time. The mean-variance problem can be embedded in a linear–quadratic (LQ) optimal stochastic control problem. A semi-Lagrangian scheme is used to solve the resulting nonlinear Hamilton–Jacobi–Bellman (HJB) PDE. This method is essentially...
Article
Full-text available
We analyze a number of techniques for pricing American options under a regime switching stochastic process. The techniques analyzed include both explicit and implicit discretizations with the focus being on methods which are unconditionally stable. In the case of implicit methods we also compare a number of iterative procedures for solving the asso...
Article
We solve the optimal asset allocation problem using a mean variance approach. The original mean variance optimization problem can be embedded into a class of auxiliary stochastic linear-quadratic (LQ) problems using the method in Zhou and Li (2000) and Li and Ng (2000). We use a finite difference method with fully implicit timestepping to solve the...
Article
Full-text available
In this paper, we propose a one-factor regime-switching model for the risk adjusted natural gas spot price and study the implications of the model on the valuation and optimal operation of natural gas storage facilities. We calibrate the model parameters to both market futures and options on futures. Calibration results indicate that the regime-swi...
Article
We propose the use of a mean–quadratic-variation criteria to determine an optimal trading strategy in the presence of price impact. We derive the Hamilton Jacobi Bellman (HJB) Partial Differential Equation (PDE) for the optimal strategy, assuming the underlying asset follows Geometric Brownian Motion (GBM). We also derive the HJB PDE assuming that...
Article
Full-text available
In this paper, we give a method for computing the fair insurance fee associated with the guaranteed minimum death benefit (GMDB) clause included in many variable annuity contracts. We allow for partial withdrawals, a common feature in most GMDB contracts, and determine how this affects the GMDB fair insurance charge. Our method models the GMDB pric...
Article
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risk by imposing delta neutrality; and (ii) minimize an objective that is a linear combination of a jump risk and transaction cost penalty function. Since reducing the jump risk is a competing goal vis-`a-vis controlling for transaction cost, the respective components in the objective must be appropriately weighted. Hedging simulations of this proc...
Article
In this article, an extensive study of the no-arbitrage fee for Guaranteed Minimum Withdrawal Benefit (GMWB) variable annuity riders is carried out. The value of the GMWB guarantee increases substantially when taking into account the separation of mutual fund fees and the fees earmarked for hedging the guarantee. In addition, the fee is also advers...
Article
Under the assumption that two financial assets evolve by correlated finite activity jumps superimposed on correlated Brownian motion, the value of a contingent claim written on these two assets is given by a two-dimensional parabolic partial integro-differential equation (PIDE). An implicit, finite difference method is derived in this paper. This a...
Article
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In this paper, we value hydroelectric power plant cash flows under a stochastic con- trol framework, taking into consideration the implication of operational constraints such as ramping and minimum flow rate constraints for the purpose of environmental protection. The power plant valuation problem under a ramping constraint is characterized as a bo...
Article
We investigate the Jacobi-Davidson algorithm for solving the time-harmonic Maxwell equation in axisymmetric vertical cavity surface emitting lasers (VCSELs). We compare various strategies in the extraction and extension phase of the algorithm and discuss ...
Article
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When modeling transport of chemicals or solute in realistic large-scale subsurface systems, numerical issues are a serious concern, even with the continual progress made over the past few decades in both simulation algorithms and computer hardware. The problem becomes even more difficult when dealing with chemical transport in a vadose zone or mult...
Article
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In this paper, we outline an impulse stochastic control formulation for pricing variable annuities with a guaranteed minimum withdrawal benefit (GMWB) assuming the policyholder is allowed to withdraw funds continuously. We develop a numerical scheme for solving the Hamilton–Jacobi–Bellman (HJB) variational inequality corresponding to the impulse co...
Article
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In order to ensure convergence to the viscosity solution, the standard method for discretizing HJB PDEs uses forward/backward differencing for the drift term. In this paper, we devise a monotone method which uses central weighting as much as possible. In order to solve the discretized algebraic equations, we have to maximize a possibly discontinuou...
Article
Full-text available
Many nonlinear option pricing problems can be formulated as optimal control problems, leading to Hamilton-Jacobi-Bellman (HJB) or Hamilton-Jacobi-Bellman-Isaacs (HJBI) equations. We show that such formulations are very convenient for developing monotone discretization methods which ensure convergence to the financially relevant solution, which in t...
Article
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We develop an implicit discretization method for pricing European and American options when the underlying asset is driven by an infinite activity Lévy process. For processes of finite variation, quadratic convergence is obtained as the mesh and time step are refined. For infinite variation processes, better than first order accuracy is achieved. T...
Article
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The valuation of a gas storage facility is characterized as a stochastic control problem, result- ing in a Hamilton-Jacobi-Bellman (HJB) equation. In this paper, we present a semi-Lagrangian method for solving the HJB equation for a typical gas storage valuation problem. The method is able to handle a wide class of spot price models that exhibit me...
Article
Hedging a contingent claim with an asset which is not perfectly correlated with the underlying asset results in unhedgeable residual risk. Even if the residual risk is considered diversifiable, the option writer is faced with the problem of uncertainty in the estimation of the drift rates of the underlying and the hedging instrument. If the residua...