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125
A stochastic mesh method for pricing highdimensional American options
 Journal of Computational Finance
, 1997
"... Highdimensional problems frequently arise in the pricing of derivative securities – for example, in pricing options on multiple underlying assets and in pricing term structure derivatives. American versions of these options, ie, where the owner has the right to exercise early, are particularly chal ..."
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Cited by 92 (7 self)
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Highdimensional problems frequently arise in the pricing of derivative securities – for example, in pricing options on multiple underlying assets and in pricing term structure derivatives. American versions of these options, ie, where the owner has the right to exercise early, are particularly challenging to price. We introduce a stochastic mesh method for pricing highdimensional American options when there is a finite, but possibly large, number of exercise dates. The algorithm provides point estimates and confidence intervals; we provide conditions under which these estimates converge to the correct values as the computational effort increases. Numerical results illustrate the performance of the method. 1
Primaldual simulation algorithm for pricing multidimensional American options
, 2001
"... This paper describes a practical algorithm based on Monte Carlo simulation for the pricing of multidimensional American (i.e., continuously exercisable) and Bermudan (i.e., discretelyexercisable) options. The method generates both lower and upper bounds for the Bermudan option price and hence gives ..."
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Cited by 78 (2 self)
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This paper describes a practical algorithm based on Monte Carlo simulation for the pricing of multidimensional American (i.e., continuously exercisable) and Bermudan (i.e., discretelyexercisable) options. The method generates both lower and upper bounds for the Bermudan option price and hence gives valid confidence intervals for the true value. Lower bounds can be generated using any number of primal algorithms. Upper bounds are generated using a new Monte Carlo algorithm based on the duality representation of the Bermudan value function suggested independently in Haugh and Kogan (2001) and Rogers (2001). Our proposed algorithm can handle virtually any type of process dynamics, factor structure, and payout specification. Computational results for a variety of multifactor equity and interest rate options demonstrate the simplicity and efficiency of the proposed algorithm. In particular, we use the proposed method to examine and verify the tightness of frequently used exercise rules in Bermudan swaption markets.
Optimal portfolio choice and the valuation of illiquid securities
 The Review of Financial Studies
, 2001
"... Traditional models of portfolio choice assume that investors can continuously trade unlimited amounts of securities. In reality, investors face liquidity constraints. I analyze a model where investors are restricted to trading strategies that are of bounded variation. An investor facing this type o ..."
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Cited by 45 (8 self)
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Traditional models of portfolio choice assume that investors can continuously trade unlimited amounts of securities. In reality, investors face liquidity constraints. I analyze a model where investors are restricted to trading strategies that are of bounded variation. An investor facing this type of illiquidity behaves very differently from an unconstrained investor. A liquidityconstrained investor endogenously acts as if facing borrowing and shortselling constraints, and one may take riskier positions than in liquid markets. I solve for the shadow cost of illiquidity and show that large price discounts can be sustained in a rational model. The brass assembled at headquarters at 7 a.m. that Sunday. One after another, LTCM's partners, calling in from Tokyo and London, reported that their markets had dried up. There were no buyers, no sellers. It was all but impossible to maneuver out of large trading bets.Wall Street Journal, November 16, 1998. 1.
Learning and Value Function Approximation in Complex Decision Processes
, 1998
"... In principle, a wide variety of sequential decision problems  ranging from dynamic resource allocation in telecommunication networks to financial risk management  can be formulated in terms of stochastic control and solved by the algorithms of dynamic programming. Such algorithms compute and sto ..."
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Cited by 38 (4 self)
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In principle, a wide variety of sequential decision problems  ranging from dynamic resource allocation in telecommunication networks to financial risk management  can be formulated in terms of stochastic control and solved by the algorithms of dynamic programming. Such algorithms compute and store a value function, which evaluates expected future reward as a function of current state. Unfortunately, exact computation of the value function typically requires time and storage that grow proportionately with the number of states, and consequently, the enormous state spaces that arise in practical applications render the algorithms intractable. In this thesis, we study tractable methods that approximate the value function. Our work builds on research in an area of artificial intelligence known as reinforcement learning. A point of focus of this thesis is temporaldifference learning  a stochastic algorithm inspired to some extent by phenomena observed in animal behavior. Given a selection of...
An analysis of a least squares regression method for American option pricing
 Finance and Stochastics
"... Recently, various authors proposed MonteCarlo methods for the computation of American option prices, based on least squares regression. The purpose of this paper is to analyze an algorithm due to Longstaff and Schwartz. This algorithm involves two types of approximation. Approximation one: replace ..."
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Cited by 36 (0 self)
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Recently, various authors proposed MonteCarlo methods for the computation of American option prices, based on least squares regression. The purpose of this paper is to analyze an algorithm due to Longstaff and Schwartz. This algorithm involves two types of approximation. Approximation one: replace the conditional expectations in the dynamic programming principle by projections on a finite set of functions. Approximation two: use MonteCarlo simulations and least squares regression to compute the value function of approximation one. Under fairly general conditions, we prove the almost sure convergence of the complete algorithm. We also determine the rate of convergence of approximation two and prove that its normalized error is asymptotically Gaussian.
A quantization algorithm for solving multidimensional Optimal Stopping problems
 Bernoulli
, 2001
"... A new grid method for computing the Snell envelop of a function of a R valued Markov chain (X k ) 0#k#n is proposed. (This problem is typically non linear and cannot be solved by the standard Monte Carlo method.) Every X k is replaced by a "quantized approximation" X k taking its valu ..."
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Cited by 32 (2 self)
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A new grid method for computing the Snell envelop of a function of a R valued Markov chain (X k ) 0#k#n is proposed. (This problem is typically non linear and cannot be solved by the standard Monte Carlo method.) Every X k is replaced by a "quantized approximation" X k taking its values in a grid # k of size N k . The n grids and their transition probability matrices make up a discrete tree on which a pseudoSnell envelop is devised by mimicking the regular dynamic programming formula. We show, using Quantization Theory of probability distributions the existence of a set of optimal grids, given the total number N of elementary R valued vector quantizers. A recursive stochastic algorithm, based on some simulations of (X k ) 0#k#n , yields the optimal grids and their transition probability matrices. Some a priori error estimates based on the quantization errors are established. These results are applied to the computation of the Snell envelop of a di#usion (assuming that it can be directly simulated or using its Euler scheme). We show how this approach yields a discretization method for Reflected Backward Stochastic Di#erential Equation. Finally, some first numerical tests are carried out on a 2dimensional American option pricing problem.
Fair valuation of life insurance liabilities: The impact of interest rate guarantees, surrender options, and bonus policies
, 2000
"... The paper analyzes one of the most common life insurance products — the socalled participating (or with profits) policy. This type of contract stands in contrast to unitlinked (UL) products in that interest is credited to the policy periodically according to some mechanism which smoothes past retu ..."
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Cited by 29 (0 self)
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The paper analyzes one of the most common life insurance products — the socalled participating (or with profits) policy. This type of contract stands in contrast to unitlinked (UL) products in that interest is credited to the policy periodically according to some mechanism which smoothes past returns on the life insurance company’s (LIC) assets. As is the case for UL products, the participating policies are typically equipped with an interest rate guarantee and possibly also an option to surrender (sellback) the policy to the LIC before maturity. The paper shows that the typical participating policy can be decomposed into a risk free bond element, a bonus option, and a surrender option. A dynamic model is constructed in which these elements can be valued separately using contingent claims analysis. The impact of various bonus policies and various levels of the guaranteed interest rate is analyzed numerically. We find that values of participating policies are highly sensitive to the bonus policy, that surrender options can be quite valuable, and that LIC solvency can be quickly jeopardized if earning opportunities deteriorate in a situation where bonus reserves are
Pricing American options: A comparison of Monte Carlo simulation approaches
 Journal of Computational Finance
, 1999
"... A number of Monte Carlo simulationbased approaches have been proposed within the past decade to address the problem of pricing Americanstyle derivatives. The purpose of this paper is to empirically test some of these algorithms on a common set of problems in order to be able to assess the strength ..."
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Cited by 29 (7 self)
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A number of Monte Carlo simulationbased approaches have been proposed within the past decade to address the problem of pricing Americanstyle derivatives. The purpose of this paper is to empirically test some of these algorithms on a common set of problems in order to be able to assess the strengths and weaknesses of each approach as a function of the problem characteristics. In addition, we introduce another simulationbased approach that parameterizes the early exercise curve and casts the valuation problem as an optimization problem of maximizing the expected payoff (under the martingale measure) with respect to the associated parameters, the optimization problem carried out using a simultaneous perturbation stochastic approximation (SPSA) algorithm.
An adaptive sampling algorithm for solving Markov decision processes
 Operations Research
, 2005
"... Based on recent results for multiarmed bandit problems, we propose an adaptive sampling algorithm that approximates the optimal value of a finite horizon Markov decision process (MDP) with infinite state space but finite action space and bounded rewards. The algorithm adaptively chooses which actio ..."
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Cited by 24 (6 self)
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Based on recent results for multiarmed bandit problems, we propose an adaptive sampling algorithm that approximates the optimal value of a finite horizon Markov decision process (MDP) with infinite state space but finite action space and bounded rewards. The algorithm adaptively chooses which action to sample as the sampling process proceeds, and it is proven that the estimate produced by the algorithm is asymptotically unbiased and the worst possible bias is bounded by a quantity that converges to zero at rate O � � H ln N N,whereHis the horizon length and N is the total number of samples that are used per state sampled in each stage. The worstcase runningtime complexity of the algorithm is O((AN) H), independent of the state space size, where A  is the size of the action space. The algorithm can be used to create an approximate receding horizon control to solve infinite horizon MDPs.
Pricing American options by simulation using a stochastic mesh with optimized weights
 in Probabilistic Constrained Optimization: Methodology and Applications
, 2000
"... This paper develops a simulation method for pricing pathdependent American options, and American options on a large number of underlying assets, such as basket options. Standard numerical procedures (lattice methods and nite difference methods) are generally inapplicable to such highdimensional pr ..."
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Cited by 15 (4 self)
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This paper develops a simulation method for pricing pathdependent American options, and American options on a large number of underlying assets, such as basket options. Standard numerical procedures (lattice methods and nite difference methods) are generally inapplicable to such highdimensional problems, and this has motivated research into simulationbased methods. The optimal stopping problem embedded in the pricing of American options makes this a nonstandard problem for simulation. This paper extends the stochastic mesh introduced in Broadie and Glasserman [5]. In its original form, the stochastic mesh method required knowledge of the transition density of the underlying process of asset prices and other state variables. This paper extends the method to settings in which the transition density is either unknown or fails to exist. We avoid the need for a transition density by choosing mesh weights through a constrained optimization problem. If the weights are constrained to correctly price su ciently many simple instruments, they can be expected to work well in pricing a more complex American option. We investigate two criteria for use in the optimization  maximum entropy and least squares. The methods are illustrated through numerical examples. 32 1