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177
Pricing american options: a duality approach
 Operations Research
, 2001
"... We develop a new method for pricing American options. The main practical contribution of this paper is a general algorithm for constructing upper and lower bounds on the true price of the option using any approximation to the option price. We show that our bounds are tight, so that if the initial ap ..."
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Cited by 153 (5 self)
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We develop a new method for pricing American options. The main practical contribution of this paper is a general algorithm for constructing upper and lower bounds on the true price of the option using any approximation to the option price. We show that our bounds are tight, so that if the initial approximation is close to the true price of the option, the bounds are also guaranteed to be close. We also explicitly characterize the worstcase performance of the pricing bounds. The computation of the lower bound is straightforward and relies on simulating the suboptimal exercise strategy implied by the approximate option price. The upper bound is also computed using Monte Carlo simulation. This is made feasible by the representation of the American option price as a solution of a properly defined dual minimization problem, which is the main theoretical result of this paper. Our algorithm proves to be accurate on a set of sample problems where we price call options on the maximum and the geometric mean of a collection of stocks. These numerical results suggest that our pricing method can be successfully applied to problems of practical interest. ∗An earlier draft of this paper was titled Pricing HighDimensional American Options: A Duality
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 137 (8 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 130 (3 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.
Option pricing under a double exponential jump diffusion model
 Management Science
, 2004
"... Analytical tractability is one of the challenges faced by many alternative models that try to generalize the BlackScholes option pricing model to incorporate more empirical features. The aim of this paper is to extend the analytical tractability of the BlackScholes model to alternative models with ..."
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Cited by 100 (4 self)
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Analytical tractability is one of the challenges faced by many alternative models that try to generalize the BlackScholes option pricing model to incorporate more empirical features. The aim of this paper is to extend the analytical tractability of the BlackScholes model to alternative models with jumps. We demonstrate a double exponential jump diffusion model can lead to an analytic approximation for Þnite horizon American options (by extending the BaroneAdesi and Whaley method) and analytical solutions for popular pathdependent options (such as lookback, barrier, and perpetual American options). Numerical examples indicate that the formulae are easy to be implemented and accurate.
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 79 (13 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.
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 57 (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
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 56 (4 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.
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 53 (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.
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 42 (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...
038 "Adaptive Simulation Algorithms for Pricing American and Bermudan Options by Local Analysis of Financial Market" by Denis Belomestny and Grigori
, 2006
"... This research was supported by the Deutsche Forschungsgemeinschaft through the SFB 649 "Economic Risk". ..."
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Cited by 39 (5 self)
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This research was supported by the Deutsche Forschungsgemeinschaft through the SFB 649 "Economic Risk".