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54
Optimal multiple stopping and valuation of swing options
 in Half? How Policy Reform and Effective Aid Can Meet International Development Goals.” World Development 29(11
, 2006
"... ABSTRACT. The connection between optimal stopping of random systems and the theory of the Snell envelope is well understood, and its application to the pricing of American contingent claims is well known. Motivated by the pricing of swing contracts (whose recall components can be viewed as contingen ..."
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Cited by 57 (7 self)
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ABSTRACT. The connection between optimal stopping of random systems and the theory of the Snell envelope is well understood, and its application to the pricing of American contingent claims is well known. Motivated by the pricing of swing contracts (whose recall components can be viewed as contingent claims with multiple exercises of American type) we investigate the mathematical generalization of these results to the case of possible multiple stopping. We prove existence of the multiple exercise policies in a fairly general setup. We then concentrate on the BlackScholes model for which we give a constructive solution in the perpetual case, and an approximation procedure in the finite horizon case. The last two sections of the paper are devoted to numerical results. We illustrate the theoretical results of the perpetual case, and in the finite horizon case, we introduce numerical approximation algorithms based on ideas of the Malliavin calculus. 1.
A regressionbased Monte Carlo method to solve backward stochastic differential equations
, 2005
"... We are concerned with the numerical resolution of backward stochastic differential equations. We propose a new numerical scheme based on iterative regressions on function bases, which coefficients are evaluated using Monte Carlo simulations. A full convergence analysis is derived. Numerical experime ..."
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Cited by 57 (6 self)
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We are concerned with the numerical resolution of backward stochastic differential equations. We propose a new numerical scheme based on iterative regressions on function bases, which coefficients are evaluated using Monte Carlo simulations. A full convergence analysis is derived. Numerical experiments about finance are included, in particular, concerning option pricing with differential interest rates.
A quantization tree method for pricing and hedging multidimensional American options
 Math. Finance
, 2005
"... Abstract We present here the quantization method which is welladapted for the pricing and hedging of American options on a basket of assets. Its purpose is to compute a large number of conditional expectations by projection of the diffusion on optimal grid designed to minimize the (square mean) pr ..."
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Cited by 53 (7 self)
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Abstract We present here the quantization method which is welladapted for the pricing and hedging of American options on a basket of assets. Its purpose is to compute a large number of conditional expectations by projection of the diffusion on optimal grid designed to minimize the (square mean) projection error ([24]). An algorithm to compute such grids is described. We provide results concerning the orders of the approximation with respect to the regularity of the payoff function and the global size of the grids. Numerical tests are performed in dimensions 2, 4, 6, 10 with American style exchange options. They show that theoretical orders are probably pessimistic.
Discrete Time Approximation and MonteCarlo Simulation of Backward Stochastic Differential Equations
, 2002
"... We suggest a discretetime approximation for decoupled forwardbackward stochastic differential equations. The L^p norm of the error is shown to be of the order of the time step. Given a simulationbased estimator of the conditional expectation operator, we then suggest a backward simulation scheme, ..."
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Cited by 20 (1 self)
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We suggest a discretetime approximation for decoupled forwardbackward stochastic differential equations. The L^p norm of the error is shown to be of the order of the time step. Given a simulationbased estimator of the conditional expectation operator, we then suggest a backward simulation scheme, and we study the induced L^p error. This estimate is more investigated in the context of the Malliavin approach for the approximation of conditional expectations. Extensions to the reflected case are also considered.
The valuation of multidimensional American real options using computerbased simulation". The 9th Annual Real Options Conference
, 2005
"... In this paper we show how a multidimensional American real option may be solved using a computerbased simulation procedure. We implement an approach originally proposed for a financial option and show how it can be used in a much more complex setting. We extend a wellknown natural resource real opt ..."
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Cited by 14 (0 self)
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In this paper we show how a multidimensional American real option may be solved using a computerbased simulation procedure. We implement an approach originally proposed for a financial option and show how it can be used in a much more complex setting. We extend a wellknown natural resource real option model, originally solved using finite difference methods, to include a more realistic 3 factor stochastic process for commodity prices, more in line with current research. We show how complexity may be reduced by adequately choosing the implementation variables. Numerical results show that the procedure may be successfully used for multidimensional models, notably expanding the applicability of the real options approach. Scope and purpose Even though there has been an increasing literature on the benefits of using the contingent claim approach to value real assets, limitations on solving procedures and computing power have often forced academics and practitioners to simplify these real option models to a level in which they loose relevance for realworld decision making. Real option models present a higher challenge than their financial option counterparts because of two main reasons: First, many real options have a longer maturity which makes risk modeling critical and may force considering many risk factors as opposed to the classic Black and Scholes onefactor model. Second, many times real investments have a more complex set of interacting American options available, making them more difficult to value. In recent years new approaches for solving American options have been proposed which, coupled with an increasing availability of computing power, have been successfully applied to solving longterm financial options and opening new hopes for increasing the use of this modeling approach for valuing real assets.
Valuing Simple MultipleExercise Real Options in Infrastructure Projects
"... Abstract: The revenue risk is considerable in infrastructure project financing arrangements such as build–operate–transfer �BOT�. A potential mitigation strategy for the revenue risk is a governmental revenue guarantee, where the government secures a minimum amount of revenue for a project. Such a g ..."
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Cited by 12 (0 self)
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Abstract: The revenue risk is considerable in infrastructure project financing arrangements such as build–operate–transfer �BOT�. A potential mitigation strategy for the revenue risk is a governmental revenue guarantee, where the government secures a minimum amount of revenue for a project. Such a guarantee is: �1 � only redeemable at distinct points in time; and �2 � more economical if the government limits the guarantee’s availability to the early portions of a BOT’s concession period. Hence, a guarantee characterized by this type of structure takes the form of either a Bermudan or a simple multipleexercise real option, depending upon the number of exercise opportunities afforded. The multileastsquares Monte Carlo technique is presented and illustrated as a promising approach to determine the fair value of this variety of real option. This method is far more flexible than prevailing approaches, so it represents an important step toward improving risk mitigation and facilitating contractual and financial negotiations in BOT projects.
A Martingale Control Variate Method for Option Pricing with Stochastic Volatility, ESAIM: Probability and Statistics
, 2007
"... A generic control variate method is proposed to price options under stochastic volatility models by Monte Carlo simulations. This method provides a constructive way to select control variates which are martingales in order to reduce the variance of unbiased option price estimators. We apply a singu ..."
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Cited by 10 (7 self)
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A generic control variate method is proposed to price options under stochastic volatility models by Monte Carlo simulations. This method provides a constructive way to select control variates which are martingales in order to reduce the variance of unbiased option price estimators. We apply a singular and regular perturbation analysis to characterize the variance reduced by martingale control variates. This variance analysis is done in the regime where time scales of associated driving volatility processes are well separated. Numerical results for European, Barrier, and American options are presented to illustrate the effectiveness and robustness of this martingale control variate method in regimes where these time scales are not so well separated. 1
Adaptive Control Variates for Pricing MultiDimensional American Options
, 2006
"... We explore a class of control variates for the American option pricing problem. We construct the control variates by using multivariate adaptive linear regression splines to approximate the option’s value function at each time step; the resulting approximate value functions are then combined to cons ..."
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Cited by 8 (1 self)
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We explore a class of control variates for the American option pricing problem. We construct the control variates by using multivariate adaptive linear regression splines to approximate the option’s value function at each time step; the resulting approximate value functions are then combined to construct a martingale that approximates a “perfect ” control variate. We demonstrate that significant variance reduction is possible even in a highdimensional setting. Moreover, the technique is applicable to a wide range of both option payoff structures and assumptions about the underlying riskneutral market dynamics. The only restriction is that one must be able to compute certain onestep conditional expectations of the individual underlying random variables. 1
Pathwise optimization for optimal stopping problems
 Management Science
"... Abstract We introduce the pathwise optimization (PO) method, a new convex optimization procedure to produce upper and lower bounds on the optimal value (the 'price') of a highdimensional optimal stopping problem. The PO method builds on a dual characterization of optimal stopping problem ..."
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Cited by 8 (1 self)
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Abstract We introduce the pathwise optimization (PO) method, a new convex optimization procedure to produce upper and lower bounds on the optimal value (the 'price') of a highdimensional optimal stopping problem. The PO method builds on a dual characterization of optimal stopping problems as optimization problems over the space of martingales, which we dub the martingale duality approach. We demonstrate via numerical experiments that the PO method produces upper bounds of a quality comparable with stateoftheart approaches, but in a fraction of the time required for those approaches. As a byproduct, it yields lower bounds (and suboptimal exercise policies) that are substantially superior to those produced by stateoftheart methods. The PO method thus constitutes a practical and desirable approach to highdimensional pricing problems. Further, we develop an approximation theory relevant to martingale duality approaches in general and the PO method in particular. Our analysis provides a guarantee on the quality of upper bounds resulting from these approaches, and identifies three key determinants of their performance: the quality of an input value function approximation, the square root of the effective time horizon of the problem, and a certain spectral measure of 'predictability' of the underlying Markov chain. As a corollary to this analysis we develop approximation guarantees specific to the PO method. Finally, we view the PO method and several approximate dynamic programming (ADP) methods for highdimensional pricing problems through a common lens and in doing so show that the PO method dominates those alternatives.
SimulationBased Pricing of Convertible Bonds
"... We propose and empirically study a pricing model for convertible bonds based on Monte Carlo simulation. The method uses parametric representations of the early exercise decisions and consists of two stages. Pricing convertible bonds with the proposed Monte Carlo approach allows us to better capture ..."
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Cited by 5 (0 self)
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We propose and empirically study a pricing model for convertible bonds based on Monte Carlo simulation. The method uses parametric representations of the early exercise decisions and consists of two stages. Pricing convertible bonds with the proposed Monte Carlo approach allows us to better capture both the dynamics of the underlying state variables and the rich set of realworld convertible bond specifications. Furthermore, using the simulation model proposed, we present an empirical pricing study of the US market, using 32 convertible bonds and 69 months of daily market prices. Our results do not confirm the evidence of previous studies that market prices of convertible bonds are on average lower than prices generated by a theoretical model. Similarly, our study is not supportive of a strong positive relationship between moneyness and mean pricing error, as argued in the literature.