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71
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 77 (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.
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 36 (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...
Forward Rate Volatilities, Swap Rate Volatilities, And The Implementation OF THE LIBOR MARKET MODEL
, 1999
"... This paper is concerned with the implementation of the LIBOR market model and its extensions. It develops and tests a simple analytic approximation for calculating the volatilities used by the market to price European swap options from the volatilities used by the market to price interest rate caps. ..."
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Cited by 36 (0 self)
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This paper is concerned with the implementation of the LIBOR market model and its extensions. It develops and tests a simple analytic approximation for calculating the volatilities used by the market to price European swap options from the volatilities used by the market to price interest rate caps. The approximation is found to be very accurate for the range of market parameters normally encountered. It enables swap option volatility skews to be implied from cap volatility skews. It also allows the LIBOR market model to be easily calibrated to broker quotes on caps and European swap options so that a wide range of nonstandard interest rate derivatives can be valued. 1 FORWARD RATE VOLATILITIES, SWAP RATE VOLATILITIES, AND THE IMPLEMENTATION OF THE LIBOR MARKET MODEL The most popular overthecounter interest rate options are interest rate caps/floors and European swap options. The standard market models for valuing these instruments are versions of Black's (1976) model. This mode...
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 34 (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.
Continuoustime methods in finance: A review and an assessment
 Journal of Finance
, 2000
"... I survey and assess the development of continuoustime methods in finance during the last 30 years. The subperiod 1969 to 1980 saw a dizzying pace of development with seminal ideas in derivatives securities pricing, term structure theory, asset pricing, and optimal consumption and portfolio choices. ..."
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Cited by 32 (0 self)
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I survey and assess the development of continuoustime methods in finance during the last 30 years. The subperiod 1969 to 1980 saw a dizzying pace of development with seminal ideas in derivatives securities pricing, term structure theory, asset pricing, and optimal consumption and portfolio choices. During the period 1981 to 1999 the theory has been extended and modified to better explain empirical regularities in various subfields of finance. This latter subperiod has seen significant progress in econometric theory, computational and estimation methods to test and implement continuoustime models. Capital market frictions and bargaining issues are being increasingly incorporated in continuoustime theory. THE ROOTS OF MODERN CONTINUOUSTIME METHODS in finance can be traced back to the seminal contributions of Merton ~1969, 1971, 1973b! in the late 1960s and early 1970s. Merton ~1969! pioneered the use of continuoustime modeling in financial economics by formulating the intertemporal consumption and portfolio choice problem of an investor in a stochastic dynamic programming setting.
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.
Application of the fast Gauss transform to option pricing
 in 5th Columbia JAFEE Conference on Mathematics of Finance
, 2002
"... In many of the numerical methods for pricing American options based on the dynamic programming approach, the most computationally intensive part can be formulated as summation of Gaussians. Though this operation usually requires O(MN) work when there are M summations to compute and the number of ter ..."
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Cited by 18 (0 self)
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In many of the numerical methods for pricing American options based on the dynamic programming approach, the most computationally intensive part can be formulated as summation of Gaussians. Though this operation usually requires O(MN) work when there are M summations to compute and the number of terms appearing in each summation is N, we can reduce the amount of work to O(M + N) by using a technique called fast Gauss transform. In this paper, we apply this technique to the multinomial method and the stochastic mesh method, and show by numerical experiments how it can speed up these methods considerably, both for the BlackScholes model and Merton’s lognormal jumpdiffusion model. We also propose some extensions to apply the fast Gauss transform to Kou’s doubleexponential jumpdiffusion model and Heston’s stochastic volatility model. 1
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 16 (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
Efficiency Improvements for Pricing American Options with a Stochastic Mesh
 In Proceedings of the 1999 Winter Simulation Conference
, 1999
"... We develop and study generalpurpose techniques for improving the efficiency of the stochastic mesh method that was recently developed for pricing American options via Monte Carlo simulation. First, we develop a meshbased, biasedlow estimator. By recursively averaging the low and high estimators ..."
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Cited by 14 (0 self)
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We develop and study generalpurpose techniques for improving the efficiency of the stochastic mesh method that was recently developed for pricing American options via Monte Carlo simulation. First, we develop a meshbased, biasedlow estimator. By recursively averaging the low and high estimators at each stage, we obtain a significantly more accurate point estimator at each of the mesh points. Second, we adapt the importance sampling ideas for simulation of European pathdependent options in Glasserman, Heidelberger, and Shahabuddin (1998a) to pricing of American options with a stochastic mesh. Third, we sketch generalizations of the mesh method and we discuss links with other techniques for valuing American options. Our empirical results show that the biasreduced point estimates are much more accurate than the standard meshmethod point estimates. Importance sampling is found to increase accuracy for a smooth optionpayoff functions, while variance increases are possible for nonsm...
Iterative construction of the optimal Bermudan stopping time. WIASPreprint No
 Finance Stoch
, 2004
"... Abstract. We present an iterative procedure for computing the optimal Bermudan stopping time, hence the Bermudan Snell envelope. The method produces an increasing sequence of approximations of the Snell envelope from below, which coincide with the Snell envelope after finitely many steps. Then, by d ..."
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Cited by 13 (7 self)
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Abstract. We present an iterative procedure for computing the optimal Bermudan stopping time, hence the Bermudan Snell envelope. The method produces an increasing sequence of approximations of the Snell envelope from below, which coincide with the Snell envelope after finitely many steps. Then, by duality, the method induces a convergent sequence of upper bounds as well. In a Markovian setting the presented iterative procedure allows to calculate approximative solutions with only a few nestings of conditionals expectations and is therefore tailormade for a plain MonteCarlo implementation. The method presented may be considered generic for all discrete optimal stopping problems. The power of the procedure is demonstrated at Bermudan swaptions in a full factor LIBOR market model.