This paper describes a practical algorithm based on Monte Carlo simulation for the pricing of multi-dimensional 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 multi-factor 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.

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 temporal-difference learning -- a stochastic algorithm inspired to some extent by phenomena observed in animal behavior. Given a selection of...

Recently, various authors proposed Monte-Carlo 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 Monte-Carlo 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 number of Monte Carlo simulation-based approaches have been proposed within the past decade to address the problem of pricing American-style 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 simulation-based 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.

I survey and assess the development of continuous-time 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 continuous-time models. Capital market frictions and bargaining issues are being increasingly incorporated in continuous-time theory. THE ROOTS OF MODERN CONTINUOUS-TIME 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 continuous-time modeling in financial economics by formulating the intertemporal consumption and portfolio choice problem of an investor in a stochastic dynamic programming setting.

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 non-standard interest rate derivatives can be valued. 1 FORWARD RATE VOLATILITIES, SWAP RATE VOLATILITIES, AND THE IMPLEMENTATION OF THE LIBOR MARKET MODEL The most popular over-the-counter 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...

We develop and study general-purpose 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 mesh-based, biased-low 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 path-dependent 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 bias-reduced point estimates are much more accurate than the standard mesh-method point estimates. Importance sampling is found to increase accuracy for a smooth optionpayoff functions, while variance increases are possible for non-sm...

A shout option may be broadly defined as a financial contract which can be modified by its holder according to specified rules. In a simple example, the holder could have the right to set the strike of an option equal to the current value of the underlying asset. In such a case, the holder effectively has the right to select when to take ownership of an at-the-money option. More generally, the holder could have multiple rights along these lines, in some cases with a limit placed on the number of rights which may be exercised within a given time period (e.g. four times per year). The value of these types of contracts can be estimated by solving a system of interdependent linear complementarity problems. This paper describes a general framework for the valuation of complex types of shout options. Numerical issues related to interpolation and choice of timestepping method are considered in detail. Some illustrative examples are provided.

This paper develops a simulation method for pricing path-dependent 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 high-dimensional problems, and this has motivated research into simulation-based 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

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 Black-Scholes model and Merton’s lognormal jump-diffusion model. We also propose some extensions to apply the fast Gauss transform to Kou’s double-exponential jump-diffusion model and Heston’s stochastic volatility model. 1