High-dimensional 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 high-dimensional 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

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.

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 pseudo-Snell 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 2-dimensional American option pricing problem.

Based on recent results for multi-armed 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 worst-case running-time complexity of the algorithm is O((|A|N) 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.

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...

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

American-Asian options are average-price options that allow early exercise. In this paper, we first derive structural properties of the optimal exercise policy for these call options in a general setting. In particular, we show that the optimal policy is a threshold policy: the option should be exercised as soon as the average asset price reaches a characterized threshold, which can be written as a function of asset price at that time. After further characterizing the exercise boundary, we parameterize it, and then derive gradient estimators with respect to the parameters of the model. Implementing these estimators in an iterative gradient-based stochastic approximation algorithm, we approximate the optimal exercise boundary and consequently obtain an estimate for the price of the American-Asian option. Numerical experiments carried out indicate that the algorithm performs extremely well.