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We consider the optimal exercise of a portfolio of American call options in an incomplete market. Options are written on a single underlying asset but may have different characteristics of strikes, maturities, and vesting dates. Our motivation is to model the decision faced by an employee who is granted options periodically on the stock of her company, and who is not permitted to trade this stock. The first part of our study considers the optimal exercise of single options. We prove results under minimal assumptions and give several counterexamples where these assumptions fail-describing the shape and nesting properties of the exercise regions. The second part of the study considers portfolios of options with differing characteristics. The main result is that options with comonotonic strike, maturity, and vesting date should be exercised in order of increasing strike. It is true under weak assumptions on preferences and requires no assumptions on prices. Potentially the exercise ordering result can significantly reduce the complexity of computations in a particular example. This is illustrated by solving the resulting dynamic programming problem in a constant absolute risk aversion utility indifference model. [ABSTRACT FROM AUTHOR]

This paper considers the problem of pricing options with early-exercise features whose pay-off depends on several sources of uncertainty. We propose a stochastic grid method for estimating the optimal exercise policy and use this policy to obtain a low-biased estimator for high-dimensional Bermudan options. The method has elements of the least-squares method (LSM) of Longstaff and Schwartz [Valuing American options by simulation: A simple least-squares approach, Rev. Finan. Stud. 3 (2001), pp. 113–147], the stochastic mesh method of Broadie and Glasserman [A stochastic mesh method for pricing high-dimensional American option, J. Comput. Finance 7 (2004), pp. 35–72], and stratified state aggregation along the pay-off method of Barraquand and Martineau [Numerical valuation of high-dimensional multivariate American securities, J. Financ. Quant. Anal. 30 (1995), pp. 383–405], with certain distinct advantages over the existing methods. We focus on the numerical results for high-dimensional problems such as max option and arithmetic basket option on several assets, with basic error analysis for a general one-dimensional problem. [ABSTRACT FROM PUBLISHER]

A new characterization of the American-style option is proposed under a very general multifactor Markovian and diffusion framework. The efficiency of the proposed pricing solutions is shown to depend only on the use of a viable valuation method for the corresponding European-style option and for the transition density of the model's state variables. Under a Gauss-Markov stochastic interest rates setup, these new American option pricing solutions are shown to offer a much better accuracy-efficiency trade-off than the approximations already available in the literature. This result is also used to price callable corporate bonds under an endogenous bankruptcy structural approach, by decomposing the option to call or default into a European put on the firm value plus two early exercise premium components. [ABSTRACT FROM AUTHOR]

We present a new approach to pricing American-style derivatives that is applicable to any Markovian setting (i,e., not limited to geometric Brownian motion) for which European call-option prices are readily available. By approximating the value function with an appropriately chosen interpolation function, the pricing of an American-style derivative with arbitrary payoff function is converted to the pricing of a portfolio of European call options, leading to analytical expressions for those cases where analytical European call prices are available (e.g., the Merton jump-diffusion process). Furthermore, in many settings, the approach yields upper and lower analytical bounds that probably converge to the true option price. We provide computational results to illustrate the convergence and accuracy of the resulting estimators. [ABSTRACT FROM AUTHOR]

Unlike European options, American options cannot be valued by closed-form formulae and require the use of numerical methods. There are two main numerical approaches: One is the probabilistic approach developed in Kushner (1977), based on the approximation of diffusions by Markov chains, which has been addressed by Boyle et al. (1989), who extended the standard binomial algorithm for pricing American options in the two-dimensional Black-Scholes setting. The other is the analytic approach related to the discretization of the variational inequality satisfied by the option value which leads to the resolution of a linear complementarity problem. From a theoretical viewpoint, the convergence of the probabilistic approach has been proved by Lamberton and Pagès (1990) and Amin and Khanna (1994), while the convergence of the analytic approach was shown by Glowinski et al. (1976). In this paper, we will discuss American options written on two stocks in the bidimensional Black-Scholes framework and we will concentrate on analytic methods. [ABSTRACT FROM AUTHOR]

This article derives the first analytical pricing formulas for American-style Asian options of the so-called floating strike type. Geometric as well as arithmetic averaging is considered. The setup is a standard Black-Scholes framework where the price of the underlying security evolves according to a geometric Brownian motion. A decomposition result that splits up the value of the floating strike American option into the price of an otherwise equivalent European option and an early exercise premium is first presented. This decomposition result is then manipulated further for the two separate types of averaging. With geometric averaging we derive an exact pricing formula, whereas with arithmetic averaging we develop an analytical approximation formula that proves to he very precise. Numerical examples are provided. [ABSTRACT FROM AUTHOR]

This paper is a compendium of results--theoretical and computational--from a series of recent papers developing a new American option valuation technique based on linear programming (LP). Some further computational results are included for completeness. A proof of the basic analytical theorem is given, as is the analysis needed to solve the inverse problem of determining local (one-factor) volatility from market data. The ideas behind a fast accurate revised simplex method, whose performance is linear in time and space discretizations, are described and the practicalities of fitting the volatility smile are discussed. Numerical results are presented which show the LP valuation technique to be extremely fast--lattice speed with PDE accuracy. American options valued in the paper range from vanilla, through exotic with constant volatility, to exotic options fitting the volatility smile. [ABSTRACT FROM AUTHOR]