American-style Options Research Articles

Forest of stochastic meshes: A new method for valuing high-dimensional swing options

Swing options generalize American-style options as they allow the holder multiple exercise rights and control over the exercise amounts. In this work, we replace the standard (binomial) trees in the forest of trees algorithm with stochastic meshes, yielding the forest of stochastic meshes; a simulation-based method for valuing high-dimensional swing options. This new method handles general price processes and payoffs, produces high- and low-biased consistent estimators and a true option price confidence interval.

OPTIMAL LIQUIDATION OF DERIVATIVE PORTFOLIOS

We consider the problem facing a risk averse agent who seeks to liquidate or exercise a portfolio of (infinitely divisible) perpetual American style options on a single underlying asset. The optimal liquidation strategy is of threshold form and can be characterized explicitly as the solution of a calculus of variations problem. Apart from a possible initial exercise of a tranche of options, the optimal behavior involves liquidating the portfolio in infinitesimal amounts, but at times which are singular with respect to calendar time. We consider a number of illustrative examples involving CRRA and CARA utility, stocks, and portfolios of options with different strikes, and a model where the act of exercising has an impact on the underlying asset price.

Fast Computation of Upper Bounds for American-Style Options Without Nested Simulation

This article presents a fast yet effective simulation-based approach for estimating the upper bounds of American-style option prices without nested simulations. We derive the analytical representation for the 'correct' martingale that is to be applied to the dual pricing formula to give the correct option price under a generic theoretical setting. This representation turns out to bear clear implication for hedging and is the main contribution of our paper. We further propose a simulation algorithm that directly builds on this analytical form for martingale construction and circumvents the time-consuming sub-simulation part that is usually required in other existing dual simulation procedures. Our approach is shown to be capable of producing viable upper bounds as well as significantly reducing the computing time allocated to upper bound estimation. We illustrate the effectiveness of our approach with a series of increasingly complex but realistic examples.

Canonical Least-Squares Monte Carlo: Empirical Evidences from S&P 100 Index and IBM Puts

This paper tests empirically the recently proposed canonical least-squares Monte-Carlo (CLM) method for pricing American options. Market data used are daily last prices for the American style S&P 100 Index puts (OEX puts) and IBM puts from 30 July 2008 to 30 January 2009, the period of which covers the US financial crisis of 2008. Unlike what have been done in many other similar empirical studies, rates interpolated from the US Treasury yield curve are used, and dividends, a continuous constant yield for the S&P 100 Index and a fixed cash amount for the IBM common stock, are taken into consideration. Compared with the benchmark prices from finite difference using historic volatilities, CLM outperforms in almost all twelve categories of moneyness and maturity for both OEX and IBM puts, and is shown to be a viable alternative for pricing American style options.

Sequential Monte Carlo pricing of American-style options under stochastic volatility models

We introduce a new method to price American-style options on underlying investments governed by stochastic volatility (SV) models. The method does not require the volatility process to be observed. Instead, it exploits the fact that the optimal decision functions in the corresponding dynamic programming problem can be expressed as functions of conditional distributions of volatility, given observed data. By constructing statistics summarizing information about these conditional distributions, one can obtain high quality approximate solutions. Although the required conditional distributions are in general intractable, they can be arbitrarily precisely approximated using sequential Monte Carlo schemes. The drawback, as with many Monte Carlo schemes, is potentially heavy computational demand. We present two variants of the algorithm, one closely related to the well-known least-squares Monte Carlo algorithm of Longstaff and Schwartz [The Review of Financial Studies 14 (2001) 113-147], and the other solving the same problem using a "brute force" gridding approach. We estimate an illustrative SV model using Markov chain Monte Carlo (MCMC) methods for three equities. We also demonstrate the use of our algorithm by estimating the posterior distribution of the market price of volatility risk for each of the three equities.

BSDEs with random terminal time under enlarged filtration. American-style options hedging by an insider

Markets with asymmetrical information are generally studied from a wealth optimization point of view. We focus here on a hedging problem for a financial agent who has an additional information on the market. We extend the results given for hedging strategies with fixed terminal time to the case of a random terminal time. In particular, we provide tools to understand the behavior of American option hedging by an insider. To achieve this aim, we prove the existence and uniqueness of backward stochastic differential equations solutions, when terminal time is random, under an initially enlarged filtration.

A dynamic programming approach for pricing CDS and CDS options

We propose a flexible framework for pricing single-name knock-out credit derivatives. Examples include Credit Default Swaps (CDSs) and European, American and Bermudan CDS options. The default of the underlying reference entity is modelled within a doubly stochastic framework where the default intensity follows a CIR++ process. We estimate the model parameters through a combination of a cross sectional calibration-based method and a historical estimation approach. We propose a numerical procedure based on dynamic programming and a piecewise linear approximation to price American-style knock-out credit options. Our numerical investigation shows consistency, convergence and efficiency. We find that American-style CDS options can complete the credit derivatives market by allowing the investor to focus on spread movements rather than on the default event.

Pseudospectral methods for pricing options

Models with two or more risk sources have been widely applied in option pricing in order to capture volatility smiles and skews. However, the computational cost of implementing these models can be large—especially for American-style options. This paper illustrates how numerical techniques called ‘pseudospectral’ methods can be used to solve the partial differential and partial integro-differential equations that apply to these multifactor models. The method offers significant advantages over finite-difference and Monte Carlo simulation schemes in terms of accuracy and computational cost.

An efficient implementation of a least squares Monte Carlo method for valuing American-style options

Several methods for valuing high-dimensional American-style options were proposed in the last years. Longstaff and Schwartz (LS) have suggested a regression-based Monte Carlo approach, namely the least squares Monte Carlo method. This article is devoted to an efficient implementation of this algorithm. First, we suggest a code for faster runs. Regression-based Monte Carlo methods are sensitive to the choice of basis functions for pricing high-dimensional American-style options and, like all Monte Carlo methods, to the underlying random number generator. For this reason, we secondly propose an optimal selection of basis functions and a random number generator to guarantee stable results. Our basis depends on the payoff of the high-dimensional option and consists of only three functions. We give a guideline for an efficient option price calculation of high-dimensional American-style options with the LS algorithm, and we test it in examples with up to 10 dimensions.

Empirical Tests of Canonical Nonparametric American Option Pricing Methods

Alcock and Carmichael (2008) introduce a nonparametric method for pricing American style options that is derived from the canonical valuation developed by Stutzer (1996). While the statistical properties of this nonparametric pricing methodology have been studied in a controlled simulation environment, no study has yet examined the empirical validity of this method.We introduce an extension to this method that incorporates information contained in a small number of observed option prices. We explore the applicability of both the original method and our extension using a large sample of OEX American index options traded on the S&P100 index. While the Alcock and Carmichael (2008) method fails to outperform a traditional implied volatility based Black-Scholes valuation or a Binomial Tree approach, our extension generates signi cantly lower pricing errors and performs comparably well to the implied-volatility Black-Scholes pricing, in particular for out-of-the-money American put options.

Accurate and efficient lattice algorithms for American-style Asian options with range bounds

Asian options are popular path-dependent options and it has been a long-standing problem to price them efficiently and accurately. Since there is no known exact pricing formula for Asian options, numerical pricing formulas like lattice models must be employed. A lattice divides a certain time interval into n time steps and the pricing results generated by the lattice (called desired option values for convenience) converge to the theoretical option value as n → ∞ . Since a brute-force lattice pricing algorithm runs in subexponential time in n, some heuristics, like interpolation method, are used to strike the balance between the efficiency and the accuracy. But the pricing results might not converge due to the accumulation of interpolation errors. For pricing European-style Asian options, the evaluation on the major part of the lattice can be done by a simple formula, and the interpolation method is only required on the minor part of the lattice. Thus polynomial time algorithms with convergence guarantee for European-style Asian options can be derived. However, such a simple formula does not exist for American-style Asian options. This paper suggests an efficient range-bound algorithm that bracket the desired option value. By taking advantages of the early exercise property of American-style options, we show that part of the lattice can be evaluated by a simple formula. The interpolation method is required on the remaining part of the lattice and the upper and the lower bounds option values produced by the proposed algorithm are essentially numerically identical. Thus the theoretical option value is said to be obtained practically when the range bound algorithm runs on a lattice with large number of time steps.

Improved lower and upper bound algorithms for pricing American options by simulation

This paper introduces new variance reduction techniques and computational improvements to Monte Carlo methods for pricing American-style options. For simulation algorithms that compute lower bounds of American option values, we apply martingale control variates and introduce the local policy enhancement, which adopts a local simulation to improve the exercise policy. For duality-based upper bound methods, specifically the primal–dual simulation algorithm, we have developed two improvements. One is sub-optimality checking, which saves unnecessary computation when it is sub-optimal to exercise the option along the sample path; the second is boundary distance grouping, which reduces computational time by skipping computation on selected sample paths based on the distance to the exercise boundary. Numerical results are given for single asset Bermudan options, moving window Asian options and Bermudan max options. In some examples the computational time is reduced by a factor of several hundred, while the confidence interval of the true option value is considerably tighter than before the improvements.

Binomial approximations of shortfall risk for game options

We show that the shortfall risk of binomial approximations of game (Israeli) options converges to the shortfall risk in the corresponding Black–Scholes market considering Lipschitz continuous path dependent payoffs for both discrete and continuous time cases. This results are new also for usual American style options. The paper continues and extends the study of [6] where estimates for binomial approximations of prices of game options were obtained. Our arguments rely, in particular, on strong invariance principle type approximations via the Skorokhod embedding, estimates from [6] and the existence of optimal shortfall hedging in the discrete time established in [2]. 1. Introduction. This paper deals with game (Israeli) options introduced in [5] sold in a standard securities market consisting of a nonrandom component bt representing the value of a savings account at time t with an interest rate r and of a random component St representing the stock price at time t. As usual, we view St, t > 0 as a stochastic process on a probability space (, F, P) and we assume that it generates a right continuous filtration {Ft}. The setup includes also two continuous stochastic payoff processes Xt ≥ Yt ≥ 0 adapted to the above filtration. Recall, that game contingent claim (GCC) or game option is defined as a contract between the seller and the buyer of the option such that both have the right to exercise it at any time up to a maturity date (horizon) T. If the buyer exercises the contract at time t then he receives the payment Yt, but if the seller exercises (cancels) the contract before the buyer then the latter receives Xt. The dif

Error estimates for binomial approximations of game options

A game or Israeli option is an American style option where both the writer and the holder have the right to terminate the contract before the expiration time. Kifer (2000) shows that the fair price for this option can be expressed as the value of a Dynkin game. In general, there are no explicit formulas for fair prices of American and game options and approximations are used for their computations. The paper by Lamberton (1998) provides error estimates for binomial approximation of American put options and here we extend the approach of Lamberton (1998) in order to obtain error estimates for binomial approximations of game put options which is more complicated as it requires us to deal with two free boundaries corresponding to the writer and to the holder of the game option.

Kernel-based Monte Carlo simulation for American option pricing

Valuation of an American option with Monte Carlo methods is one of the most important and difficult problems in pricing, since it involves the determination of optimal exercise timing in the sense that the option can be exercised at any time prior to its own maturity. Regression approaches have been widely used to price an American-style option approximately with Monte Carlo simulation. However, the conventional regression methods are very sensitive in the kind and the number of their basis functions, thereby affecting prediction accuracy. In this paper, we propose a novel kernel-based Monte Carlo simulation algorithm to overcome such shortcomings of the regression approaches and conduct a simulation on some American options with promising results on its pricing accuracy.

On the Structure of a Swing Contract's Optimal Value and Optimal Strategy

Consider a sales contract, called a swing contract, between a seller and a buyer concerning some underlying commodity, with the contract specifying that during some future time interval the buyer will purchase an amount of the commodity between some specified minimum and maximum values. The purchase price and capacity at each time point is also prespecified in the contract. Assuming a random market price process and ignoring the possibility of storage, we look for the maximal expected net gain for the buyer of such a contract, and the strategy that achieves this maximal expected net gain. We study the effects that various contract constraints and market price processes have on the optimal strategy and on the contract value. We show how we can reduce the general swing contract to a multiple exercising of American (Bermudan) style options. Also, in important special cases, we give explicit expressions for the optimal contract value function and the optimal strategy.

On the Structure of a Swing Contract's Optimal Value and Optimal Strategy

Consider a sales contract, called a swing contract, between a seller and a buyer concerning some underlying commodity, with the contract specifying that during some future time interval the buyer will purchase an amount of the commodity between some specified minimum and maximum values. The purchase price and capacity at each time point is also prespecified in the contract. Assuming a random market price process and ignoring the possibility of storage, we look for the maximal expected net gain for the buyer of such a contract, and the strategy that achieves this maximal expected net gain. We study the effects that various contract constraints and market price processes have on the optimal strategy and on the contract value. We show how we can reduce the general swing contract to a multiple exercising of American (Bermudan) style options. Also, in important special cases, we give explicit expressions for the optimal contract value function and the optimal strategy.

Analysis of exercise boundary of American interest rate option

By applying the variational inequality technique, we analyzed the behavior of the exercise boundary of the American-style interest rate option under the assumption that the interest rates obey a mean-reverting random walk as given by the Vasicek model. The monotonicity, boundedness and C ∞-smoothness of the exercise boundary are proved in this paper.

Pricing options with Green's functions when volatility, interest rate and barriers depend on time

We derive the Green's function for the Black–Scholes partial differential equation with time-varying coefficients and time-dependent boundary conditions. We provide a thorough discussion of its implementation within a pricing algorithm that also accommodates American style options. Greeks can be computed as derivatives of the Green's function. Generic handling of arbitrary time-dependent boundary conditions suggests our approach to be used with the pricing of (American) barrier options, although options without barriers can be priced equally well. Numerical results indicate that knowledge of the structure of the Green's function together with the well-developed tools of numerical integration make our approach fast and numerically stable.

Pricing Expropriation Risk in Natural Resource Contracts - A Real Options Approach

We develop a model for pricing expropriation risk in natural resource projects, in particular an oil field. The government is viewed as holding an American-style option to expropriate the oil field, but facing the following three possible expropriation costs: A state-run company may produce oil less cost-efficiently than a private firm,the government may have to pay a compensation to the firm, and an expropriation may trigger lower investor confidence negatively affecting the overall economy. The dynamics of key variables – the spot price, futures prices and volatility – is described by a model proposed and estimated in Trolle and Schwartz (2007). For reasonable parameter values and under market conditions not too different from what has been seen in recent years, the value of the expropriation option can be substantial