American option pricing with discrete and continuous time models: An empirical comparison

This paper considers discrete time GARCH and continuous time SV models and uses these for American option pricing. We first of all show that with a particular choice of framework the parameters of the SV models can be estimated using simple maximum likelihood techniques. We then perform a Monte Carlo study to examine their differences in terms of option pricing, and we study the convergence of the discrete time option prices to their implied continuous time values. Finally, a large scale empirical analysis using individual stock options and options on an index is performed comparing the estimated prices from discrete time models to the corresponding continuous time model prices. The results show that, while the overall differences in performance are small, for the in the money put options on individual stocks the continuous time SV models do generally perform better than the discrete time GARCH specifications.     

Refining the least squares Monte Carlo method by imposing structure

The least squares Monte Carlo method of Longstaff and Schwartz has become a standard numerical method for option pricing with many potential risk factors. An important choice in the method is the number of regressors to use and using too few or too many regressors leads to biased results. This is so particularly when considering multiple risk factors or when simulation is computationally expensive and hence relatively few paths can be used. In this paper we show that by imposing structure in the regression problem we can improve the method by reducing the bias. This holds across different maturities, for different categories of moneyness and for different types of option payoffs and often leads to significantly increased efficiency.     

Additive and multiplicative duals for American option pricing

We investigate and compare two dual formulations of the American option pricing problem based on two decompositions of supermartingales: the additive dual of Haugh and Kogan (Oper. Res. 52:258–270, 2004) and Rogers (Math. Finance 12:271–286, 2002) and the multiplicative dual of Jamshidian (Minimax optimality of Bermudan and American claims and their Monte- Carlo upper bound approximation. NIB Capital, The Hague, 2003). Both provide upper bounds on American option prices; we show how to improve these bounds iteratively and use this to show that any multiplicative dual can be improved by an additive dual and vice versa. This iterative improvement converges to the optimal value function. We also compare bias and variance under the two dual formulations as the time horizon grows; either method may have smaller bias, but the variance of the multiplicative method typically grows much faster than that of the additive method. We show that in the case of a discrete state space, the additive dual coincides with the dual of the optimal stopping problem in the sense of linear programming duality and the multiplicative method arises through a nonlinear duality.      

Measurement of risk perceptions in social research: a comparative analysis of ordinary least squares,ordinal and multinomial logistic regression models

Drawing on data gathered in the 2006 Monitoring the Future study of American youth (n = 2489), this investigation offers a comparative analysis of ordinary least squares (OLS), ordinal and multinomial logistic regression models in examining the effects of multiple factors on perceptions of alcohol risk. The article addresses limitations of OLS models in risk analyses and demonstrates how scholars can avoid making statistical errors when positioning vague quantifiers as ordinal dependent measures. Substantively, the article finds differential effects for (1) sex, (2) perceived attitudes of peers toward alcohol consumption, (3) frequency of intoxication, (4) teacher efforts toward alcohol education, (5) frequency of communicating with friends, and (6) newspaper exposure, as determinants of alcohol risk perceptions. Through statistical results and visual displays, the article reveals how inferences made about these effects stand to vary depending on the regression method chosen.     

American bond option pricing in one-factor dynamic term structure models

Arbitrage-tree pricing of American options on bonds in one-factor dynamic term structure models is investigated. We re-derive a general decomposition result which states that the American bond option premium can be split into the value of an otherwise equivalent European option and anearly exercise premium. This extends earlier work on American equity options by e.g. Kim (1990), Jamshidian (1992) and Carr, Jarrow, and Myneni (1992) and parallels recent work by Jamshidian (1991, 1992, 1993) and Chesney, Elliott, and Gibson (1993). We examine a Gaussian class of special cases in some detail and provide a variety of numerical valuation results.An earlier version of the paper was entitled American Bond Option Pricing in One-Factor Spot Interest Rate Models.I am grateful for many helpful comments from two anonymous referees, the participants of the Second Nordic Symposium on Contingent Claims Analysis in Finance held in Bergen, Norway in May of 1994 and from the participants of the EIASM Doctoral Tutorial held in connection with the 1994 EFA annual meeting in Bruxelles. I am particularly indebted to Krishna Ramaswamy for his help and advice during my stay as visiting doctoral fellow at the Wharton School of the University of Pennsylvania. Financial support from the Aarhus University Research Foundation (Grants # E-1994-SAM-1-1-72 & E-1995-SAM-1-59), the Danish Social Science Research Council, and the Danish Research Academy is gratefully acknowledged. All errors and omissions are my own.     

American option pricing under the double Heston model based on asymptotic expansion

This paper focuses on pricing American put options under the double Heston model proposed by Christoffersen et al. By introducing an explicit exercise rule, we obtain the asymptotic expansion of the solution to the partial differential equation for pricing American put options. We calculate American option price by the sum of the European option price and the early exercise premium. The early exercise premium is calculated by the difference between the American and European option prices based on asymptotic expansions. The European option price is obtained by the efficient COS method. Based on the obtained American option price, the double Heston model is calibrated by minimizing the distance between model and market prices, which yields an optimization problem that is solved by a differential evolution algorithm combined with the Matlab function fmincon.m. Numerical results show that the pricing approach is fast and accurate. Empirical results show that the double Heston model has better performance in pricing short-maturity American put options and capturing the volatility term structure of American put options than the Heston model.     

A Refined Binomial Lattice for Pricing American Asian Options

We present simple and fast algorithms for computing very tight upper and lower bounds on the prices of American Asian options in the binomial model. We introduce a new refined version of the Cox-Ross-Rubinstein (1979) binomial lattice of stock prices. Each node in the lattice is partitioned into nodelets, each of which represents all paths arriving at the node with a specific geometric stock price average. The upper bound uses an interpolation idea similar to the Hull-White (1993) method. From the backward-recursive upper-bound computation, we estimate a good exercise rule that is consistent with the refined lattice. This exercise rule is used to obtain a lower bound on the option price using a modification of a conditional-expectation based idea from Rogers-Shi (1995) and Chalasani-Jha-Varikooty (1998). Our algorithms run in time proportional to the number of nodelets in the refined lattice, which is smaller than n^4/20 for n > 14 periods.     

Z-Transform and preconditioning techniques for option pricing

In the present paper, we convert the usual n-step backward recursion that arises in option pricing into a set of independent integral equations by using a z-transform approach. In order to solve these equations, we consider different quadrature procedures that transform the integral equation into a linear system that we solve by iterative algorithms and we study the benefits of suitable preconditioning techniques. We show the relevance of our procedure in pricing options (such as plain vanilla, lookback, single and double barrier options) when the underlying evolves according to an exponential Lévy process.     

The behavior of risky asset prices in a symmetric least squares learning environment

This paper examines the dynamics of asset prices in a heterogeneous market. Traders are made up of learners who possess limited information and use limited models for predicting the future. The market also includes noise traders in the sense of Black, along with liquidity traders. Learners revise their prediction equations using least squares learning as defined by Marcet and Sargent. We derive the equilibrium price process and show how convergence is obtained. The price process is shown to have a number of interesting properties that are consistent with propositions outlined by Black. Numerical calculations for several examples illuminate how learning takes place in the model.     

On the use of partial least squares path modeling in accounting research

Partial least squares (PLS) is an approach to structural equation modeling (SEM) that is extensively used in the social sciences to analyze quantitative data. However, PLS has not been as readily adopted in the accounting discipline. A review of the accounting literature found 20 studies in a subset of accounting journals that used PLS as the data analysis tool. PLS allows researchers to analyze the measurement model simultaneously with the structural model and allows researchers to adopt more complex research models with both moderating and mediating relationships. This paper assists accounting researchers that may be interested in adopting PLS as an analysis tool. We explain the benefits of using PLS and compare and contrast this analysis approach with both ordinary least squares regression and covariance-based SEM. We also explain how the PLS algorithm works to derive estimates for the measurement and structural models. To further assist researchers interested in using PLS, we offer guidelines in the development of research models, analysis of the data, and the interpretation of these results with PLS. We apply these guidelines to the accounting studies that have used PLS and offer further recommendations about how researchers could apply PLS in future accounting research.     

The use of least median of squares in the estimation of land value equations

Ordinary least squares (OLS) regression is relatively sensitive to the presence of outliers in a data set. In this paper, a robust estimation method, least median of squares (LMS) is used to identify outliers in land value data. Once the outliers are identified, are the land value equations re-estimated. The results show that a few observations can have a significant effect on the estimated coefficients. Finally, the observations which were identified as outliers were examined in more detail. One cause of outliers is an omitted variable. In this case, a large fraction of the outliers were found to be observations with high development potential.     

Truncation and acceleration of the Tian tree for the pricing of American put options

We present a new method for truncating binomial trees based on using a tolerance to control truncation errors and apply it to the Tian tree together with acceleration techniques of smoothing and Richardson extrapolation. For both the current (based on standard deviations) and the new (based on tolerance) truncation methods, we test different truncation criteria, levels and replacement values to obtain the best combination for each required level of accuracy. We also provide numerical results demonstrating that the new method can be 50% faster than previously presented methods when pricing American put options in the Black–Scholes model.     

Primal–dual quasi-Monte Carlo simulation with dimension reduction for pricing American options

The pricing of American options is one of the most challenging problems in financial engineering due to the involved optimal stopping time problem, which can be solved by using dynamic programming (DP). But applying DP is not always practical, especially when the state space is high dimensional. However, the curse of dimensionality can be overcome by Monte Carlo (MC) simulation. We can get lower and upper bounds by MC to ensure that the true price falls into a valid confidence interval. During the recent decades, progress has been made in using MC simulation to obtain both the lower bound by least-squares Monte Carlo method (LSM) and the upper bound by duality approach. However, there are few works on pricing American options using quasi-Monte Carlo (QMC) methods, especially to compute the upper bound. For comparing the sample variances and standard errors in the numerical experiments, randomized QMC (RQMC) methods are usually used. In this paper, we propose to use RQMC to replace MC simulation to compute both the lower bound (by the LSM) and the upper bound (by the duality approach). Moreover, we propose to use dimension reduction techniques, such as the Brownian bridge, principal component analysis, linear transformation and the gradients based principle component analysis. We perform numerical experiments on American–Asian options and American max-call options under the Black–Scholes model and the variance gamma model, in which the options have the path-dependent feature or are written on multiple underlying assets. We find that RQMC in combination with dimension reduction techniques can significantly increase the efficiency in computing both the lower and upper bounds, resulting in better estimates and tighter confidence intervals of the true price than pure MC simulation.     

High-order accurate implicit finite difference method for evaluating American options

A numerical method is presented for valuing vanilla American options on a single asset that is up to fourth-order accurate in the log of the asset price, and second-order accurate in time. The method overcomes the standard difficulty encountered in developing high-order accurate finite difference schemes for valuing American options; that is, the lack of smoothness in the option price at the critical boundary. This is done by making special corrections to the right-hand side of the differnce equations near the boundary, so they retain their level of accuracy. These corrections are easily evaluated using estimates of the boundary location and jump in the gamma that occurs there, such as those developed by Carr and Eaguet. Results of numerical experiments are presented comparing the method with more standard finite difference methods.     

American-type basket option pricing: a simple two-dimensional partial differential equation

We consider the pricing of American-type basket derivatives by numerically solving a partial differential equation (PDE). The curse of dimensionality inherent in basket derivative pricing is circumvented by using the theory of comonotonicity. We start with deriving a PDE for the European-type comonotonic basket derivative price, together with a unique self-financing hedging strategy. We show how to use the results for the comonotonic market to approximate American-type basket derivative prices for a basket with correlated stocks. Our methodology generates American basket option prices which are in line with the prices obtained via the standard Least-Square Monte-Carlo approach. Moreover, the numerical tests illustrate the performance of the proposed method in terms of computation time, and highlight some deficiencies of the standard LSM method.     

美式期权的Monte Carlo模拟法定价理论及应用

近年来随着计算机技术的飞速发展,美式期权的Monte Carlo模拟法定价取得了实质性的突破。本文分析介绍了美式期权的Monte Carlo模拟法定价理论及在此基础上推导出的线性回归MonteCarlo模拟法定价公式及其在实际的应用。     

Covariance dependent kernels,a Q-affine GARCH for multi-asset option pricing

This paper introduces a class of multivariate GARCH models that extends the existing literature by explicitly modeling correlation dependent pricing kernels. A large subclass admits closed-form recursive solutions for the moment generating function under the risk-neutral measure, which permits efficient pricing of multi-asset options. We perform a full calibration to three bivariate series of index returns and their corresponding volatility indexes in a joint maximum likelihood estimation. The results empirically confirm the presence of correlation dependance in addition to the well known variance dependance in the pricing kernel. The model improves both the overall likelihood and the VIX-implied likelihoods, with a better fitting of marginal distributions, e.g., 15% less error on one-asset option prices. The new degree of freedom is also shown to significantly impact the shape of marginal and joint pricing kernels, and leads to up to 53% differences for out-of-the-money two-asset correlation option prices.     

An exact and explicit solution for the valuation of American put options

In this paper, an exact and explicit solution of the well-known Black–Scholes equation for the valuation of American put options is presented for the first time. To the best of the author's knowledge, a closed-form analytical formula has never been found for the valuation of American options of finite maturity, although there have been quite a few approximate solutions and numerical approaches proposed. The closed-form exact solution presented here is written in the form of a Taylor's series expansion, which contains infinitely many terms. However, only about 30 terms are actually needed to generate a convergent numerical solution if the solution of the corresponding European option is taken as the initial guess of the solution series. The optimal exercise boundary, which is the main difficulty of the problem, is found as an explicit function of the risk-free interest rate, the volatility and the time to expiration. A key feature of our solution procedure, which is based on the homotopy-analysis method, is the optimal exercise boundary being elegantly and temporarily removed in the solution process of each order, and, consequently, the solution of a linear problem can be analytically worked out at each order, resulting in a completely analytical and exact series-expansion solution for the optimal exercise boundary and the option price of American put options.