American Option Pricing Research Articles

American Option Pricing

American Option Pricing

American option pricing by a method of error correction

ABSTRACTThe real options approach often assumes that investment projects last indefinitely, which is an unrealistic assumption. When projects live finitely, valuation techniques from American option pricing are required. This article presents a method for pricing American options based on the first-passage approach to the problem. The key is to correct the error associated with the price obtained from a rough first approximation. The procedure leads to a significant reduction in error corresponding to the initial approximation. As a particular case of the method proposed, we derive a closed-form approximation of the option price. The existence of a closed-form approximating formula (that does not involve iterative methods) keeps the computational cost low. In terms of accuracy, the method can be compared to much more sophisticated methods. A tight lower bound (given in closed form) is also provided. The method is fast, accurate, flexible, and easy to implement. A spreadsheet suffices for practical implementation.

Pricing American Options Under Regime Switching Using Method of Lines

This paper considers the American option pricing problem under regime-switching by using the method-of-lines (MOL) scheme. American option prices in each regime involve prices in all other regimes. We treat the prices from other regimes implicitly, thus guaranteeing consistency. Iterative procedures are required but very few iterative steps are needed in practice. Numerical tests demonstrate the robustness, accuracy and efficiency of the proposed numerical scheme. We compare our results with Buffington and Elliott (2002)'s analytical approximation under two regimes. Our MOL scheme provides improved results especially for out-of-the money options, possibly because they use a separation of variable approach to the PDEs which cannot hold around the early exercise region. We also compare our results with those of Khaliq and Liu (2009) and suggest that their implicit scheme can be improved.

Perpetual American options in diffusion-type models with running maxima and drawdowns

We study perpetual American option pricing problems in an extension of the Black–Merton–Scholes model in which the dividend and volatility rates of the underlying risky asset depend on the running values of its maximum and maximum drawdown. The optimal exercise times are shown to be the first times at which the underlying asset hits certain boundaries depending on the running values of the associated maximum and maximum drawdown processes. We obtain closed-form solutions to the equivalent free-boundary problems for the value functions with smooth fit at the optimal stopping boundaries and normal reflection at the edges of the state space of the resulting three-dimensional Markov process. The optimal exercise boundaries for the perpetual American options on the maximum of the market depth with fixed and floating strikes are determined as the minimal solutions of certain first-order nonlinear ordinary differential equations.

A Markov Chain Approximation for American Option Pricing in Tempered Stable-GARCH Models

This paper considers the American option pricing problem under the stochastic volatility models. In particular, we introduce the GARCH model with two heavy-tailed distributions: classical tempered stable (CTS) and normal tempered stable (NTS) distribution. Then we apply the Markov chain approach to compute the prices of American style options under these two models. Minimal entropy provides a convenient way to construct equivalent martingale measure (EMM) and allows us to overcome the difficulties in incorporating the Markov chain approximation. The convergence of the approximation is also proved. Both numerical and empirical results are analyzed to show the advantages and drawbacks of our approach.

Pricing American Options Under High-Dimensional Models with Recursive Adaptive Sparse Expectations

We introduce a novel numerical framework for pricing American options in high dimensions. Such settings naturally arise for derivatives with multiple underlying assets, like basket options. They are equally important for single-asset options because high-dimensional models are best capable of capturing observed price dynamics. Yet, higher-dimensional settings come at the cost of a loss of tractability due to the accompanying exponential growth of computational complexity. Our scheme manages to alleviate the problem of dimension scaling through the use of adaptive sparse grids. We approximate the value function with a low number of points and recursively apply fast approximations of the expectation operator from an exercise period to the previous one. The algorithm copes with discretely spaced, possibly nonuniform, time grids. This makes it particularly fast for options with a limited number of exercise periods, like Bermudan options, and options for which the optimal exercise schedule is known ex ante. As compared to Monte Carlo simulations, our scheme processes an entire cross section of options in a single execution. It thereby offers an immediate solution to the estimation of hedging coefficients through finite differences and is ideal when multiple related options need to be analyzed. The algorithm is also capable of dealing with discrete dividends with no performance deterioration, thus improving on the documented inefficiency of exercise policies under continuous dividend yield approximations. We benchmark our algorithm under both the canonical model of Black and Scholes and the stochastic volatility model of Heston in the presence of discrete dividends. We illustrate the massive improvement of complexity scaling over dense grids with a basket option study including up to eight underlying assets. We show how the high degree of parallelism of our scheme makes it suitable for deployment on massively parallel computing units to scale to higher dimensions or further speed up the solution process.

Numerical Methods for Pricing American Options with Time-Fractional PDE Models

In this paper we develop a Laplace transform method and a finite difference method for solving American option pricing problem when the change of the option price with time is considered as a fractal transmission system. In this scenario, the option price is governed by a time-fractional partial differential equation (PDE) with free boundary. The Laplace transform method is applied to the time-fractional PDE. It then leads to a nonlinear equation for the free boundary (i.e., optimal early exercise boundary) function in Laplace space. After numerically finding the solution of the nonlinear equation, the Laplace inversion is used to transform the approximate early exercise boundary into the time space. Finally the approximate price of the American option is obtained. A boundary-searching finite difference method is also proposed to solve the free-boundary time-fractional PDEs for pricing the American options. Numerical examples are carried out to compare the Laplace approach with the finite difference method and it is confirmed that the former approach is much faster than the latter one.

A numerical method to estimate the parameters of the CEV model implied by American option prices: Evidence from NYSE

We develop a highly efficient procedure to forecast the parameters of the constant elasticity of variance (CEV) model implied by American options. In particular, first of all, the American option prices predicted by the CEV model are calculated using an accurate and fast finite difference scheme. Then, the parameters of the CEV model are obtained by minimizing the distance between theoretical and empirical option prices, which yields an optimization problem that is solved using an ad-hoc numerical procedure. The proposed approach, which turns out to be very efficient from the computational standpoint, is used to test the goodness-of-fit of the CEV model in predicting the prices of American options traded on the NYSE. The results obtained reveal that the CEV model does not provide a very good agreement with real market data and yields only a marginal improvement over the more popular Black–Scholes model.

A Generalization of the Recursive Integration Method for the Analytic Valuation of American Options

This article provides a general accelerated recursive integration method for pricing American options based on stochastic volatility and double jump processes. Our proposed model is a generalization of the recursive integral representation method. American option prices can be evaluated by the sum of a corresponding European option price and an early exercise premium integral. Numerical results show that our proposed method is efficient and accuracy in pricing American options with stochastic volatility and double jump processes. © 2015 Wiley Periodicals, Inc. Jrl Fut Mark 36:887–901, 2016

A hybrid approach for the implementation of the Heston model

We propose a hybrid tree-finite difference method in order to approximate the Heston model. We prove the convergence by embedding the procedure in a bivariate Markov chain and we study the convergence of European and American option prices. We finally provide numerical experiments that give accurate option prices in the Heston model, showing the reliability and the efficiency of the algorithm.

Solving partial integro-differential option pricing problems for a wide class of infinite activity Lévy processes

In this paper, numerical analysis of finite difference schemes for partial integro-differential models related to European and American option pricing problems under a wide class of Lévy models is studied. Apart from computational and accuracy issues, qualitative properties such as positivity are treated. Consistency of the proposed numerical scheme and stability in the von Neumann sense are included. Gauss–Laguerre quadrature formula is used for the discretization of the integral part. Numerical examples illustrating the potential advantages of the presented results are included.

Monte Carlo Simulation with Machine Learning for Pricing American Options and Convertible Bonds

Li, Szepesvari and Schuurmans (2009) show that reinforcement learning (RL) algorithms are superior to the classical methods (such as Longstaff and Schwartz (2001)) in pricing American options using Monte Carlo simulation. We extend their techniques to the problem of pricing convertible bonds and show that RL outperforms LS on this task. Additionally, we propose a new method, based on the random forest algorithm from machine learning [Breiman (2001)], that can be used for pricing both American options and convertible bonds with Monte Carlo simulation. We show that this algorithm outperforms LS and is also superior to RL in most cases. We demonstrate how to use Monte Carlo simulation with the methods described above for pricing a complex convertible bond trading at the Tel Aviv stock exchange. Like many Israeli convertibles, this bond exhibits the "gradually diminishing principal" feature, meaning that instead of one payment of the principal at maturity, there are multiple principal payments during the lifetime of the bond. This feature presents a challenge to existing models. We also model other exotic features of this bond, such as path-dependent conversion ratio and exchange rate indexation. The prices that we obtain using this model are close to the market prices of the bond.

Adaptive finite differences and IMEX time-stepping to price options under Bates model

In this paper, we consider numerical pricing of European and American options under the Bates model, a model which gives rise to a partial-integro differential equation. This equation is discretized in space using adaptive finite differences while an IMEX scheme is employed in time. The sparse linear systems of equations in each time-step are solved using an LU-decomposition and an operator splitting technique is employed for the linear complementarity problems arising for American options. The integral part of the equation is treated explicitly in time which means that we have to perform matrix-vector multiplications each time-step with a matrix with dense blocks. These multiplications are accomplished through fast Fourier transforms. The great performance of the method is demonstrated through numerical experiments.

A meshless method for Asian style options pricing under the Merton jump-diffusion model

In this paper, we consider the partial integro-differential equation arising when a stock follows a Poisson distributed jump process, for the pricing of Asian options. We make use of the meshless radial basis functions with differential quadrature for approximating the spatial derivatives and demonstrate that the algorithm performs effectively well as compared to the commonly employed finite difference approximations. We also employ Strang splitting with the exponential time integration technique to improve temporal efficiency. Throughout the numerical experiments covered in the paper, we show how the proposed scheme can be efficiently employed for the pricing of American style Asian options under both the Black–Scholes and the Merton jump-diffusion models.

American options under stochastic volatility: control variates, maturity randomization & multiscale asymptotics

American options are actively traded worldwide on exchanges, thus making their accurate and efficient pricing an important problem. As most financial markets exhibit randomly varying volatility, in this paper we introduce an approximation of an American option price under stochastic volatility models. We achieve this by using the maturity randomization method known as Canadization. The volatility process is characterized by fast and slow-scale fluctuating factors. In particular, we study the case of an American put with a single underlying asset and use perturbative expansion techniques to approximate its price as well as the optimal exercise boundary up to the first order. We then use the approximate optimal exercise boundary formula to price an American put via Monte Carlo. We also develop efficient control variates for our simulation method using martingales resulting from the approximate price formula. A numerical study is conducted to demonstrate that the proposed method performs better than the least squares regression method popular in the financial industry, in typical settings where values of the scaling parameters are small. Further, it is empirically observed that in the regimes where the scaling parameter value is equal to unity, fast and slow-scale approximations are equally accurate.

Optimum Weighting for the Least Squares Monte Carlo Approach to American Options Under the CEV Model

In this paper we propose the optimum weighting scheme for pricing American options under a local volatility model. American options are priced under the constant elasticity of variance volatility model using Monte Carlo simulation. The residuals obtained from regression were heteroscedastic. For spot prices deep out-of-the-money, alternate weighting methods were found to provide improved accuracy over ordinary least squares. For spot prices deep in-the-money, the residuals were also heteroscedastic, however, the number of residuals present in the regression dominated and ordinary least squares provided improved accuracy. Generalised least squares was found to proved the most accurate overall weighting method.

A predictor–corrector approach for pricing American options under the finite moment log-stable model

This paper investigates the pricing of American options under the finite moment log-stable (FMLS) model. Under the FMLS model, the price of American-style options is governed by a highly nonlinear fractional partial differential equation (FPDE) system, which is much more complicated to solve than the corresponding Black–Scholes (B–S) system, with difficulties arising from the semi-globalness of the fractional operator, in conjunction with the nonlinearity associated with the early exercise nature of American-style options. Albeit difficult, in this paper, we propose a new predictor–corrector scheme based on the spectral-collocation method to solve for the prices of American options under the FMLS model. In the current approach, the nonlinearity of the pricing system is successfully dealt with using the predictor–corrector framework, whereas the non-localness of the fractional operator is elegantly handled. We have also provided an elegant error analysis for the current approach. Various numerical experiments suggest that the current method is fast and efficient, and can be easily extended to price American-style options under other fractional diffusion models. Based on the numerical results, we have also examined quantitatively the influence of the tail index on American put options.

A mean-reverting currency model in an uncertain environment

Uncertain finance has shown great significance in managing financial cases such as stock prices and currency options. Early researchers have put up some currency models to describe the foreign exchange rate. This paper proposes a mean-reverting currency model to describe the foreign change rate in the long term, and derives its European and American option pricing formulas. Besides, it gives some numerical examples to illustrate the formulas.

Optimal Stopping of a Hilbert Space Valued Diffusion: An Infinite Dimensional Variational Inequality

A finite horizon optimal stopping problem for an infinite dimensional diffusion $X$ is analyzed by means of variational techniques. The diffusion is driven by a SDE on a Hilbert space $\mathcal{H}$ with a non-linear diffusion coefficient $\sigma(X)$ and a generic unbounded operator $A$ in the drift term. When the gain function $\Theta$ is time-dependent and fulfils mild regularity assumptions, the value function $\mathcal{U}$ of the optimal stopping problem is shown to solve an infinite-dimensional, parabolic, degenerate variational inequality on an unbounded domain. Once the coefficient $\sigma(X)$ is specified, the solution of the variational problem is found in a suitable Banach space $\mathcal{V}$ fully characterized in terms of a Gaussian measure $\mu$. This work provides the infinite-dimensional counterpart, in the spirit of Bensoussan and Lions \cite{Ben-Lio82}, of well-known results on optimal stopping theory and variational inequalities in $\mathbb{R}^n$. These results may be useful in several fields, as in mathematical finance when pricing American options in the HJM model.

Pricing American options under the constant elasticity of variance model: An extension of the method by Barone-Adesi and Whaley

We consider the problem of pricing American options on an underlying described by the constant elasticity of variance (CEV) model. Such a problem does not have an exact closed-form solution, and therefore some kind of approximation is required. In this paper we extend the approach proposed by Barone-Adesi and Whaley (1997), which allows us to obtain a direct semi-analytical approximate solution. Numerical experiments are presented showing that the proposed method is satisfactorily accurate and computationally very fast.