American-style Options Research Articles

A note on the numerical approximation of Greeks for American-style options

In this note, we consider the approximation of the Greeks Delta and Gamma of American-style options through the numerical solution of time-dependent partial differential complementarity problems (PDCPs). This approach is very attractive as it can yield accurate approximations to these Greeks at essentially no additional computational cost during the numerical solution of the PDCP for the pertinent option value function. For the temporal discretization, the Crank–Nicolson method is arguably the most popular method in computational finance. It is well-known, however, that this method can have an undesirable convergence behaviour in the approximation of the Greeks Delta and Gamma for American-style options, even when backward Euler damping (Rannacher smoothing) is employed.In this note, for the temporal discretization of the PDCP, we study an interesting family of diagonally implicit Runge–Kutta (DIRK) methods together with the two-stage Lobatto IIIC method. Through ample numerical experiments for one- and two-asset American-style options, it is shown that these methods can yield a regular second-order convergence behaviour for the option value as well as for the Greeks Delta and Gamma. A mutual comparison reveals that the DIRK method with suitably chosen parameter θ is preferable.

A deep learning method for pricing high-dimensional American-style options via state-space partition

A deep learning method for pricing high-dimensional American-style options via state-space partition

On the American style futures contracts

There is a large number of sources devoted to the American style options. On the other hand, the American futures contracts are understudied in the scientific literature. This motivated us to examine these instruments in comparison to the relevant options. Their optimal boundaries are obtained and a finite difference scheme is applied to the pricing problem. We consider separately the long and short positions.

An integral equation approach for pricing American put options under regime-switching model

Regime-switching models have been heavily studied recently, as they have some clear advantages of over other non-constant volatility model to resolve the so-called smirk effect displayed when constant volatility models are used to price financial derivatives such as options. However, due to the increased model complexity, the associated computational effort usually increases as well, particularly when they are used to price American-style options. In this paper, a novel computational approach based on integral equations is presented. A distinctive feature of our approach, in comparison with other numerical approaches, is that the coupled partial differential equations (PDEs) in a PDE system have been decoupled in the Fourier space, resulting in a completely decoupled integral equation for each economical states, and thus has greatly reduced computational effort. Some examples with preliminary results for a two-state regime-switching model are used to demonstrate our approach.

The empirical performance of option implied volatility surface-driven optimal portfolios

We apply a two-step strategy to forecast the dynamics of the volatility surface implicit in option prices to all American-style options written on the stocks that have entered the Dow Jones Industrial Average Index between 2004 and 2016. We explore whether the implied volatilities extracted through the two-step approach help improve the out-of-sample performance of minimum-variance portfolios. We find that, by using option-implied volatilities in estimating the covariance matrix, the ex-post volatility of the minimum-variance portfolio is lower when compared with the equal-weighted portfolio and a minimum-variance portfolio simply derived from the historical, sample covariance matrix estimator. Moreover, over most of our 13-year sample, the realized Sharpe, Sortino and information ratios increase when the sample covariance matrix estimator is replaced with its implied counterpart. However, the benefits of using option-implied information are countered by an increase in portfolio turnover that may imply higher (implicit) transaction costs. We also apply shrinkage methods to both the sample covariance estimator and the implied covariance estimator and note that they often lead to significant improvements in portfolio performance.

Second-Hand House Valuation Based on Binomial Option Pricing Model

More and more families choose to buy the second-hand houses as their first house. Due to a high demand in second-hand housing market, there is a continuously rising trend in the price of second-hand houses. This article mainly focus on the valuation of second-hand housing in Chinese market, based on the binomial option pricing model. Besides valuing the second-hand houses as the American-style options, the time value is another important element in the pricing model. Based on the defined pricing model, the result of empirical analysis is that the estimation of current housing price has a difference from the actual transaction prices to some extent. It can be concluded that the current price of second-hand houses in Chinese market is far from reflecting the time value, and the time value of the second-hand houses is unable to be ignored in the future. It is hoped that there will be a more precise way to measure the value of time in the second-hand housing market specifically.

Simulated Greeks for American options

This paper develops a method to estimate price sensitivities, so-called Greeks, for American style options using flexible simulation methods combined with initially dispersed state variables. The asymptotic properties of the estimators are studied, and convergence of the method is established. A 2-stage method is proposed with an adaptive choice of optimal dispersion of state variables, which controls and balances off the bias of the estimates against their variance. Numerical results show that the method compares exceptionally well to existing alternatives, works well for very reasonable choices of dispersion sizes, regressors, and simulated paths, and it is robust to choices of these parameters. We apply the method to models with time varying volatility, demonstrating that there are large differences between estimated Greeks with affine and non-affine models, that Greeks vary significantly through periods of crisis, and that the errors made when using Greeks implied from, e.g. misspecified models with constant volatility can be extremely large.

Penalty method for pricing American-style Asian option with jumps diffusion process

American-style options are important derivative contracts in today's worldwide financial markets. They trade large volumes on various underlying assets, including stocks, indices, foreign exchange rates, and futures. In this work, a penalty approach is derived and examined for use in pricing the American style of Asian option under the Merton model. The Black–Scholes equation incorporates a small non-linear penalty factor. In this approach, the free and moving boundary imposed by the contract's early exercise feature is removed in order to create a stable solution domain. By including Jump-diffusion in the models, they are able to capture the skewness and kurtosis features of return distributions often observed in several assets in the market. The performance of the schemes is investigated through a series of numerical experiments.

Implied volatility surface construction for commodity futures options traded in China

European futures options are not traded on the Chinese exchanges and that generates difficulties to calibrate fundamental market parameters, such as the implied volatilities. We propose an efficient willow tree method to resolve the problem of calibrating the implied volatility from American-style options. The proposed willow tree construction is independent of the volatility itself so as to minimize the cost of the calibration. We also apply the proposed method to calibrate the implied volatilities of most frequently traded options in the Chinese market, sugar and soybean meal, based on the daily closing prices, and construct the corresponding implied volatility surfaces. The results indicate the seasonality in the volatility of commodity spot prices and futures prices in China. Moreover, based on the implied volatility distortion close to the option maturity observed in our empirical results, we suggest a minimum tick price scheme to avoid the distortion and decrease of hedging costs.

Pricing American Options by a Fourier Transform Multinomial Tree in a Conic Market

Based on FFT, a high‐order multinomial tree is constructed, and the method to obtain the price of American style options in the Lévy conic market is studied. Firstly, the nature of the Lévy process and the pricing principle of European‐style options are introduced. Secondly, the method to construct a high‐order multinomial tree based on Fourier transform is presented. It can be proved by theoretical derivation that the multinomial tree can converge to the Lévy process. Thirdly, we introduce the conic market theory based on the concave distortion function and give the discretization method of the concave distortion expectation. Then, the American option pricing method based on reverse iteration is given. Finally, the CGMY process is used to demonstrate how to price the American put option in the Lévy conic market. We can draw conclusions that the Fourier transform multinomial tree can avoid the difficulty of parameter estimation when using traditional moment matching methods to construct multinomial trees. Because the Lévy process has the analytic form characteristic function, this method is a promising method to calculate the prices of options in the Lévy conic market.

ИСПОЛЬЗОВАНИЕ БИНОМИАЛЬНОЙ МОДЕЛИ ПРИ РАСЧЕТЕ ОПТИМАЛЬНЫХ СУБСИДИЙ ДЛЯ КИТАЙСКИХ ИНВЕСТИЦИЙ В ВОЗОБНОВЛЯЕМЫЕ ИСТОЧНИКИ ЭНЕРГИИ

This article focuses on the use of a binomial model to calculate the required subsidy, which would encourage investors to optimally immediately use the Americanstyle option laid down at the launch stage of Chinese renewable energy investment projects. In addition, this article also aims to contrast the binomial model with the more timeconsuming Monte Carlo simulation previously used to estimate the appropriate subsidy. Using the same data, but using a different method, and reducing the number of uncertain factors to one, it is assumed that these two methods have similar results, but the binomial method requires significantly less calculations and is more understandable.

Option pricing under a discrete-time Markov switching stochastic volatility with co-jump model

<p style='text-indent:20px;'>We consider option pricing using a discrete-time Markov switching stochastic volatility with co-jump model, which can capture asset price features such as leptokurtosis, skewness, volatility clustering, and varying mean-reversion speed of volatility. For pricing European options, we develop a computationally efficient method for obtaining the probability distribution of average integrated variance (AIV), which is key to option pricing under stochastic-volatility-type models. Building upon the efficiency of the European option pricing approach, we are able to price an American-style option, by converting its pricing into the pricing of a portfolio of European options. Our work also provides constructive guidance for analyzing derivatives based on variance, e.g., the variance swap. Numerical results indicate our methods can be implemented very efficiently and accurately.</p>

JDOI variance reduction method and the pricing of American-style options

This article revisits the Diffusion Operator Integral (DOI) variance reduction technique originally proposed in Heath and Platen (2002) and extends its theoretical concept to the pricing of American-style options under (time-homogeneous) Lévy stochastic differential equations. The resulting Jump Diffusion Operator Integral (JDOI) method can be combined with numerous Monte Carlo-based stopping-time algorithms, including the ubiquitous least-squares Monte Carlo (LSMC) algorithm of Longstaff and Schwartz (cf. Carriere (1996) and Longstaff and Schwartz (2001)). We exemplify the usefulness of our theoretical derivations under a concrete, though very general jump-diffusion stochastic volatility dynamics and test the resulting LSMC-based version of the JDOI method. The results provide evidence of a strong variance reduction when compared with a simple application of the LSMC algorithm and proves that applying our technique on top of Monte Carlo-based pricing schemes provides a powerful way to speed-up these methods.

Predicting Arbitrage-free American Option Prices Using Artificial Neural Network with Pseudo Inputs

Machine learning models, which have recently been applied to evaluate financial variables, have a major difficulty to accomplish arbitrage-free valuation. We propose an American style option pricing method using multilayer artificial neural networks with arbitrage-free pseudo inputs. The proposed neural network model was trained with samples composed of market data and pseudo grid points generated by the calibrated parametric models. The trained model found arbitrage-free price or nearest price for each strike price and expiration date. We compared the proposed model with a conventional multilayer neural network model in terms of model prediction using S&P 100 American put options from 2012. The proposed model achieved better prediction performance than the conventional neural network model. In addition, prices obtained from the proposed method were much closer to the arbitrage-free prices from the parametric model.

Double barrier American put option pricing under uncertain volatility model

This paper aims to study the asymptotic behavior of double barrier American-style put option prices under an uncertain volatility model, which degenerates to a single point. We give an approximation of the double barrier American-style option prices with a small volatility interval, expressed by the Black–Scholes–Barenblatt equation. Then, we propose a novel representation for the early exercise boundary of American-style double barrier options in terms of the optimal stopping boundary of a single barrier contract.

A Study on Optimal Multiple Stopping and Swing Options Pricing

In this paper, we have studied the optimal stopping of random process as well as the costing of Swing options, specially the valuation of electricity market which is considered to an American style option having multiple practicing rights. Since this type of options are widely used in investing, so it requires some methods for valuation and that should be as precise as possible. So, we discuss two numerical methods for getting swing options prices in the field of electricity market, namely Monte Carlo and Finite difference. Finally, we compare our obtained results numerically and graphically with the help of MATLAB. GANIT J. Bangladesh Math. Soc. 40.2 (2020) 145-155

JDOI Variance Reduction Method and the Pricing of American-Style Options

The present article revisits the Diffusion Operator Integral (DOI) variance reduction technique originally proposed in [HP02] and extends its theoretical concept to the pricing of American-style options under (time-homogeneous) Levy stochastic differential equations. The resulting Jump Diffusion Operator Integral (JDOI) method can be combined with numerous Monte Carlo based stopping-time algorithms, including the ubiquitous least-squares Monte Carlo (LSMC) algorithm of Longstaff and Schwartz (cf. [Car96], [LS01]). We exemplify the usefulness of our theoretical derivations under a concrete, though very general jump-diffusion stochastic volatility dynamics and test the resulting LSMC based version of the JDOI method. The results provide evidence of a strong variance reduction when compared with a simple application of the LSMC algorithm and proves that applying our technique on top of Monte Carlo based pricing schemes provides a powerful way to speed-up these methods.

FFT-network for bivariate Lévy option pricing

We propose a two-dimensional fast Fourier transform (FFT) network to retrieve the prices of options that depend on two Levy processes. Applications include, but are not limited to, the valuation of options on two stocks under the Levy processes, and options on a single stock under a random time-change Levy process. The proposed numerical scheme can be applied to different multivariate Levy constructions such as subordination and linear combination provided that the joint characteristic function is available. The proposed FFT-network can be thought of as a lattice approach implemented through the characteristic function. With the prevalent implementation of FFT, the network approach results in significant computational time reduction while maintaining satisfactory accuracy. Furthermore, we investigate option pricing on a single asset where the asset return and its volatility are driven by a pair of dependent Levy processes. Such a model is also called the random time-changed Levy process. Numerical examples are given to demonstrate the efficiency and accuracy of FFT-network applied to exotic and American-style options.

Pricing and Hedging American-Style Options with Deep Learning

In this paper we introduce a deep learning method for pricing and hedging American-style options. It first computes a candidate optimal stopping policy. From there it derives a lower bound for the price. Then it calculates an upper bound, a point estimate and confidence intervals. Finally, it constructs an approximate dynamic hedging strategy. We test the approach on different specifications of a Bermudan max-call option. In all cases it produces highly accurate prices and dynamic hedging strategies with small replication errors.

A numerical study of the utility-indifference approach for pricing American options

Utility-indifference approach is a useful approach to be adopted for pricing financial derivatives in an incomplete market and is an ongoing hot research topic in quantitative finance. One interesting question associated with this approach is whether or not it renders to the same option prices, degenerately, when the market becomes infinitesimally close to a complete market. The answer for such a question has been provided for European-style options as there is a well-documented theoretical proof in Davis et al. (1993). However, a theoretical proof for the case of pricing American-style options is unavailable at this stage and the answer for this question must be at least numerically confirmed before it can be comfortably used to price American-style options in incomplete markets. The contribution of this paper is to provide such a numerical verification.