American Option Pricing Research Articles

Sample Path Generation of the Stochastic Volatility CGMY Process and Its Application to Path-Dependent Option Pricing

This paper proposes the sample path generation method for the stochastic volatility version of the CGMY process. We present the Monte-Carlo method for European and American option pricing with the sample path generation and calibrate model parameters to the American style S&P 100 index options market, using the least square regression method. Moreover, we discuss path-dependent options, such as Asian and Barrier options.

A fractional reduced differential transform method for solving time fractional Black Scholes American option pricing equation

In this paper, fractional reduced differential transform method (FRDTM) is operated to solve time fractional Black-Scholes American option pricing equation paying no dividends.The Black-Scholes model plays a significant role in the evaluation of European or American call and put options. The advantage of the proposed method to other existing methods is that it finds the solution without discretization or transformation. While using this method, no recommended assumptions are needed and hence the computational work reduces to a greater extent. Numerical experiments prove that the proposed method is efficient and valid for obtaining the solution of time fractional Black-Scholes equation governing American options. This method proves to be powerful for solving general fractional order partial differential equations (PDEs) existing in the field of Science, Engineering and other related fields.

A Partition of unity finite element method for valuation American option under Black-Scholes model

Abstract In this paper, we present an intelligent combination of partition of unity (PU) and finite element (FE) methods for valuing American option pricing problems governed by the Black-Scholes (BS) model. The model is based on a partial differential equation (PDE) from which one can deduce the Black-Scholes formula, which gives a theoretical estimated value of options using current stock prices, expected dividends, the option’s strike price, expected interest rates, time to expiration and expected volatility. Although the finite element method (FEM) seems to be an alternative tool for pricing options with a few applications reported in the literature, this combination called the Partition of Unity Finite Element Method (PUFEM) appears to offer many of the desired properties. The main advantage of the proposed approach is its ability to locally refine the solution by adapting an incorporated specific class of enrichment in the finite element space instead of generating a new fine mesh for the problem under study. Numerical computations are carried out to show a huge reduction in the number of degrees of freedom required to achieve a fixed accuracy which confirms that the PUFE method used is very efficient and gives better accuracy than the conventional FE method.

Efficient pricing and hedging of high-dimensional American options using deep recurrent networks

We propose a deep recurrent neural network (RNN) framework for computing prices and deltas of American options in high dimensions. Our proposed framework uses two deep RNNs, where one network learns the continuation price and the other learns the delta for each timestep. Our proposed framework yields prices and deltas for the entire spacetime, not only at a given point (e.g. t = 0). The computational cost of the proposed approach is linear in N, which improves on the quadratic time seen for feedforward networks that price American options. The computational memory cost of our method is constant in N, which is an improvement over the linear memory costs seen in feedforward networks. Our numerical simulations demonstrate these contributions and show that the proposed deep RNN framework is computationally more efficient than traditional feedforward neural network frameworks in time and memory.

Fast pricing of American options under variance gamma

We investigate methods for pricing American options under the variance gamma model. The variance gamma process is a pure jump process which is constructed by replacing the calendar time by the gamma time in a Brownian motion with drift, which makes it a time-changed Brownian motion. In general, the finite difference method and the simulation method can be used for pricing under this model, but their speed is not satisfactory. So there is a need for fast but accurate approximation methods. In the case of Black-Merton-Scholes model, there are fast approximation methods, but they cannot be utilized for the variance gamma model. We develop a new fast method inspired by the quadratic approximation method, while reducing the error by making use of a machine learning technique on pre-calculated quantities. We compare the performance of our proposed method with those of the existing methods and show that this method is efficient and accurate for practical use.

Continuation value computation using Malliavin calculus under general volatility stochastic process for American option pricing

American options represent an important financial instrument but are notoriously difficult to price, especially when the volatility is not constant. We explore the conditions required to apply Malliavin calculus to price American options when the volatility follows a general stochastic differential process, and develop the expressions to compute the continuation value at any time before the expiration date, given the current asset price and volatility. The developed methodology can then be applied to price American options.

Revisiting the Risk-Neutral Approach to Optimal Policyholder Behavior: A Study of Withdrawal Guarantees in Variable Annuities

Policyholder exercise behavior presents an important risk factor for pricing Variable Annuities. However, approaches presented in the literature—building on value-maximizing strategies akin to pricing American options—do not square with observed price and exercise patterns for popular withdrawal guarantees. We show that including taxes into the valuation closes this gap between theory and practice. In particular, we develop a subjective risk-neutral valuation methodology that takes into consideration differences in the tax structure between investment opportunities. We demonstrate that accounting for tax advantages significantly affects the value of the guarantees and produces results that are in line with empirical patterns.

An efficient numerical method for pricing American put options under the CEV model

The constant elasticity of variance (CEV) model is a practical approach to option pricing by fitting to the implied volatility smile. However, pricing American options is computationally intensive because no analytical formulas are available. In this paper, we present numerical methods to find the optimal exercise boundary with respect to an American put option under the CEV model. This problem corresponds to the free boundary (also called an optimal exercise boundary) problem for a partial differential equation. We use a transformed function that has Lipschitz character near the optimal exercise boundary to determine the optimal exercise boundary. After determining the optimal exercise boundary, we calculate American put option values under the CEV model using the finite-difference method. Finally, we use the proposed numerical method to obtain several results that are then compared with the results of other methods.

American option pricing: Optimal Lattice models and multidimensional efficiency tests

AbstractWe introduce a set of lattice techniques to the Leisen‐Reimer and Tian binomial models with a view to accelerating computation time and improving accuracy of American Option valuation. A level of accuracy and efficiency combined can be achieved that surpass commonly used analytical analogues. We compare these efficient lattice models with analytical formulae for pricing different groups of options according to the deepness of American quality and moneyness. Our results reveal that counter to received wisdom, lattices constructs produce greater speed and accuracy for all option categories relative to the best performing closed form American analogues.

Option Pricing Formulas in a New Uncertain Mean-Reverting Stock Model with Floating Interest Rate

Options play a very important role in the financial market, and option pricing has become one of the focus issues discussed by the scholars. This paper proposes a new uncertain mean-reverting stock model with floating interest rate, where the interest rate is assumed to be the uncertain Cox-Ingersoll-Ross (CIR) model. The European option and American option pricing formulas are derived via the α -path method. In addition, some mathematical properties of the uncertain option pricing formulas are discussed. Subsequently, several numerical examples are given to illustrate the effectiveness of the proposed model.

Haar‐wavelet based approximation for pricing American options under linear complementarity formulations

AbstractIn this manuscript, we present a novel and highly accurate wavelet‐based approximation technique to explore the sensitivities and value of American options diagnosed by linear complementarity problems. For a detailed analysis of such financially relevant problems, we transform the actual final value problem into a dimensionless initial value problem. To avoid the unacceptable large truncation error, the unbounded domain is trimmed into a bounded domain. A remarkable observation is that to investigate the various physical and numerical aspects of the options' sensitivities; the proposed scheme is efficient as it explicitly provides the numerical approximation of all the derivatives of the solution function. The multiresolution technique of the wavelets and the convergence of the proposed wavelet scheme are comprehensively analyzed. The wavelet analysis is accompanied by illustrative examples to demonstrate the proficiency and robustness of the present method coupled with graphical representations. It has been shown that the present method is efficient to solve free boundary problems. It is worthy to note that the highly accurate and promising computational results are enough to confirm the performance of the proposed method. The simulated results of options' Greeks analyzed and discussed have vast applications in different financial institutes and trading markets.

Optimal Retirement Problem Under Partial Information

The optimal retirement decision is an optimal stopping problem when retirement is irreversible. We investigate the optimal consumption, investment, and retirement decisions when the mean return of a risky asset is unobservable and is estimated by filtering from historical prices. To ensure non-negativity of the consumption rate and the borrowing constraints on the wealth process of the representative agent, we conduct our analysis using a duality approach. We link the dual problem to American option pricing with stochastic volatility and prove that the duality gap is closed. We then apply our theory to a hidden Markov model for regime-switching mean return with Bayesian learning or the Wonham filter. We fully characterize the existence and uniqueness of variational inequality in the dual optimal stopping problem, as well as the free boundary of the problem. An asymptotic closed-form solution is derived for optimal retirement timing by small-scale perturbation. We discuss the potential applications of the results to other partial information settings.

LAPLACE BOUNDS APPROXIMATION FOR AMERICAN OPTIONS

In this paper, we develop the lower–upper-bound approximation in the space of Laplace transforms for pricing American options. We construct tight lower and upper bounds for the price of a finite-maturity American option when the underlying stock is modeled by a large class of stochastic processes, e.g. a time-homogeneous diffusion process and a jump diffusion process. The novelty of the method is to first take the Laplace transform of the price of the corresponding “capped (barrier) option” with respect to the time to maturity, and then carry out optimization procedures in the Laplace space. Finally, we numerically invert the Laplace transforms to obtain the lower bound of the price of the American option and further utilize the early exercise premium representation in the Laplace space to obtain the upper bound. Numerical examples are conducted to compare the method with a variety of existing methods in the literature as benchmark to demonstrate the accuracy and efficiency.

On a Free Boundary Problem for American Options Under the Generalized Black–Scholes Model

We consider the problem of pricing American options using the generalized Black–Scholes model. The generalized Black–Scholes model is a modified form of the standard Black–Scholes model with the effect of interest and consumption rates. In general, because the American option problem does not have an exact closed-form solution, some type of approximation is required. A simple numerical method for pricing American put options under the generalized Black–Scholes model is presented. The proposed method corresponds to a free boundary (also called an optimal exercise boundary) problem for a partial differential equation. We use a transformed function that has Lipschitz character near the optimal exercise boundary to determine the optimal exercise boundary. Numerical results indicating the performance of the proposed method are examined. Several numerical results are also presented that illustrate a comparison between our proposed method and others.

A front-fixing ETD numerical method for solving jump–diffusion American option pricing problems

American options prices under jump–diffusion models are determined by a free boundary partial integro-differential equation (PIDE) problem. In this paper, we propose a front-fixing exponential time differencing (FF-ETD) method composed of several steps. First, the free boundary is included into equation by applying the front-fixing transformation. Second, the resulting nonlinear PIDE is semi-discretized, that leads to a system of ordinary differential equations (ODEs). Third, a numerical solution of the system is constructed by using exponential time differencing (ETD) method and matrix quadrature rules. Finally, numerical analysis is provided to establish empirical stability conditions on step sizes. Numerical results show the efficiency and competitiveness of the FF-ETD method.

Thanksgiving Meal for Americans

Leif Andersen and Mark Lake took over the July 2019 issue with their high-precision Heston-based pricing method, and, in my humble opinion, their paper on high-performance American option pricing is just as important.

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.

Pricing European and American options under Heston model using discontinuous Galerkin finite elements

This paper deals with pricing of European and American options, when the underlying asset price follows Heston model, via the interior penalty discontinuous Galerkin finite element method (dGFEM). The advantages of dGFEM space discretization with Rannacher smoothing as time integrator with nonsmooth initial and boundary conditions are illustrated for European vanilla options, digital call and American put options. The convection dominated Heston model for vanishing volatility is efficiently solved utilizing the adaptive dGFEM. For fast solution of the linear complementary problem of the American options, a projected successive over relaxation (PSOR) method is developed with the norm preconditioned dGFEM. We show the efficiency and accuracy of dGFEM for option pricing by conducting comparison analysis with other methods and numerical experiments.

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.

Nonparametric predictive inference for American option pricing based on the binomial tree model

In this article, we present the American option pricing procedure based on the binomial tree from an imprecise statistical aspect. Nonparametric Predictive Inference (NPI) is implemented to infer imprecise probabilities of underlying asset movements, reflecting uncertainty while learning from data, which is superior to the constant risk-neutral probability. In a recent article, we applied the NPI method to the European option pricing procedure that gives good results when the investor has non-perfect information. We now investigate the NPI method for American option pricing, of which imprecise probabilities are considered and updated for every one-time-step path. Different from the classic models, this method is shown that it may be optimal to early exercise an American non-dividend call option because our method considers all information that occurs in the future steps. We also study the performance of the NPI pricing method for American options via simulations in two different scenarios compared to the Cox, Ross and Rubinstein binomial tree model (CRR), first where the CRR assumptions are right, and second where the CRR model uses wrong assumptions. Through the performance study, we conclude that the investor using the NPI method tends to achieve good results in the second scenario.