American Put Option Pricing Research Articles

Optimal approximations for the free boundary problems of the space-time fractional Black-Scholes equations using a combined physics-informed neural network

The combined physics-informed neural network is employed to deal with the free boundary problems of fractional Black-Scholes equations. The solution assumption and the loss function are determined, the transfer learning is borrowed, the combined neural network with data enhancement layer is designed, then the classical Black-Scholes model is numerically solved and the comparative analysis of numerical results under different neural networks is made. For further insight into the long-term memory of fluctuation, the free boundary problems of the space-time Black-Scholes equations under Caputo, Caputo-Fabrizio and Atangana-Baleanu-Caputo fractional derivatives are studied. The corresponding empirical analyses are presented and the optimal exercise boundaries of American put option are simulated. The market analysis shows that introducing fractional calculus tools and neural network algorithms into American put option pricing can yield more realistic prediction results. The work provides a viable method for subsequent researchers to study American option pricing using fractional calculus and neural networks combined with true market data and to deal with the free boundary problems in other research fields.

Lattice Boltzmann method for the linear complementarity problem arising from American option pricing

In this paper, a lattice Boltzmann method is proposed for solving the linear complementarity problem (LCP) arising in single and multi-asset American put option pricing. The LCP for American option is a variable coefficient parabolic model defined on an unbounded domain. Initially, using the far field estimate method and the penalty method respectively, the LCP could be reformulated into a nonlinear parabolic partial differential equation on a bounded domain. To construct a unified lattice Boltzmann model for the option pricing problems, the above transformation equations are rewritten into an equivalent divergence form. Then, through the incorporation of an amending function into the evolution equation, which assists in recovering the source term and eliminating the error term, the lattice Boltzmann model with spatial second-order accuracy is constructed. Finally, the present model is validated using numerical simulations, and the numerical results agree well with the option values obtained by existing methods, which indicates that the present lattice Boltzmann model is efficient for solving the American put option pricing problem.

Two-factor Heston model equipped with regime-switching: American option pricing and model calibration by Levenberg–Marquardt optimization algorithm

In this paper, we consider the pricing of American options under a regime-switching double Heston model, such that the interest rate and mean-reversion level parameters in both stochastic volatility models shift in various states. We develop a semi-analytical formula for double Heston partial differential equation by using the equivalent European put option price and standard portfolio-consumption model. Then through the moment-generating function of this particular model, the American put option price is evaluated. We employ the Levenberg–Marquardt optimization method to calibrate the regime-switching double Heston model. Numerical experiments have also been performed to demonstrate the accuracy of the proposed formula and the performance of regime change mechanism on option pricing. Ultimately, through an experimental application, we indicate the proposed model is premier to the double Heston model, which illustrates the importance of considering regime-switching factor.

The Correction of Multiscale Stochastic Volatility to American Put Option: An Asymptotic Approximation and Finite Difference Approach

It has been found that the surface of implied volatility has appeared in financial market embrace volatility “Smile” and volatility “Smirk” through the long-term observation. Compared to the conventional Black-Scholes option pricing models, it has been proved to provide more accurate results by stochastic volatility model in terms of the implied volatility, while the classic stochastic volatility model fails to capture the term structure phenomenon of volatility “Smirk.” More attempts have been made to correct for American put option price with incorporating a fast-scale stochastic volatility and a slow-scale stochastic volatility in this paper. Given that the combination in the process of multiscale volatility may lead to a high-dimensional differential equation, an asymptotic approximation method is employed to reduce the dimension in this paper. The numerical results of finite difference show that the multiscale volatility model can offer accurate explanations of the behavior of American put option price.

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.

Calibration of the double Heston model and an analytical formula in pricing American put option

This paper proposes a novel approach to pricing of American put option under double Heston model. We develop an analytical solution to the double Heston partial differential equation (double Heston PDE) using the equivalent European put option price and standard portfolio-consumption model. Then through the affine form of the model and Fourier transform method, we evaluate the American put option price. Finally, we calibrate the option prices resulting from double Heston model to a set of observed index options by employing Genetic optimization algorithm. A detailed numerical study illustrates the efficiency of the proposed method.

A simple method for extracting the probability of default from American put option prices

AbstractWe present a novel method for extracting the risk‐neutral probability of default (PD) of a firm from American put option prices. Building on the idea of a default corridor proposed by Carr and Wu, we derive a parsimonious closed‐form formula for American put option prices from which the PD can be inferred. The method is easy to implement. Our empirical results based on seven large US firms for the period 2002–2010 show that, in some cases, our option‐implied PD can provide a more accurate estimate of default probability than the estimates implied from credit default swaps.

A Simple Method for Extracting the Probability of Default from American Put Option Prices

In this paper, we present a novel method to extract the risk-neutral probability of default of a firm from American put option prices. Building on the idea of a default corridor proposed in Carr and Wu (2011), we derive a parsimonious closed-form formula for American put option prices from which the probability of default can be inferred. The proposed method is easy to implement and helps overcome the main limitation of the method used in Carr and Wu (2011), which relies on the price of one deep-out-of-the-money put option. Our empirical results are based on seven large U.S. firms for the period 2002 to 2010. These results show that, in some cases, the option-implied probability of default can provide a more accurate estimate of default probability, compared to the estimates implied from credit default swap spreads.

Numerical solution of fractional Black‐Scholes model of American put option pricing via a nonstandard finite difference method: Stability and convergent analysis

In this paper, a free boundary fractional Black‐Scholes (FBS) model of American put option pricing is investigated. To convert the free boundary FBS model to a model with known boundary condition, the quasi‐stationary method is applied, which leads to solvability of the American put option problem. Then, a nonstandard finite difference method and Grünwald‐Letnikov approximation are respectively used to approximate the derivatives with respect to stock price and time fractional derivative to get a fractional nonstandard finite difference problem. It is then shown that the proposed method is stable and convergent. The uniqueness of the approximate solution is also proved for the proposed method. Our numerical results indicate that the general physical conditions of American put option valuation formulas under the FBS model are satisfied.

Extracting Option-Implied Probability of Default: A Novel Method

In this paper, we present a novel method to extract the risk-neutral probability of default from American put option prices. Under the assumptions of Carr and Wu (2011), we derive a closed form expression for American put options from which the probability of default can be inferred. Our empirical results based on four large institutions in the US during the 2007-2009 crisis show that directly applying the methodology of Carr and Wu (2011) can overestimate the probability of default. Our model also allows us to test the effect of positive equity recovery on estimated default probabilities.

Existence and Uniqueness Solution Under Non-Lipschiz Condition of the Mixed Fractional Heston's Model

This paper focuses on a mixed fractional version of Heston model in which the volatility Brownian and price Brownian are replaced by mixed fractional Brownian motion with the Hurst parameter $H\in(\frac{3}{4},1)$ so that the model exhibits the long range dependence. The existence and uniqueness of solution of mixed fractional Heston model is established under various non-Lipschitz condition and a related Euler discretization method is discussed. An example on the American put option price using Least Squares Monte Carlo Algorithm to produce acceptable results under the mixed fractional Heston model is presented to illustrate the applicability of the theory. The numerical result obtained proves the performanceof our results.

Robust upper bounds for American put options

AbstractIn this paper, we develop robust and model‐free upper bounds for American put option prices. Our bounds have all of those appealing features of the upper bounds for European options provided in DeMarzo et al. (2016, Robust option pricing: Hannan and Blackwell meet Black and Scholes, Journal of Economic Theory, 410‐434) but cover more popular derivatives in practice. Numerical and empirical investigations illustrate the performance of our method.

A Stable and Convergent Finite Difference Method for Fractional Black–Scholes Model of American Put Option Pricing

We introduce the mathematical modeling of American put option under the fractional Black–Scholes model, which leads to a free boundary problem. Then the free boundary (optimal exercise boundary) that is unknown, is found by the quasi-stationary method that cause American put option problem to be solvable. In continuation we use a finite difference method for derivatives with respect to stock price, Grunwal Letnikov approximation for derivatives with respect to time and reach a fractional finite difference problem. We show that the set up fractional finite difference problem is stable and convergent. We also show that the numerical results satisfy the physical conditions of American put option pricing under the FBS model.

Mixed fractional Heston model and the pricing of American options

This paper presents a fractional version of the Heston model in which the volatility Brownian and price Brownian are replaced by mixed fractional Brownian motions with Hurst parameter H ∈ ( 3 4 , 1 ) so that the model exhibits a long range dependence. Then the existence and uniqueness of solution of mixed fractional Heston model are discussed as well as the error of an Euler scheme applied on this model. Finally, some numerical illustrations are given in the last section by computing American put option prices.

American Put Option Pricing Using a Hybrid Evolutionary Computation and Monte-Carlo Simulation Method

American put option pricing is a challenging, complex problem, and existing methods to address this problem are computationally intensive. In this paper, a self-adaptive evolutionary computation method is used for computing American put option price. The proposed method essentially transforms a discrete time exercisable American option to a continuous time exercisable option. The performance of the proposed method is compared with that of plain European Monte Carlo and Binomial Lattice option values. Further, in pricing American options this method exhibited better results with considerable improvements over that of conventional Monte-Carlo simulation method. It is argued that the proposed method effectively computes the upper bound on the American put options.

Computing American option price under regime switching with rationality parameter

American put option pricing under regime switching is modelled by a system of coupled partial differential equations. The proposed model combines better the reality of the market by incorporating the regime switching jointly with the emotional behaviour of traders using the rationality parameter approach recently introduced by Tågholt Gad and Lund Petersen to cope with possible irrational exercise policy. The classical rational exercise is recovered as a limit case of the rational parameter. The resulting nonlinear system of PDEs is solved by a weighted finite difference method, also known as θ-method. In order to avoid the need of an iterative method for nonlinear system, the term with rationality parameter and the coupling term are treated explicitly. Next, the resulting linear system is solved by Thomas algorithm. Stability conditions for the numerical scheme are studied by using von Neumann approach. Numerical examples illustrate the efficiency and accuracy of the proposed method.

Binomial Bias in Pricing and Early Exercising American Put Options

The binomial algorithm (Cox and Rubinstein, 1985) is an accepted theoretically justifiable standard for measuring the accuracy of American put option pricing algorithms. An important question is whether it also generates accurate estimates of the early exercise boundary (Lamberton, 1993). I show that, in addition to the non-linearity and distribution errors recognised in the literature, the algorithm systematically misprices the early exercise boundary.While convergence to the true boundary ultimately occurs, the convergence rate is slow. An analytic integral equation (Carr et al., 1992) is solved sequentially for the true boundary and is a better benchmark for American put option pricing algorithms.

A new efficient numerical method for solving American option under regime switching model

A system of coupled free boundary problems describing American put option pricing under regime switching is considered. In order to build numerical solution firstly a front-fixing transformation is applied. Transformed problem is posed on multidimensional fixed domain and is solved by explicit finite difference method. The numerical scheme is conditionally stable and is consistent with the first order in time and second order in space. The proposed approach allows the computation not only of the option price but also of the optimal stopping boundary. Numerical examples demonstrate efficiency and accuracy of the proposed method. The results are compared with other known approaches to show its competitiveness.

The Stability Analysis of Predictor–Corrector Method in Solving American Option Pricing Model

In this paper, a new technique is investigated to speed up the order of accuracy for American put option pricing under the Black---Scholes (BS) model. First, we introduce the mathematical modeling of American put option, which leads to a free boundary problem. Then the free boundary is removed by adding a small and continuous penalty term to the BS model that cause American put option problem to be solvable on a fixed domain. In continuation we construct the method of lines (MOL) in space and reach a non-linear problem and we show that the proposed MOL is more stable than the other kinds. To deal with the non-linear problem, an algorithm is used based on the predictor---corrector method which corresponds to two parameters, $$\theta $$? and $$\phi $$?. These parameters are chosen optimally using a rational approximation to determine the order of time convergence. Finally in numerical results a second order convergence is shown in both space and time variables.

The Differential Algorithm for American Put Option with Transaction Costs under CEV Model

AbstractThis paper mainly studies the American put option pricing with transaction costs in the CEV process. The specific Crank-Nicolson form of numerical solution is obtained by the finite difference method. On this basis, Hong Kong stock CKH option is selected as the object to estimate option price. Finally, by comparing with the actual price, the American put option pricing model is verified as reasonable. This paper is significant to the rational pricing and the institutional construction of the upcoming stock options in mainland China.