American Option Valuation Research Articles

American Option Valuation Under the Framework of CGMY Model with Regime-Switching Process

American Option Valuation Under the Framework of CGMY Model with Regime-Switching Process

Is Reinforcement Learning Good at American Option Valuation?

This paper investigates algorithms for identifying the optimal policy for pricing American Options. The American Option pricing is reformulated as a Sequential Decision-Making problem with two binary actions (Exercise or Continue), transforming it into an optimal stopping time problem. Both the least square Monte Carlo simulation method (LSM) and Reinforcement Learning (RL)-based methods were utilized to find the optimal policy and, hence, the fair value of the American Put Option. Both Classical Geometric Brownian Motion (GBM) and calibrated Stochastic Volatility models served as the underlying uncertain assets. The novelty of this work lies in two aspects: (1) Applying LSM- and RL-based methods to determine option prices, with a specific focus on analyzing the dynamics of “Decisions” made by each method and comparing final decisions chosen by the LSM and RL methods. (2) Assess how the RL method updates “Decisions” at each batch, revealing the evolution of the decisions during the learning process to achieve optimal policy.

A primal-dual active set approach to the valuation of American options in regime-switching models: numerical solutions and convergence analysis

A primal-dual active set approach to the valuation of American options in regime-switching models: numerical solutions and convergence analysis

American option in a market generated by interactive agents

This study introduces a new financial market model inspired by Remita and Eisele's methodology, incorporating a substantial number of interacting agents denoted by “n”. Focusing on the valuation of European and American options within this vast network of agents, and more specifically on behavior when “n” tends to infinity, the research illustrates the convergence of an initially incomplete market towards completeness. In addition, the study scrutinizes the price stability of options on the underlying risky asset, St, in the context of an increasing number of agents. Taking into account a comprehensive set of internal and external factors influencing market dynamics, this study offers a holistic analysis of option price dynamics and market completeness. By examining the complex interactions between a large number of agents and option pricing, the research provides valuable insights into the complexity of financial markets. The results not only elucidate convergence phenomena within an expanding network of agents, they also shed light on the stability of option prices under ever-changing market conditions. This comprehensive analysis underlines the multifaceted nature of financial markets, highlighting the intrinsic relationship between option price dynamics and various market influences. Overall, this study makes a significant contribution to the understanding of financial market dynamics, particularly in the context of option pricing.

Primal‐dual active set algorithm for valuating American options under regime switching

AbstractThis paper focuses on numerical algorithms to value American options under regime switching. The prices of such options satisfy a set of complementary parabolic problems on an unbounded domain. Based on our previous experience, the pricing model could be truncated into a linear complementarity problem (LCP) over a bounded domain. In addition, we transform the resulting LCP into an equivalent variational problem (VP), and discretize the VP by an Euler‐finite element method. Since the variational matrix in the discretized system is P‐matrix, a primal‐dual active set (PDAS) algorithm is proposed to evaluate the option prices efficiently. As a specialty of PDAS, the optimal exercise boundaries in all regimes are obtained without further computation cost. Finally, numerical simulations are carried out to test the performance of our proposed algorithm and compare it to existing methods.

The stochastic balance equation for the American option value function and its gradient

In the paper we consider the problem of valuation and hedging of American options written on dividend-paying assets whose price dynamics follow a multidimensional diffusion model. We derive a stochastic balance equation for the American option value function and its gradient. We prove that the latter pair is the unique solution of the stochastic balance equation as a result of the uniqueness in the related adapted future-supremum problem. The latter problem has an attractive interpretation: the given adapted stochastic process can be adjusted by a martingale in such a manner, that the observer will gain the perfect foresight of the resulting future-supremum process via the Snell envelope of the given stochastic process.

The impact of negative interest rates on the pricing of options written on equity: a technical study for a suitable estimate of early termination

This work aims to investigate the main problems that impact the pricing models and the sensitivity measures of American options written on shares without a pay-out, in the presence of negative interest rates with a specific focus on the Monte Carlo method. The first paragraph carries out a review of the anomalies caused by such an odd condition and focuses thereafter on the core topic of the research by treating a wide range of numerical models suitable for unbiased evaluation of the early exercise, thus expanding the existing literature. The two following paragraphs are dedicated to describing the models used for the correct estimation of fair value: binomial lattice models (Cox-Ross-Rubinstein - CRR Tree, Leisen Reimer - LR Tree, Jarrow-Rudd - JR Tree and Tian Tree), trinomial stochastic trees, Finite Difference Method (FDM) scheme and the Longstaff-Schwartz Monte Carlo. Particular attention is paid to this last approach which allows to combine the flexibility of traditional numerical integration schemes for stochastic processes on equity with the estimation of the convenience of exercising the American option ahead of time. After conducting quantitative tests both on pricing and on the estimation of sensitivity measures, the LR Tree was selected as the most performing deterministic algorithm to be compared with the Monte Carlo stochastic technique. The final part of the work focuses on quantifying the valuation gap introduced by negative interest rates in the valuation of American options written on an unprofitable underlying comparing the traditional valuation approach and the deterministic Leisen Reimer model and the Longstaff-Schwartz stochastic model.

Numerical Valuation of European and American Options under Fractional Black-Scholes Model

In this paper, we investigate the numerical valuation of European and American options under the time fractional Black-Scholes model. We first apply a coordinate stretching transformation to the asset price so that the spatial region can focus on the vicinity of singularities, which are usually found in the payoff function. The radial basis function finite difference method is used for the spatial discretization, and the improved L1 method is used to deal with the reduced order of convergence for the nonsmooth initial data. We use the operator splitting method for solving the linear complementary problem of American options. The proposed scheme leads to a sparse linear system which is trivial to solve. Moreover, the stability of the proposed numerical scheme is analyzed using Fourier analysis. Numerical experiments demonstrate the accuracy and efficiency of the proposed method.

Semi-implicit FEM for the valuation of American options under the Heston model

Semi-implicit FEM for the valuation of American options under the Heston model

Projection and contraction method for the valuation of American options under regime switching

This paper proposes an efficient numerical algorithm to evaluate American put options under regime switching. A set of variational inequalities could describe the pricing model, which is equivalent to a coupled parabolic free boundary problem. With variable substitutions and truncation techniques, the original pricing problem is transformed into a coupled linear complementarity problem on a bounded rectangular domain. Then, the variational problem related to the linear complementary problem is obtained. Furthermore, a finite difference method in the temporal direction and a finite element method in the spatial direction lead to a full-discretized approximation of the variational problem. Based on the positive definiteness of the discretized matrix, a projection and contraction method is adopted for the resulting discretized variational problem. Finally, several numerical simulations are carried out to illustrate the efficiency of the proposed method compared with existing methods.

Greek Factor Basis Function of the Least-Squares Monte Carlo Approach to American Option Valuation

Greek Factor Basis Function of the Least-Squares Monte Carlo Approach to American Option Valuation

Numerical valuation of European and American options under Merton's model

In this paper, a new combination of time and spatial discretization is proposed for a partial integro-differential equation (PIDE) arising in the valuation of European options under Merton's model. We first present a high-order compact (HOC) difference scheme in space based on a uniform mesh to obtain a highly accurate result, and the discontinuous Galerkin (DG) finite element method in time is introduced that can deal with the loss of the time analyticity. A penalty method is proposed for a partial integro-differential complementarity problem arising in the valuation of the American put option. Numerical experiments are performed to verify the accuracy and efficiency of the proposed method.

Inventory strategy development under supply disruption risk

This paper proposes a multi-period decision-making framework that enables procurement managers to develop an optimal supply inventory strategy under uncertain supply conditions. The framework draws on financial options valuation techniques by adopting the view of a procurement manager facing several uncertainties, while seeking to maximise its profit by exercising its flexibility to procure supplies and use inventories. Given the multitude of uncertain variables influencing inventory management decision in practice, the model is developed to remain robust to the inclusion of several underlying stochastic variables; it uses an American options valuation method and a least squares Monte Carlo simulation technique to solve the underlying dynamic programming problem. To demonstrate the application of the developed framework, a case study is presented based on data from a dairy supply chain. The case study explores the decision problem in several scenarios, showing how a decision-maker’s perception of product demand and supply price uncertainties, expectations over the timing of a supply disruption, discount rate, price shocks and disruption duration can be suitably incorporated into the decision-making framework.

Pricing formulas for perpetual American options with general payoffs

Abstract An American option gives the holder the right, but not the obligation, to buy/sell an underlying asset from/to the writer at an agreed strike price at any time on or before the expiry date. Options are mainly used for speculation and hedging. The pricing of options is a fundamental problem in mathematical finance. One of the attractions of options is that they can be used to construct a wide range of trading strategies characterized by different payoff functions. As a preliminary step in the valuation of American options for a variety of trading strategies, in this article the pricing of perpetual American options with general payoffs is considered, where the perpetual American call and put are special cases. Four broad classes of payoff functions are identified for which analytical pricing formulas can be derived by utilizing a Mellin transform technique and an optimization procedure. Depending on the class of payoff functions considered, free boundary problems with one or two boundaries are obtained. Illustrative examples are provided and benchmarked numerically with the binomial method. The characterization of different payoffs for perpetual American options considered in this article will be instrumental in the identification and pricing of new free boundary problems for (non-perpetual) American-style financial derivatives.

Deep learning-based least squares forward-backward stochastic differential equation solver for high-dimensional derivative pricing

We propose a new forward-backward stochastic differential equation solver for high-dimensional derivative pricing problems by combining a deep learning solver with a least squares regression technique widely used in the least squares Monte Carlo method for the valuation of American options. Our numerical experiments demonstrate the accuracy of our least squares backward deep neural network solver and its capability to produce accurate prices for complex early exercisable derivatives, such as callable yield notes. Our method can serve as a generic numerical solver for pricing derivatives across various asset groups, in particular, as an accurate means for pricing high-dimensional derivatives with early exercise features.

A Multilevel Monte Carlo Method for the Valuation of Swing Options

In this study, we propose a novel approach for the valuation of swing options. Swing options are a kind of American options with multiple exercise rights traded in energy markets. Longstaff and Schwartz have suggested a regression-based Monte Carlo method known as the least-squares Monte Carlo (LSMC) method to value American options. In this work, first we introduce the LSMC method for the pricing of swing options. Then, to achieve a desired accuracy for the price estimation, we combine the idea of LSMC with multilevel Monte Carlo (MLMC) method. Finally, to illustrate the proper behavior of this combination, we conduct numerical results based on the Black–Scholes model. Numerical results illustrate the efficiency of the proposed approach.

A computational weighted finite difference method for American and barrier options in subdiffusive Black–Scholes model

Subdiffusion is a well established phenomenon in physics. In this paper we apply the subdiffusive dynamics to analyze financial markets. We focus on the financial aspect of time fractional diffusion model with moving boundary i.e. American and barrier option pricing in the subdiffusive Black–Scholes (B–S) model. Two computational methods for valuing American options in the considered model are proposed - the weighted finite difference (FD) and the Longstaff–Schwartz method. In the article it is also shown how to valuate numerically wide range of barrier options using the FD approach.

PRICING MULTI-ASSET AMERICAN OPTION WITH STOCHASTIC CORRELATION COEFFICIENT UNDER VARIANCE GAMMA ASSET PRICE DYNAMIC

This paper considers a class of Levy process namely the variance gamma (VG) process to offer a more realistic way to model the dynamics of the logarithm of stock prices. Then, we verify the uniqueness and existence of the solution to the stochastic differential equation of the model. We also examine the valuation of multi-asset American options under VG model when the correlation coefficient is governed by the modified Ornstein–Uhlenbeck process. Various simulation experiments are presented and the achieved results are tested empirically for option prices using S&P 500 data.

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.

A Replicating Portfolio Approach to Valuing American Options in Closed-Form

I derive closed-form expressions for the value of American call and put options (on an asset with continuous yield) using the Feynman-Kac formula, modelling the size of early exercise premium as a function of the strike, dividend yield and time to maturity. Strategies involving European options and binary options on forwards on the asset replicate an American option exactly. Violations of Put-Call parity for American options is also explored in this framework.