American Options Research Articles

Statistical Machine Learning for Quantitative Finance

We survey the active interface of statistical learning methods and quantitative finance models. Our focus is on the use of statistical surrogates, also known as functional approximators, for learning input–output relationships relevant for financial tasks. Given the disparate terminology used among statisticians and financial mathematicians, we begin by reviewing the main ingredients of surrogate construction and the motivating financial tasks. We then summarize the major surrogate types, including (deep) neural networks, Gaussian processes, gradient boosting machines, smoothing splines, and Chebyshev polynomials. The second half of the article dives deeper into the major applications of statistical learning in finance, covering ( a) parametric option pricing, ( b) learning the implied/local volatility surface, ( c) learning option sensitivities, ( d) American option pricing, and ( e) model calibration. We also briefly detail statistical learning for stochastic control and reinforcement learning, two areas of research exploding in popularity in quantitative finance.

Option Pricing for Uncertain Stock Model Based on Optimistic Value

Option pricing plays an important role in modern finance. This paper investigates the uncertain option pricing problems based on uncertainty theory by using the method to calculate the optimistic value of uncertain returns of options instead of the method of traditional expected value in the sense of the weighted average. The pricing formulas of the European and American options are derived for Liu’s uncertain stock model and Peng’s mean-reverting stock model which are two basic and representative uncertain stock models in uncertain finance. In the end, some numerical experiments are given to illustrate the effectiveness of the obtained results.

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.

Generalized Finite Integration Method with space–time decomposition technique for solving high dimensional option pricing models

In this paper, the recently developed Generalized Finite Integration Method (GFIM) is further combined with the technique of domain decomposition to solve multiple assets option pricing problems. By taking the time as a temporal variable incorporated with the spatial variables, we apply the space–time decomposition technique to divide the time interval into disjoint union of sub-intervals. The approximate solution at each sub-domain can then be obtained by using the GFIM iteratively. The advantage of this GFIM with space–time decomposition technique will be demonstrated by the numerical examples of solving free-boundary American option and two-assets exchange options pricing problems.

Compound Option Pricing and the Roll-Geske-Whaley Formula under the Conjugate-Power Dagum Distribution

We explore the pricing of compound derivatives under the newly introduced conjugate-power Dagum distribution. Assuming a discrete-time multiplicative conjugate-power Dagum random walk, we first provide an alternative derivation of the price of a married put based on a change of measure, which is helpful for the pricing of compound options. Then, we apply these results to obtain the equivalent of the Roll-Geske-Whaley formula for the pricing of American options in presence of one known discrete dividend under this alternative distribution.

Numerical approach for coupled systems resulting from pricing of derivatives: Modeling and pricing of installment options

In this paper, we proposed Markov‐modulated models to value vanilla European‐style continuous‐installment options with a numerical approach. The flexibility in installment options to paying installments or stopping payments makes these options more attractive to holders. We consider regime‐switching models for the dynamic of the underlying asset; in this way, the pricing problem of installment options can be formulated as coupled partial differential equations with free boundary features, which in many ways is similar to the free boundary problem for vanilla American options due to the possibility of early exercise. To value the installment options in this model, we express the problem as a linear complementarity problem (LCP), then we apply a finite element method as the numerical solving procedure to the resulting variational inequality. We illustrate some numerical results to examine the suitability of the proposed model and the accuracy of the applied numerical method for the pricing problem of installment options.

Stochastic representation under g-expectation and applications: The discrete time case

In this paper, we address the stochastic representation problem in discrete time under (non-linear) g-expectation. We establish existence and uniqueness of the solution, as well as a characterization of the solution. As an application, we investigate a new approach to the optimal stopping problem under g-expectation and the related pricing of American options under Knightian uncertainty. Our results are also applied to a (non-linear) Skorokhod-type obstacle 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.

Flexibility to switch project size: A real option application for photovoltaic investment valuation

Solar energy projects contribute to smart city goals by reducing pollution and assisting cities to become self-sustaining. However, these projects, characterised by their sequential nature, require a huge amount of irreversible investment, especially in the case of large-scale photovoltaic projects (MW). This aspect could make solar projects unattractive for potential investors. To mitigate the risks of financial losses the investor could invest in small-scale solar projects and switch to large-scale investment projects only if the market conditions are favourable. This paper aims to embed this managerial flexibility of changing investment size during the project lifetime into its valuation. To accomplish this, we use a compound options approach with a switch and call options models to reliably price these projects by considering their riskiness and, their stochastic and multi-stage nature. We also consider the determination of the optimal investment time like an American option. We also propose a case study that shows that considering the managerial flexibility value in the project valuation makes them attractive and financially profitable.

MLE Evolution Equation for Fractional Diffusions and Berry-Esseen Inequality of Stochastic Gradient Descent Algorithm for American Option

We study recursive parameter estimation in fractional diffusion processes. First, stability and asymptotic properties of the global maximum likelihood estimator (MLE) of the drift parameter are obtained under some regularity conditions. Then we obtain an evolution equation for the MLE of the drift parameter in nonhomogeneous fractional stochastic differential equation (fSDE) driven by fractional Brownian motion. This equation is then modified to yield an algorithm which is consistent, asymptotically efficient and converges to the MLE. The gradient and Newton type algorithm are firstorder approximations. Finally we study the Berry-Esseen inequality for stochastic gradient descent in continuous time (SGDCT) algorithm for American option. We compare it with Longstaff-Schwartz regression based method.

Pricing American options with stochastic volatility and small nonlinear price impact: A PDE approach

In this paper, transaction costs as small nonlinear price impact are introduced into American option pricing under the Heston stochastic volatility model, forming a gap between option prices of the holder and writer. Through the use of a dynamic hedging strategy together with a known option independent of transaction costs, we derive two different nonlinear pricing partial differential equation systems for the holder and writer, respectively. A numerical algorithm is designed to solve the systems so that American option prices as well as the optimal exercise boundary can be simultaneously obtained. Examples are presented to illustrate the effect of transaction costs on both option prices and optimal exercise prices.

An analytical approximation for convertible bonds

This paper looks at adapting the method of Medvedev and Scaillet for pricing short-term American options to evaluate short-term convertible bonds. However unlike their method, we provide explicit formulae for the coefficients of our series solution. This means that we do not need to solve complicated recursive systems, and can efficiently provide fast solutions. We also compare the method with numerical solutions, and find that it performs extremely well, giving accurate bond prices as well as accurate optimal conversion prices. doi:10.1017/S1446181122000062

PENGGUNAAN METODE SIMULASI QUASI MONTE CARLO DENGAN BARISAN QUASI ACAK VAN DER COURPUT DALAM MENENTUKAN HARGA OPSI EROPA

An option is a right by a person or institution to sell or buy an investment instrument at a certain price for a certain period. The point of view for way an option is exercised, there are two types of options known today, namely European options and American options. European options are characterized by option contracts that can only be exercised at the expiration date of the option. American options are indicated by option contracts that can be exercised at any time within the validity of the option contract. There are several methods that can be used in determining option prices, including the Black Scholes method which can be used to calculate the standard price of options. Monte Carlo simulation is a method that gives all possible values of a variable using the average as an estimator of its exact value. Quasi Monte Carlo simulation is an alternative to Monte Carlo simulation which uses quasi-random sequences instead of random numbers. In this article, the quasi-random sequence used is Van der Courput. Calculate option prices with Monte Carlo simulation and Quasi Monte Carlo simulation using MATLAB.

Forecasting the behaviour of fractional Black-Scholes option pricing equation by laplace perturbation iteration algorithm

Financial derivatives plays a major role in all financial deals these days. Black–Scholes option pricing model gives a risk free analysis for investing in options. In the current work, a method called the Laplace Perturbation Iteration Algorithm is being applied on Fractional Black–Scholes Equation to obtain its fractional analytical solutions in series form to analyze its results for the European and American option pricing problem, quickly and accurately. Laplace Perturbation Iteration Algorithm incorporates Laplace transform and Perturbation Iteration Algorithm and forms an iterative scheme that derives the analytical solutions without any inconvenience. The Fractional Black–Scholes Equation analytical solution is obtained as a convergent power series of conveniently computed components. Four examples have been solved and their results have been discussed in this paper, in order to check the efficacy of Laplace Perturbation Iteration Algorithm.

On the exercise of American quanto options

American option pricing is an important and engaging area of financial economics, particularly so in the presence of negative interest rates. Quanto options offer major international hedging/investment opportunities. We provide a comprehensive description of the optimal exercise policies associated with American quanto options. We show that a non-standard exercise policy characterized by a double continuation region may be optimal in the presence of non-positive domestic interest rates. We study empirical examples of finite-maturity American quanto options for which a double continuation region surrounding a non-empty early exercise region exists even if the infinite-maturity early exercise region is empty and the value of the infinite-maturity option is unbounded. Under the assumptions underpinning such empirical examples, we carefully characterize the existence, the monotonicity properties and the close-to-maturity behaviour of the upper and lower critical prices.

An Iteration Algorithm for American Options Pricing Based on Reinforcement Learning

In this paper, we present an iteration algorithm for the pricing of American options based on reinforcement learning. At each iteration, the method approximates the expected discounted payoff of stopping times and produces those closer to optimal. In the convergence analysis, a finite sample bound of the algorithm is derived. The algorithm is evaluated on a multi-dimensional Black-Scholes model and a symmetric stochastic volatility model, the numerical results implied that our algorithm is accurate and efficient for pricing high-dimensional American options.

Solving optimal stopping problems under model uncertainty via empirical dual optimisation

In this work, we consider optimal stopping problems with model uncertainty incorporated into the formulation of the underlying objective function. Typically, the robust, efficient hedging of American options in incomplete markets may be described as optimal stopping of such kind. Based on a generalisation of the additive dual representation of Rogers (Math. Financ. 12:271–286, 2002) to the case of optimal stopping under model uncertainty, we develop a novel regression-based Monte Carlo algorithm for the approximation of the corresponding value function. The algorithm involves optimising a penalised empirical dual objective functional over a class of martingales. This formulation allows us to construct upper bounds for the optimal value with reduced complexity. Finally, we carry out a convergence analysis of the proposed algorithm and illustrate its performance by several numerical examples.

Reflected BSDEs driven by inhomogeneous simple Lévy processes with RCLL barrier

We study the solution of a backward stochastic differential equation driven by an inhomogeneous simple Lévy process with a rcll reflecting barrier. We show the existence and uniqueness of solution by means of the Snell envelope and the fixed point theorem when the coefficient is stochastic Lipschitz. In terms of application, we provide the fair price of the American option in a Lévy market.

Reconstruction of local volatility surface from American options

Abstract In this paper, we discuss the reconstruction of a local volatility surface from American option prices. First of all, the American option prices are calculated by an accurate and fast finite difference scheme. Then the local volatility is obtained by minimizing the distance between theoretical prices and market option prices, which yields an optimization problem. The Bicubic spline regularization technique is used to overcome the ill-posedness of the reconstruction problem. We solve the nonlinear optimization problem by using a gradient-based optimization algorithm. Finally, we test our model with numerical examples and real market American put option data. The results show the good performance of our method.

American options and stochastic interest rates

We study finite-maturity American equity options in a stochastic mean-reverting diffusive interest rate framework. We allow for a non-zero correlation between the innovations driving the equity price and the interest rate. Importantly, we also allow for the interest rate to assume negative values, which is the case for some investment grade government bonds in Europe in recent years. In this setting we focus on American equity call and put options and characterize analytically their two-dimensional free boundary, i.e. the underlying equity and the interest rate values that trigger the optimal exercise of the option before maturity. We show that non-standard double continuation regions may appear, extending the findings documented in the literature in a constant interest rate framework. Moreover, we contribute by developing a bivariate discretization of the equity price and interest rate processes that converges in distribution as the time step shrinks. This discretization, described by a recombining quadrinomial tree, allows us to compute American equity options’ prices and to analyze their free boundaries with respect to time and current interest rate. Finally, we document the existence of non-standard optimal exercise policies for American call options on a non-dividend-paying equity.