Early Exercise Boundary Research Articles

A Robust Numerical Simulation of a Fractional Black–Scholes Equation for Pricing American Options

After the discovery of fractal structures of financial markets, fractional partial differential equations (fPDEs) became very popular in studying dynamics of financial markets. Available research results involves two key modelling aspects; firstly, derivation of tractable asset pricing models, those that closely reflects the actual dynamics of financial markets. Secondly, the development of robust numerical solution methods. Often times, most the effective models are of a nonlinear nature, and as such, reliable analytical solution methods are seldomly available. On the other hand, the accurate value of American options strongly lies on the unknown free boundaries associated with these types of derivative contracts. The free boundaries emanates from the flexibility of the early exercise features with American options. To the best of our knowledge, the approach of pricing American options under the fractional calculus framework has not been extensively explored in literature, and an obvious wider research gap still exist on the design of robust solution methods for pricing American option problems formulated under the fractional calculus framework. Therefore, this paper serve to propose a robust numerical scheme for solving time-fractional Black–Scholes PDEs for pricing American put option problems. The proposed scheme is based on the front-fixing algorithm, under which the early exercise boundaries are transformed into fixed boundaries, allowing for a simultaneous computation of optimal exercise boundaries and their corresponding fair premiums. Results herein indicate that, the proposed numerical scheme is consistent, stable, convergent with order O(h2,k)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {O}}(h^2,k)$$\\end{document}, and also does guarantee positivity of solutions under all possible market conditions.

American strangle options with arbitrary strikes

AbstractThe so‐called American strangle options are examined in this paper. Their main characteristic is the combined put and call feature. The holder has the right to exercise prematurely choosing the option's style—put or call. We abandon the traditional assumption that the put strike is below the call one considering arbitrary values. We also assume that the put and call weights are different. The equations for the early exercise boundaries are derived in the perpetual case. After that we approximate numerically these boundaries for the finite maturity options maximizing the option holder's utility. On the basis of them we apply a Crank–Nicolson finite difference method to the corresponding Black–Scholes‐style partial differential equation to obtain the fair option price.

An Alternative Numerical Scheme to Approximate the Early Exercise Boundary of American Options

This paper deals with a new numerical method for the approximation of the early exercise boundary in the American option pricing problem. In more detail, using the mean-value theorem for integrals, we provide a flexible algorithm that allows for reaching a more accurate numerical solution with fewer calculations rather than other previously described methods.

Approximate pricing of American exchange options with jumps

AbstractThis paper presents a numerical method to price American exchange options based on jump‐diffusion processes. We first derive a closed‐form expression for the value of European exchange options, then decompose the value function of an American exchange option into a European counterpart, and an early exercise premium that is derived analytically. The early exercise boundary for an American exchange option approximately satisfies an algebraic equation that can be quickly numerically solved. Consequently, a formula is obtained for efficiently pricing American exchange options. The numerical results reveal that our pricing formula is robust and accurate.

A numerical approach to pricing exchange options under stochastic volatility and jump-diffusion dynamics

We consider a method of lines (MOL) approach to determine prices of European and American exchange options when underlying asset prices are modeled with stochastic volatility and jump-diffusion dynamics. As with any other numerical scheme for partial differential equations (PDEs), the MOL becomes increasingly complex when higher dimensions are involved, so we first simplify the problem by transforming the exchange option into a call option written on the ratio of the yield processes of the two assets. This is achieved by taking the second asset yield process as the numéraire. Under the equivalent martingale measure induced by this change of numéraire, we derive the exchange option pricing integro-partial differential equations (IPDEs) and investigate the early exercise boundary of the American exchange option. We then discuss a numerical solution of the IPDEs using the MOL, its implementation using computing software and possible alternative boundary conditions at the far limits of the computational domain. Our analytical and numerical investigation shows that the near-maturity behavior of the early exercise boundary of the American exchange option is significantly influenced by the dividend yields and the presence of jumps in the underlying asset prices. Furthermore, with the numerical results generated by the MOL, we are able to show that key jump and stochastic volatility parameters significantly affect the early exercise boundary and exchange option prices. Our numerical analysis also verifies that the MOL performs more efficiently, than other finite difference methods or simulation approaches for American options, since the MOL integrates the computation of option prices, greeks and the early exercise boundary, and does so with the least error.

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 Front-Fixing Implicit Finite Difference Method for the American Put Options Model

In this paper, we present an implicit finite difference method for the numerical solution of the Black–Scholes model of American put options without dividend payments. We combine the proposed numerical method by using a front-fixing approach where the option price and the early exercise boundary are computed simultaneously. We study the consistency and prove the stability of the implicit method by fixing the values of the free boundary and of its first derivative. We improve the accuracy of the computed solution via a mesh refinement based on Richardson’s extrapolation. Comparisons with some proposed methods for the American options problem are carried out to validate the obtained numerical results and to show the efficiency of the proposed numerical method. Finally, by using an a posteriori error estimator, we find a suitable computational grid requiring that the computed solution verifies a prefixed error tolerance.

A new approach for pricing discounted American options

The purpose of this paper is to present a new numerical approach for finding the early exercise boundary of discounted American options, which payment structure is generalized by adding an additional discount factor. We approximate this boundary by exponent of piecewise linear functions maximizing the option’s holder utility. Once we have derived the exercise boundary, the free boundary problem, which describes the American derivative price, turns to a partial differential equation (PDE) with known boundaries. This PDE is an extension of the Kolmogorov’s equation and can be converted to the heat equation, for which many numerical solvers are available. We present two different Monte Carlo methods and a finite difference approach to obtain the fair option prices and give some numerical examples. In the perpetual case we derive closed form formulas.

Pricing American options under negative rates

This paper starts by defining the criteria where the early-exercise of an American option is never optimal, under positive, or negative rates. It follows with a short analysis of the various shapes of the exercise region under negative interest rates. It then presents a new integral equation, which establishes the option price, and the two early exercise boundaries, under negative rates. It shows how to solve this new equation, through modifications of the modern and efficient algorithm of Andersen and Lake, from the initial guess of the two boundaries to more subtle changes required in their fixed point method for stability. Finally, the performance and accuracy of the resulting algorithm is assessed against a cutting edge finite difference method implementation.

The Free Boundary for the American Put Option

The free boundary of the American put option is analytically characterized via new and exact formulae. New accurate short-time asymptotics are an immediate corollary of these analytical results. The free boundary is also represented formulaically throughout all tenors by a simple two-parameter generalized Gaussian functional form. TOPICS:Options, fundamental equity analysis, statistical methods Key Findings • First exact formula for the free boundary (early exercise boundary) of the American put option in terms of the exogenous inputs: risk free rate, volatility, and a boundary dependent integral. • A simple and accurate asymptotics formula for the free boundary of the American put option near expiration. • An empirical solution (possibly exact) of the free boundary as a two-parameter generalized Gaussian functional form.

A note on the nonlinear Volterra integral equation for the early exercise boundary

Abstract In this paper, we study the nonlinear Volterra integral equation satisfied by the early exercise boundary of the American put option in the one-dimensional diffusion model for a stock price with constant interest rate and constant dividend yield and with a local volatility depending on the current time t and the current stock price S. In the classical Black–Sholes model for a stock price, Theorem 4.3 of [S. D. Jacka, Optimal stopping and the American put, Math. Finance 1 1991, 2, 1–14] states that if the family of integral equations (parametrized by the variable S) holds for all S ≤ b ⁢ ( t ) {S\leq b(t)} with a candidate function b ⁢ ( t ) {b(t)} , then this b ⁢ ( t ) {b(t)} must coincide with the American put early exercise boundary c ⁢ ( t ) {c(t)} . We generalize Peskir’s result [G. Peskir, On the American option problem, Math. Finance 15 2005, 1, 169–181] to state that if the candidate function b ⁢ ( t ) {b(t)} satisfies one particular integral equation (which corresponds to the upper limit S = b ⁢ ( t ) {S=b(t)} ), then all other integral equations (corresponding to S, S ≤ b ⁢ ( t ) {S\leq b(t)} ) will be automatically satisfied by the same function b ⁢ ( t ) {b(t)} .

Early exercise boundaries for American-style knock-out options

This paper proposes a novel representation for the early exercise boundary of American-style double knock-out options in terms of the simpler optimal stopping boundary of a nested single barrier contract. Such representation only requires the existence, continuity and monotonicity (in time) of the nested single barrier exercise boundary, and these requirements are proved for the whole class of single-factor exponential-Lévy processes. To illustrate the practical relevance of our results, a new put-call duality relation is obtained, a real options application is provided and the Fourier space time-stepping method, the COS approximation, and the static hedging portfolio approach are all adapted to the valuation of American-style double knock-out options.

American Option Pricing: Optimal Lattice Models and Multidimensional Efficiency Tests

This paper introduces a set of lattice techniques with a view to accelerating computation time and improving the accuracy of American Option valuation. Estimation speed can be enhanced through developing a parsimonious early exercise boundary search routine combined with reliance on dynamic memory and lattice truncation. Furthermore, Black-Scholes and Richardson extrapolation modifications to the lattices can also be applied individually and/or together to improve the accuracy of lattices. In this paper, we investigate the improvement introduced by obtaining the best combination of varying features. By introducing these techniques to the Leisen-Reimer and Tian binomial model, we can achieve a level of accuracy and efficiency combined that surpass analytical analogues prominent in the literature. Significantly, the Leisen-Reimer and Tian structure can accommodate arbitrary improvements in accuracy by simply increasing the density of their own mesh. Analytical methods generally do not afford much scope for optimising speed and efficiency in a granular fashion. We also compare efficient lattice models with analytical formulae for pricing different groups of options according to the deepness of American quality and the moneyness of the options. The appropriate model is recommended for pricing particular types of the options. Lattices importantly afford an explicit trade-off locus between accuracy and speed that can be navigated according to predetermined precision tolerance levels and option types. This should have practical relevance to trading platforms that require real-time estimates of implied volatility.

Bootstrapping the Early Exercise Boundary in the Least-Squares Monte Carlo Method

This paper proposes an innovative algorithm that significantly improves on the approximation of the optimal early exercise boundary obtained with simulation based methods for American option pricing. The method works by exploiting and leveraging the information in multiple cross-sectional regressions to the fullest by averaging the individually obtained estimates at each early exercise step, starting from just before maturity, in the backwards induction algorithm. With this method, less errors are accumulated, and as a result of this, the price estimate is essentially unbiased even for long maturity options. Numerical results demonstrate the improvements from our method and show that these are robust to the choice of simulation setup, the characteristics of the option, and the dimensionality of the problem. Finally, because our method naturally disassociates the estimation of the optimal early exercise boundary from the pricing of the option, significant efficiency gains can be obtained by using less simulated paths and repetitions to estimate the optimal early exercise boundary than with the regular method.

Bootstrapping the Early Exercise Boundary in the Least-Squares Monte Carlo Method

This paper proposes an innovative algorithm that significantly improves on the approximation of the optimal early exercise boundary obtained with simulation based methods for American option pricing. The method works by exploiting and leveraging the information in multiple cross sectional regressions to the fullest by averaging the individually obtained estimates at each early exercise step, starting from just before maturity, in the backwards induction algorithm. With this method less errors are accumulated, and as a result of this the price estimate is essentially unbiased even for long maturity options. Numerical results demonstrate the improvements from our method and show that these are robust to the choice of simulation setup, the characteristics of the option, and the dimensionality of the problem. Finally, because our method naturally disassociates the estimation of the optimal early exercise boundary from the pricing of the option, significant efficiency gains can be obtained by using less simulated paths and repetitions to estimate the optimal early exercise boundary than with the regular method.

Ratcheting Up: Adjusting the Incentives in the NBA Draft

Abstract: In this paper, we develop a modified salary scale for NBA rookies that adjusts player financial incentives to enter the draft early. Our proposed “ratcheting” salary schedule pays different salary amounts depending on draft pick and the number of years the player delayed entry to the draft to alter the financial incentives of declaring early. We analyze this proposed new salary scale using a put option analogy to determine early exercise boundary positions in which there is no value in the option to delay entry into the draft. The results show that under a ratchet salary system the incentives for early entry to the draft can be greatly altered, especially for freshmen and sophomores unless they are expected to be lottery picks. This information may help players make better decisions of when to enter the draft and can address unraveling in the NBA labor market.

A Method-of-lines Approach for Solving American Option Problems

The early exercise property of American option changes the original Black-Scholes equation to an inequality that cannot be solved via traditional finite difference method. Therefore, finding the early exercise boundary prior to spatial discretization is a must in each time step. This overhead slows down the computation and the accuracy of solution relies on if the early exercise boundary can be accurately located. A simple numerical method based on finite difference and method of lines is proposed here to overcome this difficulty in American option valuation. Our method averts the otherwise necessary procedure of locating the optimal exercise boundary before applying finite difference discretization. The method is efficient and flexible to all kinds of pay-off. Computations of American put, American call with dividend, American strangle, two-factor American basket put option, and two-factor convertible bond with embedded call and put options are demonstrated to show the efficiency of the current method.

Pricing and Exercising American Options: an Asymptotic Expansion Approach

This paper proposes and implements a novel asymptotic expansion approach for pricing discretely monitored American options and approximating their optimal early exercise boundaries, under a generic class of multivariate derivative pricing models incorporating both stochastic volatility and Lévy-driven jumps in asset return. The price and the critical value can be expanded up to any arbitrary order around those under a simple constant-volatility jump-diffusion model. The expansion terms are then efficiently implemented by exactly solving some backward inductions via closed-form Fourier transforms, where a substantial extension of the Hilbert transform method of Feng and Linetsky (2008a, 2009) plays an important role. The efficiency of our method is illustrated through some representative examples. As applications, we analyze the impacts of various model parameters on the optimal early exercise boundary, compare the optimal early exercise boundaries under different models, and propose a derivative-proxy based method for practically exercising American options under models with stochastic volatility and jumps.

Valuing American-style options under the CEV model: an integral representation based method

This article derives a new integral representation of the early exercise boundary for valuing American-style options under the constant elasticity of variance (CEV) model. An important feature of this novel early exercise boundary characterization is that it does not involve the usual (time) recursive procedure that is commonly employed in the so-called integral representation approach well known in the literature. Our non-time recursive pricing method is shown to be analytically tractable under the local volatility CEV process and the numerical experiments demonstrate its robustness and accuracy.

A product integration method for the approximation of the early exercise boundary in the American option pricing problem

Integral equation methods are now becoming well‐established tools in the study of financial models used in theory and practice. In this paper, we investigate the fully nonlinear weakly singular nonstandard Volterra integral equations representing the early exercise boundary of American option contracts, which gained popularity in recent years. We propose a product integration approach based on linear barycentric rational interpolation to solve the problem. The price of the option will then be computed using the obtained approximation of the early exercise boundary and a barycentric rational quadrature. The convergence of the approximation scheme will also be analyzed. Finally, some numerical experiments based on the introduced method are presented and compared with some exiting approaches.