Exercise Boundary Research Articles

Optimal stopping games in models with various information flows

We study zero-sum optimal stopping games associated with perpetual convertible bonds in an extension of the Black-Merton-Scholes model with random dividends under various information flows. In this type of contracts, the writers have the right to withdraw the bonds, before the holders convert them into assets. We derive closed-form expressions for the associated value function and optimal exercise boundaries in the model with an accessible dividend rate policy which is described by a continuous-time Markov chain with two states. We further consider the optimal stopping game in the model with inaccessible dividend rate policy and prove that the optimal exercise times are the first times at which the asset price process hits monotone boundaries depending on the running state of the filtering dividend rate estimate. We finally present the value of the optimal stopping game for the model in which the dividend rate policy is accessible to the writers but remains inaccessible to the holders of the bonds.

Between Scylla and Charybdis: The Bermudan Swaptions Pricing Odyssey

Bermudan swaptions are options on interest rate swaps which can be exercised on one or more dates before the final maturity of the swap. Because the exercise boundary between the continuation area and stopping area is inherently complex and multi-dimensional for interest rate products, there is an inherent “tug of war” between the pursuit of calibration and pricing precision, tractability, and implementation efficiency. After reviewing the main ideas and implementation techniques underlying both single- and multi-factor models, we offer our own approach based on dimension reduction via Markovian projection. Specifically, on the theoretical side, we provide a reinterpretation and extension of the classic result due to Gyöngy which covers non-probabilistic, discounted, distributions relevant in option pricing. Thus, we show that for purposes of swaption pricing, a potentially complex and multidimensional process for the underlying swap rate can be collapsed to a one-dimensional one. The empirical contribution of the paper consists in demonstrating that even though we only match the marginal distributions of the two processes, Bermudan swaptions prices calculated using such an approach appear well-behaved and closely aligned to counterparts from more sophisticated models.

Analysis of the optimal exercise boundary of American put options with delivery lags

A make-your-mind-up option is an American derivative with delivery lags. We show that its put option can be decomposed as a European put and a new type of American-style derivative. The latter is an option for which the investor receives the Greek Theta of the corresponding European option as the running payoff, and decides an optimal stopping time to terminate the contract. Based on this decomposition and using free boundary techniques, we show that the associated optimal exercise boundary exists and is a strictly increasing and smooth curve, and analyze the asymptotic behavior of the value function and the optimal exercise boundary for both large maturity and small time lag.

An investigation on the existence and uniqueness analysis of the optimal exercise boundary of American put option

In this paper, we discuss the Banach fixed point theorem conditions on the optimal exercise boundary of American put option paying continuously dividend yield, to investigate whether its existence, uniqueness, and convergence are derived. In this respect, we consider the integral representation of the optimal exercise boundary which is extracted as a consequence of the Feynman-Kac formula. In order to prove the above features, we define a nonempty closed set in Banach space and prove that the proposed mapping is contractive and onto. At final, we illustrate the ratio convergence of the mapping on the optimal exercise boundary.

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.

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.

Modelling and computation of optimal multiple investment timing in multi-stage capacity expansion infrastructure projects

<p style='text-indent:20px;'>So far, the optimal investment timing to maximize the total profit of multi-stage capacity expansion infrastructure projects is not clear. In the case of uncertain demands, the optimal multiple stopping time theory is adopted to model the optimal decision-making of investment timing for multi-stage expansion infrastructure projects in a finite time horizon. In this context, the first-stage of the project involves a dedicated asset investment for later expansion, and the capacity of the project at each stage is constrained, which makes the cash flow of the project exhibit the characteristic of bull call spread. The upwind finite difference method and multi-least squares Monte Carlo simulation are combined to solve the project value and determine the optimal exercise boundaries at all stages described by a sequence of demand thresholds. A multi-stage power plant project is taken as an example to validate the model. Through the example, the optimal investment strategies and the value of the multi-stage project are provided; the effects of the dedicated asset and capacity constraint are illustrated. This study novelly reveals the effect of the capacity constraints on the project value using the bull call spread theory.</p>

An Artificial Intelligence Approach to the Valuation of American-style Derivatives: A Use of Particle Swarm Optimization

In this paper, we evaluate American-style, path-dependent derivatives with an artificial intelligence technique. Specifically we use swarm intelligence to find the optimal exercise boundary for an American-style derivative. Swarm intelligence is particularly efficient (computation and accuracy) in solving high-dimensional optimization problems and hence perfectly suitable for valuing complex American-style derivatives (e.g. multiple-asset, path-dependent) which require a high-dimensional optimal exercise boundary.

A simple numerical method for pricing American power put options

Abstract In this paper, we present numerical methods to determine the optimal exercise boundary in case of an American power put option with non-dividend yields. The payoff of a power option is typified by its underlying share price raised to a constant power. The nonlinear payoffs of power options offer considerable flexibility to investors and can be applied in various applications. Herein, we exploit a transformed function to obtain the optimal exercise boundary of the American power put option. Employing it, we can easily determine the optimal exercise boundary. After determining the optimal exercise boundary, we calculate the American power put option values using the finite difference method. Generally, the optimal exercise boundary may not be observed at the grid points. Therefore, the interpolation method is used to determine the value of the American power put option. Furthermore, we present several numerical results obtained by comparing the proposed method and the existing methods.

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.

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 new integral equation approach for pricing American-style barrier options with rebates

In this paper, the pricing of an American-style down-and-out call option with debates is explored under the Black–Scholes model. This is a complex problem due to the combination of the knock-out feature and early-exercise feature, which requires the unknown optimal exercise boundary to be determined simultaneously with the option price. Despite all the challenges, we are still able to present a semi-analytical solution by deriving an integral equation representation for the target option price using the incomplete Fourier transform, after innovatively introducing another unknown function for the first-order derivative of the option price at the barrier. Numerical schemes are then established to implement the integral equations. Remarkably, it turns out that only one single nonlinear equation for the optimal exercise boundary needs to be solved numerically, while the introduced unknown function has an analytical form in the recursive process, which guarantees the efficiency of our approach.

Least-squares Monte-Carlo methods for optimal stopping investment under CEV models

The optimal stopping investment is a kind of mixed expected utility maximization problems with optimal stopping time. The aim of this paper is to develop the least-squares Monte-Carlo methods to solve the optimal stopping investment under the constant elasticity of variance (CEV) model. Such a problem has no closed-form solutions for the value functions, optimal strategies and optimal exercise boundaries due to the early exercised feature. The dual optimal stopping problem is first derived and then the strong duality between the dual and prime problems is established. The least-squares Monte-Carlo methods based on the dual control theory are developed and numerical simulations are provided. Both the power and non-HARA utilities are studied.

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)} .

Pricing methods for α-quantile and perpetual early exercise options based on Spitzer identities

We present new numerical schemes for pricing perpetual Bermudan and American options as well as α-quantile options. This includes a new direct calculation of the optimal exercise boundary for early-exercise options. Our approach is based on the Spitzer identities for general Lévy processes and on the Wiener–Hopf method. Our direct calculation of the price of α-quantile options combines for the first time the Dassios–Port–Wendel identity and the Spitzer identities for the extrema of processes. Our results show that the new pricing methods provide excellent error convergence with respect to computational time when implemented with a range of Lévy processes.

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.

Option Investor Rationality Revisited: The Role of Exercise Boundary Violations

The authors dispute the theory that American call options should not be exercised early. By looking at intraday pricing, they find that the best bid price available is often lower than the option’s intrinsic value and early exercise can be rational. Our empirical results overturn the well-known textbook theory that American options should not be exercised early except in very limited conditions. In the real world, the best bid available in the market is frequently below the option’s intrinsic value (e.g., nearly half of all quotes for in-the- money calls), which we call an “exercise boundary violation” (EBV). In an EBV, early exercise can be the right strategy and the “American” option characteristic has economic value. Some option holders do exercise, some sell at EBV bid prices, and our results suggest that actual exercise behavior is much less irrational than earlier studies concluded. Disclosure: The authors report no conflicts of interest. Editor’s Notes Submitted 26 May 2019 Accepted 4 November 2019 by Stephen J. Brown. This article was externally reviewed using our double-blind peer-review process. When the article was accepted for publication, the authors thanked the reviewers in their acknowledgments. Giuseppe Ballocchi, CFA, and William Fung were the reviewers for this article.

Option Investor Rationality Revisited: The Role of Exercise Boundary Violations

Do option investors rationally exercise their options? Numerous studies report evidence of irrational behavior. In this paper, we pay careful attention to intraday option quotes and reach the opposite conclusion. An exercise boundary violation (EBV) occurs when the best bid price for an American option is below the option’s intrinsic value. Far from being unusual, we show that EBVs occur very frequently. Under these conditions, the rational response of an investor liquidating an option is to exercise the option rather than sell it. Empirically, we find that the likelihood of early exercise is strongly influenced by the existence and duration of EBVs. Not only do these results reverse standard theory on American option valuation and optimal exercise strategy, but they also suggest that the ability to avoid selling at an EBV price creates an additional source of value for American options that is unrelated, and in addition to, dividend payments. This additional value may help explain why American options appear overpriced relative to European options.

Closed Form Optimal Exercise Boundary of the American Put Option

We present three models of stock price with time-dependent interest rate, dividend yield, and volatility, respectively, that allow for explicit forms of the optimal exercise boundary of the American put option. The optimal exercise boundary satisfies nonlinear integral equation of Volterra type. We choose time-dependent parameters of the model so that the integral equation for the exercise boundary can be solved in the closed form. We also define the contracts of put type with time-dependent strike price that support the explicit optimal exercise boundary. All these results can be used as approximations to the standard model with constant parameters, i.e., geometric Brownian motion process.

Executive Stock Option Exercise with Full and Partial Information on a Drift Change Point

We analyze the optimal exercise of an American call executive stock option (ESO) written on a stock whose drift parameter falls to a lower value at a change point, an exponentially distributed random time independent of the Brownian motion driving the stock. Two agents, who do not trade the stock, have differing information on the change point and seek to optimally exercise the option by maximizing its discounted payoff under the physical measure. The first agent has full information and observes the change point. The second agent has partial information and filters the change point from price observations. This scenario is designed to mimic the positions of two employees of varying seniority, a fully informed executive and a partially informed less senior employee, each of whom receives an ESO. The partial information scenario yields a model under the observation filtration $\widehat{\mathbb{F}}$ in which the stock drift becomes a diffusion driven by the innovations process, an $\widehat{\mathbb{F}}$-Brownian motion also driving the stock under $\widehat{\mathbb{F}}$, and the partial information optimal stopping value function has two spatial dimensions. We rigorously characterize the free boundary PDEs for both agents, establish shape and regularity properties of the associated optimal exercise boundaries, and prove the smooth pasting property in both information scenarios, exploiting some stochastic flow ideas to do so in the partial information case. We develop finite difference algorithms to numerically solve both agents' exercise and valuation problems and illustrate that the additional information of the fully informed agent can result in exercise patterns which exploit the information on the change point, lending credence to empirical studies which suggest that privileged information of bad news is a factor leading to early exercise of ESOs prior to poor stock price performance.