Perpetual American Options Research Articles

Geometric Compound Hawkes Process and its Applications in Finance

The authors introduce a new model for a stock price, namely, geometric compound Hawkes process, and show how this model can be applied to solving many problems in finance, including European and American option pricing (perpetual American options), and Merton portfolio optimization problem. This model is a generalization of some well-known models in finance, such as Cox-Ross-Rubinstein model (1976) (geometric binomial process), Aase model (1988) (geometric compound Poisson process) and geometric Markov renewal model (2013). numerical examples are presented as well

Perpetual American Options with Asset-Dependent Discounting

In this paper we consider the following optimal stopping problem VAω(s)=supτ∈TEs[e-∫0τω(Sw)dwg(Sτ)],\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} V^{\\omega }_{\ ext {A}}(s) = \\sup _{\ au \\in {\\mathcal {T}}} {\\mathbb {E}}_{s}[e^{-\\int _0^\ au \\omega (S_w) dw} g(S_\ au )], \\end{aligned}$$\\end{document}where the process St\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$S_t$$\\end{document} is a jump-diffusion process, T\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {T}}$$\\end{document} is a family of stopping times while g and ω\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\omega $$\\end{document} are fixed payoff function and discount function, respectively. In a financial market context, if g(s)=(K-s)+\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$g(s)=(K-s)^+$$\\end{document} or g(s)=(s-K)+\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$g(s)=(s-K)^+$$\\end{document} and E\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {E}}$$\\end{document} is the expectation taken with respect to a martingale measure, VAω(s)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$V^{\\omega }_{\ ext {A}}(s)$$\\end{document} describes the price of a perpetual American option with a discount rate depending on the value of the asset process St\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$S_t$$\\end{document}. If ω\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\omega $$\\end{document} is a constant, the above problem produces the standard case of pricing perpetual American options. In the first part of this paper we find sufficient conditions for the convexity of the value function VAω(s)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$V^{\\omega }_{\ ext {A}}(s)$$\\end{document}. This allows us to determine the stopping region as a certain interval and hence we are able to identify the form of VAω(s)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$V^{\\omega }_{\ ext {A}}(s)$$\\end{document}. We also present a put-call symmetry for American options with asset-dependent discounting. In the case when St\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$S_t$$\\end{document} is a spectrally negative geometric Lévy process we give exact expressions using the so-called omega scale functions introduced in [30]. We show that the analysed value function satisfies the HJB equation and we give sufficient conditions for the smooth fit property as well. Finally, we present a few examples for which we obtain the analytical form of the value function VAω(s)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${V}^{\\omega }_{\ ext {A}}(s)$$\\end{document}.

On perpetual American options in a multidimensional Black–Scholes model

We consider the problem of pricing perpetual American options written on dividend-paying assets whose price dynamics follow a multidimensional Black and Scholes model. For convex Lipschitz continuous reward functions, we give a probabilistic characterization of the fair price in terms of a reflected BSDE, and an analytical one in terms of an obstacle problem. We also provide the early exercise premium formula.

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.

Optimal timing of investments modeled as perpetual American options in a Levy market

Using a real option approach, this paper models an arbitrary real life investment, which typically has a long maturity date, as a perpetual American call option in a Levy market. Expressions for the moments, characteristic function and infinitesimal generator of the associated jump-diffusion Levy process, defined by two independent compound Poisson processes and two correlated standard Brownian motions, are derived and these fundamental results are employed to determine the optimal time for investment. An application of the results to a Build Operate and Transfer investment is furnished.

Competitively-Issued Convertible Bank Notes in a Theory of Finance: Earl Thompson Meets Fischer Black

Abstract In this paper, I show the validity of and the relationship between two previously unrelated claims in monetary theory. The first claim, made by Earl Thompson, is that privately-issued bank notes pay a positive rate of return in a competitive equilibrium. The second claim, made by Fischer Black, is that it is possible to have a gold standard in which the gold reserves of the central bank are near zero. I show that both of these claims are correct under the assumption of complete markets and perfect commitment. The link between these claims is the Black-Scholes equation applied to convertible bank notes. In commodity-based monetary systems, bank notes are perpetual American options. I extend the model to consider the implications of a lack of commitment on the part of the bank and incomplete markets. I show that both arguments break down when banks lack commitment to redemption or markets are incomplete. I conclude with implications for macroeconomic theory.

Explaining credit ratings through a perpetual-debt structural model

In this paper, we calibrate a perpetual-debt structural model (PDSM) by using Moody’s historical credit ratings. In the PDSM, stocks are equivalent to a portfolio that contains a perpetual American option to default, and bonds are perpetual securities whose face value plays the role of a “notional” capital used to calculate the amount of interest due in the given unit of time. The key question is whether the PDSM can generate (real-world) default probabilities consistent with those historically estimated by Moody’s, under empirically reasonable parameter choices. The answer is yes. The paper also contains an application at the level of a single listed firm: Deutsche Bank. The PDSM risk indicators are used to assign a credit rating to the firm that is consistent with Moody’s scale.

On the Pricing Formula for the Perpetual American Volatility Option Under the Mean-reverting Processes

This paper studies the properties of the parabolic free-boundary problem arising from pricing of American volatility options in mean-reverting volatility processes. When the volatility index follows the mean-reverting square root process (MRSRP), we derive a closed-form pricing formula for the perpetual American power volatility option. Moreover, an artificial neural network (ANN) approach is extended to find an approximate solution of the free boundary problem arising from pricing the perpetual American option. The comparison results demonstrates that the ANN provides an accurate approach to approximate solution for the free boundary problem.

Competitively-Issued Convertible Bank Notes in a Theory of Finance: Earl Thompson Meets Fischer Black

In this paper, I show the validity of and the relationship between two previously unrelated claims in monetary theory. The first claim, made by Earl Thompson, is that privately-issued bank notes pay a positive rate of return in a competitive equilibrium. The second claim, made by Fischer Black, is that it is possible to have a gold standard in which the gold reserves of the central bank are near zero. I first show the validity of Thompson's claim. I then show that a corollary of my result is that Black's claim is correct in a competitive market. The link between these claims is the Black-Scholes equation applied to bank notes. In commodity-based monetary systems, bank notes are perpetual American options. I extend the model to consider the implications of a lack of commitment on the part of the bank and incomplete markets. I show that both arguments break down when banks lack commitment or markets are incomplete. I conclude with implications for macroeconomic theory.

The Valuation of Equity Options in the Perpetual-Debt Structural Model

We propose some formulas for the valuation of equity options in the Perpetual-Debt Structural Model (PDSM), where stockholders have a perpetual American option to default. Our formulas are expressed in terms of binary barrier options, using the results obtained by Rubinstein and Reiner (1991). We also use the results obtained by Barone (2013), where it is proved that perpetual American options follow a geometric Brownian motion, under the standard Black-Scholes-Merton assumptions.

Diffusion transformations, Black–Scholes equation and optimal stopping

We develop a new class of path transformations for one-dimensional diffusions that are tailored to alter their long-run behaviour from transient to recurrent or vice versa. This immediately leads to a formula for the distribution of the first exit times of diffusions, which is recently characterised by Karatzas and Ruf [Probab. Theory Related Fields 164 (2016) 1027–1069] as the minimal solution of an appropriate Cauchy problem under more stringent conditions. A particular limit of these transformations also turn out to be instrumental in characterising the stochastic solutions of Cauchy problems defined by the generators of strict local martingales, which are well known for not having unique solutions even when one restricts solutions to have linear growth. Using an appropriate diffusion transformation, we show that the aforementioned stochastic solution can be written in terms of the unique classical solution of an alternative Cauchy problem with suitable boundary conditions. This in particular resolves the long-standing issue of non-uniqueness with the Black–Scholes equations in derivative pricing in the presence of bubbles. Finally, we use these path transformations to propose a unified framework for solving explicitly the optimal stopping problem for one-dimensional diffusions with discounting, which in particular is relevant for the pricing and the computation of optimal exercise boundaries of perpetual American options.

Tempered stable process, first passage time, and path-dependent option pricing

In this paper, we will discuss an approximation of the characteristic function of the first passage time for a Levy process using the martingale approach. The characteristic function of the first passage time of the tempered stable process is provided explicitly or by an indirect numerical method. This will be applied to the perpetual American option pricing and the barrier option pricing. Numerical illustrations are provided for the calibrated parameters using the market call and put prices.

Quantile Optimization Under Derivative Constraint

This paper studies a new type of quantile optimization problems arising from insurance contract design models. This type of optimization problems is characterized by a constraint of infinity-dimension, that is, the derivatives of the decision quantile functions are bounded. Such a constraint essentially comes from the incentive compatibility constraint for any optimal insurance contract to avoid the potential severe problem of moral hazard in insurance contract design models. By a further development of the author's relaxation method, this paper provides a systemic approach to solving this new type of quantile optimization problems. The optimal quantile is expressed via the solution of a free boundary problem for a second-order nonlinear ordinary differential equation (ODE), which is similar to the Black-Scholes ODE for perpetual American options and has been well studied in literature theoretically and numerically.

Timing options for a startup with early termination and competition risks

This paper analyzes the timing options embedded in a startup firm, and the associated market entry and exit timing decisions under the exogenous risks of early termination and competitor's entry. Our valuation approach leads to the analytical study of a non-standard perpetual American installment option nested with an optimal sequential stopping problem. Explicit formulas are derived for the firm's value functions. Analytically and numerically, we show that early termination risk leads to earlier voluntary entry or exit, and the threat of competition has a non-trivial effect on the firm's entry and abandonment strategies.

Perpetual American options in diffusion-type models with running maxima and drawdowns

We study perpetual American option pricing problems in an extension of the Black–Merton–Scholes model in which the dividend and volatility rates of the underlying risky asset depend on the running values of its maximum and maximum drawdown. The optimal exercise times are shown to be the first times at which the underlying asset hits certain boundaries depending on the running values of the associated maximum and maximum drawdown processes. We obtain closed-form solutions to the equivalent free-boundary problems for the value functions with smooth fit at the optimal stopping boundaries and normal reflection at the edges of the state space of the resulting three-dimensional Markov process. The optimal exercise boundaries for the perpetual American options on the maximum of the market depth with fixed and floating strikes are determined as the minimal solutions of certain first-order nonlinear ordinary differential equations.

Timing Options for a Startup with Early Termination and Competition Risks

This paper analyzes the timing options embedded in a startup firm, and the associated market entry and exit timing decisions under the exogenous risks of early termination and competitor's entry. Our valuation approach leads to the analytical study of a non-standard perpetual American installment option nested with an optimal sequential stopping problem. Explicit formulas are derived for the firm's value functions. Analytically and numerically, we show that early termination risk leads to earlier voluntary entry or exit, and the threat of competition has a non-trivial effect on the firm's entry and abandonment strategies.

尤拉方程的两个自由边界问题的相容性

本文将永久美式期权的自由边界问题归结为在半无界区域具有多个（或单个）奇异点的边值问题来研究，引入广义特征函数法获得了多个奇异点的数学模型的精确解。只有一个奇异点的情形，所得到的解函数在奇异点处取最大值；并得到了左、右自由边界问题同时有一致解的相容性条件。证明了在相容性条件下，左、右自由边界点与奇异点三点合一，从而左、右自由边界点与奇异点都是永久美式期权最佳实施边界点。具有多个奇异点的情形，获得了判断左、右自由边界点成为最佳或较佳实施边界点的条件。 In this paper, we model the free boundary problem of the Perpetual American Option as boundary value problem with multiple (or single) singular points in the semi infinite domain, and introduce the generalized characteristic function method to be able to obtain the exact solution of the mathematical model of multiple singular point. In the single singular point case, our solution function takes the maximum value at the singular point. We deduce the consistency condition of the left and right free boundary problem. Under the compatibility condition, the three points, the left and right free boundary points and singular point are the same, so that they all are the optimal implementation point of the Perpetual American Option. In the case of multiple singular points, the conditional judgment of the left and right free boundary points to be the optimal or nearly optimal implementation point is obtained.

Pricing Perpetual American Compound Options under a Matrix-Exponential Jump-Diffusion Model

ABSTRACTThis paper considers the problem of pricing perpetual American compound options under a matrix-exponential jump-diffusion model. The rational prices of these options are defined as the value functions of the corresponding optimal stopping problems. The general optimal stopping theory and the averaging method for solving the optimal stopping problems are applied to find the value functions and the optimal stopping times and thereby to determine the rational prices and the optimal boundaries of these perpetual American compound options. Explicit formulae for the rational prices and the optimal boundaries are also obtained for hyper-exponential jump-diffusion models.

Pricing perpetual American options under multiscale stochastic elasticity of variance

This paper studies pricing the perpetual American options under a constant elasticity of variance type of underlying asset price model where the constant elasticity is replaced by a fast mean-reverting Ornstein–Ulenbeck process and a slowly varying diffusion process. By using a multiscale asymptotic analysis, we find the impact of the stochastic elasticity of variance on the option prices and the optimal exercise prices with respect to model parameters. Our results enhance the existing option price structures in view of flexibility and applicability through the market prices of elasticity risk.

Optimal Stopping Problems in Diffusion-Type Models with Running Maxima and Drawdowns

We study optimal stopping problems related to the pricing of perpetual American options in an extension of the Black-Merton-Scholes model in which the dividend and volatility rates of the underlying risky asset depend on the running values of its maximum and maximum drawdown. The optimal stopping times of the exercise are shown to be the first times at which the price of the underlying asset exits some regions restricted by certain boundaries depending on the running values of the associated maximum and maximum drawdown processes. We obtain closed-form solutions to the equivalent free-boundary problems for the value functions with smooth fit at the optimal stopping boundaries and normal reflection at the edges of the state space of the resulting three-dimensional Markov process. We derive first-order nonlinear ordinary differential equations for the optimal exercise boundaries of the perpetual American standard options.