Search Results (14)

In this paper, we present a fast and accurate numerical approach applied to specific American-style derivatives, namely American power call and put options, whose main feature is that the underlying asset is raised to a power. The study is set in the Black–Scholes framework, and we consider continuously paying dividends assets and arbitrary positive values for the power. It is important to note that although a log-normal process raised to a power is again log-normal, the resulting change in variables may lead to a negative dividend rate, and this case remains largely understudied in the literature. We derive closed-form formulas for the perpetual options’ optimal boundaries and for the fair prices. For finite maturities, we approximate the optimal boundary using some first-hitting properties of the Brownian motion. As a consequence, we obtain the option price quickly and with relatively high accuracy—the error is at the third decimal position. We further provide a comprehensive analysis of the impact of the parameters on the options’ value, and discuss ordinary European and American capped options. Various numerical examples are provided. Full article

In this paper, we study perpetual American put options with a generalized standard put payoff and establish sufficient conditions for the existence and uniqueness of the solution to the associated pricing problem. As a key tool, we express the Black–Scholes operator in terms of elasticity. This formulation enables us to demonstrate that the considered pricing problem admits a unique solution when the payoff function exhibits strictly decreasing elasticity with respect to the underlying asset. Furthermore, this approach allows us to derive closed-form solutions for option pricing. Full article

Risk is often the most concerning factor in financial transactions. Option Greeks provide valuable insights into the risks inherent in option trading and serve as tools for risk mitigation. Traditionally, scholars compute option Greeks through extensive calculations. This article introduces an alternative method that bypasses conventional derivative computation using the Mellin transform and its properties. Specifically, we derive Greeks for perpetual American options with general payoffs, represented as piecewise linear functions, and examine higher-order risk metrics for practical implications. Examples are provided to illustrate the effectiveness of our approach, offering a novel perspective on calculating and interpreting option Greeks. Full article

The impact of racial healthcare disparities has been well documented. Adverse social determinants of health, such as poverty, inadequate housing, and limited access to healthcare, are intricately linked to these disparities and negative health outcomes, highlighting the profound impact that social and economic factors have on individuals’ overall well-being. Recent evidence underscores the role of residential location on individual health outcomes. Despite the importance of a healthy lifestyle, regular physical activity, balanced nutrition, and stress management for favorable health outcomes, individuals living in socioeconomically disadvantaged areas may face obstacles in achieving these practices. Adverse living conditions, environmental factors, and systemic biases against Black Americans perpetuate allostatic load. This, compounded by decreased physical activity and limited healthy food options, may contribute to increased oxidative stress and inflammation, fundamental drivers of morbidities such as cardiovascular disease and cancer. Herein, we perform a narrative review of associations between healthcare disparities, chronic stress, allostatic load, inflammation, and cancer in Black Americans, and we discuss potential mechanisms and solutions. Additional research is warranted in the very important area of cancer disparities. Full article

The aim of this paper is to examine some American-style financial instruments that lead to two-sided optimal hitting problems. We pay particular attention to derivatives that are similar to strangle options but have a quadratic payoff function. We consider these derivatives in light of much more general payoff structures under certain conditions which guarantee that the optimal strategy is an exit from a strip. Closed-form formulas for the optimal boundaries and the fair price are derived when the contract has no maturity constraints. We obtain the form of the optimal boundaries under the finite maturity horizon and approximate them by maximizing the financial utility of the derivative holder. The Crank–Nicolson finite difference method is applied to the pricing problem. The importance of these novel financial instruments is supported by several features that are very useful for financial practice. They combine the characteristics of the power options and the ordinary American straddles. Quadratic strangles are suitable for investors who need to hedge strongly, far from the strike positions. In contrast, the near-the-money positions offer a relatively lower payoff than the ordinary straddles. Note that the usual options pay exactly the overprice; no more, no less. In addition, the quadratic strangles allow investors to hedge the positions below and above the strike together. This is very useful in periods of high volatility when large market movements are expected but their direction is unknown. Full article

A new explicit form is provided for the solution of optimal stopping problems involving a multidimensional geometric Brownian motion. A free-boundary value approach is adopted and the value function is obtained via fundamental solution methods. There are many applications for the valuation of perpetual options of American style, which are of interest for finance and managerial decisions. Full article

We derive the explicit price of the perpetual American put option canceled at the last-passage time of the underlying above some fixed level. We assume that the asset process is governed by a geometric spectrally negative Lévy process. We show that the optimal exercise time is the first moment when the asset price process drops below an optimal threshold. We perform numerical analysis considering classical Black–Scholes models and the model where the logarithm of the asset price has additional exponential downward shocks. The proof is based on some martingale arguments and the fluctuation theory of Lévy processes. Full article

The main objective of this paper is to present an algorithm of pricing perpetual American put options with asset-dependent discounting. The value function of such an instrument can be described as $V_{{\mathrm{A}}^{\mathrm{Put}}}^{\omega }(s)={sup}_{\tau \in \mathcal{T}}{\mathbb{E}}_s[e^{-{\int }_0^{\tau }\omega (S_w)dw}{(K-S_{\tau })}^{+}],$ where $\mathcal{T}$ is a family of stopping times, $\omega$ is a discount function and $\mathbb{E}$ is an expectation taken with respect to a martingale measure. Moreover, we assume that the asset price process $S_t$ is a geometric Lévy process with negative exponential jumps, i.e., $S_t=s{e}^{\zeta t+\sigma B_t-{\sum }_{i=1}^{N_t}{Y}_i}$. The asset-dependent discounting is reflected in the $\omega$ function, so this approach is a generalisation of the classic case when $\omega$ is constant. It turns out that under certain conditions on the $\omega$ function, the value function $V_{{\mathrm{A}}^{\mathrm{Put}}}^{\omega }(s)$ is convex and can be represented in a closed form. We provide an option pricing algorithm in this scenario and we present exact calculations for the particular choices of $\omega$ such that $V_{{\mathrm{A}}^{\mathrm{Put}}}^{\omega }(s)$ takes a simplified form. Full article

We derive explicit solutions to the perpetual American cancellable standard put and call options in an extension of the Black–Merton–Scholes model. It is assumed that the contracts are cancelled at the last hitting times for the underlying asset price process of some constant upper or lower levels which are not stopping times with respect to the observable filtration. We show that the optimal exercise times are the first times at which the asset price reaches some lower or upper constant levels. The proof is based on the reduction of the original optimal stopping problems to the associated free-boundary problems and the solution of the latter problems by means of the smooth-fit conditions. Full article

Managing unemployment is one of the key issues in social policies. Unemployment insurance schemes are designed to cushion the financial and morale blow of loss of job but also to encourage the unemployed to seek new jobs more proactively due to the continuous reduction of benefit payments. In the present paper, a simple model of unemployment insurance is proposed with a focus on optimality of the individual’s entry to the scheme. The corresponding optimal stopping problem is solved, and its similarity and differences with the perpetual American call option are discussed. Beyond a purely financial point of view, we argue that in the actuarial context the optimal decisions should take into account other possible preferences through a suitable utility function. Some examples in this direction are worked out. Full article

As decarbonisation progresses and conventional thermal generation gradually gives way to other technologies including intermittent renewables, there is an increasing requirement for system balancing from new and also fast-acting sources such as battery storage. In the deregulated context, this raises questions of market design and operational optimisation. In this paper, we assess the real option value of an arrangement under which an autonomous energy-limited storage unit sells incremental balancing reserve. The arrangement is akin to a perpetual American swing put option with random refraction times, where a single incremental balancing reserve action is sold at each exercise. The power used is bought in an energy imbalance market (EIM), whose price we take as a general regular one-dimensional diffusion. The storage operator’s strategy and its real option value are derived in this framework by solving the twin timing problems of when to buy power and when to sell reserve. Our results are illustrated with an operational and economic analysis using data from the German Amprion EIM. Full article

Oral health is an important yet often neglected component of overall health, linked to heart disease, stroke, and diabetic complications. Disparities exist for many groups, including racial and ethnic minorities such as African Americans. The purpose of this study was to examine the potential factors that perpetuate oral health care disparities in African American children in the United States. A systematic search of three literature databases produced 795 articles; 23 articles were included in the final review. Articles were analyzed using a template coding approach based on the social ecological model. The review identified structural, sociocultural, and familial factors that impact the ability of African Americans to utilize oral care services, highlighting the importance of the parent/caregiver role and the patient–provider relationship; policy-level processes that impact access to quality care; the value of autonomy in treatment and prevention options; and the impact of sociocultural factors on food choices (e.g., food deserts, gestures of affection). In conclusion, oral health care remains an underutilized service by African American children, despite increasing access to oral care secondary to improvements in insurance coverage and community-based programs. Full article

We present closed-form solutions to the perpetual American dividend-paying put and call option pricing problems in two extensions of the Black–Merton–Scholes model with random dividends under full and partial information. We assume that the dividend rate of the underlying asset price changes its value at a certain random time which has an exponential distribution and is independent of the standard Brownian motion driving the price of the underlying risky asset. In the full information version of the model, it is assumed that this time is observable to the option holder, while in the partial information version of the model, it is assumed that this time is unobservable to the option holder. The optimal exercise times are shown to be the first times at which the underlying risky asset price process hits certain constant levels. The proof is based on the solutions of the associated free-boundary problems and the applications of the change-of-variable formula. Full article

In this short paper, we study the asymptotics for the price of call options for very large strikes and put options for very small strikes. The stock price is assumed to follow the Black–Scholes models. We analyze European, Asian, American, Parisian and perpetual options and conclude that the tail asymptotics for these option types fall into four scenarios. Full article