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On the Laplace Transforms of the First Hitting Times for Drawdowns and Drawups of Diffusion-Type Processes
Get PDFWe obtain closed-form expressions for the value of the joint Laplace transform of therunning maximum and minimum of a diffusion-type process stopped at the first time at which theassociated drawdown or drawup process hits a constant level before an independent exponentialrandom time. It is assumed that the coefficients of the diffusion-type process are regular functionsof the current values of its running maximum and minimum. The proof is based on the solution tothe equivalent inhomogeneous ordinary differential boundary-value problem and the applicationof the normal-reflection conditions for the value function at the edges of the state space of theresulting three-dimensional Markov process. The result is related to the computation of probabilitycharacteristics of the take-profit and stop-loss values of a market trader during a given time period.</jats:p
Optimal stopping problems in mathematical finance
No full textThis thesis is concerned with the pricing of American-type contingent claims. First, the explicit solutions to the perpetual American compound option pricing problems in the Black-Merton-Scholes model for financial markets are presented. Compound options are financial contracts which give their holders the right (but not the obligation) to buy or sell some other options at certain times in the future by the strike prices given. The method of proof is based on the reduction of the initial two-step optimal stopping problems for the underlying geometric Brownian motion to appropriate sequences of ordinary one-step problems. The latter are solved through their associated one-sided free-boundary problems and the subsequent martingale verification for ordinary differential operators. The closed form solution to the perpetual American chooser option pricing problem is also obtained, by means of the analysis of the equivalent two-sided free-boundary problem. Second, an extension of the Black-Merton-Scholes model with piecewise-constant dividend and volatility rates is considered. The optimal stopping problems related to the pricing of the perpetual American standard put and call options are solved in closed form. The method of proof is based on the reduction of the initial optimal stopping problems to the associated free-boundary problems and the subsequent martingale verification using a local time-space formula. As a result, the explicit algorithms determining the constant hitting thresholds for the underlying asset price process, which provide the optimal exercise boundaries for the options, are presented. Third, the optimal stopping games associated with perpetual convertible bonds in an extension of the Black-Merton-Scholes model with random dividends under different information flows are studied. In this type of contracts, the writers have a right to withdraw the bonds before the holders can exercise them, by converting the bonds into assets. The value functions and the stopping boundaries' expressions are derived in closed-form in the case of observable dividend rate policy, which is modelled by a continuous-time Markov chain. The analysis of the associated parabolic-type free-boundary problem, in the case of unobservable dividend rate policy, is also presented and the optimal exercise times are proved to be the first times at which the asset price process hits boundaries depending on the running state of the filtering dividend rate estimate. Moreover, the explicit estimates for the value function and the optimal exercise boundaries, in the case in which the dividend rate is observable by the writers but unobservable by the holders of the bonds, are presented. Finally, the 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 its maximum drawdown, are studied. The latter process represents the difference between the running maximum and the current asset value. The optimal stopping times for exercising 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. The closed-form solutions to the equivalent free-boundary problems for the value functions are obtained 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 of the perpetual American call, put and strangle options are obtained as solutions of arithmetic equations and first-order nonlinear ordinary differential equations
Optimal stopping problems in mathematical finance
No full textThis thesis is concerned with the pricing of American-type contingent claims. First, the explicit solutions to the perpetual American compound option pricing problems in the Black-Merton-Scholes model for financial markets are presented. Compound options are financial contracts which give their holders the right (but not the obligation) to buy or sell some other options at certain times in the future by the strike prices given. The method of proof
is based on the reduction of the initial two-step optimal stopping problems for the underlying geometric Brownian motion to appropriate sequences of ordinary one-step problems. The latter are solved through their associated one-sided free-boundary problems and the subsequent martingale verification for ordinary differential operators. The closed form solution to the perpetual
American chooser option pricing problem is also obtained, by means of the analysis of the equivalent two-sided free-boundary problem. Second, an extension of the Black-Merton-Scholes model with piecewise-constant dividend
and volatility rates is considered. The optimal stopping problems related to the pricing of the perpetual American standard put and call options are solved in closed form. The method of proof is based on the reduction of the initial optimal stopping problems to the associated free-boundary problems and the subsequent martingale verification using a local time-space formula. As a result, the explicit algorithms determining the constant hitting thresholds for the underlying asset price process, which provide the optimal exercise boundaries for the options,
are presented. Third, the optimal stopping games associated with perpetual convertible bonds in an extension of the Black-Merton-Scholes model with random dividends under different information flows are studied. In this type of contracts, the writers have a right to withdraw the bonds
before the holders can exercise them, by converting the bonds into assets. The value functions and the stopping boundaries' expressions are derived in closed-form in the case of observable dividend rate policy, which is modelled by a continuous-time Markov chain. The analysis of the associated parabolic-type free-boundary problem, in the case of unobservable dividend rate policy, is also presented and the optimal exercise times are proved to be the first times at which the asset price process hits boundaries depending on the running state of the filtering dividend rate estimate. Moreover, the explicit estimates for the value function and the optimal exercise boundaries, in the case in which the dividend rate is observable by the writers but unobservable by the holders of the bonds, are presented. Finally, the 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 its maximum
drawdown, are studied. The latter process represents the difference between the running maximum and the current asset value. The optimal stopping times for exercising 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. The closed-form solutions to the equivalent free-boundary problems for the value functions are obtained 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 of the perpetual American call, put and strangle options are obtained as solutions of arithmetic equations and first-order nonlinear ordinary differential equations
On the Singular Control of a Diffusion and Its Running Infimum or Supremum
No full textFerrari G, Rodosthenous N. On the Singular Control of a Diffusion and Its Running Infimum or Supremum. Center for Mathematical Economics Working Papers. Vol 745. Bielefeld: Center for Mathematical Economics; 2025.We study a class of singular stochastic control problems for a one-dimensional
diffusion in which the performance criterion to be optimised depends explicitly on the running
infimum (or supremum ) of the controlled process. We introduce two novel integral operators
that are consistent with the Hamilton-Jacobi-Bellman equation for the resulting two-dimensional
singular control problems. The first operator involves integrals where the integrator is the
control process of the two-dimensional process or ; the second operator concerns
integrals where the integrator is the running infimum or supremum process itself. Using these
definitions, we prove a general verification theorem for problems involving two-dimensional
state-dependent running costs, costs of controlling the process, costs of increasing the running
infimum (or supremum) and exit times. Finally, we apply our results to explicitly solve an
optimal dividend problem in which the manager’s time-preferences depend on the company’s
historical worst performance.MSC2020 subject classification: 93E20; 60J60; 49L12; 91B7
Optimal management of debt-to-GDP ratio with regime-switching interest rate
Get PDFWe consider the problem of a government that wants to manage the country's debt-to- GDP (gross domestic product) ratio. The latter evolves stochastically in continuous time, and its drift is given by the interest rate on government debt, net of the growth rate of GDP. We further allow the interest rate to be affected by an exogenous macroeconomic risk process modelled by a continuous-time Markov chain with N states. The debt-to-GDP ratio level can be reduced by the government, e.g. through austerity policies in the form of spending cuts, or increased, e.g. by public investments. The aim of the government is to choose a policy which minimises the total expected cost of having debt plus the total expected cost of austerity policies, counterbalanced by the total expected gain from public investments. We model this as a bounded-variation stochastic control problem over an infinite time-horizon with regime switching, and we provide its explicit solution. To the best of our knowledge, such a problem has not been previously solved in the literature. We show that it is optimal for the government to adopt a policy that keeps the debt-to-GDP ratio in an interval, whose boundaries are depending on the states of the risk process, and are given through a zero-sum optimal stopping game with regime switching. We completely characterise these boundaries as solutions to a system of nonlinear algebraic equations with constraints. Finally, we put in practice our methodology in a case study of a Markov chain with N = 2 states; we provide a thorough analysis and we complement our theoretical results by a detailed numerical study on the sensitivity of the optimal debt ratio management policy with respect to the model's parameters
Optimal stopping problems in diffusion-type models with running maxima and drawdowns
No full textWe 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
On the drawdowns and drawups in diffusion-type models with running maxima and minima
Get PDFWe obtain closed-form expressions for the values of joint Laplace transforms of the running maximum and minimum of a diffusion-type process stopped at the first time at which the associated drawdown and drawup processes hit constant levels. It is assumed that the coefficients of the diffusion-type process are regular functions of the running values of the process itself, its maximum and minimum, as well as its maximum drawdown and maximum drawup processes. The proof is based on the solution to the equivalent boundary-value problems and application of the normal-reflection conditions for the value functions at the edges of the state space of the resulting five-dimensional Markov process. We show that the joint Laplace transforms represent linear combinations of solutions to the systems of first-order partial differential equations arising from the application of the normal-reflection conditions
Going Beyond Counting First Authors in Author Co-citation Analysis
Get PDFThe present study examines one of the fundamental aspects of author co-citation analysis (ACA) - the way co-citation
counts are defined. Co-citation counting provides the data on which all subsequent statistical analyses and mappings
are based, and we compare ACA results based on two different types of co-citation counting - the traditional type that
only counts the first one among a cited work's authors on the one hand and a non-traditional type that takes into
account the first 5 authors of a cited work on the other hand. Results indicate that the picture produced through this non-traditional author co-citation counting contains more coherent author groups and is therefore considerably clearer. However, this picture represents fewer specialties in the research field being studied than that produced through the traditional first-author co-citation counting when the same number of top-ranked authors is selected and analyzed. Reasons for these effects are discussed
Variations on the Author
Get PDF“Variations on the Author” discusses two of Eduardo Coutinho’s recent films (Um Dia na Vida, from 2010, and Últimas Conversas, posthumously released in 2015) and their contribution to the general question of documentary authorship. The director’s filmography is characterized by a consistent yet self-effacing form of authorial self-inscription: Coutinho often features as an interviewer that rather than express opinions propels discourses; an interviewer that is good at listening. This mode of self-inscription characterizes him as an author who is not expressive but who is nonetheless markedly present on the screen. In Um Dia na Vida, however, Coutinho is completely absent form the image, while Últimas Conversas, on the contrary, includes a confessional prologue that moves the director from the margins to the center of his films. This article examines the ways in which these works stand out in the filmography of a director who offers new insights into the notion of cinematic authorship
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