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76
Russian and American put options under exponential phasetype Lévy models
, 2002
"... Consider the American put and Russian option [33, 34, 17] with the stock price modeled as an exponential Lévy process. We find an explicit expression for the price in the dense class of Lévy processes with phasetype jumps in both directions. The solution rests on the reduction to the first passage ..."
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Cited by 96 (4 self)
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Consider the American put and Russian option [33, 34, 17] with the stock price modeled as an exponential Lévy process. We find an explicit expression for the price in the dense class of Lévy processes with phasetype jumps in both directions. The solution rests on the reduction to the first passage time problem for (reflected) Lévy processes and on an explicit solution of the latter in the phasetype case via martingale stopping and WienerHopf factorisation. Also the first passage time problem is studied for a regime switching Lávy process with phasetype jumps. This is achieved by an embedding into a a semiMarkovian regime switching Brownian motion.
Smoothness of scale functions for spectrally negative Lévy processes
, 2006
"... Scale functions play a central role in the fluctuation theory of spectrally negative Lévy processes and often appear in the context of martingale relations. These relations are often complicated to establish requiring excursion theory in favour of Itô calculus. The reason for the latter is that stan ..."
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Cited by 86 (19 self)
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Scale functions play a central role in the fluctuation theory of spectrally negative Lévy processes and often appear in the context of martingale relations. These relations are often complicated to establish requiring excursion theory in favour of Itô calculus. The reason for the latter is that standard Itô calculus is only applicable to functions with a sufficient degree of smoothness and knowledge of the precise degree of smoothness of scale functions is seemingly incomplete. The aim of this article is to offer new results concerning properties of scale functions in relation to the smoothness of the underlying Lévy measure. We place particular emphasis on spectrally negative Lévy processes with a Gaussian component and processes of bounded variation. An additional motivation is the very intimate relation of scale functions to renewal functions of subordinators. The results obtained for scale functions have direct implications offering new results concerning the smoothness of such renewal functions for which there seems to be very little existing literature on this topic.
Exit Problems for Spectrally Negative Lévy Processes and Applications to Russian, American and Canadized Options
 Ann. Appl. Probab
"... this paper we consider the class of spectrally negative L'evy processes. These are real valued random processes with stationary independent increments which have no positive jumps. Amongst others Emery [11], Suprun [23], Bingham [4] and Bertoin [3] have all considered fluctuation theory for thi ..."
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Cited by 78 (24 self)
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this paper we consider the class of spectrally negative L'evy processes. These are real valued random processes with stationary independent increments which have no positive jumps. Amongst others Emery [11], Suprun [23], Bingham [4] and Bertoin [3] have all considered fluctuation theory for this class of processes. Such processes are often considered in the context of the theories of dams, queues, insurance risk and continuous branching processes; see for example [6, 4, 5, 19]. Following the exposition on two sided exit problems in Bertoin [3] we study first exit from an interval containing the origin for spectrally negative L'evy processes reflected in their supremum (equivalently spectrally positive L'evy processes reflected in their infimum). In particular we derive the joint Laplace transform of the time to first exit and the overshoot. The aforementioned stopping time can be identified in the literature of fluid models as the time to buffer overflow (see for example [1, 13]). Together Universit'e de Pau, email: Florin.Avram@univpau.fr y Utrecht University, email: kyprianou@math.uu.nl z Utrecht University, email: pistorius@math.uu.nl 1 with existing results on exit problems we apply our results to certain optimal stopping problems that are now classically associated with mathematical finance. In sections 2 and 3 we introduce notation and discuss and develop existing results concerning exit problems of spectrally negative L'evy processes. In section 4 an expression is derived for the joint Laplace transform of the exit time and exit position of the reflected process from an interval containing the origin. This Laplace transform can be written in terms of scale functions that already appear in the solution to the two sided exit problem. In Section 5 we ou...
Risk vs. ProfitPotential; A Model for Corporate Strategy
 J. Econ. Dynam. Control
, 1996
"... A firm whose net earnings are uncertain, and that is subject to the risk of bankruptcy, must choose between paying dividends and retaining earnings in a liquid reserve. Also, different operating strategies imply different combinations of expected return and variance. We model the firm's cash re ..."
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Cited by 48 (1 self)
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A firm whose net earnings are uncertain, and that is subject to the risk of bankruptcy, must choose between paying dividends and retaining earnings in a liquid reserve. Also, different operating strategies imply different combinations of expected return and variance. We model the firm's cash reserve as the difference between the cumulative net earnings and the cumulative dividends. The first is a diffusion (additive), whose drift/volatility pair is chosen dynamically from a finite set, A. The second is an arbitrary nondecreasing process, chosen by the firm. The firm's strategy must be nonclairvoyant. The firm is bankrupt at the first time, T , at which the cash reserve falls to zero (T may be infinite), and the firm's objective is to maximize the expected total discounted dividends from 0 to T , given an initial reserve, x; denote this maximum by V (x). We calculate V explicitly, as a function of the set A and the discount rate. The optimal policy has the form: (1) pay no dividends if ...
043 "An Iteration Procedure for Solving Integral Equations Related to Optimal Stopping Problems" by Denis Belomestny and Pavel
, 2006
"... We present an iterative algorithm for computing values of optimal stopping problems for onedimensional diffusions on finite time intervals. The method is based on a time discretisation of the initial model and a construction of discretised analogues of the associated integral equation for the valu ..."
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Cited by 35 (1 self)
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We present an iterative algorithm for computing values of optimal stopping problems for onedimensional diffusions on finite time intervals. The method is based on a time discretisation of the initial model and a construction of discretised analogues of the associated integral equation for the value function. The proposed iterative procedure converges in a finite number of steps and delivers in each step a lower or an upper bound for the discretised value function on the whole time interval. We also give remarks on applications of the method for solving the integral equations related to several optimal stopping problems. 1
The Russian option: Finite horizon
 Finance Stoch
, 2005
"... We show that the optimal stopping boundary for the Russian option with finite horizon can be characterized as the unique solution of a nonlinear integral equation arising from the early exercise premium representation (an explicit formula for the arbitragefree price in terms of the optimal stopping ..."
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Cited by 25 (6 self)
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We show that the optimal stopping boundary for the Russian option with finite horizon can be characterized as the unique solution of a nonlinear integral equation arising from the early exercise premium representation (an explicit formula for the arbitragefree price in terms of the optimal stopping boundary having a clear economic interpretation). The results obtained stand in a complete parallel with the best known results on the American put option with finite horizon. The key argument in the proof relies upon a local timespace formula. 1.
A note on optimal stopping of regular diffusions under random discounting. Theory of Probability and its Applications
, 2002
"... Summary. Let X be a onedimensional regular diffusion, A a positive continuous additive functional of X, and h a measurable realvalued function. A method is proposed to determine a stopping rule T ∗ that maximizes E{e−ATh(XT)1{T<∞}} over all stopping times T of X. Several examples, some related ..."
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Cited by 23 (1 self)
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Summary. Let X be a onedimensional regular diffusion, A a positive continuous additive functional of X, and h a measurable realvalued function. A method is proposed to determine a stopping rule T ∗ that maximizes E{e−ATh(XT)1{T<∞}} over all stopping times T of X. Several examples, some related to Mathematical Finance, are discussed. AMS 1991 subject classifications. 60G40, 60J60.
Some Optimal Stopping Problems With NonTrivial Boundaries for Pricing Exotic Options
 J. Appl. Probab
, 2001
"... We solve the following three optimal stopping problems for dierent kinds of options, based on the BlackScholes model of stock uctuations: (i) The perpetual lookback American option for the running maximum of the stock price during the life of the option. This problem is more dicult than the closely ..."
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Cited by 20 (1 self)
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We solve the following three optimal stopping problems for dierent kinds of options, based on the BlackScholes model of stock uctuations: (i) The perpetual lookback American option for the running maximum of the stock price during the life of the option. This problem is more dicult than the closely related one for the Russian option and we show that for a class of utility functions the free boundary is governed by a nonlinear ordinary dierential equation. (ii) A new type of stock option for a company, where the company provides a guaranteed minimum as an added incentive in case the market appreciation of the stock is low, thereby making the option more attractive to the employee. We show that the value of this option is given by solving a nonalgebraic equation. (iii) A new call option for the option buyer who is riskaverse and gets to choose, a priori, a xed constant l as a \hedge" on a possible downturn of the stock price, where the buyer gets the maximum of l and the price at ...
Inside Information And Stock Fluctuations
, 1999
"... A model of an incomplete market with the incorporation of a new notion of "inside information" is posed. The usual assumption that the stock price is Markovian is modified by adjoining a hidden Markov process to the BlackScholes exponential Brownian motion model for stock fluctuations. ..."
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Cited by 13 (5 self)
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A model of an incomplete market with the incorporation of a new notion of "inside information" is posed. The usual assumption that the stock price is Markovian is modified by adjoining a hidden Markov process to the BlackScholes exponential Brownian motion model for stock fluctuations. The drift and volatility parameters take different values when the hidden Markov process is in different states. For example, it is 0 when there is no subset of the market which has or which believes it has, extra information. However, the hidden process is in state 1 when information is not equally shared by all, and then the behavior of the members in the subset causes increased fluctuations in the stock price. This model
Discounted optimal stopping problems for the maximum process
 J. Appl. Probab
, 2000
"... The maximality principle [6] is shown to be valid in some examples of discounted optimal stopping problems for the maximum process. In each of these examples explicit formulas for the value functions are derived and the optimal stopping times are displayed. In particular, in the framework of the Bla ..."
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Cited by 13 (0 self)
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The maximality principle [6] is shown to be valid in some examples of discounted optimal stopping problems for the maximum process. In each of these examples explicit formulas for the value functions are derived and the optimal stopping times are displayed. In particular, in the framework of the BlackScholes model, the fair prices of two lookback options with infinite horizon are calculated. The main aim of the paper is to show that in each considered example the optimal stopping boundary satisfies the maximality principle and that the value function can be determined explicitly. 1.