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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 89 (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.
Optimal stopping and perpetual options for Lévy processes
, 2000
"... Solution to the optimal stopping problem for a L'evy process and reward functions (e x \Gamma K) + and (K \Gamma e x ) + , discounted at a constant rate is given in terms of the distribution of the overall supremum and infimum of the process killed at this rate. Closed forms of this sol ..."
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Cited by 65 (7 self)
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Solution to the optimal stopping problem for a L'evy process and reward functions (e x \Gamma K) + and (K \Gamma e x ) + , discounted at a constant rate is given in terms of the distribution of the overall supremum and infimum of the process killed at this rate. Closed forms of this solutions are obtained under the condition of positive jumps mixedexponentially distributed. Results are interpreted as admissible pricing of perpetual American call and put options on a stock driven by a L'evy process, and a BlackScholes type formula is obtained. Keywords and Phrases: Optimal stopping, L'evy process, mixtures of exponential distributions, American options, Derivative pricing. JEL Classification Number: G12 Mathematics Subject Classification (1991): 60G40, 60J30, 90A09. 1 Introduction and general results 1.1 L'evy processes Let X = fX t g t0 be a real valued stochastic process defined on a stochastic basis(\Omega ; F ; F = (F t ) t0 ; P ) that satisfy the usual conditions. A...
WienerHopf factorization for Lévy processes having positive jumps with rational transforms
 Journal of Applied Probability Vol.45
"... We give the closed form of the ruin probability for a Lévy processes, possibly killed at a constant rate, with completely arbitrary positive distributed jumps, and finite intensity negative jumps with distribution characterized by having a rational Laplace or Fourier transform. Abbreviated Title: ..."
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Cited by 20 (0 self)
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We give the closed form of the ruin probability for a Lévy processes, possibly killed at a constant rate, with completely arbitrary positive distributed jumps, and finite intensity negative jumps with distribution characterized by having a rational Laplace or Fourier transform. Abbreviated Title: WHfactors of Lévy processes with rational jumps. 1
The distribution of the maximum of a Lévy process with positive jumps of phasetype
 Theory Stoch. Proc
, 2002
"... Consider a Lévy process with finite intensity positive jumps of the phasetype and arbitrary negative jumps. Assume that the process either is killed at a constant rate or drifts to −∞. We show that the distribution of the overall maximum of this process is also of phasetype, and find the distribu ..."
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Cited by 12 (1 self)
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Consider a Lévy process with finite intensity positive jumps of the phasetype and arbitrary negative jumps. Assume that the process either is killed at a constant rate or drifts to −∞. We show that the distribution of the overall maximum of this process is also of phasetype, and find the distribution of this random variable. Previous results (hyperexponential positive jumps) are obtained as a particular case.
Perpetual options for Lévy processes in the Bachelier model
 Proceedings of the Steklov Mathematical Institute
, 2001
"... Solution to the optimal stopping problem V (x) = sup τ Ee−δτg(x+Xτ) is given, where X = {Xt}t≥0 is a Lévy process, τ is an arbitrary stopping time, δ ≥ 0 is a discount rate, and the reward function g takes the form gc(x) = (x−K)+ or gp(x) = (K−x)+ Results, interpreted as option prices of perpetu ..."
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Cited by 1 (1 self)
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Solution to the optimal stopping problem V (x) = sup τ Ee−δτg(x+Xτ) is given, where X = {Xt}t≥0 is a Lévy process, τ is an arbitrary stopping time, δ ≥ 0 is a discount rate, and the reward function g takes the form gc(x) = (x−K)+ or gp(x) = (K−x)+ Results, interpreted as option prices of perpetual options in Bachelier’s model are expressed in terms of the distribution of the overall supremum in case g = gc and overall infimum in case g = gp of the process X killed at rate δ. Closed form solutions are obtained under mixed exponentially distributed positive jumps with arbitrary negative jumps for gc, and under arbitrary positive jumps and mixed exponentially distributed negative jumps for gp. In case g = gc a prophet inequality comparing prices of perpetual lookback call options and perpetual call options is obtained.
Convexity of Ruin Probability and Optimal Dividend Strategies for a General Lévy Process
, 2011
"... ..."
Russian and American put options under
"... Consider the American put and Russian option [46,47,22] 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 phase–type jumps in both directions. The solution rests on the reduction to the first passage ..."
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Consider the American put and Russian option [46,47,22] 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 phase–type 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 phase–type case via martingale stopping and WienerHopf factorisation. The same type of approach is also applied to the more general class of regime switching Lévy processes with phasetype jumps.
unknown title
"... the unified form of pollaczek–khinchine formula for lévy processes with matrixexponential negative jumps D. Gusak, Ie. Karnaukh For Lévy processes with matrixexponential negative jumps, the unified form of the PollaczekKhinchine formula is established. 1 ..."
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the unified form of pollaczek–khinchine formula for lévy processes with matrixexponential negative jumps D. Gusak, Ie. Karnaukh For Lévy processes with matrixexponential negative jumps, the unified form of the PollaczekKhinchine formula is established. 1