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28
OLD AND NEW EXAMPLES OF SCALE FUNCTIONS FOR SPECTRALLY Negative Lévy Processes
, 2009
"... We give a review of the state of the art with regard to the theory of scale functions for spectrally negative Lévy processes. From this we introduce a general method for generating new families of scale functions. Using this method we introduce a new family of scale functions belonging to the Gaussi ..."
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Cited by 27 (11 self)
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We give a review of the state of the art with regard to the theory of scale functions for spectrally negative Lévy processes. From this we introduce a general method for generating new families of scale functions. Using this method we introduce a new family of scale functions belonging to the Gaussian Tempered Stable Convolution (GTSC) class. We give particular emphasis to special cases as well as crossreferencing their analytical behaviour against known general considerations.
On American options under the Variance Gamma process
 Applied Mathematical Finance
, 2004
"... We consider American options in a market where the underlying asset follows a Variance Gamma process. We give a sufficient condition for the failure of the smooth fit principle for finite horizon call options. We also propose a second order accurate finitedifference method to find the American opti ..."
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Cited by 14 (4 self)
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We consider American options in a market where the underlying asset follows a Variance Gamma process. We give a sufficient condition for the failure of the smooth fit principle for finite horizon call options. We also propose a second order accurate finitedifference method to find the American option price and the exercise boundary. The problem is formulated as a Linear Complementarity Problem and numerically solved by a convenient splitting. Computations have been accelerated with the help of the Fast Fourier Transform. A stability analysis shows that the scheme is conditionally stable, with a mild stability condition of the form k = O(log(h)  −1). The theoretical results are verified numerically throughout a series of numerical experiments. Keywords: Integrodifferential equations, Variance Gamma, finite differences, FFT.
Pathwise Inequalities for Local Time: Applications to Skorokhod Embeddings and Optimal Stopping. Annals of Applied Probability
, 2008
"... We develop a class of pathwise inequalities of the form H(Bt) ≥ Mt + F(Lt), whereBt is Brownian motion, Lt its local time at zero and Mt alocal martingale. The concrete nature of the representation makes the inequality useful for a variety of applications. In this work, we use the inequalities to d ..."
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Cited by 11 (1 self)
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We develop a class of pathwise inequalities of the form H(Bt) ≥ Mt + F(Lt), whereBt is Brownian motion, Lt its local time at zero and Mt alocal martingale. The concrete nature of the representation makes the inequality useful for a variety of applications. In this work, we use the inequalities to derive constructions and optimality results of Vallois ’ Skorokhod embeddings. We discuss their financial interpretation in the context of robust pricing and hedging of options written on the local time. In the final part of the paper we use the inequalities to solve a class of optimal stopping problems of the form sup τ E[F(Lτ) − ∫ τ 0 β(Bs)ds]. The solution is given via a minimal solution to a system of differential equations and thus resembles the maximality principle described by Peskir. Throughout, the emphasis is placed on the novelty and simplicity of the techniques. 1. Introduction. The
A Wiener– Hopf Monte Carlo simulation technique for Lévy processes
 Ann. Appl. Probab
, 2011
"... We develop a completely new and straightforward method for simulating the joint law of the position and running maximum at a fixed time of a general Lévy process with a view to application in insurance and financial mathematics. Although different, our method takes lessons from Carr’s socalled “Can ..."
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Cited by 10 (5 self)
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We develop a completely new and straightforward method for simulating the joint law of the position and running maximum at a fixed time of a general Lévy process with a view to application in insurance and financial mathematics. Although different, our method takes lessons from Carr’s socalled “Canadization ” technique as well as Doney’s method of stochastic bounds for
Meromorphic Lévy processes and their fluctuation identities
 Annals of Applied Probability
, 2011
"... The last couple of years has seen a remarkable number of new, explicit examples of the Wiener–Hopf factorization for Lévy processes where previously there had been very few. We mention, in particular, the many cases of spectrally negative Lévy processes in [Sixth Seminar on Stochastic Analysis, ..."
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Cited by 9 (3 self)
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The last couple of years has seen a remarkable number of new, explicit examples of the Wiener–Hopf factorization for Lévy processes where previously there had been very few. We mention, in particular, the many cases of spectrally negative Lévy processes in [Sixth Seminar on Stochastic Analysis,
Principle of smooth fit and diffusions with angles
 Stochastics
, 2007
"... We show that there exists a regular diffusion process X and a differentiable gain function G such that the value function V of the optimal stopping problem V (x) = sup ExG(Xτ) τ fails to satisfy the smooth fit condition V ′(b) = G ′(b) at the optimal stopping point b. On the other hand, if the sca ..."
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Cited by 6 (1 self)
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We show that there exists a regular diffusion process X and a differentiable gain function G such that the value function V of the optimal stopping problem V (x) = sup ExG(Xτ) τ fails to satisfy the smooth fit condition V ′(b) = G ′(b) at the optimal stopping point b. On the other hand, if the scale function S of X is differentiable at b, then the smooth fit condition V ′(b) = G ′(b) holds (whenever X is regular and G is differentiable at b). We give an example showing that the latter can happen even when d + G/dS < d + V/dS < d − V/dS < d − G/dS at b. 1.
ACCURATE EVALUATION OF EUROPEAN AND AMERICAN OPTIONS UNDER THE CGMY PROCESS
"... A finitedifference method for integrodifferential equations arising from Lévy driven asset processes in finance is discussed. The equations are discretized in space by the collocation method and in time by an explicit backward differentiation formula. The discretization is shown to be secondorder ..."
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Cited by 5 (0 self)
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A finitedifference method for integrodifferential equations arising from Lévy driven asset processes in finance is discussed. The equations are discretized in space by the collocation method and in time by an explicit backward differentiation formula. The discretization is shown to be secondorder accurate independently of the degree of the singularity in the Lévy measure. The singularity is dealt with by means of an integration by parts technique. An application of the fast Fourier transform gives the overall amount of work O(MN log N), rendering the method fast.
A proof of the smoothness of the finite time horizon american put option for jump diffusions
, 2008
"... We give a new proof of the fact that the value function of the finite time horizon American put option for a jump diffusion, when the jumps are from a compound Poisson process, is the classical solution of a free boundary equation. We also show that the value function is C¹ across the optimal stopp ..."
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Cited by 3 (3 self)
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We give a new proof of the fact that the value function of the finite time horizon American put option for a jump diffusion, when the jumps are from a compound Poisson process, is the classical solution of a free boundary equation. We also show that the value function is C¹ across the optimal stopping boundary. Our proof, which only uses the classical theory of parabolic partial differential equations of [7, 8], is an alternative to the proof that uses the theory of vicosity solutions (see [14]). This new proof relies on constructing a monotonous sequence of functions, each of which is a value function of an optimal stopping problem for a geometric Brownian motion, converging to the value function of the American put option for the jump diffusion uniformly and exponentially fast. This sequence is constructed by iterating a functional operator that maps a certain class of convex functions to classical solutions of corresponding free boundary equations. On the other hand, since the approximating sequence converges to the value function exponentially fast, it naturally leads to a good numerical scheme. We also show that the assumption that [14] makes on the parameters of the problem, in order to guarantee that the value function is the unique classical solution of the corresponding free boundary equation, can be dropped.