The distribution of refracted Lévy processes with jumps having rational Laplace transforms

Abstract We consider a refracted jump diffusion process having two-sided jumps with rational Laplace transforms. For such a process, by applying a straightforward but interesting approach, we derive formulae for the Laplace transform of its distribution. Our formulae are presented in an attractive form and the approach is novel. In particular, the idea in the application of an approximating procedure is remarkable. In addition, the results are used to price variable annuities with state-dependent fees.

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A general approach to the integral functionals of epidemic processes

Journal of Applied Probability ◽

10.1017/jpr.2018.37 ◽

2018 ◽

Vol 55 (2) ◽

pp. 593-609

Author(s):

Claude Lefèvre ◽

Philippe Picard

Keyword(s):

Laplace Transform ◽

Epidemic Model ◽

Special Class ◽

Laplace Transforms ◽

Integral Functionals ◽

New Approach ◽

Arbitrary State ◽

State Dependent ◽

The Laplace Transform ◽

Recursive Relations

Abstract In this paper we consider the integral functionals of the general epidemic model up to its extinction. We develop a new approach to determine the exact Laplace transform of such integrals. In particular, we obtain the Laplace transform of the duration of the epidemic T, the final susceptible size ST, the area under the trajectory of the infectives AT, and the area under the trajectory of the susceptibles BT. The method relies on the construction of a family of martingales and allows us to solve simple recursive relations for the involved parameters. The Laplace transforms are then expanded in terms of a special class of polynomials. The analysis is generalized in part to Markovian epidemic processes with arbitrary state-dependent rates.

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ANALYTICAL PRICING OF DOUBLE-BARRIER OPTIONS UNDER A DOUBLE-EXPONENTIAL JUMP DIFFUSION PROCESS: APPLICATIONS OF LAPLACE TRANSFORM

International Journal of Theoretical and Applied Finance ◽

10.1142/s0219024904002402 ◽

2004 ◽

Vol 07 (02) ◽

pp. 151-175 ◽

Cited By ~ 73

Author(s):

ARTUR SEPP

Keyword(s):

Laplace Transform ◽

Diffusion Process ◽

Asset Price ◽

Option Price ◽

Jump Diffusion ◽

Time Dependent ◽

Barrier Option ◽

Double Exponential ◽

Jump Diffusion Process ◽

Risk Parameters

We derive explicit formulas for pricing double (single) barrier and touch options with time-dependent rebates assuming that the asset price follows a double-exponential jump diffusion process. We also consider incorporating time-dependent volatility. Assuming risk-neutrality, the value of a barrier option satisfies the generalized Black–Scholes equation with the appropriate boundary conditions. We take the Laplace transform of this equation in time and solve it explicitly. Option price and risk parameters are computed via the numerical inversion of the corresponding solution. Numerical examples reveal that the pricing formulas are easy to implement and they result in accurate prices and risk parameters. Proposed formulas allow fast computing of smile-consistent prices of barrier and touch options.

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