scholarly journals Correlating Lévy processes with self-decomposability: applications to energy markets

2021 ◽  
Author(s):  
Matteo Gardini ◽  
Piergiacomo Sabino ◽  
Emanuela Sasso
Keyword(s):  
Monte Carlo ◽  
Closed Form ◽  
Lévy Processes ◽  
Systematic Risk ◽  
Levy Processes ◽  
Energy Markets ◽  
Characteristic Functions ◽  
Stochastic Delay ◽  
Spread Options

AbstractBased on the concept of self-decomposability, we extend some recent multidimensional Lévy models built using multivariate subordination. Our aim is to construct multivariate Lévy processes that can model the propagation of the systematic risk in dependent markets with some stochastic delay instead of affecting all the markets at the same time. To this end, we extend some known approaches keeping their mathematical tractability, study the properties of the new processes, derive closed-form expressions for their characteristic functions and detail how Monte Carlo schemes can be implemented. We illustrate the applicability of our approach in the context of gas, power and emission markets focusing on the calibration and on the pricing of spread options written on different underlying commodities.

Scale functions of Lévy processes and busy periods of finite-capacity M/GI/1 queues

2004 ◽  
Vol 41 (4) ◽  
pp. 1145-1156 ◽  
Author(s):  
Parijat Dube ◽  
Fabrice Guillemin ◽  
Ravi R. Mazumdar
Keyword(s):  
Closed Form ◽  
Lévy Processes ◽  
Busy Period ◽  
Exit Time ◽  
Levy Processes ◽  
Finite Capacity ◽  
Scale Functions ◽  
Period Distribution ◽  
Finite Capacity Queues ◽  
Time Theory

In this paper we use the exit time theory for Lévy processes to derive new closed-form results for the busy period distribution of finite-capacity fluid M/G/1 queues. Based on this result, we then obtain the busy period distribution for finite-capacity queues with on–off inputs when the off times are exponentially distributed.

scholarly journals Multilevel Monte Carlo simulation for Lévy processes based on the Wiener–Hopf factorisation

2014 ◽  
Vol 124 (2) ◽  
pp. 985-1010 ◽  
Author(s):  
A. Ferreiro-Castilla ◽  
A.E. Kyprianou ◽  
R. Scheichl ◽  
G. Suryanarayana
Keyword(s):  
Monte Carlo Simulation ◽  
Monte Carlo ◽  
Lévy Processes ◽  
Levy Processes ◽  
Multilevel Monte Carlo

scholarly journals Randomisation and recursion methods for mixed-exponential Lévy models, with financial applications

2015 ◽  
Vol 52 (04) ◽  
pp. 1076-1096
Author(s):  
Aleksandar Mijatović ◽  
Martijn R. Pistorius ◽  
Johannes Stolte
Keyword(s):  
Monte Carlo ◽  
Lévy Processes ◽  
Variance Reduction ◽  
First Passage Time ◽  
Passage Time ◽  
Levy Processes ◽  
Weak Approximation ◽  
Occupation Times ◽  
First Passage ◽  
Exponential Lévy Models

We develop a new Monte Carlo variance reduction method to estimate the expectation of two commonly encountered path-dependent functionals: first-passage times and occupation times of sets. The method is based on a recursive approximation of the first-passage time probability and expected occupation time of sets of a Lévy bridge process that relies in part on a randomisation of the time parameter. We establish this recursion for general Lévy processes and derive its explicit form for mixed-exponential jump-diffusions, a dense subclass (in the sense of weak approximation) of Lévy processes, which includes Brownian motion with drift, Kou's double-exponential model, and hyperexponential jump-diffusion models. We present a highly accurate numerical realisation and derive error estimates. By way of illustration the method is applied to the valuation of range accruals and barrier options under exponential Lévy models and Bates-type stochastic volatility models with exponential jumps. Compared with standard Monte Carlo methods, we find that the method is significantly more efficient.

scholarly journals Conditional Characteristic Functions of Molchan-Golosov Fractional Lévy Processes with Application to Credit Risk

2013 ◽  
Vol 50 (4) ◽  
pp. 983-1005 ◽  
Author(s):  
Holger Fink
Keyword(s):  
Brownian Motion ◽  
Characteristic Function ◽  
Fractional Calculus ◽  
Hazard Rate ◽  
Lévy Processes ◽  
Levy Processes ◽  
Characteristic Functions ◽  
Infinitely Divisible Distributions ◽  
Infinitely Divisible ◽  
Fractional Stochastic Differential Equations

Molchan-Golosov fractional Lévy processes (MG-FLPs) are introduced by way of a multivariate componentwise Molchan-Golosov transformation based on an n-dimensional driving Lévy process. Using results of fractional calculus and infinitely divisible distributions, we are able to calculate the conditional characteristic function of integrals driven by MG-FLPs. This leads to important predictions concerning multivariate fractional Brownian motion, fractional subordinators, and general fractional stochastic differential equations. Examples are the fractional Lévy Ornstein-Uhlenbeck and Cox-Ingersoll-Ross models. As an application we present a fractional credit model with a long range dependent hazard rate and calculate bond prices.

Scale functions of Lévy processes and busy periods of finite-capacity M/GI/1 queues

2004 ◽  
Vol 41 (04) ◽  
pp. 1145-1156 ◽  
Author(s):  
Parijat Dube ◽  
Fabrice Guillemin ◽  
Ravi R. Mazumdar
Keyword(s):  
Closed Form ◽  
Lévy Processes ◽  
Busy Period ◽  
Exit Time ◽  
Levy Processes ◽  
Finite Capacity ◽  
Scale Functions ◽  
Period Distribution ◽  
Finite Capacity Queues ◽  
Time Theory

In this paper we use the exit time theory for Lévy processes to derive new closed-form results for the busy period distribution of finite-capacity fluid M/G/1 queues. Based on this result, we then obtain the busy period distribution for finite-capacity queues with on–off inputs when the off times are exponentially distributed.

scholarly journals Error Bounds for Small Jumps of Lévy Processes

2013 ◽  
Vol 45 (1) ◽  
pp. 86-105
Author(s):  
E. H. A. Dia
Keyword(s):  
Monte Carlo ◽  
Brownian Motion ◽  
Error Bounds ◽  
Monte Carlo Methods ◽  
Lévy Process ◽  
Lévy Processes ◽  
Levy Processes ◽  
Levy Process ◽  
Lévy Measure ◽  
Exponential Lévy Models

The pricing of options in exponential Lévy models amounts to the computation of expectations of functionals of Lévy processes. In many situations, Monte Carlo methods are used. However, the simulation of a Lévy process with infinite Lévy measure generally requires either truncating or replacing the small jumps by a Brownian motion with the same variance. We will derive bounds for the errors generated by these two types of approximation.

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