scholarly journals Asymmetrically tempered stable distributions with applications to finance

2019 ◽  
Vol 39 (1) ◽  
pp. 85-98
Author(s):  
A. Arefi ◽  
R. Pourtaheri
Keyword(s):  
Lévy Processes ◽  
Stable Distributions ◽  
Levy Processes ◽  
Infinitely Divisible ◽  
Financial Modeling ◽  
New Family ◽  
Tempered Stable Distributions ◽  
Exponential Lévy Models

In this paper, we introduce a technique to produce a new family of tempered stable distributions. We call this family asymmetrically tempered stable distributions.We provide two examples of this family named asymmetrically classical modified tempered stable ACMTS and asymmetrically modified classical tempered stable AMCTS distributions. Since the tempered stable distributions are infinitely divisible, Levy processes can be induced by the ACMTS and AMCTS distributions. The properties of these distributions will be discussed along with the advantages in applying them to financial modeling. Furthermore, we develop exponential Levy models for them. To demonstrate the advantages of the exponential Levy ACMTS and AMCTS models, we estimate parameters for the S&P 500 Index.

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.

FRACTIONAL LÉVY PROCESSES AND NOISES ON GEL′FAND TRIPLE

2010 ◽  
Vol 10 (01) ◽  
pp. 37-51 ◽  
Author(s):  
ZHIYUAN HUANG ◽  
XUEBIN LÜ ◽  
JIANPING WAN
Keyword(s):  
Integral Method ◽  
Lévy Processes ◽  
Fractional Integral ◽  
Levy Processes ◽  
Infinitely Divisible Distributions ◽  
Infinitely Divisible ◽  
White Noises

In this paper, we construct a class of infinitely divisible distributions on Gel′fand triple. Based on this construction, we define Lévy processes on Gel′fand triple and give their Lévy–Itô decompositions. Then, we construct the general Lévy white noises on Gel′fand triple. By using the Riemann–Liouville fractional integral method, we define the general fractional Lévy noises on Gel′fand triple and investigate their distribution properties.

scholarly journals Connecting discrete and continuous lookback or hindsight options in exponential Lévy models

2011 ◽  
Vol 43 (4) ◽  
pp. 1136-1165 ◽  
Author(s):  
E. H. A. Dia ◽  
D. Lamberton
Keyword(s):  
Lévy Processes ◽  
Jump Diffusion ◽  
Levy Processes ◽  
Diffusion Models ◽  
Lookback Options ◽  
Exponential Lévy Models ◽  
The Difference

Motivated by the pricing of lookback options in exponential Lévy models, we study the difference between the continuous and discrete supremums of Lévy processes. In particular, we extend the results of Broadie, Glasserman and Kou (1999) to jump diffusion models. We also derive bounds for general exponential Lévy models.

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