scholarly journals Memory-Reduction Method for Pricing American-Style Options under Exponential Lévy Processes

2011 ◽  
Vol 1 (1) ◽  
pp. 20-34 ◽  
Author(s):  
Raymond H. Chan ◽  
Tao Wu
Keyword(s):  
Lévy Processes ◽  
Computational Cost ◽  
Simulation Method ◽  
Levy Processes ◽  
Storage Requirement ◽  
Variance Gamma Process ◽  
The Monte Carlo Method ◽  
Exponential Lévy Models ◽  
American Style ◽  
Memory Reduction

AbstractThis paper concerns the Monte Carlo method in pricing American-style options under the general class of exponential Lévy models. Traditionally, one must store all the intermediate asset prices so that they can be used for the backward pricing in the least squares algorithm. Therefore the storage requirement grows like , where m is the number of time steps and n is the number of simulated paths. In this paper, we propose a simulation method where the storage requirement is only . The total computational cost is less than twice that of the traditional method. For machines with limited memory, one can now enlarge m and n to improve the accuracy in pricing the options. In numerical experiments, we illustrate the efficiency and accuracy of our method by pricing American options where the log-prices of the underlying assets follow typical Lévy processes such as Brownian motion, lognormal jump-diffusion process, and variance gamma process.

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 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.

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.

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

2011 ◽  
Vol 43 (04) ◽  
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.

Levy Processes Simulation for Modeling in Bioinformatics

2020 ◽  
Vol 10 (1) ◽  
pp. 55-64
Author(s):  
Anna V. Kuzmina
Keyword(s):  
Process Simulation ◽  
Efficient Algorithm ◽  
Stock Price ◽  
Lévy Processes ◽  
Applied Research ◽  
Levy Processes ◽  
Gamma Process ◽  
Variance Gamma ◽  
R Language ◽  
Variance Gamma Process

This article discusses the capabilities of the R language for modeling Levy processes, processes that currently most closely correspond to the nature of the evolution of stock price movements. The efficient algorithm of the CGMY process simulation as a difference of the tempered stable independent Levy is processed and programmed with the R language. The efficient algorithm of variance gamma process simulation using variance gamma random variables is processed and programmed with the R language. The article is focused on an entirely new area relevant to the scope of the International Journal of Applied Research in Bioinformatics (IJARB).

Sharing and Avatar-Based Innovation Tools on Digital Economy Perspectives Using Levy Processes Simulation

2020 ◽  
Vol 11 (3) ◽  
pp. 52-63
Author(s):  
Vardan Mkrttchian ◽  
Yulia Vertakova
Keyword(s):  
Process Simulation ◽  
Efficient Algorithm ◽  
Stock Price ◽  
Lévy Processes ◽  
Sharing Economy ◽  
Levy Processes ◽  
Digital Economy ◽  
Variance Gamma ◽  
R Language ◽  
Variance Gamma Process

This article is the Enhancement of the Mkrttchian and Vertakova article “Digital Sharing Economy” published in the International Journal of Innovation in Digital Economy (IJIDE, Volume 10, issue 2) and the chapter “Avatar-Based Innovation Tools for Managerial Perspectives on Digital Sharing Economy” in the book “Avatar-Based Models, Tools, and Innovation in the Digital Economy,” focused on an entirely new area relevant to the scope of IJIDE. The article discusses the capabilities of the R language for modeling Levy processes - processes that currently closely correspond to the nature of the evolution of stock price movements. The efficient algorithm of the CGMY process simulation as a difference of the tempered stable independent Levy is processed and programmed at R language. The efficient algorithm of variance gamma process simulation using variance gamma random variables is processed and programmed at R language, as Modelling in the Digital Globalization Era.

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

2015 ◽  
Vol 52 (4) ◽  
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 Error Bounds for Small Jumps of Lévy Processes

2013 ◽  
Vol 45 (01) ◽  
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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