scholarly journals Portfolio Strategy of Financial Market with Regime Switching Driven by Geometric Lévy Process

2014 ◽  
Vol 2014 ◽  
pp. 1-9
Author(s):  
Liuwei Zhou ◽  
Zhijie Wang
Keyword(s):  
Financial Market ◽  
Stock Prices ◽  
Regime Switching ◽  
Lévy Process ◽  
Parabolic Type ◽  
Levy Process ◽  
Portfolio Strategy ◽  
Set Up ◽  
S Model ◽  
Geometric Levy Process

The problem of a portfolio strategy for financial market with regime switching driven by geometric Lévy process is investigated in this paper. The considered financial market includes one bond and multiple stocks which has few researches up to now. A new and general Black-Scholes (B-S) model is set up, in which the interest rate of the bond, the rate of return, and the volatility of the stocks vary as the market states switching and the stock prices are driven by geometric Lévy process. For the general B-S model of the financial market, a portfolio strategy which is determined by a partial differential equation (PDE) of parabolic type is given by using Itô formula. The PDE is an extension of existing result. The solvability of the PDE is researched by making use of variables transformation. An application of the solvability of the PDE on the European options with the final data is given finally.

scholarly journals A fuzzy approach to option pricing in a Levy process setting

2013 ◽  
Vol 23 (3) ◽  
pp. 613-622 ◽  
Author(s):  
Piotr Nowak ◽  
Maciej Romaniuk
Keyword(s):  
Option Pricing ◽  
Lévy Process ◽  
Fuzzy Numbers ◽  
Levy Process ◽  
Martingale Measure ◽  
Fuzzy Arithmetic ◽  
European Option ◽  
European Call Option ◽  
Log Price ◽  
Geometric Levy Process

Abstract In this paper the problem of European option valuation in a Levy process setting is analysed. In our model the underlying asset follows a geometric Levy process. The jump part of the log-price process, which is a linear combination of Poisson processes, describes upward and downward jumps in price. The proposed pricing method is based on stochastic analysis and the theory of fuzzy sets.We assume that some parameters of the financial instrument cannot be precisely described and therefore they are introduced to the model as fuzzy numbers. Application of fuzzy arithmetic enables us to consider various sources of uncertainty, not only the stochastic one. To obtain the European call option pricing formula we use the minimal entropy martingale measure and Levy characteristics.

Variance Optimal Stopping for Geometric Lévy Processes

2015 ◽  
Vol 47 (01) ◽  
pp. 128-145 ◽  
Author(s):  
Kamille Sofie Tågholt Gad ◽  
Jesper Lund Pedersen
Keyword(s):  
Optimal Stopping ◽  
Lévy Process ◽  
Lévy Processes ◽  
Levy Processes ◽  
Levy Process ◽  
Stopping Times ◽  
Optimal Stopping Problem ◽  
Maximum Variance ◽  
Geometric Levy Process

The main result of this paper is the solution to the optimal stopping problem of maximizing the variance of a geometric Lévy process. We call this problem the variance problem. We show that, for some geometric Lévy processes, we achieve higher variances by allowing randomized stopping. Furthermore, for some geometric Lévy processes, the problem has a solution only if randomized stopping is allowed. When randomized stopping is allowed, we give a solution to the variance problem. We identify the Lévy processes for which the allowance of randomized stopping times increases the maximum variance. When it does, we also solve the variance problem without randomized stopping.

Variance Optimal Stopping for Geometric Lévy Processes

2015 ◽  
Vol 47 (1) ◽  
pp. 128-145 ◽  
Author(s):  
Kamille Sofie Tågholt Gad ◽  
Jesper Lund Pedersen
Keyword(s):  
Optimal Stopping ◽  
Lévy Process ◽  
Lévy Processes ◽  
Levy Processes ◽  
Levy Process ◽  
Stopping Times ◽  
Optimal Stopping Problem ◽  
Maximum Variance ◽  
Geometric Levy Process

The main result of this paper is the solution to the optimal stopping problem of maximizing the variance of a geometric Lévy process. We call this problem the variance problem. We show that, for some geometric Lévy processes, we achieve higher variances by allowing randomized stopping. Furthermore, for some geometric Lévy processes, the problem has a solution only if randomized stopping is allowed. When randomized stopping is allowed, we give a solution to the variance problem. We identify the Lévy processes for which the allowance of randomized stopping times increases the maximum variance. When it does, we also solve the variance problem without randomized stopping.

PRICING COVARIANCE SWAPS FOR BARNDORFF–NIELSEN AND SHEPHARD PROCESS DRIVEN FINANCIAL MARKETS

2016 ◽  
Vol 11 (03) ◽  
pp. 1650012 ◽  
Author(s):  
SEMERE HABTEMICAEL ◽  
INDRANIL SENGUPTA
Keyword(s):  
Financial Markets ◽  
Real Time ◽  
Error Estimation ◽  
Lévy Process ◽  
Levy Process ◽  
Computational Accuracy ◽  
Practical Applications ◽  
Analytic Expressions ◽  
S Model

The objective of this paper is to study the arbitrage free pricing of the covariance swap for Barndorff–Nielsen and Shephard (BN–S) type Lévy process driven financial markets. One of the major challenges in arbitrage free pricing of swap is to obtain an accurate pricing expression which can be used with good computational accuracy. In this paper, we obtain analytic expressions for the pricing of the covariance swap. We show that with the analytic expressions obtained from the BN–S model, the error estimation in fitting the delivery price is much less than the existing models with comparable parameters. The models and pricing formulas proposed in this paper are computable in real time and hence can be efficiently used in practical applications.

Volatility Smile Surface for Levy Option Pricing Model

2004 ◽  
Vol 12 (1) ◽  
pp. 73-86
Author(s):  
Jun Hui Lee ◽  
Kook Hyun Chang
Keyword(s):  
Stock Prices ◽  
Lévy Process ◽  
Implied Volatility ◽  
Diffusion Processes ◽  
Adjoint Operator ◽  
Gaussian Model ◽  
Levy Process ◽  
Forward Equation ◽  
Option Pricing Model ◽  
Variance Gamma Model

This paper discusses theoretical extensions of the implied volatility method of Dupire (1994) when the stock prices follow the Geometric Levy process. For the extensions of Kolmogorov forward equation for Levy process, this paper uses adjoint operator in L² spaces. This paper obtains similar results of Dupire (1994) and Andersen and Andreasan (2001). However, our results can be applied to more general semi-martingale processes such as well-known VG (Variance Gamma) model and NIG (Normal Inverse Gaussian) model with diffusion processes. This paper also applies the approach to the case of stochastic time changed Levy process, which generates the stochastic volatility models.

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