Option pricing and hedging under a stochastic volatility Lévy process model (original) (raw)

Stochastic Volatility for Levy Processes

Mathematical Finance, 2003

Three processes re°ecting persistence of volatility are formulated by evaluating three L ¶ evy processes at a time change given by the integral of a square root process. A positive stock price process is then obtained by exponentiating and mean correcting these processes, or alternatively by stochastically exponentiating the processes. The characteristic functions for the log price can be used to yield option prices via the fast Fourier transform. Our empirical results on index options and single name options suggest advantages to employing higher dimensional L ¶ evy systems for index options and lower dimensional structures for single names. In general, mean corrected exponentiation performs better than employing the stochastic exponential. Martingale laws for the mean corrected exponential are also studied and two new concepts termed L ¶ evy and martingale marginals are introduced. ¤ We would like to thank George Panayotov for assistance with the computations reported in this paper. Dilip Madan would like to thank Ajay Khanna for important discussions and perspectives on the problems studied here. Errors are our own responsibility.

Lévy processes for financial modeling

EIGHTH INTERNATIONAL CONFERENCE NEW TRENDS IN THE APPLICATIONS OF DIFFERENTIAL EQUATIONS IN SCIENCES (NTADES2021), 2022

The article considers different types of Lévy processes, their properties, methodology for their analysis, and their applications for financial modeling. Lévy processes admit jumps. Financial models based on Lévy processes with jumps are mainly two types. In the first type, called jump-diffusion models, the normal evolution of prices is given by a diffusion process, punctuated by jumps at random intervals. Here the jumps represent rare events. The second type consists of models with infinite number of jumps in every interval which we will call infinite activity Lévy processes. In these models, one does not need to introduce a Gaussian (Brownian) component since the dynamics of jumps is rich enough to generate nontrivial small time behavior. So very often the Gaussian part is not considered and the processes are called non-Gaussian Lévy processes. The use of the exponential Lévy processes for option pricing is considered as well as some alternatives based on telegraph processes. Some new proposals, including ST-tempering and ST-subordinating, are presented.

Option pricing with time-changed Lévy processes

Applied Financial Economics, 2013

In this paper, we introduce two new six-parameter processes based on time-changing tempered stable distributions and develop an option pricing model based on these processes. This model provides a good fit to observed option prices. To demonstrate the advantages of the new processes, we conduct two empirical studies to compare their performance to other processes that have been used in the literature.

Time-Changed Levy Process and Option Pricing

SSRN Electronic Journal, 2000

The classic Black-Scholes option pricing model assumes that returns follow Brownian motion, but return processes differ from this benchmark in at least three important ways. First, asset prices jump, leading to non-normal return innovations. Second, return volatilities vary stochastically over time. Third, returns and their volatilities are correlated, often negatively for equities. Time-changed Lévy processes can simultaneously address these three issues. We show that our framework encompasses almost all of the models proposed in the option pricing literature, and it is straightforward to select and test a particular option pricing model through the use of characteristic function technology.

Time-changed Lévy processes and option pricing

Journal of Financial Economics, 2004

The classic Black-Scholes option pricing model assumes that returns follow Brownian motion, but return processes differ from this benchmark in at least three important ways. First, asset prices jump, leading to non-normal return innovations. Second, return volatilities vary stochastically over time. Third, returns and their volatilities are correlated, often negatively for equities. Time-changed L! evy processes can simultaneously address these three issues. We show that our framework encompasses almost all of the models proposed in the option pricing literature, and it is straightforward to select and test a particular option pricing model through the use of characteristic function technology.

Specification analysis of option pricing models based on time‐changed Lévy processes

2005

We analyze the specifications of option pricing models based on time-changed Lévy processes. We classify option pricing models based on the structure of the jump component in the underlying return process, the source of stochastic volatility, and the specification of the volatility process itself. Our estimation of a variety of model specifications indicates that to better capture the behavior of the S&P 500 index options, we need to incorporate a high frequency jump component in the return process and generate stochastic volatilities from two different sources, the jump component and the diffusion component.

Option Pricing with Log-stable Lévy Processes

New Economic Windows

We model the logarithm of the price (log-price) of a financial asset as a random variable obtained by projecting an operator stable random vector with a scaling index matrix E onto a non-random vector. The scaling index E models prices of the individual financial assets (stocks, mutual funds, etc.). We find the functional form of the characteristic function of real powers of the price returns and we compute the expectation value of these real powers and we speculate on the utility of these results for statistical inference. Finally we consider a portfolio composed of an asset and an option on that asset. We derive the characteristic function of the deviation of the portfolio, D (t) t , defined as a temporal change of the portfolio diminished by the the compound interest earned. We derive pseudo-differential equations for the option as a function of the logstock-price and time and we find exact closed-form solutions to that equation. These results were not known before. Finally we discuss how our solutions correspond to other approximate results known from literature,in particular to the well known Black & Scholes equation.

Option pricing in stochastic volatility models driven by fractional Lévy processes

International Journal of Financial Markets and Derivatives, 2016

In this paper, we propose a continuous time fractional stochastic volatility model which extends the Barndorff-Nielsen and Shephard (2001) (BNS) model. Our model is the fractional BNS model, where we model the volatility as a fractional Lévy-driven Ornstein-Uhlenbeck process. We allow the memory parameter to be flexible so that our model can potentially produce short-or long-memory in volatility. We derive the analytical formula for option pricing using Fourier inversion technique. We numerically study the effect of memory parameter on the option prices and the calibration result indicates that the fractional model significantly improves the performance of the original BNS model.

Pricing and Hedging in Exponential Lévy Models: Review of Recent Results

Lecture Notes in Mathematics, 2011

These lecture notes cover a major part of the crash course on financial modeling with jump processes given by the author in Bologna on May 21-22, 2009. After a brief introduction, we discuss three aspects of exponential Lévy models: absence of arbitrage, including more recent results on the absence of arbitrage in multidimensional models, properties of implied volatility, and modern approaches to hedging in these models.