Application of Homotopy Analysis Method to Option Pricing Under Lévy Processes

Takayuki Sakuma and Yuji Yamada ()

Asia-Pacific Financial Markets, 2014, vol. 21, issue 1, 14 pages

Abstract: Option pricing under the Lévy process has been considered an important research direction in the field of financial engineering, where a closed-form expression for the standard European option is available due to the existence of analytically tractable characteristic function according to the Lévy–Khinchin representation. However, this approach cannot be applied to exotic derivatives (such as barrier options) directly, although a large volume of exotic derivatives are actively traded in the current options market. An alternative approach is to solve the corresponding partial integro-differential equation (PIDE) numerically, which is, in fact, time-consuming and is not computationally tractable in general. In this paper, we apply the so-called homotopy analysis method (HAM) to solve the corresponding PIDE in a semi analytic form, being obtained from the following three steps: (1) Apply the Fourier transform to convert the PIDE to an ordinal differential equitation (ODE), and construct a differential system of ODEs. (2) Solve the system of ODEs, where each differential equation is shown to have an analytical solution. (3) Express the option price using the sum of infinite series, where each term may be expressed analytically and derived by applying Steps (1) and (2) recursively. To illustrate our technique more precisely, we take the variance gamma model as an example and provide the semi-analytic form. Numerical examples demonstrate a fast convergence of our proposed method to the prices of European and down-and-out call options with a few number of terms. Note that this method is easy to implement and can be applied to other types of options under general Lévy processes. Copyright Springer Japan 2014

Keywords: Barrier options; Homotopy analysis method; Lévy processes; Variance gamma model (search for similar items in EconPapers)
Date: 2014
References: View references in EconPapers View complete reference list from CitEc
Citations:

Downloads: (external link)
http://hdl.handle.net/10.1007/s10690-013-9175-2 (text/html)
Access to full text is restricted to subscribers.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:kap:apfinm:v:21:y:2014:i:1:p:1-14

Ordering information: This journal article can be ordered from
http://www.springer.com/finance/journal/10690/PS2

DOI: 10.1007/s10690-013-9175-2

Access Statistics for this article

Asia-Pacific Financial Markets is currently edited by Jiro Akahori

More articles in Asia-Pacific Financial Markets from Springer, Japanese Association of Financial Economics and Engineering
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().