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Approximation Methods of European Option Pricing in Multiscale Stochastic Volatility Model
Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics. (MAM)ORCID iD: 0000-0002-0835-7536
Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics. (MAM)ORCID iD: 0000-0001-8361-4152
Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics. (MAM)ORCID iD: 0000-0002-0139-0747
Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics. (MAM)ORCID iD: 0000-0003-4554-6528
2017 (English)In: INCPAA 2016 Proceedings: 11th International Conference on Mathematical Problems in Engineering, Aerospace, and Sciences, 5 - 8 July 2016. Location: La Rochelle, France / [ed] S. Sivasundaram, 2017, 020112Conference paper (Refereed)
Abstract [en]

In the classical Black-Scholes model for financial option pricing, the asset price follows a geometric Brownian motion with constant volatility. Empirical findings such as volatility smile/skew, fat-tailed asset return distributions have suggested that the constant volatility assumption might not be realistic. A general stochastic volatility model, e.g. Heston model, GARCH model  and SABR volatility model , in which the variance/volatility itself follows typically a mean-reverting stochastic process, has shown to be superior in terms of capturing the empirical facts. However in order to capture more features of the volatility smile a two-factor, of double Heston type, stochastic volatility model is more useful as shown by Christoffersen, Heston and Jacobs.  We consider one specific type of such two-factor volatility models in which the volatility has multiscale mean-reversion rates. Our model contains two mean-reverting volatility processes with a fast and a slow reverting rate respectively. We consider the European option pricing problem under one type of the multiscale stochastic volatility model where the two volatility processes act as independent factors in the asset price process.  The novelty in this chapter is an approximating analytical solution using asymptotic expansion method which extends the authors earlier research in Canhanga et al. In addition we propose a numerical approximating solution using Monte-Carlo simulation. For completeness and for comparison we also implement the semi-analytical solution by Chiarella and Ziveyi using method of characteristics, Fourier and bivariate Laplace transforms.

Place, publisher, year, edition, pages
2017. 020112
Keyword [en]
Black-Scholes model, option pricing, Brownian motion, stochastic volatility, asymptotic expansion
National Category
Probability Theory and Statistics
Identifiers
URN: urn:nbn:se:mdh:diva-33474DOI: 10.1063/1.4972704ISI: 000399203000111ScopusID: 2-s2.0-85013633019ISBN: 9780735414648 (print)OAI: oai:DiVA.org:mdh-33474DiVA: diva2:1040247
Conference
11th International Conference on Mathematical Problems in Engineering, Aerospace, and Sciences, 5 - 8 July 2016. Location: La Rochelle, France
Funder
Sida - Swedish International Development Cooperation Agency
Available from: 2016-10-26 Created: 2016-10-26 Last updated: 2017-05-12Bibliographically approved
In thesis
1. Asymptotic Methods for Pricing European Option in a Market Model With Two Stochastic Volatilities
Open this publication in new window or tab >>Asymptotic Methods for Pricing European Option in a Market Model With Two Stochastic Volatilities
2016 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

Modern financial engineering is a part of applied mathematics that studies market models. Each model is characterized by several parameters. Some of them are familiar to a wide audience, for example, the price of a risky security, or the risk free interest rate. Other parameters are less known, for example, the volatility of the security. This parameter determines the rate of change of security prices and is determined by several factors. For example, during the periods of stable economic growth the prices are changing slowly, and the volatility is small. During the crisis periods, the volatility significantly increases. Classical market models, in particular, the celebrated Nobel Prize awarded Black–Scholes–Merton model (1973), suppose that the volatility remains constant during the lifetime of a financial instrument. Nowadays, in most cases, this assumption cannot adequately describe reality. We consider a model where both the security price and the volatility are described by random functions of time, or stochastic processes. Moreover, the volatility process is modelled as a sum of two independent stochastic processes. Both of them are mean reverting in the sense that they randomly oscillate around their average values and never escape neither to very small nor to very big values. One is changing slowly and describes low frequency, for example, seasonal effects, another is changing fast and describes various high frequency effects. We formulate the model in the form of a system of a special kind of equations called stochastic differential equations. Our system includes three stochastic processes, four independent factors, and depends on two small parameters. We calculate the price of a particular financial instrument called European call option. This financial contract gives its holder the right (but not the obligation) to buy a predefined number of units of the risky security on a predefined date and pay a predefined price. To solve this problem, we use the classical result of Feynman (1948) and Kac (1949). The price of the instrument is the solution to another kind of problem called boundary value problem for a partial differential equation. The resulting equation cannot be solved analytically. Instead we represent the solution in the form of an expansion in the integer and half-integer powers of the two small parameters mentioned above. We calculate the coefficients of the expansion up to the second order, find their financial sense, perform numerical studies, and validate our results by comparing them to known verified models from the literature. The results of our investigation can be used by both financial institutions and individual investors for optimization of their incomes.

Place, publisher, year, edition, pages
Mälardalen University, Västerås, Sweden, 2016
Series
Mälardalen University Press Dissertations, ISSN 1651-4238 ; 219
Keyword
Asymptotic Expansion, European Options, Stochastic Volatilities
National Category
Mathematics
Research subject
Mathematics/Applied Mathematics
Identifiers
urn:nbn:se:mdh:diva-33475 (URN)978-91-7485-300-1 (ISBN)
Public defence
2016-12-07, Kappa, Mälardalens högskola, Västerås, 13:15 (English)
Opponent
Supervisors
Available from: 2016-10-28 Created: 2016-10-26 Last updated: 2016-11-10Bibliographically approved
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