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Numerical Studies on Asymptotics of European Option under Multiscale Stochastic Volatility
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)
Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics. (MAM)ORCID iD: 0000-0002-0835-7536
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2015 (English)In: ASMDA 2015 Proceedings: 16th Applied Stochastic Models and Data Analysis International Conference with 4th Demographics 2015 Workshop / [ed] Christos H Skiadas, ISAST: International Society for the Advancement of Science and Technology , 2015, 53-66 p.Conference paper, Published paper (Refereed)
Abstract [en]

Multiscale stochastic volatilities models relax the constant volatility assumption from Black-Scholes option pricing model. Such model can capture the smile and skew of volatilities and therefore describe more accurately the movements of the trading prices. Christoffersen et al. [3] presented a model where the underlying priceis governed by two volatility components, one changing fast and another changing slowly. Chiarella and Ziveyi [2] transformed Christoffersen’s model and computed an approximate formula for pricing American options. They used Duhamel’s principle to derive an integral form solution of the boundary value problem associated to the option price. Using method of characteristics, Fourier and Laplace transforms, they obtained with good accuracy the American options prices. In a previous research of the authors (Canhanga et al. [1]), a particular case of Chiarella and Ziveyi [2] model is used for pricing of European options. The novelty of this earlier work is to present an asymptotic expansion for the option price. The present paper provides experimental and numerical studies on investigating the accuracy of the approximation formulae given by this asymptotic expansion. We present also a procedure for calibrating the parameters produced by our first-order asymptotic approximation formulae. Our approximated option prices will be compared to the approximation obtained by Chiarella and Ziveyi [2].

1. Canhanga B., Malyarenko, A., Ni, Y. and Silvestrov S. Perturbation methods for pricing European options in a model with two stochastic volatilities. 3rd SMTDA Conference Proceedings. 11-14 June 2014, Lisbon Porturgal, C. H. Skiadas (Ed.) 489-500 (2014).

2. Chiarella, C, and Ziveyi, J. American option pricing under two stochastic volatility processes. J. Appl. Math. Comput. 224:283–310 (2013).

3. Christoffersen, P.; Heston, S.; Jacobs, K. The shape and term structure of the index option smirk: why multifactor stochastic volatility models work so well. Manage. Sci. 55 (2) 1914-1932; (2009).

Place, publisher, year, edition, pages
ISAST: International Society for the Advancement of Science and Technology , 2015. 53-66 p.
Keyword [en]
financial market, mean reversion volatility, asymptotic expansion, stochastic volatilities, regular perturbation, singular perturbation, European option.
National Category
Probability Theory and Statistics
Research subject
Mathematics/Applied Mathematics
Identifiers
URN: urn:nbn:se:mdh:diva-29936ISBN: 978-618-5180-05-8 (print)OAI: oai:DiVA.org:mdh-29936DiVA: diva2:882730
Conference
16th Applied Stochastic Models and Data Analysis International Conference (ASMDA2015) with Demographics 2015 Workshop, 30 June – 4 July 2015, University of Piraeus, Greece
Funder
Sida - Swedish International Development Cooperation Agency
Available from: 2015-12-15 Created: 2015-12-15 Last updated: 2016-10-28Bibliographically approved
In thesis
1.
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2. 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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