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Numerical Methods on European Options Second Order Asymptotic Expansions for 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: 0000-0002-0835-7536
Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics. (MAM)ORCID iD: 0000-0001-9635-0301
Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics. (MAM)ORCID iD: 0000-0002-0139-0747
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2017 (English)In: INCPAA 2016 Proceedings: 11th International Conference on Mathematical Problems in Engineering, Aerospace, and Sciences, ICNPAA 2016, La Rochelle, France, 4 - 8 July 2016. / [ed] S. Sivasundaram, 2017, Vol. 1798, 020035-1-020035-10 p., 020035Conference paper, Published paper (Refereed)
Abstract [en]

After Black-Scholes proposed a model for pricing European Option in 1973, Cox, Ross and Rubinstein in 1979, and Heston in 1993, showed that the constant volatility assumption in the Black-Scholes model was one of the main reasons for the model to be unable to capture some market details. Instead of constant volatilities, they introduced non-constant volatilities to the asset dynamic modeling. In 2009, Christoffersen empirically showed "why multi-factor stochastic volatility models work so well". Four years later, Chiarella and Ziveyi solved the model proposed by Christoffersen. They considered an underlying asset whose price is governed by two factor stochastic volatilities of mean reversion type. Applying Fourier transforms, Laplace transforms and the method of characteristics they presented an approximate formula for pricing American option.The huge calculation involved in the Chiarella and Ziveyi approach motivated us to investigate another approach to compute European option prices on a Christoffersen type model. Using the first and second order asymptotic expansion method we presented a closed form solution for European option, and provided experimental and numerical studies on investigating the accuracy of the approximation formulae given by the first order asymptotic expansion. In the present chapter we will perform experimental and numerical studies for the second order asymptotic expansion and compare the obtained results with results presented by Chiarella and Ziveyi.

Place, publisher, year, edition, pages
2017. Vol. 1798, 020035-1-020035-10 p., 020035
Keyword [en]
stochastic volatilities, European option, asymptotic expansion
National Category
Probability Theory and Statistics
Research subject
Mathematics/Applied Mathematics
Identifiers
URN: urn:nbn:se:mdh:diva-33473DOI: 10.1063/1.4972627ISI: 000399203000035Scopus ID: 2-s2.0-85013660646ISBN: 9780735414648 (print)OAI: oai:DiVA.org:mdh-33473DiVA: diva2:1040245
Conference
11th International Conference on Mathematical Problems in Engineering, Aerospace, and Sciences, ICNPAA 2016, La Rochelle, France, 4 - 8 July 2016.
Funder
Sida - Swedish International Development Cooperation Agency
Available from: 2016-10-26 Created: 2016-10-26 Last updated: 2017-09-03Bibliographically 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: 2017-09-28Bibliographically approved
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