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Perturbation Methods for Pricing European Options in a Model with Two Stochastic Volatilities
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: 0000-0002-0835-7536
Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics. (MAM)ORCID iD: 0000-0003-4554-6528
2015 (English)In: New Trends in Stochastic Modelling and Data Analysis / [ed] Raimondo Manca, Sally McClean, Christos H Skiadas, ISAST , 2015, 199-210 p.Chapter in book (Refereed)
Abstract [en]

Financial models have to reflect the characteristics of markets in which they are developed to be able to predict the future behavior of a financial system. The nature of most trading environments is characterized by uncertainties which are expressed in mathematical models in terms of volatilities. In contrast to the classical Black-Scholes model with constant volatility, our model includes one fast-changing and another slow-changing stochastic volatilities of mean-reversion type. The different changing frequencies of volatilities can be interpreted as the effects of weekends and effects of seasons of the year (summer and winter) on the asset price.

We perform explicitly the transition from the real-world to the risk-neutral probability measure by introducing market prices of risk and applying Girsanov Theorem. To solve the boundary value problem for the partial differential equation that corresponds to the case of a European option, we perform both regular and singular multiscale expansions in fractional powers of the speed of mean-reversion factors. We then construct an approximate solution given by the two-dimensional Black-Scholes model plus some terms that expand the results obtained by Black and Scholes.

Place, publisher, year, edition, pages
ISAST , 2015. 199-210 p.
Keyword [en]
financial market, mean reversion volatility, risk-neutral measure, partial differential equation, regular perturbation, singular perturbation, European option
National Category
Probability Theory and Statistics
Research subject
Mathematics/Applied Mathematics
Identifiers
URN: urn:nbn:se:mdh:diva-33472ISBN: 978-618-5180-06-5 (print)ISBN: 978-618-5180-10-2 (print)OAI: oai:DiVA.org:mdh-33472DiVA: diva2:1040244
Funder
Sida - Swedish International Development Cooperation Agency
Available from: 2016-10-26 Created: 2016-10-26 Last updated: 2016-12-13Bibliographically 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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