Change search
ReferencesLink to record
Permanent link

Direct link
Asymptotic Methods for Pricing European Option in a Market Model With Two Stochastic Volatilities
Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.ORCID iD: 0000-0001-8361-4152
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 [en]
Asymptotic Expansion, European Options, Stochastic Volatilities
National Category
Mathematics
Research subject
Mathematics/Applied Mathematics
Identifiers
URN: urn:nbn:se:mdh:diva-33475ISBN: 978-91-7485-300-1OAI: oai:DiVA.org:mdh-33475DiVA: diva2:1040251
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
List of papers
1. Pricing European Options Under Stochastic Volatilities Models
Open this publication in new window or tab >>Pricing European Options Under Stochastic Volatilities Models
2016 (English)In: Engineering Mathematics I: Electromagnetics, Fluid Mechanics, Material Physics and Financial Engineering / [ed] Sergei Silvestrov; Milica Rancic, Springer, 2016, 315-338 p.Chapter in book (Refereed)
Abstract [en]

Interested by the volatility behavior, different models have been developed for option pricing. Starting from constant volatility model which did not succeed on capturing the effects of volatility smiles and skews; stochastic volatility models appearas a response to the weakness of the constant volatility models. Constant elasticity of volatility, Heston, Hull and White, Schöbel-Zhu, Schöbel-Zhu-Hull-Whiteand many others are examples of models where the volatility is itself a random process. Along the chapter we deal with this class of models and we present the techniques of pricing European options. Comparing single factor stochastic volatility models to constant factor volatility models it seems evident that the stochastic volatility models represent nicely the movement of the asset price and its relations with changes in the risk. However, these models fail to explain the large independent fluctuations in the volatility levels and slope. Christoffersen et al. in [4] proposed a model with two-factor stochastic volatilities where the correlation between the underlying asset price and the volatilities varies randomly. In the last section of this chapter we introduce a variation of Chiarella and Ziveyi model, which is a subclass of the model presented in [4] and we use the first order asymptotic expansion methods to determine the price of European options.

Place, publisher, year, edition, pages
Springer, 2016
Series
, Springer Proceedings in Mathematics and Statistics, ISSN 2194-1009 ; 178
Keyword
option pricing, European options, stochastic volatilitie, asymptotic expansion
National Category
Probability Theory and Statistics
Research subject
Mathematics/Applied Mathematics
Identifiers
urn:nbn:se:mdh:diva-33388 (URN)10.1007/978-3-319-42082-0 (DOI)978-3-319-42081-3 (ISBN)978-3-319-42082-0 (ISBN)
Funder
Sida - Swedish International Development Cooperation Agency
Available from: 2016-10-11 Created: 2016-10-11 Last updated: 2016-12-05Bibliographically approved
2. Perturbation Methods for Pricing European Options in a Model with Two Stochastic Volatilities
Open this publication in new window or tab >>Perturbation Methods for Pricing European Options in a Model with Two Stochastic Volatilities
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
Keyword
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:nbn:se:mdh:diva-33472 (URN)978-618-5180-06-5 (ISBN)978-618-5180-10-2 (ISBN)
Funder
Sida - Swedish International Development Cooperation Agency
Available from: 2016-10-26 Created: 2016-10-26 Last updated: 2016-10-27
3. Numerical Studies on Asymptotics of European Option under Multiscale Stochastic Volatility
Open this publication in new window or tab >>Numerical Studies on Asymptotics of European Option under Multiscale Stochastic Volatility
Show others...
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 (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
Keyword
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:nbn:se:mdh:diva-29936 (URN)978-618-5180-05-8 (ISBN)
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
4. Second Order Asymptotic Expansion for Pricing European Options in a Model with Two Stochastic Volatilities
Open this publication in new window or tab >>Second Order Asymptotic Expansion for Pricing European Options in a Model with Two Stochastic Volatilities
2015 (English)In: ASMDA 2015 Proceedings: 16th Applied Stochastic Models and Data Analysis International Conference with 4th Demographics 2015 Workshop, 30 June – 4 July 2015 University of Piraeus, Greece / [ed] C. H. Skiadas, ISAST: International Society for the Advancement of Science and Technology , 2015, 37-52 p.Conference paper (Refereed)
Abstract [en]

Asset price processes with stochastic volatilities have been actively used by researchers in financial mathematics for valuing derivative securities. This type of models allows characterizing the uncertainties in the asset price process in financial markets. In a recent paper Chiarella and Ziveyi analyzed a model with two stochastic volatilities of mean reversion type with one variable changing fast and the other changing slowly. They used method of characteristics to solve the obtained partial differential equation and determine the price of an American option. Fouque et al presented also a similar model in which the volatility of the underlying asset is governed by two diffusion processes which are not of mean reversion type. They developed a first-order asymptotic expansion for the European option price via a perturbation method.

In this chapter we consider the model given in Chiarella and Ziveyi. Instead of pricing American options we price European options by generalizing the techniques presented in Fouque et al to a more complex model with mean reverting stochastic volatility factors. We analyse both regular and singular perturbations to obtain an asymptotic expansion up to second order which can serve as an approximation for the price of non-path-dependent European options. Similar work is done in authors earlier work Canhanga et al where a first-order asymptotic expansion has been developed. Involving the second order terms has the advantage of capturing more accurately the effects of volatility smile and skew on the option pricing. Analytical approximation formula for pricing European Option is presented.

Place, publisher, year, edition, pages
ISAST: International Society for the Advancement of Science and Technology, 2015
Keyword
financial market, mean reversion volatility, asymptotic expansion, stochastic volatilities, regular perturbation, singular perturbation, european option
National Category
Mathematics
Identifiers
urn:nbn:se:mdh:diva-33471 (URN)978-618-5180-05-8 (ISBN)
Conference
16th ASMDA Conference
Funder
Sida - Swedish International Development Cooperation Agency
Available from: 2016-10-26 Created: 2016-10-26 Last updated: 2016-10-27
5. Numerical Methods on European Options Second Order Asymptotic Expansions for Multiscale Stochastic Volatility
Open this publication in new window or tab >>Numerical Methods on European Options Second Order Asymptotic Expansions for Multiscale Stochastic Volatility
Show others...
2016 (English)In: INCPAA 2016 Proceedings: 11th Internatinal Conference on Mathematical Problems in Engineering, Aerospace, and Sciences, INCPAA 2016, La Rochelle, France, 05-08 July 2016 / [ed] S. Sivasundaram, 2016Conference 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.

Keyword
stochastic volatilities, European option, asymptotic expansion
National Category
Probability Theory and Statistics
Research subject
Mathematics/Applied Mathematics
Identifiers
urn:nbn:se:mdh:diva-33473 (URN)
Conference
11th Internatinal Conference on Mathematical Problems in Engineering, Aerospace, and Sciences, INCPAA 2016, La Rochelle, France, 05-08 July 2016
Funder
Sida - Swedish International Development Cooperation Agency
Available from: 2016-10-26 Created: 2016-10-26 Last updated: 2016-12-09Bibliographically approved
6. Approximation Methods of European Option Pricing in Multiscale Stochastic Volatility Model
Open this publication in new window or tab >>Approximation Methods of European Option Pricing in Multiscale Stochastic Volatility Model
2016 (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, 2016Conference 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.

Keyword
Black-Scholes model, option pricing, Brownian motion, stochastic volatility, asymptotic expansion
National Category
Probability Theory and Statistics
Identifiers
urn:nbn:se:mdh:diva-33474 (URN)
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: 2016-12-09Bibliographically approved

Open Access in DiVA

fulltext(494 kB)14 downloads
File information
File name FULLTEXT03.pdfFile size 494 kBChecksum SHA-512
4667867830573bc1f3683a9cca93dd5d973b95b3b61b94a15a5183fe277e53d7fbda5d09ff663523e4452f74b96b325f5e54290c0dbceebbdc9578575437662a
Type fulltextMimetype application/pdf

Search in DiVA

By author/editor
Canhanga, Betuel
By organisation
Educational Sciences and Mathematics
Mathematics

Search outside of DiVA

GoogleGoogle Scholar
Total: 14 downloads
The number of downloads is the sum of all downloads of full texts. It may include eg previous versions that are now no longer available

Total: 93 hits
ReferencesLink to record
Permanent link

Direct link