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Risk-Neutral and Physical Estimation of Equity Market Volatility
Linköping University, Department of Management and Engineering, Production Economics. Linköping University, The Institute of Technology.
2013 (English)Licentiate thesis, comprehensive summary (Other academic)
Abstract [en]

The overall purpose of the PhD project is to develop a framework for making optimal decisions on the equity derivatives markets. Making optimal decisions refers e.g. to how to optimally hedge an options portfolio or how to make optimal investments on the equity derivatives markets. The framework for making optimal decisions will be based on stochastic programming (SP) models, which means that it is necessary to generate high-quality scenarios of market prices at some future date as input to the models. This leads to a situation where the traditional methods, described in the literature, for modeling market prices do not provide scenarios of sufficiently high quality as input to the SP model. Thus, the main focus of this thesis is to develop methods that improve the estimation of option implied surfaces from a cross-section of observed option prices compared to the traditional methods described in the literature. The estimation is complicated by the fact that observed option prices contain a lot of noise and possibly also arbitrage. This means that in order to be able to estimate option implied surfaces which are free of arbitrage and of high quality, the noise in the input data has to be adequately handled by the estimation method.

The first two papers of this thesis develop a non-parametric optimization based framework for the estimation of high-quality arbitrage-free option implied surfaces. The first paper covers the estimation of the risk-neutral density (RND) surface and the second paper the local volatility surface. Both methods provide smooth and realistic surfaces for market data. Estimation of the RND is a convex optimization problem, but the result is sensitive to the parameter choice. When the local volatility is estimated the parameter choice is much easier but the optimization problem is non-convex, even though the algorithm does not seem to get stuck in local optima. The SP models used to make optimal decisions on the equity derivatives markets also need generated scenarios for the underlying stock prices or index levels as input. The third paper of this thesis deals with the estimation and evaluation of existing equity market models. The third paper gives preliminary results which show that, out of the compared models, a GARCH(1,1) model with Poisson jumps provides a better fit compared to more complex models with stochastic volatility for the Swedish OMXS30 index.

Abstract [sv]

Det övergripande syftet med doktorandprojektet är att utveckla ett ramverk för att fatta optimala beslut på aktiederivatmarknaderna. Att fatta optimala beslut syftar till exempel på hur man optimalt ska hedga en optionsportfölj, eller hur man ska göra optimala investeringar på aktiederivatmarknaderna. Ramverket för att fatta optimala beslut kommer att baseras på stokastisk programmerings-modeller (SP-modeller), vilket betyder att det är nödvändigt att generera högkvalitativa scenarier för marknadspriser för en framtida tidpunkt som indata till SP-modellen. Detta leder till en situation där de traditionella metoderna, som finns beskrivna i litteraturen, för att modellera marknadspriser inte ger scenarier av tillräckligt hög kvalitet för att fungera som indata till SP-modellen. Följaktligen är huvudfokus för denna avhandling att utveckla metoder som, jämfört med de traditionella metoderna som finns beskrivna i litteraturen, förbättrar estimeringen av ytor som impliceras av en given mängd observerade optionspriser. Estimeringen kompliceras av att observerade optionspriser innehåller mycket brus och möjligen också arbitrage. Det betyder att för att kunna estimera optionsimplicerade ytor som är arbitragefria och av hög kvalitet, så behöver estimeringsmetoden hantera bruset i indata på ett adekvat sätt.

De första två artiklarna i avhandlingen utvecklar ett icke-parametriskt optimeringsbaserat ramverk för estimering av högkvalitativa och arbitragefria options-implicerade ytor. Den första artikeln behandlar estimeringen av den risk-neutrala täthetsytan (RND-ytan) och den andra artikeln estimeringen av den lokala volatilitetsytan. Båda metoderna ger upphov till jämna och realistiska ytor för marknadsdata. Estimeringen av RND-ytan är ett konvext optimeringsproblem men resultatet är känsligt för valet av parametrar. När den lokala volatilitetsytan estimeras är parametervalet mycket enklare men optimeringsproblemet är icke-konvext, även om algoritmen inte verkar fastna i lokala optima. SP-modellerna som används för att fatta optimala beslut på aktiederivatmarknaderna behöver också indata i form av genererade scenarier för de underliggande aktiepriserna eller indexnivåerna. Den tredje artikeln i avhandlingen behandlar estimering och evaluering av existerande modeller för aktiemarknaden. Den tredje artikeln tillhandahåller preliminära resultat som visar att, av de jämförda modellerna, ger en GARCH(1,1)-modell med Poissonhopp en bättre beskrivning av dynamiken för det svenska aktieindexet OMXS30 jämfört med mer komplicerade modeller som innehåller stokastisk volatilitet.

Place, publisher, year, edition, pages
Linköping: Linköping University Electronic Press, 2013. , 19 p.
Series
Linköping Studies in Science and Technology. Thesis, ISSN 0280-7971 ; 1601
National Category
Economics and Business Probability Theory and Statistics
Identifiers
URN: urn:nbn:se:liu:diva-94360ISBN: 978-91-7519-583-4 (print)OAI: oai:DiVA.org:liu-94360DiVA: diva2:632478
Presentation
2013-06-18, Sal ACAS A-huset, Campus Valla, Linköpings universitet, Linköping, 10:15 (Swedish)
Opponent
Supervisors
Available from: 2013-06-25 Created: 2013-06-25 Last updated: 2013-06-26Bibliographically approved
List of papers
1. Non-parametric estimation of the option implied risk-neutral density surface
Open this publication in new window or tab >>Non-parametric estimation of the option implied risk-neutral density surface
(English)Manuscript (preprint) (Other academic)
Abstract [en]

Accurate pricing of exotic or illiquid derivatives which is consistent with noisy market prices presents a major challenge. The pricing accuracy will crucially depend on using arbitrage free inputs to the pricing engine. This paper develops a general optimization based framework for estimation of the option implied risk-neutral density (RND), while satisfying no-arbitrage constraints. Our developed framework is a generalization of the RNDs implied by existing parametric models such as the Heston model. Thus, the method considers all types of realistic surfaces and is hence not constrained to a certain function class. When solving the problem the RND is discretized making it possible to use general purpose optimization algorithms. The approach leads to an optimization model where it is possible to formulate the constraints as linear constraints making the resulting optimization problem convex. We show that our method produces smooth local volatility surfaces that can be used for pricing and hedging of exotic derivatives. By perturbing input data with random errors we demonstrate that our method gives better results than the Heston model in terms of yielding stable RNDs.

Keyword
Risk-neutral density surface, Non-parametric estimation, Optimization, No-arbitrage constraints, Implied volatility surface, Local volatility
National Category
Economics and Business Probability Theory and Statistics
Identifiers
urn:nbn:se:liu:diva-94357 (URN)
Available from: 2013-06-25 Created: 2013-06-25 Last updated: 2013-06-26Bibliographically approved
2. Non-parametric estimation of local variance surfaces
Open this publication in new window or tab >>Non-parametric estimation of local variance surfaces
(English)Manuscript (preprint) (Other academic)
Abstract [en]

In this paper we develop a general optimization based framework for estimation of the option implied local variance surface. Given a specific level of consistency with observed market prices there exist an infinite number of possible surfaces. Instead of assuming shape constraints for the surface, as in many traditional methods, we seek the solution in the subset of realistic surfaces. We select local volatilities as variables in the optimization problem since it makes it easy to ensure absence of arbitrage, and realistic local volatilities imply realistic risk-neutral density- (RND), implied volatility- and price surfaces. The objective function combines a measure of consistency with market prices, and a weighted integral of the squared second derivatives of local volatility in the strike and the time-to-maturity direction. Derivatives prices in the optimization model are calculated efficiently with a finite difference scheme on a non-uniform grid. The framework has previously been successfully applied to the estimation of RND surfaces. Compared to when modeling the RND, it is for local volatility much easier to choose the parameters in the model. Modeling the RND produces a convex optimization problem which is not the case when modeling local volatility, but empirical tests indicate that the solution does not get stuck in local optima. We show that our method produces local volatility surfaces with very high quality and which are consistent with observed option quotes. Thus, unlike many methods described in the literature, our method does not produce a local volatility surface with irregular shape and many spikes or a non-smooth and multimodal RND for input data with a lot of noise.

Keyword
Local volatility surface; Non-parametric estimation; Optimization; No-arbitrage conditions
National Category
Economics and Business Probability Theory and Statistics
Identifiers
urn:nbn:se:liu:diva-94358 (URN)
Available from: 2013-06-25 Created: 2013-06-25 Last updated: 2013-06-26Bibliographically approved
3. Statistical tests for selected equity market models
Open this publication in new window or tab >>Statistical tests for selected equity market models
(English)Manuscript (preprint) (Other academic)
Abstract [en]

In this paper we evaluate which of four candidate equity market models that provide the best fit to observed closing data for the OMXS30 index from 30 September 1986 to 6 May 2013. The candidate models are two GARCH type models and two stochastic volatility models. The stochastic volatility models are estimated with the help of Markov Chain Monte Carlo methods. We provide the full derivations of the posterior distributions for the two stochastic volatility models, which to our knowledge have not been provided in the literature before. With the help of statistical tests we conclude that, out of the four candidate models, a GARCH model which includes jumps in the index level provides the best fit to the observed OMXS30 closing data.

Keyword
GARCH models, stochastic volatility models, Markov Chain Monte Carlo methods, statistical tests
National Category
Economics and Business Probability Theory and Statistics
Identifiers
urn:nbn:se:liu:diva-94359 (URN)
Available from: 2013-06-25 Created: 2013-06-25 Last updated: 2013-06-26Bibliographically approved

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