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Barkhagen, Mathias
Publications (10 of 11) Show all publications
Barkhagen, M. & Blomvall, J. (2016). Modeling and evaluation of the option book hedging problem using stochastic programming. Paper presented at 13th International Conference of Stochastic Programming. Quantitative finance (Print), 16(2), 259-273
Open this publication in new window or tab >>Modeling and evaluation of the option book hedging problem using stochastic programming
2016 (English)In: Quantitative finance (Print), ISSN 1469-7688, E-ISSN 1469-7696, Vol. 16, no 2, p. 259-273Article in journal (Refereed) Published
Abstract [en]

Hedging of an option book in an incomplete market with transaction costs is an important problem in finance that many banks have to solve on a daily basis. In this paper, we develop a stochastic programming (SP) model for the hedging problem in a realistic setting, where all transactions take place at observed bid and ask prices. The SP model relies on a realistic modeling of the important risk factors for the application, the price of the underlying security and the volatility surface. The volatility surface is unobservable and must be estimated from a cross section of observed option quotes that contain noise and possibly arbitrage. In order to produce arbitrage-free volatility surfaces of high quality as input to the SP model, a novel non-parametric estimation method is used. The dimension of the volatility surface is infinite and in order to be able solve the problem numerically, we use discretization and principal component analysis to reduce the dimensions of the problem. Testing the model out-of-sample for options on the Swedish OMXS30 index, we show that the SP model is able to produce a hedge that has both a lower realized risk and cost compared with dynamic delta and delta-vega hedging strategies.

Place, publisher, year, edition, pages
Routledge, 2016
Keywords
Option hedging, Stochastic programming, Simulation, Local volatility surface, Empirical evaluation
National Category
Probability Theory and Statistics Economics and Business
Identifiers
urn:nbn:se:liu:diva-130323 (URN)10.1080/14697688.2015.1114358 (DOI)000378169900009 ()
Conference
13th International Conference of Stochastic Programming
Note

At the time for thesis presentation publication was in status: Manuscript

At the time for thesis presentation manuscript was named: Hedging of an Option Book at Actual Market Prices Using Stochastic Programming

Available from: 2016-07-29 Created: 2016-07-28 Last updated: 2017-11-28Bibliographically approved
Barkhagen, M., Blomvall, J. & Platen, E. (2016). Recovering the Real-World Density and Liquidity Premia from Option Data. Quantitative finance (Print), 16(7), 1147-1164
Open this publication in new window or tab >>Recovering the Real-World Density and Liquidity Premia from Option Data
2016 (English)In: Quantitative finance (Print), ISSN 1469-7688, E-ISSN 1469-7696, Vol. 16, no 7, p. 1147-1164Article in journal (Refereed) Published
Abstract [en]

In this paper we develop a methodology for simultaneous recovery of the real-world probability density and liquidity premia from observed S&P500 index option prices. Assuming the existence of a numeraire portfolio for the US equity market, fair prices of derivatives under the benchmark approach can be obtained directly under the real-world measure. Under this modeling framework there exists a direct link between observed call option prices on the index and the real-world density for the underlying index. We use a novel method for estimation of option implied volatility surfaces of high quality which enables the subsequent analysis. We show that the real-world density that we recover is consistent with the observed realized dynamics of the underlying index. This admits the identication of liquidity premia embedded in option price data. We identify and estimate two separate liquidity premia embedded in S&P500 index options that are consistent with previous findings in the literature.

Place, publisher, year, edition, pages
Taylor & Francis, 2016
Keywords
Real-world density; Liquidity premia; Local volatility model; No-nparametric estimation; Simulated Maximum Likelihood
National Category
Economics and Business
Identifiers
urn:nbn:se:liu:diva-117104 (URN)10.1080/14697688.2015.1128117 (DOI)000379836500011 ()
Available from: 2015-04-16 Created: 2015-04-16 Last updated: 2017-12-04Bibliographically approved
Barkhagen, M. (2015). Optimal Decisions in the Equity Index Derivatives Markets Using Option Implied Information. (Doctoral dissertation). Linköping: Linköping University Electronic Press
Open this publication in new window or tab >>Optimal Decisions in the Equity Index Derivatives Markets Using Option Implied Information
2015 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This dissertation is centered around two comprehensive themes: the extraction of information embedded in equity index option prices, and how to use this information in order to be able to make optimal decisions in the equity index option markets. These problems are important for decision makers in the equity index options markets, since they are continuously faced with making decisions under uncertainty given observed market prices. The methods developed in this dissertation provide robust tools that can be used by practitioners in order to improve the quality of the decisions that they make.

In order to be able to extract information embedded in option prices, the dissertation develops two different methods for estimation of stable option implied surfaces which are consistent with observed market prices. This is a difficult and ill-posed inverse problem which is complicated by the fact that observed option prices contain a large amount of noise stemming from market micro structure effects. Producing estimated surfaces that are stable over time is important since otherwise risk measurement of derivatives portfolios, pricing of exotic options and calculation of hedge parameters will be prone to include significant errors. The first method that we develop leads to an optimization problem which is formulated as a convex quadratic program with linear constraints which can be solved very efficiently. The second estimation method that we develop in the dissertation makes it possible to produce local volatility surfaces of high quality, which are consistent with market prices and stable over time. The high quality of the surfaces estimated with the second method is the crucial input to the research which has resulted in the last three papers of the dissertation.

The stability of the estimated local volatility surfaces makes it possible to build a realistic dynamic model for the equity index derivatives market. This model forms the basis for the stochastic programming (SP) model for option hedging that we develop in the dissertation. We show that the SP model, which uses generated scenarios for the squared local volatility surface as input,  outperforms the traditional hedging methods that are described in the literature. Apart from having an accurate view of the variance of relevant risk factors, it is when building a dynamic model also important to have a good estimate of the expected values, and thereby risk premia, of those factors. We use a result from recently published research which lets us recover the real-world density from only a cross-section of observed option prices via a local volatility model. The recovered real-world densities are then used in order to identify and estimate liquidity premia that are embedded in option prices.

We also use the recovered real-world densities in order to test how well the option market predicts the realized statistical characteristics of the underlying index. We compare the results with the performance of commonly used models for the underlying index. The results show that option prices contain a premium in the tails of the distribution. By removing the estimated premia from the tails, the resulting density predicts future realizations of the underlying index very well.

Place, publisher, year, edition, pages
Linköping: Linköping University Electronic Press, 2015. p. 103
Series
Linköping Studies in Science and Technology. Dissertations, ISSN 0345-7524 ; 1657
Keywords
Option implied information; Optimal decisions; Equity index derivatives; Stochastic programming; Local volatility surface; Real-world density
National Category
Economics and Business
Identifiers
urn:nbn:se:liu:diva-117106 (URN)10.3384/diss.diva-117106 (DOI)978-91-7519-081-5 (ISBN)
Public defence
2015-05-12, ACAS, Hus A, Campus Valla, Linköping, 10:15 (English)
Opponent
Supervisors
Available from: 2015-04-16 Created: 2015-04-16 Last updated: 2017-01-09Bibliographically approved
Barkhagen, M. (2013). Risk-Neutral and Physical Estimation of Equity Market Volatility. (Licentiate dissertation). Linköping: Linköping University Electronic Press
Open this publication in new window or tab >>Risk-Neutral and Physical Estimation of Equity Market Volatility
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. p. 19
Series
Linköping Studies in Science and Technology. Thesis, ISSN 0280-7971 ; 1601
National Category
Economics and Business Probability Theory and Statistics
Identifiers
urn:nbn:se:liu:diva-94360 (URN)978-91-7519-583-4 (ISBN)
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
Barkhagen, M. & Blomvall, J.An Improved Convex Model for Efficient Estimation of Option Implied Surfaces.
Open this publication in new window or tab >>An Improved Convex Model for Efficient Estimation of Option Implied Surfaces
(English)Manuscript (preprint) (Other academic)
Abstract [en]

Estimation of option implied surfaces that are consistent with observed market prices and stable over time is a fundamental problem in finance. This paper develops a general optimization based framework for estimation of the option implied risk-neutral density (RND) surface, while satisfying no-arbitrage constraints. Our developed framework considers all types of realistic surfaces and is hence not constrained to a certain function class. When solving the problem the RND is discretized, which leads to an optimization model where it is possible to formulate the constraints as linear constraints, making the resulting large-scale optimization problem convex and the solution a global optimum. This is a major advantage of our method compared to most estimation algorithms described in the literature, which are typically cast as non-convex optimization problems with multiple local optima. We show that our method produces smooth local volatility surfaces that can be used for pricing and hedging of exotic derivatives. The stability of our method is demonstrated through a time series study based on historical prices of S&P 500 index options.

Keywords
Risk-neutral density surface; Non-parametric estimation; Optimization; No-arbitrage constraints; Implied volatility surface; Local volatility surface
National Category
Economics and Business
Identifiers
urn:nbn:se:liu:diva-117101 (URN)
Available from: 2015-04-16 Created: 2015-04-16 Last updated: 2015-04-21
Barkhagen, M. & Blomvall, J.Hedging of an Option Book at Actual Market Prices Using Stochastic Programming.
Open this publication in new window or tab >>Hedging of an Option Book at Actual Market Prices Using Stochastic Programming
(English)Manuscript (preprint) (Other academic)
Abstract [en]

Hedging of an option book in an incomplete market with transaction costs is an important problem in finance that many banks have to solve on a daily basis. In this paper we develop a stochastic programming (SP) model for the hedging problem in a realistic setting, where all transactions take place at observed bid and ask prices. The SP model relies on a realistic modelling of the important risk factors for the application, the price of the underlying security and the volatility surface. The volatility surface is unobservable and must be estimated from a cross-section of observed option quotes that contain noise and possibly arbitrage. In order to produce arbitrage-free volatility surfaces with high quality as input to the SP model a novel non-parametric estimation method is used. The dimension of the volatility surface is infinite and in order to be able solve the problem numerically we use discretization and PCA to reduce the dimensions of the problem. Testing the model out-of-sample for options on the Swedish OMXS30 index, we show that the SP model is able to produce a hedge that has both a lower realized risk and cost compared with dynamic delta and delta-vega hedging strategies.

Keywords
Option hedging; Stochastic programming; Local volatility surface; Empirical evaluation
National Category
Economics and Business
Identifiers
urn:nbn:se:liu:diva-117103 (URN)
Available from: 2015-04-16 Created: 2015-04-16 Last updated: 2017-01-09
Barkhagen, M. & Blomvall, J.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.

Keywords
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
Barkhagen, M. & Blomvall, J.Non-Parametric Estimation of Stable Local Volatility Surfaces.
Open this publication in new window or tab >>Non-Parametric Estimation of Stable Local Volatility 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 volatility surface. We show that our method produces local volatility surfaces with very high quality and which are consistent with observed S&P 500 index 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 for input data which contains a lot of noise. Through a time series study we show that our optimization based framework produces squared local volatility surfaces that are stable over time. Given a specic level of consistency with observed market prices there exist an innite 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 squared 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- , 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 resulting optimization problem is non-convex, but extensive empirical tests indicate that the solution does not get stuck in local optima.

Keywords
Local volatility surface; Non-parametric estimation; Optimization; No-arbitrage conditions; Principal Component Analysis
National Category
Economics and Business
Identifiers
urn:nbn:se:liu:diva-117102 (URN)
Available from: 2015-04-16 Created: 2015-04-16 Last updated: 2015-04-21
Barkhagen, M. & Blomvall, J.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.

Keywords
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
Barkhagen, M. & Blomvall, J.Option Market Prediction of the S&P 500 Index Return Distribution.
Open this publication in new window or tab >>Option Market Prediction of the S&P 500 Index Return Distribution
(English)Manuscript (preprint) (Other academic)
Abstract [en]

In this paper we evaluate the density forecasts obtained from a cross-section of S&P 500 index option prices. The option implied density forecasts rely on a result derived by Heath and Platen (2006), which under certain assumptions allows us to transform risk-neutral densities into real-world densities. In order to remove liquidity premia from the real-world densities we use a  transformation into densities implied by the Minimal Market Model. The accuracy of the estimated real-world density forecasts relies on using a recently developed method for estimation of risk-neutral densities of high quality. We find that our recovered real-world densities explains the realized return distribution for S&P 500 better than historical GARCH densities for a forecasting horizon of two days. This can be contrasted to the findings in two recent papers in the literature, who find that historical densities estimated from intra-day data performs as least as well as option implied densities for a forecasting horizon of one day.

Keywords
Option implied information; Density forecast evaluation; Real-world density; Local volatility model; Non-parametric estimation
National Category
Economics and Business
Identifiers
urn:nbn:se:liu:diva-117105 (URN)
Available from: 2015-04-16 Created: 2015-04-16 Last updated: 2015-04-21Bibliographically approved
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