| 1. |
- Barenthin, Märta, et al.
(författare)
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Data-driven methods for L2-gain estimation
- 2009
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Ingår i: IFAC Proceedings Volumes (IFAC-PapersOnline). ; , s. 1597-1602
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Konferensbidrag (refereegranskat)abstract
- In this paper we present and discuss some data-driven methods for estimation of the L2-gain of dynamical systems. Partial results on convergence and statistical properties are provided. The methods are based on multiple experiments on the system. The main idea is to directly estimate the maximizing input signal by using iterative experiments on the true system. We study such a data-driven method based on a stochastic gradient method. We show that this method is very closely related to the so-called power iteration method based on the power method in numerical analysis. Furthermore, it is shown that this method is applicable for linear systems with noisy measurements. We will also study L2-gain estimation of Hammerstein systems. The stochastic gradient method and the power iteration method are evaluated and compared in simulation examples. © 2009 IFAC.
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| 2. |
- Barkhagen, Mathias, et al.
(författare)
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An Improved Convex Model for Efficient Estimation of Option Implied Surfaces
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Annan publikation (övrigt vetenskapligt/konstnärligt)abstract
- 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.
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| 3. |
- Barkhagen, Mathias, et al.
(författare)
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Hedging of an Option Book at Actual Market Prices Using Stochastic Programming
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Annan publikation (övrigt vetenskapligt/konstnärligt)abstract
- 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.
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| 4. |
- Barkhagen, Mathias, et al.
(författare)
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Modeling and evaluation of the option book hedging problem using stochastic programming
- 2016
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Ingår i: Quantitative finance (Print). - : Routledge. - 1469-7688 .- 1469-7696. ; 16:2, s. 259-273
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Tidskriftsartikel (refereegranskat)abstract
- 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.
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| 5. |
- Barkhagen, Mathias, et al.
(författare)
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Non-parametric estimation of local variance surfaces
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Annan publikation (övrigt vetenskapligt/konstnärligt)abstract
- 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.
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| 6. |
- Barkhagen, Mathias, et al.
(författare)
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Non-Parametric Estimation of Stable Local Volatility Surfaces
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Annan publikation (övrigt vetenskapligt/konstnärligt)abstract
- 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.
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| 7. |
- Barkhagen, Mathias, et al.
(författare)
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Non-parametric estimation of the option implied risk-neutral density surface
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Annan publikation (övrigt vetenskapligt/konstnärligt)abstract
- 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.
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| 8. |
- Barkhagen, Mathias, 1972-
(författare)
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Optimal Decisions in the Equity Index Derivatives Markets Using Option Implied Information
- 2015
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- 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.
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| 9. |
- Barkhagen, Mathias, et al.
(författare)
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Option Market Prediction of the S&P 500 Index Return Distribution
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Annan publikation (övrigt vetenskapligt/konstnärligt)abstract
- 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.
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| 10. |
- Barkhagen, Mathias, et al.
(författare)
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Recovering the Real-World Density and Liquidity Premia from Option Data
- 2016
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Ingår i: Quantitative finance (Print). - : Taylor & Francis. - 1469-7688 .- 1469-7696. ; 16:7, s. 1147-1164
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Tidskriftsartikel (refereegranskat)abstract
- 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.
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