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Non-Parametric Estimation of Stable Local Volatility Surfaces
Linköping University, Department of Management and Engineering, Production Economics. Linköping University, The Institute of Technology.
Linköping University, Department of Management and Engineering, Production Economics. Linköping University, The Institute of Technology.
(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.

Keyword [en]
Local volatility surface; Non-parametric estimation; Optimization; No-arbitrage conditions; Principal Component Analysis
National Category
Economics and Business
URN: urn:nbn:se:liu:diva-117102OAI: diva2:805718
Available from: 2015-04-16 Created: 2015-04-16 Last updated: 2015-04-21
In thesis
1. Optimal Decisions in the Equity Index Derivatives Markets Using Option Implied Information
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. 103 p.
Linköping Studies in Science and Technology. Dissertations, ISSN 0345-7524 ; 1657
Option implied information; Optimal decisions; Equity index derivatives; Stochastic programming; Local volatility surface; Real-world density
National Category
Economics and Business
urn:nbn:se:liu:diva-117106 (URN)10.3384/diss.diva-117106 (DOI)978-91-7519-081-5 (print) (ISBN)
Public defence
2015-05-12, ACAS, Hus A, Campus Valla, Linköping, 10:15 (English)
Available from: 2015-04-16 Created: 2015-04-16 Last updated: 2015-04-21Bibliographically approved

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Barkhagen, MathiasBlomvall, Jörgen
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