Volatility and Correlation

Reducing the number of factors in a model by reducing the rank of a correlation matrix is a problem that often arises in finance, for instance in pricing interest rate derivatives with Libor market models. A simple iterative algorithm for correlation rank reduction is introduced, the eigenvalue zeroing by iteration, EZI, algorithm. Its convergence is investigated and extension presented with particular optimality properties. The performance of EZI is compared with those of other common methods. Different data sets are considered including empirical data from the interest rate market, different possible market cases and criteria, and a calibration case. The EZI algorithm is extremely fast even in computationally complex situations, and achieves a very high level of precision. From these results, the EZI algorithm for financial application has superior performance to the main methods in current use.Correlation matrix, rank reduction, market models,

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“…This form was introduced for forward Libor rates in Rebonato (1999). Brigo (2002) uses r 2 to give comparisons between EZN and OAP.…”

Section: Numerical Results On Financial Datamentioning

confidence: 99%

An EZI Method to Reduce the Rank of a Correlation Matrix in Financial Modelling

Morini

Webber

2006

Applied Mathematical Finance

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“…Rebonato, 2004). Prior to the 1987 crash the implied volatility curve for a given maturity was relatively flat, implying that the market participants believed that the Black-Scholes model with a constant volatility and a log-normally distributed index level was a fairly accurate description of the index dynamics.…”

Section: Empirical Properties Of the Equity Index Options Marketmentioning

confidence: 99%

Optimal Decisions in the Equity Index Derivatives Markets Using Option Implied Information

Barkhagen¹

2015

Linköping Studies in Science and Technology. Dissertations

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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 i Optimal Decisions in the Equity Index Derivatives Markets Using Option Implied Information 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 resu...

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“…The need in calibrating LMM correlations has long been known (see e.g. [39,Chapters 19,20]). In order to understand the strategies used in calibrating LMM correlations, let us restate a result of Micchelli.…”

Section: A New Applicationmentioning

confidence: 99%

“…One of the interesting choices is from Rebonato [39] λ(T i , T j ) = β exp {−α max{i, j}} , β > 0 and α ∈ IR.…”

Section: A New Applicationmentioning

confidence: 99%

Conditional Quadratic Semidefinite Programming: Examples and Methods

2014

J. Oper. Res. Soc. China

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Abstract:The conditional Quadratic Semidefinite Programming (cQSDP) refers to a class of matrix optimization problems whose matrix variables are required to be positive semidefinite on a subspace and the objectives are quadratic. The chief purpose of this paper is to focus on two primal examples of cQSDP: the problem of matrix completion/approximation on a subspace and the Euclidean distance matrix problem.For the latter problem, we review some classical contributions and establish certain links among them. Moreover, we develop a semismooth Newton method for a special class of cQSDP and establish its quadratic convergence under the condition of constraint nondegeneracy.We also include an application in calibrating the correlation matrix in Libor market models. We hope this work will stimulate new research in cQSDP. Response to Reviewers:The author would like to thank the referees and the associate editor for their efforts in their speedy review. Powered by Editorial Manager® and ProduXion Manager® from Aries Systems Corporation Conditional Quadratic Semidefinite Programming: Examples and MethodsHou-Duo Qi * April 14, 2014; Revised May 3, 2014 AbstractThe conditional Quadratic Semidefinite Programming (cQSDP) refers to a class of matrix optimization problems whose matrix variables are required to be positive semidefinite on a subspace and the objectives are quadratic. The chief purpose of this paper is to focus on two primal examples of cQSDP: the problem of matrix completion/approximation on a subspace and the Euclidean distance matrix problem. For the latter problem, we review some classical contributions and establish certain links among them. Moreover, we develop a semismooth Newton method for a special class of cQSDP and establish its quadratic convergence under the condition of constraint nondegeneracy. We also include an application in calibrating the correlation matrix in Libor market models. We hope this work will stimulate new research in cQSDP.

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Volatility and Correlation

Cited by 202 publications

References 0 publications

An EZI Method to Reduce the Rank of a Correlation Matrix in Financial Modelling

An EZI Method to Reduce the Rank of a Correlation Matrix in Financial Modelling

Optimal Decisions in the Equity Index Derivatives Markets Using Option Implied Information

Conditional Quadratic Semidefinite Programming: Examples and Methods

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