Uniqueness, stability and numerical methods for the inverse problem that arises in financial markets

Bouchouev, Ilia; Isakov, Victor

doi:10.1088/0266-5611/15/3/201

Cited by 155 publications

(122 citation statements)

References 18 publications

Supporting

Mentioning

118

Contrasting

Unclassified

Order By: Relevance

“…Problems of this kind arise in the description of a multitude of scientific and engineering applications including optimal design, control, and parameter identification [8]. Examples of PDE-constrained optimization problems arise in aerodynamics [30,36], mathematical finance [10,15,16], medicine [4,26], and geophysics and environmental engineering [2,1,27]. PDE-constrained optimization problems are infinite-dimensional and often ill-posed in nature, and their discretization invariably leads to systems of equations that are typically very large and difficult to solve.…”

Section: Introductionmentioning

confidence: 99%

A preconditioning technique for a class of PDE-constrained optimization problems

Benzi

Haber

Taralli³

2011

Adv Comput Math

View full text Add to dashboard Cite

We investigate the use of a preconditioning technique for solving linear systems of saddle point type arising from the application of an inexact Gauss-Newton scheme to PDE-constrained optimization problems with a hyperbolic constraint. The preconditioner is of block triangular form and involves diagonal perturbations of the (approximate) Hessian to insure nonsingularity and an approximate Schur complement. We establish some properties of the preconditioned saddle point systems and we present the results of numerical experiments illustrating the performance of the preconditioner on a model problem motivated by image registration.

show abstract

Section: Introductionmentioning

confidence: 99%

A preconditioning technique for a class of PDE-constrained optimization problems

Benzi

Haber

Taralli³

2011

Adv Comput Math

View full text Add to dashboard Cite

show abstract

“…One example of a parametrization of the squared local volatility surface is given in Carmona and Nadtochiy (2009) who use 9 parameters in order to parameterize the surface. As pointed out in Bouchouev and Isakov (1999) the inverse problem when assuming a parametric form for the surface is generally over-determined which means that the optimal parameters cannot be found in a unique and stable way. Assuming a parametric form will typically give rise to non-convex optimization problems with many local minima.…”

Section: Estimation Methods Described In the Literaturementioning

confidence: 99%

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

Barkhagen¹

2015

Linköping Studies in Science and Technology. Dissertations

View full text Add to dashboard Cite

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...

show abstract

“…Some detailed treatments of problems in these areas can be found in [4][5][6][7][8][9][10][11][12][13][14][15][16]. In [5,9,15], Bouchouev and Isakov reduce the identification of volatility to an inverse problem with the final observation, and the local uniqueness and stability of volatility are proved under certain assumptions. Lu and Yi in [12] obtain a Fredholm integral equation from the Dupire equation.…”

Section: (S a T) = V (S B T) = (1 + R 1 T)e -R(t-t) (0 ≤ T < T)mentioning

confidence: 99%

Identifying the implied volatility using the total variation regularization

Liu

2018

Bound Value Probl

View full text Add to dashboard Cite

This paper studies an inverse problem of determining the implied volatility in the financial products linked with gold price, which has important application in financial derivatives pricing. Based on the total variation regularization strategy, the existence and necessary condition of the minimum for the control function are addressed, and the local uniqueness of the solution is also given by a modified case. Furthermore, the stability and convergence for the regularized approach are discussed. The results obtained in this paper may be useful for those who engage in hedging or risk management.

show abstract

Uniqueness, stability and numerical methods for the inverse problem that arises in financial markets

Cited by 155 publications

References 18 publications

A preconditioning technique for a class of PDE-constrained optimization problems

A preconditioning technique for a class of PDE-constrained optimization problems

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

Identifying the implied volatility using the total variation regularization

Contact Info

Product

Resources

About