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26
A convergence rates result for Tikhonov regularization in Banach spaces with nonsmooth operators
 Inverse Problems
"... There exists a vast literature on convergence rates results for Tikhonov regularized minimizers. We are concerned with the solution of nonlinear illposed operator equations. The first convergence rates results for such problems have been developed by Engl, Kunisch and Neubauer in 1989. While these ..."
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Cited by 50 (13 self)
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There exists a vast literature on convergence rates results for Tikhonov regularized minimizers. We are concerned with the solution of nonlinear illposed operator equations. The first convergence rates results for such problems have been developed by Engl, Kunisch and Neubauer in 1989. While these results apply for operator equations formulated in Hilbert spaces, the results of Burger and Osher from 2004, more generally, apply to operators formulated in Banach spaces. Recently, Resmerita et al. presented a modification of the convergence rates result of Burger and Osher which turns out a complete generalization of the rates result of Engl et. al. In all these papers relatively strong regularity assumptions are made. However, it has been observed numerically, that violations of the smoothness assumptions of the operator do not necessarily affect the convergence rate negatively. We take this observation and weaken the smoothness assumptions on the operator and prove a novel convergence
Spectral calibration of exponential Lévy models
 Finance and Stochastics
"... This research was supported by the Deutsche ..."
Tikhonov regularization applied to the inverse problem of option pricing: convergence analysis and rates
 Inverse Problems
, 2005
"... Abstract. This paper investigates the stable identification of local volatility surfaces σ(S, t) in the BlackScholes/Dupire equation from market prices of European Vanilla options. Based on the properties of the parametertosolution mapping, which assigns option prices to given volatilities, we sh ..."
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Abstract. This paper investigates the stable identification of local volatility surfaces σ(S, t) in the BlackScholes/Dupire equation from market prices of European Vanilla options. Based on the properties of the parametertosolution mapping, which assigns option prices to given volatilities, we show stability and convergence of approximations gained by Tikhonov regularization. In case of a known termstructure of the volatility surface, in particular if the volatility is assumed to be constant in time (σ(S, T) = σ(S)), we prove convergence rates under simple smoothness and decay conditions on the true volatility. The convergence rate analysis sheds light onto the importance of an appropriate apriori guess for the unknown volatility and the nature for the illposedness of the inverse problem, caused by smoothing properties and the nonlinearity of the direct problem. Finally, the theoretical results are illustrated by numerical experiments. 1.
On the nature of illposedness of an inverse problem arising in option pricing
 Inverse Problems
, 2003
"... Abstract Inverse problems in option pricing are frequently regarded as simple and resolved if a formula of BlackScholes type defines the forward operator. However, precisely because the structure of such problems is straightforward, they may serve as benchmark problems for studying the nature of i ..."
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Cited by 7 (4 self)
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Abstract Inverse problems in option pricing are frequently regarded as simple and resolved if a formula of BlackScholes type defines the forward operator. However, precisely because the structure of such problems is straightforward, they may serve as benchmark problems for studying the nature of illposedness and the impact of data smoothness and no arbitrage on solution properties. In this paper, we analyse the inverse problem (IP) of calibrating a purely timedependent volatility function from a termstructure of option prices by solving an illposed nonlinear operator equation in spaces of continuous and powerintegrable functions over a finite interval. The forward operator of the IP under consideration is decomposed into an inner linear convolution operator and an outer nonlinear Nemytskii operator given by a BlackScholes function. The inversion of the outer operator leads to an illposedness effect localized at small times, whereas the inner differentiation problem is ill posed in a global manner. Several aspects of regularization and their properties are discussed. In particular, a detailed analysis of local illposedness and Tikhonov regularization of the complete IP including convergence rates is given in a Hilbert space setting. A brief numerical case study on synthetic data illustrates and completes the paper.
An infinite dimensional stochastic analysis approach to local volatility dynamic models
 COMMUNICATIONS ON STOCHASTIC ANALYSIS
, 2008
"... The difficult problem of the characterization of arbitrage free dynamic stochastic models for the equity markets was recently given a new life by the introduction of market models based on the dynamics of the local volatility. Typically, market models are based on Itô stochastic differential equatio ..."
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Cited by 5 (2 self)
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The difficult problem of the characterization of arbitrage free dynamic stochastic models for the equity markets was recently given a new life by the introduction of market models based on the dynamics of the local volatility. Typically, market models are based on Itô stochastic differential equations modeling the dynamics of a set of basic instruments including, but not limited to, the option underliers. These market models are usually recast in the framework of the HJM philosophy originally articulated for Treasury bond markets. In this paper we streamline some of the recent results on the local volatility dynamics by employing an infinite dimensional stochastic analysis approach as advocated by the pioneering work of L. Gross and his students.
Matching Statistics of an Itô Process by a Process of Diffusion Type
, 2010
"... Suppose we are given a multidimensional Itô process, which can be regarded as a model for an underlying asset price together with related stochastic processes, e.g., volatility. The drift and diffusion terms for this Itô process are permitted to be arbitrary adapted processes. We construct a weak s ..."
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Suppose we are given a multidimensional Itô process, which can be regarded as a model for an underlying asset price together with related stochastic processes, e.g., volatility. The drift and diffusion terms for this Itô process are permitted to be arbitrary adapted processes. We construct a weak solution to a diffusiontype equation that matches the distribution of the Itô process at each fixed time. Moreover, we show how to also match the distribution at each fixed time of statistics of the Itô process, including the running maximum and running average of one of the components of the process. A consequence of this result is that a wide variety of exotic derivative securities have the same prices when written on the original Itô process as when written on the mimicking process. 1 Partially supported by the National Science Foundation under Grants No. DMS0404682 1.1 Contribution of this work In this paper, we show that it is possible to construct a process that mimics
A Bayesian Approach to Financial Model Calibration, Uncertainty Measures and Optimal Hedging
"... Michaelmas 2009This thesis is dedicated to the late ..."
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Michaelmas 2009This thesis is dedicated to the late
Robust Calibration of Financial Models Using Bayesian Estimators
, 2012
"... We consider a general calibration problem for derivative pricing models, which we reformulate into a Bayesian framework to attain posterior distributions for model parameters. It is then shown how the posterior distribution can be used to estimate prices for exotic options. We apply the procedure to ..."
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Cited by 2 (1 self)
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We consider a general calibration problem for derivative pricing models, which we reformulate into a Bayesian framework to attain posterior distributions for model parameters. It is then shown how the posterior distribution can be used to estimate prices for exotic options. We apply the procedure to a discrete local volatility model and work in great detail through numerical examples to clarify the construction of Bayesian estimators and their robustness to the model specification, number of calibration products, noisy data and misspecification of the prior. 1