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93
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
StaticArbitrage optimal subreplicating strategies
, 2004
"... In this paper we investigate the possible values of basket options. ..."
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Cited by 13 (1 self)
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In this paper we investigate the possible values of basket options.
Nine Ways to Implement the Binomial Method for Option Valuation
 SIAM Rev
, 2002
"... Abstract. In the context of a reallife application that is of interest to many students, we illustrate how the choices made in translating an algorithm into a highlevel computer code can affect the execution time. More precisely, we give nine MATLAB programs that implement the binomial method for ..."
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Abstract. In the context of a reallife application that is of interest to many students, we illustrate how the choices made in translating an algorithm into a highlevel computer code can affect the execution time. More precisely, we give nine MATLAB programs that implement the binomial method for valuing a European put option. The first program is a straightforward translation of the pseudocode in Figure 10.4 of The Mathematics of Financial Derivatives, by P. Wilmott, S. Howison, and J. Dewynne, Cambridge University Press, 1995. Four variants of this program are then presented that improve the efficiency by avoiding redundant computation, vectorizing, and accessing subarrays via MATLAB’s colon notation. We then consider reformulating the problem via a binomial coefficient expansion. Here, a straightforward implementation is seen to be improved by vectorizing, avoiding overflow and underflow, and exploiting sparsity. Overall, the fastest of the binomial method programs has an execution time that is within a factor 2 of direct evaluation of the Black–Scholes formula. One of the vectorized versions is then used as the basis for a program that values an American put option. The programs show how execution times in MATLAB can be dramatically reduced by using highlevel operations on arrays rather than computing with individual components, a principle that applies in many scientific computing environments. The relevant files are downloadable from the World Wide Web.
Characterization of optimal stopping regions of American Asian and lookback options
 Mathematical Finance
, 2006
"... A general framework is developed to analyze the optimal stopping (exercise) regions of American path dependent options with either Asian feature or lookback feature. We examine the monotonicity properties of the option values and stopping regions with respect to the interest rate, dividend yield and ..."
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Cited by 11 (3 self)
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A general framework is developed to analyze the optimal stopping (exercise) regions of American path dependent options with either Asian feature or lookback feature. We examine the monotonicity properties of the option values and stopping regions with respect to the interest rate, dividend yield and time. From the ordering properties of the values of American lookback options and American Asian options, we deduce the corresponding nesting relations between the exercise regions of these American options. We illustrate how some properties of the exercise regions of the American Asian options can be inferred from those of the American lookback options.
On coordinate transformation and grid stretching for sparse grid pricing of basket options
 Journal of Computational and Applied Mathematics
"... of basket options ..."
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ACCURATE EVALUATION OF EUROPEAN AND AMERICAN OPTIONS UNDER THE CGMY PROCESS
"... A finitedifference method for integrodifferential equations arising from Lévy driven asset processes in finance is discussed. The equations are discretized in space by the collocation method and in time by an explicit backward differentiation formula. The discretization is shown to be secondorder ..."
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Cited by 11 (1 self)
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A finitedifference method for integrodifferential equations arising from Lévy driven asset processes in finance is discussed. The equations are discretized in space by the collocation method and in time by an explicit backward differentiation formula. The discretization is shown to be secondorder accurate independently of the degree of the singularity in the Lévy measure. The singularity is dealt with by means of an integration by parts technique. An application of the fast Fourier transform gives the overall amount of work O(MN log N), rendering the method fast.
Cacheoptimal algorithms for option pricing
, 2008
"... Today computers have several levels of memory hierarchy. To obtain good performance on these processors it is necessary to design algorithms that minimize I/O traffic to slower memories in the hierarchy. In this paper, we study the computation of option pricing using the binomial and trinomial model ..."
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Today computers have several levels of memory hierarchy. To obtain good performance on these processors it is necessary to design algorithms that minimize I/O traffic to slower memories in the hierarchy. In this paper, we study the computation of option pricing using the binomial and trinomial models on processors with a multilevel memory hierarchy. We derive lower bounds on memory traffic between different levels of hierarchy for these two models. We also develop algorithms for the binomial and trinomial models that have nearoptimal memory traffic between levels. We have implemented these algorithms on an UltraSparc IIIi processor with a 4level of memory hierarchy and demonstrated that our algorithms outperform algorithms without cache blocking by a factor of up to 5 and operate at 70 % of peak performance.
The numerical solution of nonlinear Black–Scholes equations

, 2008
"... Nonlinear Black–Scholes equations have been increasingly attracting interest over the last two decades, since they provide more accurate values by taking into account more realistic assumptions, such as transaction costs, illiquid markets, risks from an unprotected portfolio or large investor’s pref ..."
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Nonlinear Black–Scholes equations have been increasingly attracting interest over the last two decades, since they provide more accurate values by taking into account more realistic assumptions, such as transaction costs, illiquid markets, risks from an unprotected portfolio or large investor’s preferences, which may have an impact on the stock price, the volatility, the drift and the option price itself. In this work we will be concerned with several models from the most relevant class of nonlinear Black–Scholes equations for European and American options with a volatility depending on different factors, such as the stock price, the time, the option price and its derivatives, where the nonlinearity results from the presence of transaction costs. In the European case we will consider a European Call option and analytically approach the option price by transforming the problem into a forward convectiondiffusion equation with a nonlinear term. In case of American options we will consider an American Call option and transform this free boundary problem into a fully nonlinear parabolic equation defined on a fixed domain following Sevčovič’s idea [72]. Finally, we will present the numerical results of different discretization schemes for European and American options for various volatility models including Leland’s model, Barles’ and Soner’s model and the Risk Adjusted
Strong i/o lower bounds for binomial and fft computation graphs
, 2010
"... Abstract. Processors on most of the modern computing devices have several levels of memory hierarchy. To obtain good performance on these processors it is necessary to design algorithms that minimize I/O traffic to slower memories in the hierarchy. In this paper, we propose a new technique, the boun ..."
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Abstract. Processors on most of the modern computing devices have several levels of memory hierarchy. To obtain good performance on these processors it is necessary to design algorithms that minimize I/O traffic to slower memories in the hierarchy. In this paper, we propose a new technique, the boundary flow technique, for deriving lower bounds on the memory traffic complexity of problems in multilevel memory hierarchy architectures. The boundary flow technique relies on identifying subcomputation structure corresponding to equal computations with a minimum number of boundary vertices, which in turn is related to the vertex isoperimetric parameter of a computation graph. We demonstrate that this technique results in stronger lower bounds for memory traffic on memory hierarchy architectures for wellknown computation structures: the binomial computation graphs and FFT computation graphs. For binomial computation we improve the lower bound by a factor of three. This reduces the gap between the lower and upper bound from a factor of 4 to a factor of 4/3. For FFT computation, past work has mostly focused on asymptotic lower bounds. We improve the best known previous lower bound for FFT computation by a factor of 8. The lower bound established is almost optimal as there exists a simple FFT algorithm that nearly achieves this bound. 1