An Ingenious, Piecewise Linear Interpolation Algorithm for Pricing Arithmetic Average Options

Dai, Tian‐Shyr; Wang, Jr‐Yan; Wei, Hui-Shan

doi:10.1007/978-3-540-72870-2_25

Cited by 3 publications

(3 citation statements)

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“…The factor that prevalently motivates the use of this type of models is the simplicity of their structure which let them be efficiently implemented in both algorithms and hardware. In general, piecewise-linear models looks very appealing for graphical tasks, like curve fitting, interpolation or extrapolation, where a function is constructed to fit or determine new values within or outside the range of a discrete set of known data points (Bian and Menz 1998; Dai et al 2007; Magnani and Boyd 2009; Misener and Floudas 2010; Jimenez-Fernandez et al 2014). However, a notorious shortcoming can be distinguished in this type of models when function derivatives are of interest.…”

Section: Introductionmentioning

confidence: 99%

Transforming the canonical piecewise-linear model into a smooth-piecewise representation

et al. 2016

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A smoothed representation (based on natural exponential and logarithmic functions) for the canonical piecewise-linear model, is presented. The result is a completely differentiable formulation that exhibits interesting properties, like preserving the parameters of the original piecewise-linear model in such a way that they can be directly inherited to the smooth model in order to determine their parameters, the capability of controlling not only the smoothness grade, but also the approximation accuracy at specific breakpoint locations, a lower or equal overshooting for high order derivatives in comparison with other approaches, and the additional advantage of being expressed in a reduced mathematical form with only two types of inverse functions (logarithmic and exponential). By numerical simulation examples, this proposal is verified and well-illustrated.

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Section: Introductionmentioning

confidence: 99%

Transforming the canonical piecewise-linear model into a smooth-piecewise representation

et al. 2016

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“…The piecewise linear function (1.4) is also popular for graphic analyses, including curve fitting, interpolation and extrapolation (e.g. Bian and Menz 1998, Dai et al 2007, Magnani and Boyd 2009, Misener and Floudas 2010, Jimenez-Fernandez et al 2014. Although appealing due to its simple structure, a major limitation of the piecewise linear model is that the first derivative of the function is discontinuous at the breakpoint (i.e.…”

Section: Piecewise Linear Modelsmentioning

confidence: 99%

Generalized bent-cable methodology for changepoint data: a Bayesian approach

Kar

2017

Journal of Applied Statistics

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The choice of the model framework in a regression setting depends on the nature of the data.The focus of this study is on changepoint data, exhibiting three phases: incoming and outgoing, both of which are linear, joined by a curved transition. These types of data can arise in many applications, including medical, health and environmental sciences. Piecewise linear models have been extensively utilized to characterize such changepoint trajectories in different scientific fields.However, although appealing due to its simple structure, a piecewise linear model is not realistic in many applications where data exhibit a gradual change over time.The most important aspect of characterizing a changepoint trajectory involves identifying the transition zone accurately. It is not only because the location of the transition zone is of particular interest in many areas of study, but also because it plays an important role in adequately describing the incoming and the outgoing phases of a changepoint trajectory. Note that once the transition is detected, the incoming and the outgoing phases can be modeled using linear functions. Overall, it is desirable to formulate a model in such a way that it can capture all the three phases satisfactorily, while being parsimonious with greatly interpretable regression coefficients. Since data may exhibit an either gradual or abrupt transition, it is also important for the transition model to be flexible.Bent-cable regression is an appealing statistical tool to characterize such trajectories, quantifying the nature of the transition between the two linear phases by modeling the transition as a quadratic phase with unknown width. We demonstrate that a quadratic function may not be appropriate to adequately describe many changepoint data. In practice, the quadratic function of the bent-cable model may lead to a wider or narrower interval than what could possibly be necessary to adequately describe a transition phase. We propose a generalization of the bent-cable model by relaxing the assumption of the quadratic bend. Specifically, an additional transition parameter is included in the bent-cable model to provide sufficient flexibility so that inference about the transition zone (i.e., shape and width of the bend) can be data driven, rather than pre-assumed as a specific type.We discuss the properties of the generalized model, and then propose a Bayesian approach for statistical inference. The generalized model is then demonstrated with applications to three data ii sets taken from environmental science and economics. We also consider a comparison among the quadratic bent-cable, generalized bent-cable and piecewise linear models in terms of goodness of fit in analyzing both real-world and simulated data. Moreover, we supplement the motivation for our generalized bent-cable methodology via extensive simulations -we simulate changepoint data under some realistic assumptions, and then fit the quadratic bent-cable, generalized bent-cable and piecewise linear models to each of the simulate...

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“…While this is adding to the computational complexity, the most important limitation of the linear interpolation model is that it is a simplistic approach that may be inappropriate for complex problems. Linear interpolation has been used successfully on many varied problems, including pricing and stock market [35,16], medical science [15] and digital imaging [39].…”

Section: Linear Interpolationmentioning

confidence: 99%

Large scale material science data analysis

Belisle

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Material Science, the science of studying materials and their properties, involves many aspects such as performing experiments to calculate certain physical properties. Scientists are always looking to utilise the collected experimental data in order to make predictions for new points, where the studied property is unknown. Using a computer model to make these predictions, whether it is via a machine learning or mathematical approach, is the desirable option, since doing actual experiments have proven to be very costly and time consuming. We are therefore looking at utilising the vast quantity of pre-collected data in the literature in order to build models for making future predictions. We already know that the Gaussian process regression interpolation technique gives accurate predictions for some physical properties. However, it is also the slowest of the machine learning algorithms and not suitable for on-line applications. For on-line learning, making quick and accurate predictions is essential. In this research we propose a novel strategy, including batch query processing and co-clustering, to achieve a scalable and efficient Gaussian process regression. This new approach, called the scalable Gaussian process (SGP), allows the use of large databases and makes it suitable for on-line applications. The proposed strategy is applied to a real application involving the prediction of materials properties. Results demonstrate the high accuracy and efficiency of our approach. We test and compare SGP with five different machine learning models on materials properties databases and make recommendations accordingly, also demonstrating that prior knowledge of the problem is essential when choosing a machine learning model.As one could expect, databases consisting of experimental data are noisy since they rely on human measurements, and also because they are an amalgamation of various independent sources (research papers). Therefore, some conflicting information can be found between the various sources. In our research we also introduce a novel truth discovery approach to reduce the amount of noise and filter the incorrect conflicting information hidden in scientific databases. Our method ranks the multiple data sources by considering the relationships between them, i.e., the amount of conflicting information and the amount of agreement, and as well eliminates the conflicting information. Our previously introduced technique, SGP, is ii then applied to the clean dataset to make predictions. We compare the prediction accuracy before and after pruning the databases. With our new approach, we are able to highly improve the accuracy of SGP predictions and provide a more reliable database. Our results also prove the extreme robustness of SGP, as we demonstrate that a relatively high amount of noise is handled very well by this technique.iii

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An Ingenious, Piecewise Linear Interpolation Algorithm for Pricing Arithmetic Average Options

Cited by 3 publications

References 10 publications

Transforming the canonical piecewise-linear model into a smooth-piecewise representation

Transforming the canonical piecewise-linear model into a smooth-piecewise representation

Generalized bent-cable methodology for changepoint data: a Bayesian approach

Large scale material science data analysis

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