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# 数学代写|随机过程Stochastic Porcesses代考|MA53200 Calibration

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## 数学代写|随机过程代写Stochastic Porcesses代考|Calibration

The calibration of advanced stochastic processes in our context is in some sense not a trivial task. As the previously discussed models are mostly envisaged for the pricing of market traded options, existing literature propagates the calibration of these processes to a basket of such options with different strike prices and maturities to capture information about the implied volatility surface. In particular, it is common to algorithmically minimise some measure of distance (e.g. the root mean squared error) between option prices in the basket and model prices subject to the input parameters. This procedure directly yields the risk-neutral parametrisation of the process, based upon which the pricing model is build. While the calibration to option prices has the advantage to be forward looking in the sense that prices reflect the aggregated opinion of all market participants on future payoffs, data availability on a basket of commodity options may resemble an obstacle for corporate finance practitioners in non-financial institutions. As a result, I presume that in a real options context it is much more convenient and flexible to be able to back out the parameters from a series of historical commodity prices directly. Along these lines a number of methods have been suggested. Gibson and Schwartz (1990) propose a seemingly unrelated regression model to fit mean-reverting models, Schwartz (1997) and Manoliu and Tompaidis (2002) employ a state space representation to estimate a number of mean-reverting models with the Kalman filter, Brigo et al. (2007) apply MLE for the Merton and VG model, Seneta (2004) uses moment matching to fit the VG process, and Brooks and Prokopczuk (2013) employ the Markov chain Monte Carlo method to fit various stochastic volatility models. While, I am sure, the list can be continued for a while, none of the above methods appears satisfactory for our purposes as they are either not general enough to work for all our processes or require mathematically involved adaptations in each case.

Instead, I propose to minimise an appropriate measure of distance between the kernel density estimate of historical returns and the analytical probability density of the model under appropriate constraints and with a suitable algorithm. The idea is to choose the process parameters in such a way that the model implied return density mimics the empirical return distribution as closely as possible. As suggested, for instance, by Schoutens (2003) or Brigo et al. (2007), this is what characterises a good model fit. For easier reference, I will call the proposed method Algorithmic Probability Density Fitting (APDF) in the remainder of this document. To familiarise the reader with the idea of APDF, we will next consider the choice of an objective function and an appropriate algorithm. Thereafter, the calibration results are benchmarked against MLE estimates and some implementation details are considered.

## 数学代写|随机过程代写Stochastic Porcesses代考|Goodness of fit

In this subsection, it is discussed which model is likely to yield the best fit to historical commodity returns. To do so, we will draw on the previously developed logic and compare the ASE between the empirical distribution estimate and the model implied distribution with APDF-fitted parameters. Since complex models with many parameters such as the Bates or NIG-CIR process generally outper- form simpler models in in-sample tests, but not in out-of-sample tests (Bakshi, Cao, \& Zhiwu, 1997), two different test set-ups are created. First, an in-sample test, where the ASE is calculated with historical density and parameter estimates based on the full sample period from 1993 to 2013 . Second, an out-of-sample test, where model parameters are calibrated over the period 1993 – 2004 and the resulting model distribution is compared to the empirical density of the period 1995 2013. Clearly, the choice of calibration and analysis period in the out-of-sample test is drastic as several extreme events occurred during the analysis period (ascent of prices after 2005 and financial crisis), which are very different from events observed during the calibration period. An additional way of out-of-sample testing could involve a leave-one-out cross validation as suggested by De Laurentis et al. (2010), however, this exercise is left to further study as the main purpose is equally well accomplished by the here adopted simple alternative. This is to understand if the nuances in the shape of return distributions that more complex models are able to capture persist over time so that it is worthwhile to account for them. In order to conserve space, I report the calibrated process parameters after some sanity checks in appendix A4.

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。