Option Pricing With Modular Neural Networks

This paper investigates a nonparametric modular neural network (MNN) model to price the S&P-500 European call options. The modules are based on time to maturity and moneyness of the options. The option price function of interest is homogeneous of degree one with respect to the underlying index price and the strike price. When compared to an array of parametric and nonparametric models, the MNN method consistently exerts superior out-of-sample pricing performance. We conclude that modularity improves the generalization properties of standard feedforward neural network option pricing models (with and without the homogeneity hint).     

Study on semiparametric Wilcoxon fuzzy neural networks

Fuzzy neural network (FNN) has long been recognized as an efficient and powerful learning machine for general machine learning problems. Recently, Wilcoxon fuzzy neural network (WFNN), which generalizes the rank-based Wilcoxon approach for linear parametric regression problems to nonparametric neural network, was proposed aiming at improving robustness against outliers. FNN and WFNN are nonparametric models in the sense that they put no restrictions, except possibly smoothness, on the functional form of the regression function. However, they may be difficult to interpret and, even worse, yield poor estimates with high computational cost when the number of predictor variables is large. To overcome this drawback, semiparametric models have been proposed in statistical regression theory. A semiparametric model keeps the easy interpretability of its parametric part and retains the flexibility of its nonparametric part. Based on this, semiparametric FNN and semiparametric WFNN will be proposed in this paper. The learning rules are based on the backfitting procedure frequently used in semiparametric regression. Simulation results show that the semiparametric models perform better than their nonparametric counterparts.     

Isogeometric analysis in option pricing

ABSTRACT

Isogeometric analysis is a recently developed computational approach that integrates finite element analysis directly into design described by non-uniform rational B-splines (NURBS). In this paper, we show that price surfaces that occur in option pricing can be easily described by NURBS surfaces. For a class of stochastic volatility models, we develop a methodology for solving corresponding pricing partial integro-differential equations numerically by isogeometric analysis tools and show that a very small number of space discretization steps can be used to obtain sufficiently accurate results. Presented solution by finite element method is especially useful for practitioners dealing with derivatives where closed-form solution is not available.     

Improving option price forecasts with neural networks and support vector regressions

Options are important financial derivatives that allow investors to control their investment risks in the securities market. Determining the theoretical price for an option, or option pricing, is regarded as one of the most important issues in financial research; a number of parametric and nonparametric option pricing approaches have been presented. While the objective of option pricing is to find the current fair price, for decision making, in contrast, the forecasting activity has to accurately predict the future option price without advance knowledge of the underlying asset price. In this paper, a simple and effective nonparametric method of forecasting option prices based on neural networks (NNs) and support vector regressions (SVRs) is presented. We first modified the improved conventional option pricing methods, allowing them to forecast the option prices. Second, we employed the NNs and SVRs to further decrease the forecasting errors of the parametric methods. Since the conventional methods mimic the trends of movement of the real option prices, using these methods in a first stage allows the NNs and SVRs to concentrate their power in nonlinear curve approximation to further reduce the forecasting errors in a second stage. Finally, extensive experimental studies with data from the Hong Kong option market demonstrated the ability of NNs and SVRs to improve forecast accuracy.     

A hybrid option pricing model using a neural network for estimating volatility

The Black–Scholes (BS) model is the standard approach used for pricing financial options. However, although being theoretically strong, option prices valued by the model often differ from the prices observed in the financial markets. This paper applies a hybrid neural network which preprocesses financial input data for improving the estimation of option market prices. This model is comprised of two parts. The first part is a neural network developed to estimate volatility. The second part is an additional neural network developed to value the difference between the BS model results and the actual market option prices. The resulting option price is then a summation between the BS model and the network response. The hybrid system with a neural network for estimating volatility provides better performance in terms of pricing accuracy than either the BS model with historical volatility (HV), or the BS model with volatility valued by the neural network.     

Option valuation based on the neural regression model

In this article, a neural regression (NR) model, which produces nonlinear coefficients of multiple regression model based on neural networks, is introduced to capture the option valuation’s nonlinear characteristics effectively. The traditional linear regression uses the least-squares estimator to estimate the coefficient of a linear regression and thus may only produce suboptimal solutions. However, Applying neural networks to forecast volatility in option pricing has increased in popularity in recent years since many studies have indicated that the conventional option pricing models are not accurate enough. Our proposed neural regression model devotes to evaluate option values to improve on the tracking error in the measurement of hedging capability. The NR model uses the variables introduced by the Black–Scholes Model and applies the multiple regressions (MR) model to re-price option values. It is worth noting that each corresponding weight coefficient in MR is constructed by a complete neural network rather than by a scalar value. By capturing the nonlinear behaviors of option pricing, our proposed NR model has lower tracking error and better hedging capability than the BS model and other studies.     

Robust Artificial Neural Networks for Pricing of European Options

The option pricing ability of Robust Artificial Neural Networks optimized with the Huber function is compared against those optimized with Least Squares. Comparison is in respect to pricing European call options on the S&P 500 using daily data for the period April 1998 to August 2001. The analysis is augmented with the use of several historical and implied volatility measures. Implied volatilities are the overall average, and the average per maturity. Beyond the standard neural networks, hybrid networks that directly incorporate information from the parametric model are included in the analysis. It is shown that the artificial neural network models with the use of the Huber function outperform the ones optimized with least squares. JEL Classification: G13, G14     

Using neural network for forecasting TXO price under different volatility models

This study applies backpropagation neural network for forecasting TXO price under different volatility models, including historical volatility, implied volatility, deterministic volatility function, GARCH and GM-GARCH models. The sample period runs from 2008 to 2009, and thus contains the global financial crisis stating in October 2008. Besides RMSE, MAE and MAPE, this study introduces the best forecasting performance ratio (BFPR) as a new performance measure for use in option pricing. The analytical result reveals that forecasting performances are related to the moneynesses, volatility models and number of neurons in the hidden layer, but are not significantly related to activation functions. Implied and deterministic volatility function models have the largest and second largest BFPR regardless of moneyness. Particularly, the forecasting performance in 2008 was significantly inferior to that in 2009, demonstrating that the global financial crisis during October 2008 may have strongly influenced option pricing performance.     

Research on wave image analysis in the implied volatility of stock options

Multimedia Tools and Applications - The study of volatility rarely starts from the perspective of implied volatility, and rarely combines stochastic volatility models with option pricing, which is...     

Forecasting the volatility of stock price index

Accurate volatility forecasting is the core task in the risk management in which various portfolios’ pricing, hedging, and option strategies are exercised. Prior studies on stock market have primarily focused on estimation of stock price index by using financial time series models and data mining techniques. This paper proposes hybrid models with neural network and time series models for forecasting the volatility of stock price index in two view points: deviation and direction. It demonstrates the utility of the hybrid model for volatility forecasting. This model demonstrates the utility of the neural network forecasting combined with time series analysis for the financial goods.     

Nonparametric Estimation of Volatility and Its Parametric Analogs

This paper suggests a nonparametric method for stochastic volatility estimation and its comparison with other widespread econometric algorithms. A major advantage of this approach is that the volatility can be estimated even in the case of its completely unknown probability distribution. As demonstrated below, the new method has better characteristics against the popular parametric algorithms based on the GARCH model and Kalman filter.     

Nonparametric modeling of neural point processes via stochastic gradient boosting regression

Statistical nonparametric modeling tools that enable the discovery and approximation of functional forms (e.g., tuning functions) relating neural spiking activity to relevant covariates are desirable tools in neuroscience. In this article, we show how stochastic gradient boosting regression can be successfully extended to the modeling of spiking activity data while preserving their point process nature, thus providing a robust nonparametric modeling tool. We formulate stochastic gradient boosting in terms of approximating the conditional intensity function of a point process in discrete time and use the standard likelihood of the process to derive the loss function for the approximation problem. To illustrate the approach, we apply the algorithm to the modeling of primary motor and parietal spiking activity as a function of spiking history and kinematics during a two-dimensional reaching task. Model selection, goodness of fit via the time rescaling theorem, model interpretation via partial dependence plots, ranking of covariates according to their relative importance, and prediction of peri-event time histograms are illustrated and discussed. Additionally, we use the tenfold cross-validated log likelihood of the modeled neural processes (67 cells) to compare the performance of gradient boosting regression to two alternative approaches: standard generalized linear models (GLMs) and Bayesian P-splines with Markov chain Monte Carlo (MCMC) sampling. In our data set, gradient boosting outperformed both Bayesian P-splines (in approximately 90% of the cells) and GLMs (100%). Because of its good performance and computational efficiency, we propose stochastic gradient boosting regression as an off-the-shelf nonparametric tool for initial analyses of large neural data sets (e.g., more than 50 cells; more than 10(5) samples per cell) with corresponding multidimensional covariate spaces (e.g., more than four covariates). In the cases where a functional form might be amenable to a more compact representation, gradient boosting might also lead to the discovery of simpler, parametric models.     

An algorithm for nonparametric GARCH modelling

A simple iterative algorithm for nonparametric first-order GARCH modelling is proposed. This method offers an alternative to fitting one of the many different parametric GARCH specifications that have been proposed in the literature. A theoretical justification for the algorithm is provided and examples of its application to simulated data from various stationary processes showing stochastic volatility, as well as empirical financial return data, are given. The nonparametric procedure is found to often give better estimates of the unobserved latent volatility process than parametric modelling with the standard GARCH(1,1) model, particularly in the presence of asymmetry and other departures from the standard GARCH specification. Extensions of the basic iterative idea to more complex time series models combining ARMA or GARCH features of possibly higher order are suggested.     

PDE formulation of some SABR/LIBOR market models and its numerical solution with a sparse grid combination technique

SABR models have been used to incorporate stochastic volatility to LIBOR market models (LMM) in order to describe interest rate dynamics and price interest rate derivatives. From the numerical point of view, the pricing of derivatives with SABR/LIBOR market models (SABR/LMMs) is mainly carried out with Monte Carlo simulation. However, this approach could involve excessively long computational times. For first time in the literature, in the present paper we propose an alternative pricing based on partial differential equations (PDEs). Thus, we pose original PDE formulations associated to the SABR/LMMs proposed by Hagan and Lesniewsk (2008), Mercurio and Morini (2009) and Rebonato and White (2008). Moreover, as the PDEs associated to these SABR/LMMs are high dimensional in space, traditional full grid methods (like standard finite differences or finite elements) are not able to price derivatives over more than three or four underlying interest rates. In order to overcome this curse of dimensionality, a sparse grid combination technique is proposed. A comparison between Monte Carlo simulation results and the ones obtained with the sparse grid technique illustrates the performance of the method.     

非完整移动机器人的神经网络鲁棒自适应控制

在非完整移动机器人轨迹跟踪问题中,针对机器人运动学与动力学模型的参数和非参数不确定性,提出了一种混合神经网络鲁棒自适应轨迹跟踪控制器,该控制器由运动学控制器和动力学控制器两部分组成;其中,采用了参数自适应的径向基神经网络对运动学模型的未知部分进行了建模,并采用权值在线调整的单层神经网络和自适应鲁棒控制项构成了动力学控制器;基于Lyapunov方法的设计过程保证了系统的稳定性和收敛性,仿真结果证明了算法的有效性。     

LSM Algorithm for Pricing American Option Under Heston–Hull–White’s Stochastic Volatility Model

In this paper, we present American option pricing under Heston–Hull–White’s stochastic volatility and stochastic interest rate model. To do this, we first discretize the stochastic processes with Euler discretization scheme. Then, we price American option by using least-squares Monte Carlo algorithm. We also compare the numerical results of our model with the Heston-CIR model. Finally, numerical results show the efficiency of the proposed algorithm for pricing American option under the Heston–Hull–White model.     

Consumption Utility-Based Pricing and Timing of the Option to Invest with Partial Information

This paper extends real options theory to consider the situation where the mean appreciation rate of the value of an irreversible investment project is not observable and governed by an Ornstein–Uhlenbeck process. Our main purpose is to analyze the impact of the uncertainty of the mean appreciation rate on the pricing and investment timing of the option to invest under incomplete markets with partial information. We assume that an investor aims to maximize expected discounted utility of lifetime consumption. Based on consumption utility indifference pricing method, stochastic control and filtering theory, we obtain under CARA utility the implied values and the optimal investment thresholds of the option to invest, which are determined by a semi-closed-form solution to a free-boundary partial differential equation (PDE) problem. The solution is independent of the utility time-discount rate. We provide numerical results by finite difference methods and compare the results with those under a fully observable case. Numerical calculations show that partial information leads to a significant loss of the implied value of the option to invest. This loss, called implied information value, IIV increases quickly with the uncertainty of the mean appreciation rate. A high volatility of project values might decrease the IIV, as well as the implied value of the option.     

Pricing the financial Heston–Hull–White model with arbitrary correlation factors via an adaptive FDM

This paper is concerned with the pricing procedure of one of the most challenging models known as the Heston–Hull–White partial differential equation (PDE) in option pricing, at which the model is a time-dependent 3D linear PDE including three mixed derivative terms. The model comes from the fact that the price, the volatility and the interest rate are assumed to be stochastic processes. To contribute and avoid huge discretized systems, an adaptive distribution of the nodes (viz, nonuniform nodes) is taken into account with emphasis on the hot area of the solution curve. New adaptive finite difference (FD) formulas of higher orders are constructed on these meshes. Then, a set of semi-discretized equations is attained which is tackled by a time-stepping method. Several financial tests are discussed in detail to reveal the superiority of the proposed approach.     

Calibration of a path-dependent volatility model: Empirical tests

The Hobson and Rogers model for option pricing is considered. This stochastic volatility model preserves the completeness of the market and can potentially reproduce the observed smile and term structure patterns of implied volatility. A calibration procedure based on ad-hoc numerical schemes for hypoelliptic PDEs is proposed and used to quantitatively investigate the pricing performance of the model. Numerical results based on S&P500 option prices are discussed.     

Monte Carlo option pricing with asymmetric realized volatility dynamics

What are the advances introduced by realized volatility models in pricing options? In this short paper we analyze a simple option pricing framework based on the dually asymmetric realized volatility model, which emphasizes extended leverage effects and empirical regularity of high volatility risk during high volatility periods. We conduct a brief empirical analysis of the pricing performance of this approach against some benchmark models using data from the S&P 500 options in the 2001-2004 period. The results indicate that as expected the superior forecasting accuracy of realized volatility translates into significantly smaller pricing errors when compared to models of the GARCH family. Most importantly, our results indicate that the presence of leverage effects and a high volatility risk are essential for understanding common option pricing anomalies.