Model Risk for European-Style Stock Index Options

In empirical modeling, there have been two strands for pricing in the options literature, namely the parametric and nonparametric models. Often, the support for the nonparametric methods is based on a benchmark such as the Black-Scholes (BS) model with constant volatility. In this paper, we study the stochastic volatility (SV) and stochastic volatility random jump (SVJ) models as parametric benchmarks against feedforward neural network (FNN) models, a class of neural network models. Our choice for FNN models is due to their well-studied universal approximation properties of an unknown function and its partial derivatives. Since the partial derivatives of an option pricing formula are risk pricing tools, an accurate estimation of the unknown option pricing function is essential for pricing and hedging. Our findings indicate that FNN models offer themselves as robust option pricing tools, over their sophisticated parametric counterparts in predictive settings. There are two routes to explain the superiority of FNN models over the parametric models in forecast settings. These are nonnormality of return distributions and adaptive learning     

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.     

A neural network model for estimating option prices

A neural network model that processes financial input data is developed to estimate the market price of options at closing. The network's ability to estimate closing prices is compared to the Black-Scholes model, the most widely used model for the pricing of options. Comparisons reveal that the mean squared error for the neural network is less than that of the Black-Scholes model in about half of the cases examined. The differences and similarities in the two modeling approaches are discussed. The neural network, which uses the same financial data as the Black-Scholes model, requires no distribution assumptions and learns the relationships between the financial input data and the option price from the historical data. The option-valuation equilibrium model of Black-Scholes determines option prices under the assumptions that prices follow a continuous time path and that the instantaneous volatility is nonstochastic.     

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.     

Normalized particle swarm optimization for complex chooser option pricing on graphics processing unit

An option is a financial instrument that derives its value from an underlying asset, for example, a stock. There are a wide range of options traded today from simple and plain (European options) to exotic (chooser options) that are very difficult to evaluate. Both buyers and sellers continue to look for efficient algorithms and faster technology to price options for better profit and to beat the competition. There are mathematical models like the Black–Scholes–Merton model used to price options approximately for simple and plain options in the form of closed form solution. However, the market is flooded with various styles of options, which are difficult to price, and hence there are many numerical techniques proposed for pricing. The computational cost for pricing complex options using these numerical techniques is exorbitant for reasonable accuracy in pricing results. Heuristic approaches such as particle swarm optimization (PSO) have been proposed for option pricing, which provide same or better results for simple options than that of numerical techniques at much less computational cost. In this study, we first map the PSO parameters to option pricing parameters. Analyzing the characteristics of PSO and option pricing, we propose a strategy to normalize some of the parameters, which helps in better understanding of the sensitivity of these and other parameters on option pricing results. We then avail of the inherent concurrency of the PSO algorithm while searching for an optimum solution, and design an algorithm for implementation on a modern state-of-the-art graphics processor unit (GPU). Our implementation makes use of the architectural features of GPU in accelerating the computing performance while maintaining accuracy on the pricing results.     

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.     

Functional link neural network approach to solve structural system identification problems

System identification problems are generally inverse vibration problems. Sometimes it is difficult to handle the inverse problems by traditional methods and classical artificial neural network. As such, the objective of this paper is to identify structural parameters by developing a novel functional link neural network (FLNN) model. FLNN model is more efficient than multi-layer neural network (MNN) as computation is less because hidden layer is not required. Here, single-layer neural network with multi-input and multi-output with feed-forward neural network model and principle of error back propagation has been used to identify structural parameters. The hidden layer is excluded by enlarging the input patterns with the help of Legendre and Hermite polynomials. Comparison of results among MNN, Legendre neural network, Hermite neural network and desired is considered and it is found that FLNN models are more effective than MNN.

     

Improving option pricing with the product constrained hybrid neural network

In the past decade, many studies across various financial markets have shown conventional option pricing models to be inaccurate. To improve their accuracy, various researchers have turned to artificial neural networks (ANNs). In this work a neural network is constrained in such a way that pricing must be rational at the option-pricing boundaries. The constraints serve to change the regression surface of the ANN so that option pricing accuracy is improved in the locale of the boundaries. These constraints lead to statistically and economically significant out-performance, relative to both the most accurate conventional and nonconventional option pricing models.     

一种利用多神经网络结构建立非线性软测量模型的方法

采用多种神经网络结构建立软测量模型。建模所用的样本数据经过Ｋｍｅａｎｓ聚类方法分成多组训练数据,每组数据建立一个单个神经网络模型,再通过ＰＣＲ方法连接起来得到整个模型的输出,从而显著地提高了模型的精确度和鲁棒性。仿真研究表明,采用该建模方法能够达到较好的建模效果。     

Exotic put options at the diffusion bond market

The direct approach is used to obtain formulas that determine the option price, portfolios (hedging strategies) and capitals for the European put options with guaranteed income for the option holder and limited payment for the investor at the diffusion (B,P)-bond market. The properties of the solution are studied. The results are specified for the bond pricing models known as the Ho-Lee and Vasicek models.     

基于GPU的最小二乘蒙特卡罗算法期权定价

期权是以金融产品作为行权品种的交易合约。随着期权交易规模和交易量的迅速增长,期权定价的计算量越来越大,在传统CPU平台上对期权进行定价变得越来越困难。图形处理器（GPU）平台的出现和发展为解决期权定价计算提供了解决方案。在GPU上使用最小二乘蒙特卡罗算法（Least Squares Monte Carlo,LSM）实现了对一维和四维美式期权定价计算：首先利用CURAND库产生大量随机数,然后并行化期权标的价格变化路径,最后对最小二乘法和贴现定价进行并行化。为提高GPU平台上LSM方法的计算效率,对整个过程进行了优化。实际测试结果表明,在CPU+GPU上实现一维和四维美式期权定价相对CPU平台的加速比最高分别达到20.275和47.538,且比其他文献的方法整体性能有较大的提升。     

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     

Fuzzy pricing of geometric Asian options and its algorithm

Owing to the fluctuations of the financial market, input data in the options pricing formula cannot be expected to be precise. This paper discusses the problem of pricing geometric Asian options under the fuzzy environment. We present the fuzzy price of the geometric Asian option under the assumption that the underlying stock price, the risk-free interest rate and the volatility are all fuzzy numbers. This assumption makes the financial investors to pick any geometric Asian option price with an acceptable belief degree. In order to obtain the belief degree, the interpolation search algorithm has been proposed. Some numerical examples are presented to illustrate the rationality and practicability of the model and the algorithm. Finally, an empirical study is performed based on the real data. The empirical study results indicate that the proposed fuzzy pricing model of geometric Asian option is a useful tool for modeling the imprecise problem in the real world.     

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.     

Online option price forecasting by using unscented Kalman filters and support vector machines

This study develops a hybrid model that combines unscented Kalman filters (UKFs) and support vector machines (SVMs) to implement an online option price predictor. In the hybrid model, the UKF is used to infer latent variables and make a prediction based on the Black–Scholes formula, while the SVM is employed to model the nonlinear residuals between the actual option prices and the UKF predictions. Taking option data traded in Taiwan Futures Exchange, this study examined the forecasting accuracy of the proposed model, and found that the new hybrid model is superior to pure SVM models or hybrid neural network models in terms of three types of options. This model can help investors for reducing their risk in online trading.     

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.     

Pricing fuzzy vulnerable options and risk management

The assumption of unrealistic “identical rationality” in classic option pricing theory is released in this article to amend Klein’s [Klein, P. (1996). Pricing Black–Scholes options with correlated credit risk. Journal of Banking Finance, 1211–1129] vulnerable option pricing formula. Through this formula, default risk and liquidity risk are both well-explained when the investment behaviors and market expectations of the participants are heterogeneous. The numerical results show that when the investing decisions of each market participant come from their individual rationality and use their own subjective price to trade, the option price becomes a boundary. The upper boundary becomes an absolutely safe line and the lower boundary becomes an absolutely unsafe line for investors who want to invest in some financial securities with default risk. The proposed model suggests a more realistic pricing mechanism for the issuers and traders who want to value options with default risk.     

Approximate Option Pricing

As increasingly large volumes of sophisticated options are traded in world financial markets, determining a ``fair' price for these options has become an important and difficult computational problem. Many valuation codes use the binomial pricing model, in which the stock price is driven by a random walk. In this model, the value of an n -period option on a stock is the expected time-discounted value of the future cash flow on an n -period stock price path. Path-dependent options are particularly difficult to value since the future cash flow depends on the entire stock price path rather than on just the final stock price. Currently such options are approximately priced by Monte Carlo methods with error bounds that hold only with high probability and which are reduced by increasing the number of simulation runs. In this article we show that pricing an arbitrary path-dependent option is \#-P hard. We show that certain types of path-dependent options can be valued exactly in polynomial time. Asian options are path-dependent options that are particularly hard to price, and for these we design deterministic polynomial-time approximate algorithms. We show that the value of a perpetual American put option (which can be computed in constant time) is in many cases a good approximation of the value of an otherwise identical n -period American put option. In contrast to Monte Carlo methods, our algorithms have guaranteed error bounds that are polynomially small (and in some cases exponentially small) in the maturity n . For the error analysis we derive large-deviation results for random walks that may be of independent interest. Received August 13, 1996; revised April 2, 1997.     

A modified Black–Scholes pricing formula for European options with bounded underlying prices

In this paper, a modified Black–Scholes (B–S) model is proposed, based on a revised assumption that the range of the underlying price varies within a finite zone, rather than being allowed to vary in a semi-infinite zone as presented in the classical B–S theory. This is motivated by the fact that the underlying price of any option can never reach infinity in reality; a trader may use our new formula to adjust the option price that he/she is willing to long or short. To develop this modified option pricing formula, we assume that a trader has a view on the realistic price range of a particular asset and the log-returns follow a truncated normal distribution within this price range. After a closed-form pricing formula for European call options has been successfully derived, some numerical experiments are conducted. To further demonstrate the meaning of the proposed model, empirical studies are carried out to compare the pricing performance of our model and that of the B–S model with real market data.     

Hedging an option portfolio with minimum transaction lots: A fuzzy goal programming problem

Options are designed to hedge against risks to their underlying assets such as stocks. One method of forming option-hedging portfolios is using stochastic programming models. Stochastic programming models depend heavily on scenario generation, a challenging task. Another method is neutralizing the Greek risks derived from the Black–Scholes formula for pricing options. The formula expresses the option price as a function of the stock price, strike price, volatility, risk-free interest rate, and time to maturity. Greek risks are the derivatives of the option price with respect to these variables. Hedging Greek risks requires no human intervention for generating scenarios. Linear programming models have been proposed for constructing option portfolios with neutralized risks and maximized investment profit. However, problems with these models exist. First, feasible solutions that can perfectly neutralize the Greek risks might not exist. Second, models that involve multiple assets and their derivatives were incorrectly formulated. Finally, these models lack practicability because they consider no minimum transaction lots. Considering minimum transaction lots can exacerbate the infeasibility problem. These problems must be resolved before option hedging models can be applied further. This study presents a revised linear programming model for option portfolios with multiple underlying assets, and extends the model by incorporating it with a fuzzy goal programming method for considering minimum transaction lots. Numerical examples show that current models failed to obtain feasible solutions when minimum transaction lots were considered. By contrast, while the proposed model solved the problems efficiently.