Fuzzy pricing of American options on stocks with known dividends and its algorithm

The path‐dependent property of American options leads to the complexity of its pricing. Based on the analysis of American options' characteristics and the influence of the stock dividend, the American call option fuzzy pricing method is discussed in this paper. Under the assumption that the price of stock, discount rate, the volatility, and interest rate are all fuzzy numbers, the fuzzy pricing formula of American option is proposed by using the Black–Scholes pricing model. Then the interpolation search algorithm is designed to solve the proposed pricing model. Finally, the validity and accuracy of this model and its algorithm have to be tested with some numerical examples. © 2010 Wiley Periodicals, Inc.     

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.     

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.     

European option pricing under fuzzy environments

The application of fuzzy sets theory to the Black–Scholes formula is proposed in this article. Owing to the vague fluctuation of financial markets from time to time, the risk‐free interest rate, volatility, and the price of underlying assets may occur imprecisely. In this case, it is natural to consider the fuzzy interest rate, fuzzy volatility, and fuzzy stock price. The form of “Resolution Identity” in fuzzy sets theory will be invoked to propose the fuzzy price of European options. Under these assumptions, the European option price at time t will turn into a fuzzy number. This will allow a financial analyst to choose the European price at his (her) acceptable degree of belief. To obtain the belief degree, the optimization problems have to be solved. © 2005 Wiley Periodicals, Inc. Int J Int Syst 20: 89–102, 2005.     

An exact subexponential-time lattice algorithm for Asian options

Asian options are popular financial derivative securities. Unfortunately, no exact pricing formulas exist for their price under continuous-time models. Asian options can also be priced on the lattice, which is a discretized version of the continuous- time model. But only exponential-time algorithms exist if the options are priced on the lattice without approximations. Although efficient approximation methods are available, they lack accuracy guarantees in general. This paper proposes a novel lattice structure for pricing Asian options. The resulting pricing algorithm is exact (i.e., without approximations), converges to the value under the continuous-time model, and runs in subexponential time. This is the first exact, convergent lattice algorithm to break the long-standing exponential-time barrier. An early version of this paper appeared in the Proceedings of the 15th Annual ACM-SIAM Symposium on Discrete Algorithms, 2004. T.-S. Dai was supported in part by NSC grant 94-2213-E-033-024. Y.-D. Lyuu was supported in part by NSC grant 94-2213-E-002-088.     

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.     

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.     

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.     

Using computational methodology to price European options with actual payoff distributions

Most option pricing methods use mathematical distributions to approximate underlying asset behavior. However, pure mathematical distribution approaches have difficulty approximating the real distribution. This study first introduces an innovative computational method for pricing European options based on the real payoff distribution of the underlying asset. This computational approach can also be applied to applications related to expected value that require real distributions rather than mathematical distributions. This study makes the following contributions: (a) solving the risk neutral issue related to price options with real payoff distributions; (b) proposing a simple method for adjusting standard deviation based on the need to apply short term volatility to real world applications; (c) demonstrating an option pricing algorithm that is easy to apply to cross field applications.     

Pricing European options based on the fuzzy pattern of Black–Scholes formula

The application of fuzzy sets theory to the Black–Scholes formula is proposed in this paper. Owing to the fluctuation of financial market from time to time, some input parameters in the Black–Scholes formula cannot always be expected in the precise sense. Therefore, it is natural to consider the fuzzy interest rate, fuzzy volatility and fuzzy stock price. The fuzzy pattern of Black–Scholes formula and put–call parity relationship are then proposed in this paper. Under these assumptions, the European option price will turn into a fuzzy number. This makes the financial analyst who can pick any European option price with an acceptable belief degree for the later use. In order to obtain the belief degree, an optimization problem has to be solved.     

DeepOption: A novel option pricing framework based on deep learning with fused distilled data from multiple parametric methods

The remarkable performance of deep learning is based on its ability to learn high-level features by processing large amounts of data. This exceptionally superior performance has attracted the attention of researchers studying option pricing. However, option data are more expensive and less accessible than other types of data and are imbalanced because of the liquidity of options. This motivated us to propose a new option pricing and delta-hedging framework called DeepOption. This framework, which is based on deep learning, can improve the performance even when applying imbalanced real option data. In particular, the framework fuses simulated big data, known as distilled data, obtained using various traditional parametric methods. The proposed model employs the following three-stage training approach: Our model is pre-trained using big distilled data after it is fine-tuned using real option data through transfer learning. Finally, a delta branch is added to the model and trained. We experimentally evaluated the proposed method using three sets of real option data, namely S&P 500 European call options, EuroStoxx50 call options, and Hang Seng Index put options. Our experimental results on option pricing demonstrate that our proposed model outperforms parametric methods and other machine learning methods. Specifically, our model, which uses pre-training with distilled data, reduces the overall mean absolute percentage error (MAPE) by more than 50%, compared with that of a deep learning model using only real option data without pre-training.     

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 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.     

基于FCM的文本迁移学习算法

传统的机器学习方法是在训练数据和测试数据分布一致的前提下进行的。然而,在一些现实世界中的应用,训练数据和测试数据来自不同的领域。在不考虑数据分布的情况下,传统的机器学习算法可能会失效,针对这一问题,提出一种基于模糊C均值(FCM)的文本迁移学习算法。首先,通过简单分类器对测试样本分类,接着,利用自然邻算法构建样本初始模糊隶属度;然后,利用FCM算法通过迭代更新样本模糊隶属度,修正样本标签;最后,对样本孤立点进行处理,得到最终分类结果。实验结果表明,该算法具有较好的正确率,有效的解决了在训练数据和测试数据分布不一致的情况下的文本分类问题。     

A stochastic approximation algorithm for option pricing model calibration with a switchable market

This paper is concerned with option pricing under a regime-switching model. The switching process takes two different modes, and the underlying stock price evolves in accordance with the two modes dictated by a continuous-time, finite-state Markov chain. At any given instance, the price follows either a geometric Brownian motion model or a mean-reversion model, depending on its market mode. Stochastic approximation/optimization algorithms are developed for model calibration. Convergence of the algorithm is proved; rate of convergence is also provided. Option market data are used to predict the future market mode.     

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).     

Pricing Bermudan options under Merton jump-diffusion asset dynamics

In this paper, a recently developed regression-based option pricing method, the Stochastic Grid Bundling Method (SGBM), is considered for pricing multidimensional Bermudan options. We compare SGBM with a traditional regression-based pricing approach and present detailed insight in the application of SGBM, including how to configure it and how to reduce the uncertainty of its estimates by control variates. We consider the Merton jump-diffusion model, which performs better than the geometric Brownian motion in modelling the heavy-tailed features of asset price distributions. Our numerical tests show that SGBM with appropriate set-up works highly satisfactorily for pricing multidimensional options under jump-diffusion asset dynamics.     

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。     

A hybrid fuzzy regression-fuzzy cognitive map algorithm for forecasting and optimization of housing market fluctuations

This paper presents a hybrid algorithm based on fuzzy linear regression (FLR) and fuzzy cognitive map (FCM) to deal with the problem of forecasting and optimization of housing market fluctuations. Due to the uncertainty and severe noise associated with the housing market, the application of crisp data for forecasting and optimization purposes is insufficient. Hence, in order to enable the decision-makers to make decisions with respect to imprecise/fuzzy data, FLR is used in the proposed hybrid algorithm. The best-fitted FLR model is then selected with respect to two indicators including Index of Confidence (IC) and Mean Absolute Percentage Error (MAPE). To achieve this objective, analysis of variance (ANOVA) for a randomized complete block design (RCBD) is employed. The primary objective of this study is to utilize imprecise/fuzzy data in order to improve the analysis of housing price fluctuations, in accordance with the factors obtained through the best-fitted FLR model. The secondary objective of this study is the exhibition of the resulted values in a schematic way via FCM. Hybridization of FLR and FCM provides a decision support system (DSS) for utilization of historical data to predict housing market fluctuation in the future and identify the influence of the other parameters. The proposed hybrid FLR-FCM algorithm enables the decision-makers to utilize imprecise and ambiguous data and represent the resulted values of the model more clearly. This is the first study that utilizes a hybrid intelligent approach for housing price and market forecasting and optimization.     

An efficient acceleration Monte Carlo simulation for pricing Asian option under variance gamma process by splitting

ABSTRACT

This paper proposes a hybrid acceleration method for pricing Asian options with arithmetic average under variance gamma process. We split the payoff of Asian option into two parts, the first part is a small probability event, so the importance sampling method is naturally used to reduce variance of simulation efficiently. The second part can be simplified as a semi-closed form, which also be considered as a conditional expectation formula, so it may reduce variance of simulations. Then the importance sampling method is also adapted to reduce variance of simulations further. To save computation cost for determining the optimal parameters in the importance sampling, a moment matching-based technique is proposed. Numerical results are given to show the high efficiency of our method. The idea used in this paper may also be applicable to price Asian option and Basket option under other time-changed Brownian motion and Heston models.