Pricing Exotic Option Under Jump-Diffusion Models by the Quadrature Method

This paper extends the quadrature method to price exotic options under jump-diffusion models. We compute the transition density of jump-extended models using convolution integrals. Furthermore, a simpler and more efficient lattice grid is introduced to implement the recursion more directly in matrix form. It can be shown that a lot of running time can be saved. At last, we apply the developed approach to the different jump-extended models to demonstrate its universality and provide a detailed comparison for the discrete path-dependent options to demonstrate its advantages in terms of speed and accuracy.

     

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 Dimension Reduction Shannon-Wavelet Based Method for Option Pricing

We present a robust and highly efficient dimension reduction Shannon-wavelet method for computing European option prices and hedging parameters under a general jump-diffusion model with square-root stochastic variance and multi-factor Gaussian interest rates. Within a dimension reduction framework, the option price can be expressed as a two-dimensional integral that involves only (i) the value of the variance at the terminal time, and (ii) the time-integrated variance process conditional on this value. A Shannon wavelet inverse Fourier technique is developed to approximate the conditional density of the time-integrated variance process. Furthermore, thanks to the excellent approximation properties of Shannon wavelets, the overall pricing procedure is reduced to the evaluation of just a single integral that involves only the density of the terminal variance value. This single integral can be accurately evaluated, since the density of the variance at the terminal time is known in closed-form. We develop sharp approximation error bounds for the option price and hedging parameters. Numerical experiments confirm the robustness and impressive efficiency of the method.     

Agent-Based Approach to Option Pricing Anomalies

Psychological studies on decision making under uncertainty, which have been inspired by Kahneman and Tversky's study, have attracted considerable interest in financial research as key factors to solve anomalies that cannot be explained by the traditional models. Recently, we proposed an agent-based prospect theoretical model and demonstrated that the loss-aversion feature of investors is capable of explaining a large number of financial stylized facts. This paper aims to extend the previous work to the field of option pricing. Two important anomalies in the field-the implied volatility smile and the skewness premium-will be analyzed. This paper can be considered as an attempt to integrate the behavioral financial theory and the option pricing theory by using the agent-based approach.     

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).     

An Agent-Based Approach to Option Pricing Anomalies

Psychological studies on decision-making under uncertainty, which have been inspired by Kahneman and Tversky's study, have attracted considerable interest in financial research as key factors to solve anomalies that cannot be explained by the traditional models. Recently, we proposed an agent-based prospect theoretical model and demonstrated that the loss-aversion feature of investors is capable of explaining a large number of financial stylized facts. This paper aims to extend the previous work to the field of option pricing. Two important anomalies in the field—the implied volatility smile and the skewness premium—will be analyzed. This study can be considered as an attempt to integrate the behavioral financial theory and the option pricing theory by using the agent-based approach.      

An Implementation of Bouchouev's Method for a Short Time Calibration of Option Pricing Models

We analyse the Bouchouev integral equation for the deterministic volatility function in the Black–Scholes option pricing model. We areable to reduce Bouchouev's original triple integral equation to a single integral equation and describe its numerical solution. Moreover we show empirically that the most complex term in the equation may often be safely ignored for the purposes of numerical calculations. We present a selection of numerical examples indicating the range of time values for which we would expect the equation to be valid.     

一类地产期权的微分方程定价模型及其计算

该文以北京西奥中心写字楼为例,分析“以租待售”型房地产营销工具具有的分期付款期权特性,运用Δ-对冲技巧和Ito引理,构造了美式分期付款地产期权的微分方程定价模型,并确定了定价模型中各个变量的内涵,包括标的资产价格、波动率、期限和执行价等。针对北京西奥中心写字楼的具体市场数据,应用有限差分策略进行数值计算,得到了相应的期权价值。     

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。     

期权定价理论在风险投资决策中的应用

在研究了风险投资的含义和基本特点的基础上，介绍了传统的投资决策方法即净现值法和内部收益率法的基本原理，并且指出现有的建立在净现值法基础之上的评价方法由于难以操作或不符合风险投资的特点，很难对风险投资项目进行有效的评价。在对风险投资行为特征进行深入分析的基础上，认为风险投资具有期权性质，提出了一种基于B1ack-Scholes定价公式的风险投资项目评价方法，从而有效克服了传统净现值法的局限，增加了风险项目投资决策的合理性和科学性。     

Legendre矩的一种有效算法

Ｌｅｇｅｎｄｒｅ正交矩的模式识别、图像分析等许多领域有成功的应用,然而,由于正交矩的复杂性,目前有关正交矩快速算法的研究很少,从而在一定程度上影响了它的应用,对此,作者对Ｌｅｇｅｎｄｒｅ多项式进行了研究,获得了一些新的有效的性质,它们能够显著地减少矩计算中的运算量。     

AzureOP:基于Windows Azure云并行计算的期权定价SaaS

计算速度对于期权交易者至关重要,关系到如何有效地制定价格并评估相应的风险,而云并行计算提供的随收随付制（Pay-as-You-Go）可以实现低成本运行。在微软云平台Windows Azure的基础上,开发了基于云并行计算的期权定价试点云软件AzureOP,该软件以较低的费用提供了低风险和高速度,并给出了AzureOP对于美式期权价格的模拟结果,绘制了对应的期权价格定价曲线和定价曲面。最后,对云并行计算在金融应用上的优势和不足进行了总结和讨论,同时举例说明了试点云软件AzureOP的具体细节。     

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.     

A New Stable Local Radial Basis Function Approach for Option Pricing

In this paper, we develop a new local meshless approach based on radial basis functions (RBFs) to solve the Black–Scholes equation. The global RBF approximations derived from conventional global collocation method usually lead to ill-conditioned matrices. The new scheme employs the idea of the finite difference method to localize them. It removes the difficulty of ill-conditioning of the original method. The new proposed approach is unconditionally stable as it is shown by Von-Neumann stability analysis. As well as it is fast and it produces accurate results as shown in numerical experiments.     

一种新的快速计算Legendre矩的方法

正交矩在模式识别，图像分析等领域有成功的应用，但由于正交矩的复杂性，有关正交矩的快速算法研究尚未得到很好的解决，该文提出一种 新的快速计算Ｌegendre矩的方法，该方法把基于像素点的二维Ｌegendre矩转换为线段的形式来计算，在计算出所有线段的积分后，使用扩展的Ｈatamian滤波方法来计算一维的Ｌegendre矩。结果显示新的算法有效地降低了计算的复杂度，并且，该方法能用于处理任意形状的物体。     

一种新的基于Legendre矩的fMRI图像配准方法

提出了一种脑功能磁共振图像配准的方法。Legendre正交矩可以用来作为图像配准的策略，它的快速计算至关重要，边界的拟合精度和速度对Legendre矩的计算影响很大。根据推导出的Legendre矩的边界特点，提出了采用四连通链码法和改进的矢量斜率法进行边界拟台，从而解决了Legendre矩快速计算的问题。采用遗传算法进行多参数配准策略优化，避免了局部极值的干扰。改进的Legendre矩配准方法是一种快速脑功能图像配准方法。     

Using Pseudo-Parabolic and Fractional Equations for Option Pricing in Jump Diffusion Models

In mathematical finance a popular approach for pricing options under some Lévy model would be to consider underlying that follows a Poisson jump diffusion process. As it is well known this results in a partial integro-differential equation (PIDE) that usually does not allow an analytical solution, while a numerical solution also faces some problems. In this paper we develop a new approach on how to transform the PIDE into a class of so-called pseudo-parabolic equations which are well known in mathematical physics but are relatively new for mathematical finance. As an example we will discuss several jump-diffusion models which Lévy measure allows such a transformation.