An Asymptotic Expansion Approach to Pricing Financial Contingent Claims

We propose a new methodology for the valuation problem of financial contingent claims when the underlying asset prices follow a general class of continuous Itô processes. Our method can be applicable to a wide range of valuation problems including contingent claims associated with stocks, foreign exchange rates, the term structure of interest rates, and even their combinations. We illustrate our method by discussing the Black-Scholes economy when the underlying asset prices follow the continuous diffusion processes, which are not necessarily time-homogeneous. The standard Black-Scholes model on stocks and the Cox-Ingersoll-Ross model on the spot interest rate are simple examples. Then we shall give a series of examples on the valuation formulae including plain vanilla options, average options, and other contingent claims. We shall also give some numerical evidence of the accuracy of the approximations we have obtained for practical purposes. Our approach can be rigorously justified by an infinite dimensional mathematics, the Malliavin-Watanabe-Yoshida theory recently developed in stochastic analysis.     

A Note: Natural Generalization of Black-Scholes in the Presence of Skewness,Using Stable Processes

The Lognormal price model is generalized to the class of Log-Stable Processes, a family which possesses self-similarity properties usually only associated with the Lognormal, but which, more generally, can model negatively skewed distributions of return. This generalization appears to explain several discrepancies between the Black-Scholes Model and observed market phenomena, such as the variation of implied volatility of option price with exercise price and term to expiry, and the nonzero probability of bankruptcy or ‘crash’. It will be argued that the class of maximally negatively skewed Stable distributions (a class which, paradoxically, contains the normal) may be utilized to produce models which imply these phenomena naturally.     

Hedging derivative securities with genetic programming

One of the most recent applications of GP to finance is to use genetic programming to derive option pricing formulas. Earlier studies take the Black–Scholes model as the true model and use the artificial data generated by it to train and to test GP. The aim of this paper is to provide some initial evidence of the empirical relevance of GP to option pricing. By using the real data from S&P 500 index options, we train and test our GP by distinguishing the case in-the-money from the case out-of-the-money. Unlike most empirical studies, we do not evaluate the performance of GP in terms of its pricing accuracy. Instead, the derived GP tree is compared with the Black–Scholes model in its capability to hedge. To do so, a notion of tracking error is taken as the performance measure. Based on the post-sample performance, it is found that in approximately 20% of the 97 test paths GP has a lower tracking error than the Black–Scholes formula. We further compare our result with the ones obtained by radial basis functions and multilayer perceptrons and one-stage GP. Copyright © 1999 John Wiley & Sons, Ltd.     

负风险溢价对我国权证市场价格的影响

文章利用B—S期权定价公式对我国权证市场价格进行检验发现,大多数的权证存在着严重的价格泡沫,而少数认购权证的价格可能低于理论价格。文章认为对投资者的同质信念假设排除了股票的负的风险溢价,这就难以利用卖空约束解释期权和权证价格偏离完全市场的理论价格。股票市场的负的风险溢价来自于投资者的异质信念,将负风险溢价与卖空约束相结合,可以解释我国权证市场存在的一部分价格异象。     

经理人股权薪酬合约结构的理论与启示

经理人股权薪酬合约中限制性股票和股票期权如何配置与其激励特性及经理人股票期权价值相关。本文使用布莱克——斯科尔斯模型衡量经理人期杈价值时存在没有考虑经理人异质风险态度、风险构成对期权定价的影响。介绍了3个不同模型支撑下的股权薪酬合约结构的理论,并联系我国的实际情况提出了对我国上市公司制定股权薪酬合约计划的启示和建议。     

The QLBS Q-Learner goes NuQLear: fitted Q iteration,inverse RL,and option portfolios

The QLBS model is a discrete-time option hedging and pricing model that is based on Dynamic Programming (DP) and Reinforcement Learning (RL). It combines the famous Q-Learning method for RL with the Black–Scholes (–Merton) (BSM) model's idea of reducing the problem of option pricing and hedging to the problem of optimal rebalancing of a dynamic replicating portfolio for the option, which is made of a stock and cash. Here we expand on several NuQLear (Numerical Q-Learning) topics with the QLBS model. First, we investigate the performance of Fitted Q Iteration for an RL (data-driven) solution to the model, and benchmark it versus a DP (model-based) solution, as well as versus the BSM model. Second, we develop an Inverse Reinforcement Learning (IRL) setting for the model, where we only observe prices and actions (re-hedges) taken by a trader, but not rewards. Third, we outline how the QLBS model can be used for pricing portfolios of options, rather than a single option in isolation, thus providing its own, data-driven and model-independent solution to the (in)famous volatility smile problem of the Black–Scholes model.     

Error Calculus and Path Sensitivity in Financial Models

In the framework of risk management, for the study of the sensitivity of pricing and hedging in stochastic financial models to changes of parameters and to perturbations of the stock prices, we propose an error calculus that is an extension of the Malliavin calculus based on Dirichlet forms. Although useful also in physics, this error calculus is well adapted to stochastic analysis and seems to be the best practicable in finance. This tool is explained here intuitively and with some simple examples.     

实物期权理论分析与应用

随着世界经济全球化的进程进一步加快,企业面临市场中不确定性越来越大,传统的投资决策理论（以净现值法为代表）已越来越不能适应公司投资决策的需要。本文首先指出传统的风险投资项目价值评估方法的局限性,然后对实物期权理论加以介绍,最后介绍实物期权的应用框架。     

基于历史和隐含波动率的权证定价效率研究

该文选取了2008年12月在沪深股票交易所交易的所有处于实值状态的权证和2只处于虚值状态的权证、2只过期权证,分别运用历史波动率和隐含波动率,计算了基于B-S定价方法的理论价格,结果发现基于历史波动率的B-S理论价格倾向于低估市场价格,对于处于深度虚值状态的权证,其已失去对价格的解释和预测能力;而基于隐含波动率的B-S理论价格对市场价格的短期解释和预测能力较好,在本文测试的样本内偏误均在1%以内;通过对两只过期权证的测算,结果表明基于历史波动率的B-S理论价格对权证在整个存续区间的运行趋势起到了决定性的作用,但市场价格的运行要高于理论价格且背离理论价格的运行具有一定的规律性,我国权证市场存在较大的投机成分.