Options and efficiency: Some experimental evidence

This paper examines the information aggregation role of options when agents possess diverse information about possible asset returns. We construct two identical experimental markets: one with and one without options. We find that options speed the information aggregation process. Asset markets that have parallel option markets aggregate traders' diverse information faster than markets where options trading is not available. Implied ranges were calculated from asset and option prices and compared to the actual ranges. These comparisons suggest that options may provide a means for agents to coordinate beliefs about asset values.     

Options with Constant Underlying Elasticity in Strikes

Closed-form solutions are derived and interpreted for European options, with stochastic strike prices, that maintain constant elasticity of the strike with respect to the price of the underlying asset. We refer to such options as CUES. CUES preserve the relative shares of exercise price risk for both the buyer and writer of the option, regardless of whether the price of the underlying asset moves up or down. The relevance of the CUES concept is established through applications in two distinct fields. First, it is established that CUES-like options are embedded in private equity investments. This concept is then used in a novel application to determine the equity share of a private company corresponding to a given level of investment. Secondly, the advantages that CUES would provide over traditional executive stock option grants are considered and it is shown that CUES can provide enhanced incentive-alignment without increasing options expense to the company. JEL Classification: G130     

Incentives of Stock Option Based Compensation

We introduce explicitly the effort as a choice variable in a continuous time utility maximisation framework of an executive who is partly compensated with stock options. We solve the model in the case where the executive is not allowed to trade in the company’s stock but is able to achieve a partial insurance through trading in a correlated market portfolio. We define the executive’s value of the options through a certainty equivalence approach both in the case of European call options and non-standard capped stock options and study the behaviour of the reservation price as relevant parameters change.JEL Classification: G13, G30, G32, J33, M12     

On the pricing of forward starting options in Heston’s model on stochastic volatility

We consider the problem of pricing European forward starting options in the presence of stochastic volatility. By performing a change of measure using the asset price at the time of strike determination as a numeraire, we derive a closed-form solution within Hestons stochastic volatility framework applying distribution properties of the volatility process. In this paper we develop a new and more suitable formula for pricing forward starting options. This formula allows to cover the smile effects observed in a Black-Scholes environment, in which the extreme exposure of forward starting options to volatility changes is ignored.Received: July 2004, Mathematics Subject Classification (2000): 91B28, 60G44, 60H30, 60E10JEL Classification: G13It is a pleasure to thank the anonymous referee for his valuable comments and suggestions on this paper. Furthermore, we would like to thank Holger Kraft, University of Kaiserslautern, and Alexander Giese, HypoVereinsbank AG Munich, for fruitful discussions and suggestions.     

VIX期权定价与校正

VIX期权作为波动率衍生品能为金融机构提供有效的市场风险对冲工具。文献中对VIX期权定价的实证分析误差都很大,原因在于模型的选取误差以及校正方法和样本选取不妥。通过在VIX模型中加入均值回复因素和跳因素,可以使VIX过程更加合理,也可以使VIX期权定价精度更高。通过对VIX期权市场中间报价进行校正,得到了4个文献模型的参数估计,并比较4个模型的定价精度和正向隐含波动率偏斜拟合效果。     

国际碳贸易价格波动对可再生能源投资的影响机制——基于实物期权理论的分析

有效利用可再生能源是促进节能减排实现绿色循环经济的重要手段。目前,可再生能源技术成本仍高于传统能源技术,因而需要获得额外的经济激励以增加其投资。《京都议定书》所建立的国际碳贸易体系是支持发展中国家实现碳减排的重要机制,但该贸易体系发展前景不明确,这将深刻影响我国可再生能源投资。本文分析和揭示了国际碳贸易体系的不确定性对可再生能源投资决策的影响,在此基础上提出了分别存在于可再生能源项目前期规划阶段和项目建设阶段的增长期权和延迟期权;通过构建两阶段期权模型研究国际碳价格波动下企业延迟投资的灵活性,并量化确定可再生能源项目投资期权价值,采用Monte-Carlo仿真分析法进一步验证模型,推导得出国际碳价格波动对可再生能源项目投资的作用机制。     

我国自然生态环境治理质量测度及其高质量发展的政策启示

党的二十大对推动绿色发展作出战略部署,明确提出到2035年,我国生态环境根本好转,绿色逐步成为高质量发展的鲜明底色。为此,构建测评指标和指数权重体系,运用2003-2019年的面板数据,通过指数分析、TOPSIS评价模型和等距划分法评级综合测度我国自然生态环境治理质量。研究发现：2003-2019年自然生态环境治理质量明显提升,其中,自然状况环境治理质量提速最快,从2003年的0.3606升至2019年的0.9608;大气环境治理质量提速相对缓慢,从2003年的0.2448提至2019年的0.6570;治理质量等级2008年实现“较差”到“中等”的飞跃,2011年达到良好。因此,提出高质量发展的政策启示。     

Error analysis of finite difference and Markov chain approximations for option pricing

Mijatovi? and Pistorius proposed an efficient Markov chain approximation method for pricing European and barrier options in general one‐dimensional Markovian models. However, sharp convergence rates of this method for realistic financial payoffs, which are nonsmooth, are rarely available. In this paper, we solve this problem for general one‐dimensional diffusion models, which play a fundamental role in financial applications. For such models, the Markov chain approximation method is equivalent to the method of lines using the central difference. Our analysis is based on the spectral representation of the exact solution and the approximate solution. By establishing the convergence rate for the eigenvalues and the eigenfunctions, we obtain sharp convergence rates for the transition density and the price of options with nonsmooth payoffs. In particular, we show that for call‐/put‐type payoffs, convergence is second order, while for digital‐type payoffs, convergence is generally only first order. Furthermore, we provide theoretical justification for two well‐known smoothing techniques that can restore second‐order convergence for digital‐type payoffs and explain oscillations observed in the convergence for options with nonsmooth payoffs. As an extension, we also establish sharp convergence rates for European options for a rich class of Markovian jump models constructed from diffusions via subordination. The theoretical estimates are confirmed using numerical examples.     

Functional Itô calculus†

We extend some results of the Itô calculus to functionals of the current path of a process to reflect the fact that often the impact of randomness is cumulative and depends on the history of the process, not merely on its current value. We express the differential of the functional in terms of adequately defined partial derivatives to obtain an Itô formula. We develop an extension of the Feynman-Kac formula to the functional case and an explicit expression of the integrand in the Martingale Representation Theorem. We establish that under certain conditions, even path dependent options prices satisfy a partial differential equation in a local sense. We exploit this fact to find an expression of the price difference between two models and compute variational derivatives with respect to the volatility surface.     

European quanto option pricing in presence of liquidity risk

In this paper, we study the pricing problems of the European quanto options in which the underlying foreign asset is in imperfectly liquid markets. First, we assume that the dynamics of the underlying foreign asset price are affected by market liquidity and propose a liquidity-adjusted quanto model. This allows for the effects of market liquidity on European quanto option pricing. And then we derive the analytical pricing formulas for four different types of European quanto options. Finally, we empirically investigate the pricing performance of our proposed model with a European quanto construction involving the SSE 50 ETF, as the underlying asset, and the CNY/HKD exchange rate. Empirical results demonstrate that the pricing accuracy of the proposed model is markedly superior to that of the Black-Scholes quanto model. In other words, allowing for liquidity risk in the framework of European quanto option pricing can make markedly improvements in fitting the real market data. Particularly, the improvement rate is high for medium-term and out-of-the-money options. Moreover, these results are robust for different liquidity measures.