结合新产品扩散特征的实物期权定价模型研究

现有实物期权定价模型未能考虑新产品销售量所呈现的扩散特征,模型的分析结果与实际情况有较大差距。文章假设新产品价格服从几何布朗运动,并以Bass模型预测新产品的销售过程,运用二叉树方法,构建了离散时间状态下结合新产品扩散特征的实物期权定价模型,并得到新技术采用的最优时机。     

Skew Brownian Motion and Pricing European Options

Abstract

The volatility smile and systematic mispricing of the Black–Scholes option pricing model are the typical motivation for examining stochastic processes other than geometric Brownian motion to describe the underlying stock price. In this paper a new stochastic process is presented, which is a special case of the skew-Brownian motion of Itô and McKean. The process in question is the sum of a standard Brownian motion and an independent reflecting Brownian motion that is similar in construction to the stochastic representation of a skew-normal random variable. This stochastic process is taken in its exponential form to price European options. The derived option price nests the Black–Scholes equation as a special case and is flexible enough to accommodate stochastic volatility as well as stochastic skewness.     

Achieving smooth asymptotics for the prices of European options in binomial trees

A new binomial approximation to the Black–Scholes model is introduced. It is shown that, for digital options and vanilla European call and put options, a complete asymptotic expansion of the error in powers of n ^?1 exists. This is the first binomial tree for which an asymptotic expansion has been shown to exist.     

Applying free random variables to random matrix analysis of financial data. Part I: The Gaussian case

We apply the concept of free random variables to doubly correlated (Gaussian) Wishart random matrix models, appearing, for example, in a multivariate analysis of financial time series, and displaying both inter-asset cross-covariances and temporal auto-covariances. We give a comprehensive introduction to the rich financial reality behind such models. We explain in an elementary way the main techniques of free random variables calculus, with a view to promoting them in the quantitative finance community. We apply our findings to tackle several financially relevant problems, such as a universe of assets displaying exponentially decaying temporal covariances, or the exponentially weighted moving average, both with an arbitrary structure of cross-covariances.     

Model-driven statistical arbitrage on LETF option markets

In this paper, we study the statistical properties of the moneyness scaling transformation, which adjusts the moneyness coordinate of the implied volatility smile in an attempt to remove the discrepancy between the IV smiles for levered and unlevered ETF options. We construct bootstrap uniform confidence bands which indicate that the implied volatility smiles are statistically different after moneyness scaling has been performed. An empirical application shows that there are trading opportunities possible on the LETF market. A statistical arbitrage type strategy based on a dynamic semiparametric factor model is presented. This strategy presents a statistical decision algorithm which generates trade recommendations based on comparison of model and observed LETF implied volatility surface. It is shown to generate positive returns with a high probability. Extensive econometric analysis of the LETF implied volatility process is performed including out-of-sample forecasting based on a semiparametric factor model and a uniform confidence bands' study. These provide new insights into the latent dynamics of the implied volatility surface. We also incorporate Heston stochastic volatility into the moneyness scaling method for better tractability of the model.     

Evaluating the economy embedded in the Brazilian ethanol–gasoline flex-fuel car: a Real Options approach

The introduction of the flex-fuel cars in the Brazilian market in 2003 changed considerably the consumer decision-making process. Prior to this date, it was necessary to choose the automobile type only by gasoline or by ethanol fuel; today it is possible to choose a car type with both fuel options. This flexibility generates economic advantages for his owner, but what are the financial benefits of a flex-fuel car in comparison with a car using only gasoline? Geographically, where is the owner of the benefits from this flexibility located? This article presents an empirical application of the Real Options Theory in the analysis of the flex-fuel car option for five geographic Brazilian regions: Northern, Northeastern, Central-Western, Southeastern and Southern. The regional price differences as well as the consumer preferences of these regions were met. For this purpose, historical fuel prices were considered stochastic and following a Mean Reverting Stochastic process. The prediction and option values were generated by a Monte Carlo simulation. The results indicated that the option embedded on the Brazilian flex-fuel car adds considerable value to the owner in all regions and car models considered, with the Southeastern Region receiving most benefits by the flex option.     

Strategic Investment Decisions under Uncertainty,Tacit Collusion and Product Differentiation

This paper examines the effects of uncertainty, tacit collusion and product differentiation on strategic investment policy. The model demonstrates the equilibrium under competitive investments and under cooperative investments. In competitive investments, two firms compete by investing to preempt each other, and ultimately one firm invests earlier than its rival. In cooperative investments, two firms invest simultaneously and collude on their outputs. Cooperative investments are likely to be sustainable only under conditions of high volatility of future demand, low probability of market extinction and high degree of product differentiation, while competitive investments are sustainable for any set of these parametric values.     

Algorithmic market making for options

In this article, we tackle the problem of a market maker in charge of a book of options on a single liquid underlying asset. By using an approximation of the portfolio in terms of its vega, we show that the seemingly high-dimensional stochastic optimal control problem of an option market maker is in fact tractable. More precisely, when volatility is modeled using a classical stochastic volatility model—e.g. the Heston model—the problem faced by an option market maker is characterized by a low-dimensional functional equation that can be solved numerically using a Euler scheme along with interpolation techniques, even for large portfolios. In order to illustrate our findings, numerical examples are provided.     

Valuing foreign exchange rate derivatives with a bounded exchange process

Foreign exchange rates have been subjected to periods of tighter or looser controls as various political and economic forces have waxed and waned. When currencies were backed by gold there were fixed exchange rates. In 1973 floating exchange rates were adopted though many countries did try to keep their currency values within certain ranges. More recently the European Economic Community formalized this practice. Free-floating exchange rates might be well characterized by the lognormal distribution which is standard in option pricing. However, this is probably a poor approximation for exchange rates which are kept within some range by the actions of one or both governments or central banks. This paper develops a model which can be used to value options and other derivative contracts when the underlying exchange rate is bounded in a fixed range (a, b). Methods for pricing both European and American style options are developed.The author would like to thank Ken French and Geert Rouwenhorst for their comments and suggestions.     

Spanning with American options

In this paper, we investigate the existence of multiperiod American options generating dynamically complete markets. We show that if a primitive security separates states at the terminal date, then generically there exist multiperiod American options on that security generating dynamically complete markets. We also provide an example of an economy in which multiperiod American options on a primitive security generate dynamically complete markets, while multiperiod European options do not.