Early-stage ?rms usually have a single large Research and Development (R&D) project that requires multi-stage investment. Firms? volatility can dramatically change due to the evolvement of R&D e¤orts and stage clearing. First, the success (failure) of R&D e¤orts within each stage (jump risk) decreases (increases) the un- certainty (i.e. volatility) level of the ?rms?future returns ?"jump e¤ect". Second, at the end of each stage, ?rms decide whether to continue next stage investment upon re-evaluating the project prospect conditional on the resolution of technical uncertainty and other information; as ?rms survive each investment stage and are becoming mature, the uncertainty level of their future returns should eventually decrease in later investment stages that lead to maturity ?"stage-clearing e¤ect". Ignoring these e¤ects results in incorrect estimation of ?rms?future volatility, an important element for early-stage ?rm valuation. In this paper, I develop a gener- alized Markov-Switching EARCH methodology for early-stage ?rms with discrete stage-clearing and jumps. My methodology can identify structural changes in the idiosyncratic volatility and also explore the relation between price changes and future volatility. Using a hand-collected dataset of early-stage biotech ?rms, I con?rmed the existence of the "stage-clearing e¤ect" and the "jump e¤ect". In the second part of my paper, I model early-stage ?rms as sequences of nested call options with jumps that lead to mature ?rms. "Jump e¤ect" arises because the early-stage ?rms are modeled as compound call options with jumps on the underly- ing cash ?ows, the volatility of the early-stage ?rms at each stage is determined by the compound call option elasticity to the underlying cash ?ows. If the downside (upside) jump happens, the value of the underlying cash ?ows decreases (increases), which makes the compound call option elasticity go up (down). As a result, the compound call option becomes riskier (less risky). "Stage-clearing e¤ect" arises because as ?rms exercise their option to continue investment, the new options that ?rms enter into will eventually become a less risky option.

Abstract: In this paper, we assume the R&D research for new products and new business models of an entrepreneur will get some cash-flow in the future. And because of the patent protection, the entrepreneur will become a monopolist in that defined market. The successful R&D research represents the growth value for the entrepreneur. We assume that the value of growth is the value of the entrepreneur. The total market value of entrepreneur can be understood as a real growth option plus a discounted real exchange option. The first time in the theory, this paper gets the formula of the real compound option under three stochastic parameters. 关键词：创业企业，增长价值，定价，复合实物期权 Keywords: entrepreneur, growth value, pricing, compound real option

This paper deals with the option-pricing problem. In the first part of the paper we study in more details the discrete setting of the option-pricing problem usually referred to as the binomial scheme. We highlight basic differences between the old and the new approaches. The main ualitative distinction of the new pricing approach from either binomial or Black Scholes’s is that it represents the option price as a stochastic process. This stochastic interpretation can not give straight forward advantage for an investor due to stochastic setting of the pricing problem. The new approach explicitly states that the option price is more risky than it is customary represented by binomial scheme or Black Scholes theory. Continuous setting will be considered in the second part of the paper following [1]. One significant conclusion follows from the new model. It states that there is no sense in using either neutral probabilities or ‘neutral world’ applications for options valuation either theoretically or numerically. Recall that after the Black Scholes’ publication [2] the ‘simplified’ approach named later binomial scheme was introduced in [3]. In this paper referring to the historical tradition, we first represent discrete scheme. In several examples we discuss two-period plain vanilla option valuation. Then we extend the discrete scheme applications to an exotic option-pricing referred to as a compound option. The compound option in Black Scholes setting was first studied in [4] and then in [5,6].

本文首先对过去几十年来实物期权方法应用研究进展进行了回顾和总结，并指出了目前在实物期权方法应用中经常出现的一些模糊概念和误用的问题。通过一些实例的分析和讨论，对现有的一些研究结论提出了质疑，指出：由于问题构模及参数定义上的差异，一些在金融期权中成立的定理，如期权价值随着波动率增加、无风险利率增大或期权期限延长时，是递增的这一规律对于实物期权的而言可能会失效。并以实物期权在企业R&D 投资项目中为例，对这一问题以及与之相关的实物期权构模和参数选择等问题进行了详细深入的讨论和分析。文章还在对Penning and Lint（1997）及Agliardi Elettra（2003）等人的结果扩展的基础上，给出了当波动率、无风险利率和期权持有成本为时变函数的条件下四种复合期权的解析解，并利用数值计算结果证实了作者的观点。 Abstract: This study first reviews the literature of real options research in the past decades, and points out the problem of obscure concepts and misuse of real option frequently appeared in the application fields. By analyzing and discussing some cases, the author put forward doubt on some research results, and indicate that because of the discrepancy of modeling and parameter definition, some theorems which true for financial options, such as the value of options will increase when the volatility and risk-free rate increase or expiration date suspension, will not true for the real options. Using the application of real options in firm’s R&D project as an example, the paper analyses and discusses the problem in detail how to applying real options from aspects of modeling, parameter estimation and sensitivity analysis correctly, etc. By extending the results of Penning and Lint (1997) as well as Agliardi Elettra (2003), a close-form solution for a generalized of the Geske formula is derived for four types compound real options: call on call, call on put, put on call, put on put in the case of time-dependent volatility and risk-free rate and option-holding cost. The author’s findings are proven by numerical results in the last.

本文采用连续时间的多重复合期权Geske及其扩展模型来解决多阶段投资决策问题，在基本Geske公式基础上，给出了具有时变参数的连续时间N重复合期权的扩展Geske公式，并对采用Geske公式和离散期权模型得出的数值结果进行了比较。实例结果分析比较表明，利用已发表的算法和目前的普通PC计算机和数学工具软件，如DATAPLOT、MATHEMATICA等，对连续时间的N重复合期权模型(N ≤ 10)的数值解求解不再具有困难，并且可以得到较其他方法更高精度的计算结果。 Abstract: This paper make uses of N-fold continuous compound option formula to resolve multi-stages investment decision problem, and give an expansion of basic Geske formula to N-fold continuous time option with variable parameters, and then make a comparison of the results of adoption expanded Geske formula with other discrete option formulas such as binomial and trinomial formula. The results show, by means of the popular home PC with mathematic software, such as DATAPLOT, MATHEMATICA etc., the solution procedure if n ≤ 10 is quite easy with a no difficult, and can get the results with higher accuracy than other solution method.

This paper deals with the option-pricing problem. In the first part of the paper we study in more details the discrete setting of the option-pricing problem usually referred to as the binomial scheme. We highlight basic differences between the old and the new approaches. The main qualitative distinction of the new pricing approach from either binomial or Black Scholes’s is that it represents the option price as a stochastic process. This stochastic interpretation can not give straightforward advantage for an investor due to stochastic setting of the pricing problem. The new approach explicitly states that the options price is more risky than represented by binomial scheme or Black Scholes theory. Continuous setting will be considered in the second part of the paper following [1]. One significant conclusion follows from the new model. It states that there is no sense in using either neutral probabilities or ‘neutral world’ applications for options valuation either theoretically or numerically. Recall that after the Black Scholes’ publication [2] the ‘simplified’ approach named later binomial scheme was introduced in [3]. In this paper referring to the historical tradition we first represent discrete scheme. In several examples we discuss two-period plain vanilla option valuation. Then we extend the discrete scheme applications to an exotic option-pricing referred to as a compound option. The compound option in Black Scholes setting was first studied in [4] and then in [5,6]. To highlight the difference between stochastic and deterministic option price definitions note that if a deterministic value is interpreted as a perfect or fair price we can comment that the stochastic interpretation provides this number or any other with the probability that real world option value at maturity will be bellow chosen number. This probability is a pricing risk of the option. Thus with an investor’s motivation of the option pricing the stochastic approach gives information about the risk taking. The investor analyzing option price and corresponding risk makes a decision to purchase the option or not. As far as this paper presents alternative point on option pricing it might be useful to present a short history of this development. Recall that according the US law institutions must provide clients by the risk information regarding client’s prospective on their investments. This circumstance implies importance new approach measuring risk of investments. Different parts of this paper were submitted and sent to journals, conferences, and prominent professors. The third part of the paper was sent to Federal Reserve from the Congressman office and simple examples showing drawbacks of the benchmark option valuation method were submitted to SEC in August 2002.