Businesses often face the problem of providing incentives for agents to work effectively together on projects that develop over time. The agents' costly and unobservable effort jointly affects the survival of the project and thus the expected value of its cash now. A key feature of many contracting problems with multiple agents is that the agents exert effort at different times: some at the outset and some over time. The optimal timing of compensation reflects the timing of effort with payment for up-front effort preceding compensation for continuous effort. Deferring payment for agents exerting effort over time improves their incentives without impairing incentives for the up-front effort because this effort is sunk once the project is set up. The exact pattern of compensation between the agents with continuous effort depends on the relative severity of their moral hazard problems. In a special case where moral hazards are equally severe, the agents equally split the cash flow once it becomes available. This study suggests an approach to understanding a broad set of contracting problems in economics and finance. It rationalizes business conventions such as deferred compensation for top executives, the 50:50 split between law firm partners, and profit shares of influential directors (or lead actors) and residual claims of producers in the movie industry. Furthermore, the model predicts business failures such as the crisis in the mortgage industry due to the lack of characteristics suggested in the optimal contract.

This paper studies sequential portfolio choices by MPS-risk-averse investors in a continuous time jump-diffusion framework. It is shown that the optimal trading strategies for MPS risk averse investors, if they exist, must be located on a so-called `temporal efficient frontier' (t.e.f.). The t.e.f. is found not to coincide with the local instantaneous frontier --- the continuous time analogue of Markowitz's mean-variance frontier. This observation is potentially useful in understanding the existence of documented financial anormally in empirical finance --- MPS risk averse investors may not wish to invest along the local instantaneous Markowitz's mean-variance frontier, but instead hold portfolios on the t.e.f.. The optimal portfolio on the t.e.f. could well fall strictly within the instantaneous local Markowitz's efficient frontier. Our observations on mutual fund separation are also profound and interesting. In contrast to the classical two-fund separation along the line of Black (1972) and Tobin (1958), our study shows that MPS-risk-averse investors' optimal trading strategy is target rate specific. Precisely, investors with different target rates may end up investing into different managed mutual funds, each involving a specific set of separating portfolios. Our theoretic findings are, nevertheless, much in line with the real world phenomena on the existence of various types of mutual funds offered by different financial institutes, each aiming to attract demand from some specific groups of investors --- a picture that is in sharp contrast to the theoretical prediction made by Black (1972) and Tobin (1958). Finally, our study sheds light on the difference between expected utility and MPS-risk-averse investors concerning their trading behavior in sequential time frame. Even though these two groups of investors may end up holding a common risky portfolio in each spot market, the differences between their trading behaviors are most reflected through the portfolio weights assigned to each of the separating portfolios within the time frame and across states. Precisely, the portfolio weights corresponding to investors respectively from the two groups are associated with recognizable different time patterns. We showed that such difference in trading behavior would be also reflected from the time patterns of the instantaneous returns and the volatilities of the funds respectively managed by investors from these two groups.

In this paper we present a critical viewpoint on interpretation of one of the most important innovation in the recent world economy. This is erivatives’ market, the options segment in particular. The standard options such as plain vanilla, nonstandard exotics or hybrid options and more recent specification called credit derivatives are actively traded around the world absorbing a significant volume of cash flows. The goal of the paper is to present the misunderstanding of the core problems in this field. This is an option price discovery. The modern probability and statistics theories are applied to provide investors and institutions information regarding the cost of the investment risk and on the other hand develop a better proximity between given historical data and analytical theory. We will show bellow that critical arguments are related to the basic fundamentals of the investment sciences that unfortunately are still difficult to comprehend by theoretical researchers, supervisory organizations, and investors.

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.

Motivated by the behavior of internet stock prices in 1998-2000, we present a continuous time equilibrium model of bubbles where overconfidence generates disagreements among agents regarding asset fundamentals. With shortsale constraints, an asset owner has an option to sell the asset to other agents who have more optimistic beliefs. This re-sale option has a recursive structure, that is, a buyer of the asset gets the option to resell it. This causes a significant bubble component in asset prices even when small dierences of beliefs are sucient to generate a trade. The model generates prices that are above fundamentals, excessive trading, excess volatility, and predictable returns. However, our analysis shows that while Tobin’s tax can substantially reduce speculative trading when transaction costs are small, it has only a limited impact on the size of the bubble or on price volatility. We give an example where the price of a subsidiary is larger than its parent firm. Finally, we show how overconfidence can justify the use of corporate strategies that would not be rewarding in a “rational” environment.