Probability and Finance: Flows of Conditional Prices, Liquidity Issues, and Impulse Control AMC-SS

在这个建议中，我们建议研究数学金融理论中不完全市场的数学框架，特别是在股票市场（以及具有相同数学结构的市场）。不完全市场的特点是存在无限数量的风险中性指标，因此对于典型的或有债权没有唯一的价格。在过去20年左右的时间里，人们提出了各种各样的方法，以提供一种合理的方法来选择一种风险中性的度量，而不是其他度量，但它们总是不令人满意，并且最终是武断的。目前流行的一种方法是使用“无差异定价”，在选择（仍然是武断的）风险中性指标时，涉及到经济学方面的争论。相反，我们建议尝试“完善”不完整的市场，对希思、贾罗和莫顿的想法进行修改，他们在债券市场上做了类似的壮举。其理念是利用金融衍生品（如期权）的连续时间定价，这些衍生品在市场上交易，（根据理论）必须是局部鞅。因此，我们需要证明存在风险中性措施，同时使标的及其交易衍生品全部鞅（或局部鞅，甚至sigma鞅）；这可能是相当技术性的，特别是在衍生品到期日早于交易期限的情况下。在股票等股票的交易中，金融衍生品（如期权）已成为重要的基本要素。期权允许一个人付费将风险（例如投资组合风险或外汇风险）从一方转移到另一方。这样，它就像房屋的火灾保险，房主将失去房屋的财务风险转移给愿意承担风险的保险公司，当然，要收取费用。然而，与火灾保险或人寿保险不同，计算金融衍生品的公平价格在数学上是相当复杂的。（有时，某些类型的房屋保险也很难定价，比如佛罗里达州和美国墨西哥湾沿岸的房屋风暴保险，保险公司越来越依赖“再保险”，这通常可以用类似期权定价的方式建模。）布莱克、斯科尔斯和默顿的模型解释了如何在简单情况下计算公平价格，后两人因此获得了诺贝尔奖。我们现在知道如何在稍微复杂一点的模型中计算公平价格，这种模型被称为“完全市场”，然而，人们普遍认为，世界比完全市场情况更复杂，实际上是“不完整的”。迄今提出的在不完全市场中计算公平价格的所有方法都是武断的，通常既不被学者接受，也不被从业者接受。这个提议可能会对解决这个问题大有帮助，通过使用期权本身的市场价格，加上潜在的市场股票价格，从本质上把一个不完整的市场变成一个完整的市场。

In this proposal we propose to study the mathematical framework of incomplete markets in Mathematical Finance Theory, and in particular in the equity markets (and markets with the same mathematical structure). Incomplete markets are characterized by there being an infinite number of risk neutral measures, and therefore there is not a unique price, for the typical contingent claim. Various methods have been proposed over the last 20 years or so to give a reasonable method to choose one risk neutral measure over the others, but they have always been unsatisfying, and ultimately arbitrary. A current popular approach is to use "indifference pricing," which involves economics arguments when choosing (still arbitrarily) risk neutral measure. We propose instead to try to "complete" incomplete markets, using a modification of the idea of Heath, Jarrow and Morton, who did an analogous feat for the bond market. The idea is to use the continuous time pricing of financial derivatives, such as options, which are traded in the markets, and which (by the theory) have to be local martingales. Therefore we need to show that there exist risk neutral measures that simultaneously make the underlying and its traded derivatives all martingales (or local martingales, or even sigma martingales); this can get quite technical, especially in cases where the maturity of the derivative is before the trading horizon.In the trading of equities, such as stocks, an element which has become of fundamental importance are financial derivatives, such as options. An option allows one to transfer risk, such as (for example) portfolio exposure, or foreign currency risk, from one party to another, for a fee. In this way, it is like fire insurance on a home, where the home owner transfers the financial risk of losing his home to an insurance company willing to bear the risk, for of course, a fee. However unlike fire insurance or life insurance, it is quite complicated mathematically to calculate a fair price for a financial derivative. (It is also difficult to price some types of home insurance too, at times, such as storm insurance for homes in Florida and along the Gulf Coast of the US, and insurance companies are increasingly relying on "re-insurance," which can often be modeled in ways analogous to option pricing.) The models of Black, Scholes, and Merton, for which the latter two were awarded a Nobel prize, explain how to calculate a fair price in simple situations. We now know how to calculate fair prices in slightly more sophisticated models, known as "complete markets", however it is widely believed that the world is more complicated than the complete market case, and is in fact "incomplete." All of the many methods proposed to date to calculate a fair price in incomplete markets have been arbitrary, and in general not accepted either by academics, nor practitioners. This proposal might go a long way towards solving that problem, by using the market prices of the options themselves, together with the underlying market stock prices, essentially to make an incomplete market into a complete one.