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）中。 不完全市场的特点是存在无限多的风险中性测度，因此对于典型的未定权益来说，不存在唯一的价格。 在过去的20年里，已经提出了各种方法来给出一个合理的方法来选择一个风险中性度量，但它们总是不能令人满意，最终是任意的。 目前流行的方法是使用“无差别定价”，这涉及经济学参数时，选择（仍然任意）风险中性措施。 相反，我们建议尝试“完善”不完整的市场，使用希思、贾罗和莫顿的想法的修改，他们为债券市场做了类似的壮举。 这个想法是使用金融衍生品的连续时间定价，例如在市场上交易的期权，并且（根据理论）必须是局部鞅。 因此，我们需要证明存在风险中性测度，使标的资产及其交易的衍生品都是鞅（或局部鞅，甚至sigma鞅）;这可能会变得相当技术性，特别是在衍生品的到期日在交易期限之前的情况下。在诸如股票的股票交易中，已经变得具有根本重要性的元素是金融衍生品，比如期权。 期权允许一个人将风险，如（例如）投资组合风险或外汇风险，从一方转移到另一方，收取费用。 这样，它就像一个家庭的火灾保险，在那里房主转移的财务风险，失去他的家到一个保险公司愿意承担风险，当然，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。 (It有时也很难为某些类型的房屋保险定价，例如为佛罗里达和沿着美国墨西哥湾沿岸的房屋提供的风暴保险，保险公司越来越依赖于“再保险”，这通常可以用类似于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的情况下计算公平的价格。 我们现在知道如何在稍微复杂的模型中计算公平价格，称为“完全市场”，但人们普遍认为，世界比完全市场更复杂，实际上是“不完整的”。“迄今为止，在不完全市场中计算公平价格的所有方法都是武断的，一般来说，学术界和实践者都不接受。 这一建议可能会大大有助于解决这一问题，通过使用期权本身的市场价格，以及基础市场股票价格，基本上使一个不完整的市场成为一个完整的市场。

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.