Theoretical and Applied Probability on Stochastic Calculus, Numerical Methods, and Mathematical Finance

0202958 Protter首席研究员和共同首席研究员将调查随机微积分和蒙特卡罗模拟子领域内的概率的几个主题。所选问题的动机主要是在金融数学中的应用，但结果将具有更普遍的理论意义。在随机微积分中，关于停止时间的新结果将导致更好的信用风险模型，并更深入地理解是什么使成功的金融风险对冲成为可能。随机微积分的新进展将有助于扩展不完全市场的理论，即市场包含的风险不能被完全对冲，因为在实践中是真实的。一个蒙特卡罗问题要解决的是最近在期权定价中流行的算法的最佳使用。另一条研究线将涉及改进与模拟有关的随机微分方程解的数值技术。第三个蒙特卡洛主题是一些方差缩减技术的最佳应用，这些技术非常适合核物理问题，稀有事件概率估计和一些期权定价问题。最近开发的一种新的股票价格模型，将交易规模纳入其中，这将导致金融衍生品的新定价技术。 概率论在金融领域的应用彻底改变了一个行业。在过去20年里，数万亿美元的衍生证券市场的建立促进了资本的全球流动，从而提高了国际商业和生产力。如果没有为衍生证券提供可靠定价的数学模型（例如，股票期权）并指导其相关风险的管理，这些市场就不可能存在。数学成功的根本主题是精确计算金融衍生品的价格，这些衍生品使公司能够通过购买金融工具来降低风险，这些金融工具保护它们免受不太可能但可能是灾难性事件的影响。同样重要的是，如果不是更重要的话，说明了票据出卖人为保护自己免受其通过销售接受的风险而应遵循的方法。在一个完整的市场中，理论从原则上解释了如何做到这一点，在这样的市场中，理论通常为实施这一配方提供了明确的指导。换句话说，在完全市场中，一种新的保险类型被创造出来了，而现有的概率理论使这种保险成为可能。这种“风险保险”引发了上述革命。然而，一个真实的问题是，在现实中，市场是不完整的，因此需要新的数学技术来扩展理论，使其更真正适用。这已经开始了，但还处于起步阶段，理论的这种扩展将是拟议项目的一个大焦点。此外，最近有人提出了新的模型，以更好地考虑流动性问题和市场摩擦（如实施股票交易时的交易成本），部分是由PI自己提出的。这些模型将继续得到开发、校准和统计验证。

0202958Protter The principal investigator and co-principal investigator will investigate several topics in probability within the subfields of stochastic calculus and Monte Carlo simulation. The problems selected are motivated primarily by applications in mathematical finance, but the results will be of more general theoretical significance. In stochastic calculus, new results about stopping times will lead to better models of credit risk and a deeper understanding of what makes possible successful hedging of financial risks. New progress in stochastic calculus will help extend theories of incomplete markets, i.e. markets containing risks that can not be perfectly hedged, as is true in practice. One Monte Carlo issue to be addressed is the optimal use of an algorithm that has recently become popular in option pricing. Another line of research will involve improving numerical techniques for solutions of stochastic differential equations in connection with simulation. A third Monte Carlo topic is the optimal application of some variance reduction techniques that are well suited to problems in nuclear physics, estimation of rare events probabilities, and some option pricing problems. The recent development of a new model for stock prices that incorporates sizes of trade will lead to new pricing technology for financial derivatives. The application of probability to finance has revolutionized an industry. In the past 20 years the creation of multi-trillion dollar derivative security markets has facilitated the world-wide flow of capital and thereby enhanced international commerce and productivity. Without the mathematical models which provide reliable pricing of derivative securities (e.g., stock options) and guide the management of their associated risk, these markets could not exist. The underlying theme of the mathematical success has been to compute precisely the price of financial derivatives which enable companies to lay off risk by buying financial instruments that protect them from unlikely but possibly disastrous events. Equally if not more important has been the description of a recipe for the seller of the instrument to follow in order to protect himself from the risk he accepts through the sale. A complete market is one in which the theory explains how to do this in principle, and in such a market the theory often provides an explicit guide to implementation of this recipe. In other words, in complete markets a new type of insurance has been created, and this has been made possible by existing probability theory. This type of "risk insurance" generated the revolution mentioned above. A real problem, however, is that in reality markets are not complete, and thus new mathematical techniques are needed to extend the theory and to make it more truly applicable. This has already begun, but it is in its infancy, and this extension of the theory will be a large focus of the proposed project. In addition, recently new models have been proposed to better incorporate liquidity issues and market frictions (such as transaction costs when implementing stock trades), in part by the PI himself. These models will continue to be developed, calibrated, and statistically verified.