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

普罗特，首席研究员和联合首席研究员，将在随机微积分和蒙特卡罗模拟的子领域内研究概率的几个主题。选择的问题主要是数学金融学中的应用，但其结果将具有更广泛的理论意义。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对冲金融风险成为可能。随机微积分的新进展将有助于扩展不完全市场的理论，即包含无法完全对冲的风险的市场，这在实践中是正确的。一个需要解决的蒙特卡罗问题是对最近在期权定价中流行的一种算法的最佳使用。另一项研究将涉及改进与模拟有关的随机微分方程解的数值技术。第三个蒙特卡罗主题是一些非常适合于核物理问题、稀有事件概率估计和一些期权定价问题的方差减少技术的最佳应用。最近，一种结合了交易规模的股票价格新模型的发展，将导致金融衍生品的新定价技术。概率在金融中的应用使一个行业发生了革命性的变化。在过去20年中，数万亿美元衍生证券市场的建立促进了资本在世界范围内的流动，从而提高了国际商业和生产力。如果没有数学模型为衍生证券(如股票期权)提供可靠的定价并指导其相关风险的管理，这些市场就不可能存在。在数学上取得成功的根本主题是精确计算金融衍生品的价格，这些衍生品使企业能够通过购买金融工具来规避风险，这些金融工具可以保护企业免受不太可能、但可能是灾难性事件的影响。同样，即使不是更重要，也是对票据出卖人为保护自己免受其通过出售而承担的风险而应遵循的配方的说明。一个完整的市场是一个理论在原则上解释如何做到这一点的市场，在这样的市场中，该理论通常为这一秘诀的实施提供了明确的指导。换句话说，在完全市场中，一种新的保险类型已经被创造出来，这是现有的概率理论使之成为可能的。这种“风险保险”引发了上述革命。然而，一个真正的问题是，在现实中，市场并不是完全的，因此需要新的数学技术来扩展这一理论，使其更加适用。这已经开始了，但还处于初级阶段，这一理论的扩展将是拟议项目的一大重点。此外，最近提出了新的模型，以更好地纳入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本人提出的。这些模型将继续开发、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.