Stochastic Analysis of Large Investors

罗宾逊1312419 研究者和他的同事们使用随机分析的渐近方法对衍生品市场的大型投资者进行了全面的研究。 具体目标是首先确定投资者和市场的特征，这些特征内生地导致大量持有，其次确定大量头寸对定价、价格影响、投资组合优化和风险管理的影响。 由于大头寸会对罕见的意外事件产生极端的敏感性，因此大偏差理论非常适合这种分析。 特别是，使用大偏差与效用为基础的理论，最佳的位置大小，调查研究时，如果有的话，一个风险厌恶的代理人在一个不完整的市场应采取大的立场，在非交易的风险资产。 关于定价和套期保值问题，众所周知，在不完全市场中，投资者的偏好会影响人们愿意购买一项权利的价格。 通常，基于效用的价格的封闭式表达式是不可用的，需要一些近似值。 现有的近似值只对小仓位有效，因此大仓位近似值提供了一个自然的对应。 除了确定渐近价格，研究者还确定了驱动这种价格的投资者效用的关键组成部分。 最后，例子表明，大的位置出现在与渐近完全的市场：调查人员因此试图将这个概念扩展到目前的一个以外，它侧重于弱收敛的资产价格法，一个更自然的概念，在渐近对冲能力。 一旦扩展，有关市场完整性的连续性问题被考虑。 从严格的数学角度来看，调查人员试图了解过去二十年来场外衍生品市场的快速增长。 鉴于许多此类产品的复杂性，我们不清楚为什么银行或其他金融机构会希望在一项索赔中持有大量头寸，特别是在缺乏完善的对冲策略或流动性市场的情况下。 因此，调查人员研究金融机构何时（如果有的话）应该拥有大量衍生品合约。 此外，鉴于这种情况，调查人员的目标是确定金融机构面临的定价、对冲和整体风险方面的影响。

Robinson1312419 The investigator and his colleagues use asymptotic methods from stochastic analysis to provide a comprehensive study of large investors in derivatives markets. The specific goals are first to identify characteristics of both the investor and the market that endogenously lead to large holdings, and second to determine the effects of large positions on pricing, price impact, portfolio optimization, and risk management. Because large positions induce extreme sensitivity to rare unexpected events, the theory of Large Deviations is well suited for this analysis. In particular, using Large Deviations in conjunction with utility-based theory on optimal position sizes, the investigator studies when, if ever, a risk-averse agent acting in an incomplete market should take a large position in a non-traded risky asset. Regarding the issue of pricing and hedging, it is well known that in incomplete markets investor preferences affect the price at which one is willing to buy a claim. Typically, closed-form expressions for utility based prices are unavailable and some approximation is necessary. Existing approximations are only valid for small position sizes and thus large holdings approximations provide a natural counterpart. In addition to determining asymptotic prices, the investigator identifies the key components of the investor's utility that drive such prices. Lastly, examples show that large positions arise in conjunction with asymptotically complete markets: the investigator thus seeks to extend this notion beyond the current one, which focuses on weak convergence of asset price laws, to a more natural notion in terms of the asymptotic ability to hedge. Once extended, questions regarding continuity with respect to market completeness are considered. From a rigorous mathematical perspective, the investigator seeks to understand the rapid growth in over-the-counter derivatives markets during the previous two decades. Given the complexity of many of these products, it is not clear why a bank, or some other financial institution, would wish to take on a large position in a claim, especially given the lack of perfect hedging strategies or liquid markets. The investigator thus studies when, if ever, a financial institution should own a significant notional amount of a derivative contract. Additionally, given such a position, the investigator aims to determine the impacts in terms of pricing, hedging, and over-all risk that a financial institution faces.