Stochastic Analysis of Large Investors

研究者和他的同事使用随机分析的渐近方法对衍生品市场中的大型投资者进行了全面研究。具体目标首先是确定投资者和市场的特征，这些特征内在地导致大量持有，其次是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。因为大仓位会导致对罕见的意外事件极度敏感，所以大偏差理论非常适合这种分析。特别地，研究者将大偏差与基于效用的最优头寸规模理论结合起来，研究了在不完全市场中行事的风险厌恶者何时（如果有的话）应该在非交易风险资产中持有大量头寸。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他们愿意购买索赔的价格。通常，基于效用的价格的封闭形式表达式是不可用的，需要进行一些近似。现有的近似只适用于小的头寸规模，因此大的持仓近似提供了一个自然的对应。除了确定渐近价格外，研究者还确定了驱动此类价格的投资者效用的关键组成部分。最后，示例表明，大型头寸与渐近完全市场一起出现：因此，研究者试图将这一概念扩展到当前的概念之外，该概念侧重于资产价格规律的弱收敛，以渐近对冲能力为基础的更自然的概念。一旦扩展，就需要考虑市场完整性方面的连续性问题。从严格的数学角度来看，研究者试图理解在过去的二十年中场外衍生品市场的快速增长。考虑到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索赔中持有大量头寸，尤其是在缺乏完美对冲策略或流动性市场的情况下。因此，调查人员研究了金融机构何时（如果有的话）应该拥有大量衍生品合约的名义金额。此外，鉴于这样的立场，调查员的目的是确定在定价，对冲和金融机构面临的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.