Stochastic Analysis of Large Investors

研究人员罗宾森1312419和他的同事使用随机分析中的渐近方法，对衍生品市场上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，这些特征内在地导致了大量持有，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。由于大额仓位会导致对罕见意外事件的极度敏感，因此大偏差理论非常适合这种分析。特别是，结合基于效用的最优头寸大小理论，研究人员研究了在不完全市场中行动的避险机构何时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一个人愿意购买债权的价格。通常，基于公用事业的价格的封闭形式表达式是不可用的，需要某种近似。现有的近似值只适用于小仓位，因此大额持仓量的近似值提供了一个天然的对应物。除了确定渐近价格外，调查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，大量头寸与渐近完备的市场一起出现：因此，研究者试图将这一概念从当前的关注资产价格定律弱收敛的概念扩展到一个更自然的概念，即渐近对冲能力。一旦扩展，将考虑与市场完备性有关的连续性问题。从严格的数学角度，这位调查者试图了解过去20年场外衍生品市场的快速增长。鉴于许多此类产品的复杂性，目前尚不清楚为什么一家银行或其他金融机构会希望在债权中持有大量头寸，特别是在缺乏完美的对冲策略或流动性市场的情况下。因此，调查者研究一家金融机构何时应该拥有一份衍生品合同的大量名义金额。此外，考虑到这样的地位，调查者的目标是确定金融机构面临的定价、对冲和总体风险方面的影响。

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.