Stochastic Analysis of Large Investors

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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.