Title: Mathematical Modeling of Insider Trading
Description: In this thesis, we study insider trading and consider a financial market and an enlarged financial market whose sets of information are respectively represented by the filtrations F and G. The filtration G is obtained by initially expanding the filtration F. We also consider that we have a finite trading horizon. First, we show that under certain conditions the enlarged market satisfies no free lunch with vanishing risk (NFLVR) locally and therefore satisfies no arbitrage with respect to admissible simple predictable trading strategies. In addition, we generalize the structure of all the G local martingale deflators and find sufficient conditions under which the enlarged market satisfies NFLVR. We apply our results to some recent examples of insider trading that have appeared in newspapers and by doing so, show the limitations of some previous works that have studied the stability of the NFLVR property under an initial expansion. \newline Second, assuming the enlarged market satisfies NFLVR and markets are incomplete, we define a notion of risk and compare the risk of a market or liquidity trader to the risk of an insider trader.

We prove that the risk of an insider is smaller than the risk of a market/liquidity trader under some sufficient conditions that involve their respective trading strategies. We find a relationship between the trading strategies of a market trader and of an insider when the risk neutral measure of the market is used. If an insider trades using the market risk neutral measure and not her own, then her trading strategy should involve not only the stock but also the volatility of the stock. \newline Finally, assuming that the enlarged market satisfies NFLVR locally, we provide a way for an insider to price her financial claims. We also define a new type of process that we call a quasi-local martingale and prove that the stock price process under local NFLVR is one such process.