Stochastic Methods in Finance

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The objectives of the proposed research are to develop the mathematical models and theory for financial applications and to apply the results to financial markets. Most finance theory has the implicit assumption that traders can buy or sell as many shares of securities as they wish by immediate transactions. But in actual markets, this assumption is not satisfied, and thus the subject "liquidity risk" has received wide attention. Liquidity risk is the additional risk in the market due to the timing and size of a trade. The price process may depend on the activities of traders, especially the trading volume. The optimal trading strategy for the investor and the optimal liquidation problems need to be investigated from the aspects of practice as well as mathematics. Many of the theoretical developments allow financial institutions (such as banks) to design specialized derivatives, find a "fair" price, and sell them to their clients. Barrier options are a widely used class of path-dependent financial derivatives. Chained option is viewed as barrier options which are chained together in a sense that another barrier option becomes active after underlying asset price crosses a primary barrier. These chained-type barrier options have become popular in the over-the-counter equity and foreign exchange derivatives market, and pricing and hedging problems are certainly important research questions. The research on chained option is originated by myself (co-authored with D. Jun), and it is a highly significant discovery. Our theoretical advances would help market participants enlarge the range of their choice and understand the economic benefits of such products. Efficient risk management is essential to the success of financial institutions such as investment banks and insurance companies. Mathematical research in this area is extremely important and should be encouraged.