Stochastic Methods in Finance

本研究的目标是发展金融应用的数学模型和理论，并将结果应用于金融市场。大多数金融理论都有一个隐含的假设，即交易员可以通过即时交易买卖任意数量的证券。但在实际市场中，这一假设并不满足，因此“流动性风险”这一主题受到了广泛关注。流动性风险是由于交易的时机和规模而在市场上产生的额外风险。价格过程可能取决于交易者的活动，尤其是交易量。投资者的最优交易策略和最优平仓问题需要从实践和数学两个方面进行研究。许多理论的发展允许金融机构（如银行）设计专门的衍生品，找到一个“公平”的价格，然后卖给他们的客户。障碍期权是一种广泛使用的路径依赖金融衍生品。链式期权被视为障碍期权，在某种意义上，另一个障碍期权在标的资产价格越过主要障碍后变得活跃。这些链式障碍期权已在场外股票和外汇衍生品市场流行起来，定价和对冲问题无疑是重要的研究问题。链式期权的研究是我本人（与D. Jun合著）提出的，是一个非常有意义的发现。我们的理论进展将有助于市场参与者扩大他们的选择范围，并了解这些产品的经济效益。有效的风险管理对投资银行和保险公司等金融机构的成功至关重要。这方面的数学研究是极其重要的，应该予以鼓励。

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.