This is the first collective work in the series of research efforts undertaken together by scholars in different disciplines who met at The Graduate Department of Financial Engineering, Ajou University over the summers of 2010 and 2011. Financial economists, mathematicians and control theorists sat together and started to raise questions and formulate problems. The main objective has been to find new research directions in financial engineering, which mainly consisted of quantitative tools for derivative pricing, hedging, and portfolio selection. The big questions we have been dealing with are applications of financial engineering such as real options – particularly emphasizing analysis of competition with real option opportunities, how to deal with model uncertainty in the face of the infinite array of models which can be chosen for pricing and hedging, emerging new techniques such as forward backward stochastic differential equations, current issues such as agency problems in financial engineering, and the theory of banking in relation to financial engineering.

Ajou University is a unique recipient of the World Class University grant in financial engineering administered by the Ministry of Education, Science and Technology of the Korean Government. The mission given to the university has been to attain class of excellence in research and education in the designated field. Under the help of the government grant, the university has succeeded in attracting scholars of the finest caliber such as Alain Bensoussan, Abel Cadenillas, Zengjing Chen, Kiseop Lee, and Shanjian Tang from the US, Canada, and China.

According to the definition provided by International Association of Financial Engineers “Financial engineering is the application of mathematical methods to the solution of problems in finance”. Financial engineering has advanced to a remarkable degree by this definition. Fanciful tools of mathematics, Markov chains, forward and backward stochastic differential equations, variational inequalities, functional analysis and optimal control theory have been successfully applied to derivative pricing, hedging, portfolio management, and the development of trading strategies. Furthermore, the availability and success of the mathematical tools have helped to attract talented people to the financial services industry and thereby deepened the human resource pool of the industry. Financial innovations which occurred for the last four decades owe much to the development of financial engineering techniques and the human resources attracted to the field by the discipline.

A serious challenge, however, confronts the discipline and the people who engage in it. The recent financial crisis has just made this challenge more apparent. The world seems to have suffered from financially engineered products such as collateralized debt obligations and credit default swaps. The risk management techniques developed by financial engineers did not seem to work when they were desperately needed. In general, current financial engineering practices are criticized for being narrowly focused with technical aspects without providing the connection with other aspects of reality. Thus, the challenge is to develop models and methods for financial engineers to better manage real-world financial problems. Our team, consisting of both economists and mathematicians, tried to identify problems and relevant issues to meet the challenge, and attempted to discover new research trends which have been being pursued by frontier researchers. The book is a first fruit of our joint efforts to tackle the challenge as well as to further advance innovations in financial engineering.

In the first part of this volume we review recent developments of real options. Real options are an application of the theory of financial options to capital investments, particularly emphasizing flexibility. They have wide applications, ranging from the development of natural resources such as oil fields, research and development decisions, venture capital evaluations, and more recently, to the development of alternative energy sources, desalination, and to environmental protection. In this sense the theory of real options is one of the most beneficial products of financial engineering in that it addresses the most impending economic and environmental issues.

In the first chapter Alain Bensoussan and coauthors explain how the technique of variational inequalities can provide a highly nontrivial solution to a problem of oligopolistic competition when investments involve sunk costs. Alain Bensoussan initiated applications of variational inequalities to the theory of optimal control in his joint work with Jaques Louis Lions. In this chapter he and coauthors explain how this innovative technique solves the most challenging problems in the theory of real options.

In the next chapter, Tang and Koo review recent developments of the theory of real options by using forward and backward stochastic differential equations. Backward stochastic differential equations have been very useful as models of stochastic differential utility, derivative pricing, portfolio optimization, and risk measures. Backward stochastic differential equations coupled with forward stochastic differential equations provide flexible tools for modeling state variables and value processes. Tang, an associate editor of the SIAM Journal on Control and Optimization, is a renowned researcher in the application of forward and backward stochastic differential equations to the theory of optimal control, and Koo is a financial economist who specializes in optimal portfolio theory. They explain applications of forward backward stochastic differential equations to real options with their economic intuition.

In the third chapter, Shim and Koo provide explanation about a more concrete application of the real options approach to a consumption and portfolio selection problem. The problems are mixtures of problems regarding consumption/investment and those regarding choices of stopping times. The models they explain encompass the following: a model in which an investor's investment opportunity gets enlarged at a threshold wealth level, and models of consumption/investment with a choice of optimal retirement time. Shim is an expert on the mixture problem of optimal consumption and investment and optimal stopping. Shim and Koo discuss economic implications for optimal consumption and investment implied by the existence of a stopping option.

In the second part of the volume we present new topics. In the fourth chapter Abel Cadenillas discusses a firm's problem of controlling dividend payments and risk management simultaneously. Dividend policy has been a long-standing research topic in finance, but has attracted relatively little attention by financial engineers. Cadenillas, a renowned financial mathematician who has been working on optimal dividend policy in relationship with risk management. He explains his work with coauthors which models dividend policy as a continuous-time optimization problem involving taxes and transaction costs.

In the fifth chapter Zengjing Chen and coauthors explain a new topic of risk measures and nonlinear probability models. Chen has written a seminal paper in asset pricing with Knightian uncertainty in joint work with Larry Epstein (Chen and Epstein 2002). Nonlinear probability models arise naturally in the context of Knightian uncertainty under which economics agents are not able to assess probabilities about future states precisely. They explain the law of large numbers, the law of iterated logarithm, and the central limit theorem under nonlinear probability measures. This chapter further discusses potential estimation and hypothesis testing under nonlinear probability measures.

In the sixth chapter Kiseop Lee gives a survey on recent studies on market microstructure. Market microstructure has attracted attention by financial engineers as the popularity of high-frequency trading increases. Lee is a reputed scholar in the theory and empirical study of market microstructure with emphasis on applications to financial engineering. He reviews both empirical and theoretical research on microstructure from the perspective of financial engineers.

In the last part of the volume we propose a new perspective. In the 7th chapter Jaeyoung Sung explains agency problems in financial markets and economy. Jaeyoung Sung is a leading expert in the theory of principal-agent relationship in continuous time. He explains why moral hazard problems are important for financial engineering problems.

In the last chapter Koo and Hyun review the history and theory of banking and financial markets in an attempt to draw lessons for theory and practice of financial engineering. In particular they review how development of financial engineering changed banking and financial markets and their incentive structure. There has been a very fruitful development of theories of banking and financial markets as applications of information economics. They discuss the possibility of linking the theories to financial engineering.

I feel privileged to participate in the historically meaningful WCU program and to work as the editor of the present volume. I thank all our members who have made invaluable contributions to make the program successful, and especially thank them for contributing chapters to this volume. We also thank Professor Yoon Dokko and Professor Dong Chul Won and Ms. Younah Song and Ms. Hyeran Yoon for their continuous encouragement and professional administrative jobs, without which our program would not have been sustainable. With regard to this book project, we thank Se Ryoong Ahn, Bon Chun Koo, Byung Lim Koo, and Jung Lim Koo for their very careful editorial help.

Hyeng Keun Koo

The research culminated in this book was supported by WCU (World Class University) program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (R31-20007).

Graduate Department of Financial Engineering, Ajou University