This book is the second volume in the WCU financial engineering series by the financial engineering program of Ajou University, supported by the Korean Government under the world-class-university (WCU) project grant. Ajou University is the unique recipient of the grant in Korea to establish a world class university in financial engineering. The main objective of the series is to disseminate, faster than textbooks, recent developments of important issues in financial engineering to graduate students and researchers, providing surveys or pedagogical expositions of published important papers in broad perspectives, or analyses of recent important financial news on financial-engineering research, practices or regulations.

The first volume was published by the IOS press in 2011 under the title of “New Trends in Financial Engineering”, containing articles to introduce recent topics in financial engineering, contributed by WCU-project participants. This volume focuses on important topics in financial engineering such as ambiguity, real options, and credit risk and insurance, and has 12 chapters organized in three parts. These chapters are contributed by globally recognized active researchers in mathematical finance mostly outside the WCU-project participants.

Part I consists of five chapters. Real options analysis addresses the issue of investment decisions in complex, innovative, risky projects. This approach extends considerably the traditional NPV approach, much too limited to deal with the complexity of real situations. In preparing the investment decision, a project manager should determine which project to choose, when to choose it, and in what scale. He/She should incorporate flexibility in order to benefit from acquiring later on important information about all aspects of uncertainties related to the investment. Consequently, during the project life, the manager still faces further decisions on how to manage, contract, expand, or abandon and to meet industrial competition, not to mention performing basic managerial functions and making financial decisions. Towards the end of the project, the manager faces closure decisions such as sale, reorganization or liquidation. Flexibility is not the unique characteristics of real options. One additional idea is to take advantage of valuation techniques developed in context of financial products, in order to define properly the value of industrial projects. This is more and more possible in the context of commodities with an organized market. The energy sector is an important example. An important difference between real and financial options concerns the issue of competition. For complex investment projects, there are generally few possible players. For financial products, the number of players is very large and therefore each of them does not change dramatically the context (it may be possible of course). The decision making with competition introduces challenging problems.

Villeneuve and Décamps examine the optimal investment policy for a cashconstrained firm which has no access to external financing, and show that an increase in the volatility of the underlying asset can actually decrease the value of the growth option value. Huisman, Kort and Plasmans apply the real option theory to analyze a real life case, and show that negative NPV projects are optimally undertaken (when discount rates are high and technology progresses fast) in the hope of new opportunities or growth options for the firm. Thijssen enriches real options analysis by introducing industrial competition into standard real option problems and argues competition can be bad for welfare in a dynamic setting. Hugonnier and Morellec consider a real options problem for a risk averse decision maker with undiversifiable risks and show that the risk aversion can make him/her delay investment, reducing the (market) value of the project. Finally, Bensoussan and Chevalier-Roignant consider capital budgeting decisions on not only timing but also scale of a project and show how optimal trigger policy integrates the two aspects.

Part II has three chapters on ambiguity. We believe that the notion of ambiguity is one of major breakthroughs in the expected utility theory. Ambiguity arises as uncertainties cannot be precisely described in the probability space. The objective is to understand rational decision making behaviors of an economic agent when his decision making environment is subject to ambiguity. Mathematics underlying those economics problems can be very challenging, imposing great obstacles to the economic analysis of the problems. Chen, Tian and Zhao survey recent developments on problems of optimal stopping under ambiguity, and develop the theory of optimal stopping under ambiguity in a general framework. Ji and Wei review the principal-agent literature in continuous time, and apply to the optimal insurance design problem in the presence of ambiguity. Shige Peng provides a survey of recent significant and systematic progress in the area of G-expectations: new central limit theorems under sublinear expectations, Brownian motions under ambiguity (G-Brownian motions), its related stochastic calculus of Itô's types and some typical pricing models. He further shows that prices of contingent claims in the world of ambiguity can be expressed as g-expectation (nonlinear expectation) of future claims, and that the method of the nonlinear expectation turns out to be powerful in characterizing these prices in general.

In Part III, four chapters are devoted to risk and insurance. In particular, this part covers mutual insurance for non-traded risks, downside risk management, and credit risk in fixed income markets. Liu, Taksar and Yuan introduce mutual insurance which can be viewed as a mutual reserve system for homogeneous mutual members, such as P&I Clubs in marine mutual insurance and Federal Reserve reserve banks in the U.S., and explain why many mutual insurance companies, which were once quite popular in the financial markets, are either disappeared or converted to non-mutual ones.

The importance of downside risk minimization has attracted lots of attention from both practitioners and academics in light of recent experience of the Subprime Mortgage Crisis. Nagai discusses the large deviation estimates of the probability of falling below a given target growth rate for controlled semi-martingales, in relation to certain ergodic risk-sensitive stochastic control problems in the risk averse case. Portfolio insurance techniques are related to the downside risk minimization problem. Sekine reviews several dynamic portfolio insurance techniques such as generalized CPPI (Constant Proportion Portfolio Insurance) methods, American OBPI (Option-Based Portfolio Insurance) method, and DFP (Dynamic Fund Protection) method, and applies these techniques to solve the long-term risk-sensitive growth rate maximization problem subject to the floor constraint or the generalized drawdown constraint. Credit risk is also an important topic for both practitioners and academics, being particularly important to the determination of subprime mortgage rates. Ahn and Sung provide a pedagogical review of literature focusing on determinants of credit risk spreads with emphasis on methodological aspects of structural models.

This broad spectrum of concepts and methods shows the richness of the domain of mathematical/engineering finance. We hope this volume will be useful to both graduate students and researchers in understanding relatively new areas in economics and finance and challenging aspects of mathematics. In this manner, we think contributing to the expectations of the WCU project.

Alain Bensoussan

Shige Peng

Jaeyoung Sung

Graduate Department of Financial Engineering, Ajou University. 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 (R31-2009-000-20007-0).