# Ebook: Real Options, Ambiguity, Risk and Insurance

**Description**

Financial engineering has become the focus of widespread media attention as a result of the worldwide financial crisis of recent years. This book is the second in a series dealing with financial engineering from Ajou University in Korea. The main objective of the series is to disseminate recent developments and important issues in financial engineering to graduate students and researchers, and to provide surveys or pedagogical exposition of important published papers in a broad perspective, as well as analyses of important financial news concerning financial engineering research, practices or regulations.

Real Options, Ambiguity, Risk and Insurance, comprises 12 chapters and is divided into three parts. In Part I, five chapters deal with real options analysis, which addresses the issue of investment decisions in complex, innovative or risky projects. Part II presents three chapters on ambiguity. The notion of ambiguity is one of the major breakthroughs in the expected utility theory; ambiguity arises as uncertainties cannot be precisely described in the probability space. Part III consists of four chapters devoted to risk and insurance, and covers mutual insurance for non-traded risks, downside risk management, and credit risk in fixed income markets.

This volume will be useful to both graduate students and researchers in understanding relatively new areas in economics and finance, as well as challenging aspects of mathematics.

**Preface**

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).

**Abstract**

This chapter considers a representative firm taking investment decisions in a high-tech environment where different generations of production facilities are invented over time. First, we develop a general real options investment model for high-tech industries in which, according to standard practice, the sales price and the unit production cost both satisfy a geometric Brownian motion (GBM) process. Second, we use the developed model to analyze actual investment decisions in the LCD industry. Real life data is used to fit the parameters of the model and to discuss the actual investments of the two largest companies in the LCD industry: Samsung Display and LG Display. We conclude that their investments in the 8th generation LCD production facilities are have negative NPVs. We present several reasons how these investments can be justified.

**Abstract**

This chapter covers the basic mathematics needed to value the investment opportunities of firms that operate in an oligopolistic market. It combines the tools of (financial) option pricing and industrial organization. At a mathematical level the model presented here is a combination of optimal stopping theory and game theory. Most of the game theoretic real options literature is based on the notion of equilibrium introduced by Fudenberg and Tirole (1984, Review of Economic Studies). This chapter, however, builds on recent work by Thijssen ([1], mimeo), which exploits the strong Markovian nature of diffusions. The theory is applied to a simple duopoly where it is shown, numerically, that competition in a dynamic setting may be bad for welfare.

**Abstract**

In the standard real options approach to investment under uncertainty, agents formulate optimal policies under the assumptions of risk neutrality or complete financial markets. Although these assumptions are crucial to the implications of the approach, they are not particularly relevant to most real world environments where agents face incomplete markets and are exposed to undiversifiable risks. In this chapter we extend the real options approach to incorporate risk aversion for a general class of utility functions. We show that risk aversion provides an incentive for the investor to delay investment and leads to a significant erosion in project values.

**Abstract**

The modeling of investment problems as being analogous to the exercise of perpetual American call options has become commonplace in economics and finance since [16]. By exploiting the analogy with traded options, management's flexibility to decide on scale at the time of investment is generally unaccounted for; this assumption is at odds with business practice. In this paper, we study a situation in which an incumbent firm has leeway in choosing when and by how much to raise capital. We consider a general setting and prove the unicity and optimality of a threshold policy under certain conditions. The literature on real options analysis typically considers the timing of lump-sum investments wherein the change in scale is known beforehand. In another stream of the economic literature, stochastic models of capital accumulation deal with situations where, at each instant, the firm decides on its optimal level of capital goods with the aim to maximize its expected discounted revenues netted of capital expenditures; fixed adjustment costs are ignored in this perspective. We consider fixed and variable adjustment costs and allow for the optimal time of investment and choice of scale. We thus reconciliate these two distinct approaches in a unified theory of investment under uncertainty with time and scale flexibility.

**Abstract**

This chapter is a survey to the recent developments on problems of optimal stopping under ambiguity. This chapter develops a theory of optimal stopping under ambiguity in a fairly general framework. The characterization of the value process and the optimal stopping rule are presented. Moreover, the value function is leaded to a free boundary problem in a Markov setting.

**Abstract**

This chapter reviews four principal-agent models in continuous time. The first model is about a contract problems with full information, which is known as a risk-sharing problem. The second model is concerned with optimal contract problems with hidden actions and the payment to the agent is lump-sum at the end of the contract. The third model is similar as the second one, but the payment to the agent is continuous. The last model is about a concrete problem–the optimal insurance design problem, in which both the insurer and the insured are subject to Knightian uncertainty about the loss distribution, and the Knightian uncertainty is modeled in a g-expectation framework. In this problem, the endogenous characterization of the optimal indemnity extends the classical theorems of Arrow and Raviv in the classical situation. In the presence of Knightian uncertainty, the optimal insurance contract is shown to be not only contingent on the realized loss but also on another source of uncertainty coming from the ambiguity.

**Abstract**

We review the developments in the theory of Backward Stochastic Differential Equations during the last 20 years, including the solutions' existence and uniqueness, comparison theorem, nonlinear Feynman-Kac formula, g-expectation and many other important results in BSDE theory and their applications to dynamic pricing and hedging in an incomplete financial market. We present our new framework of nonlinear expectation and its applications to financial risk measures under uncertainty of probability distributions. The generalized form of the law of large numbers and central limit theorem under sublinear expectation shows that the limit distribution is a sublinear G-normal distribution. A new type of Brownian motion, G-Brownian motion, is constructed which is a continuous stochastic process with independent and stationary increments under a sublinear expectation (or a nonlinear expectation). The corresponding robust version of Itô's calculus turns out to be a basic tool for problems of risk measures in finance and, more general, for decision theory under uncertainty. We also discuss a type of “fully nonlinear” BSDE under nonlinear expectation.

**Abstract**

In this chapter, we investigate the optimization of mutual proportional reinsurance — a mutual reserve system that is intended for the collective reinsurance needs of homogeneous mutual members, such as P&I Clubs in marine mutual insurance and reserve banks in the U.S. Federal Reserve. Compared to general (non-mutual) insurance models, which involve one-sided impulse control (i.e., either downside or upside impulse) of the underlying insurance reserve process that is required to be positive, a mutual insurance differs in allowing two-sided impulse control (i.e., both downside and upside impulse), coupled with the classical proportional control of reinsurance. We prove that a special band-type impulse control (a,A,B,b) with a = 0 and a < A < B < b, coupled with a proportional reinsurance policy (classical control), is optimal when the objective is to minimize the total maintenance cost. That is, when the reserve position reaches a lower boundary of a = 0, the reserve should immediately be raised to level A; when the reserve reaches an upper boundary of b, it should immediately be reduced to a level B. An interesting finding produced by the study reported in this chapter is that there exists a situation such that if the upside fixed cost is relatively large in comparison to a finite threshold, then the optimal band control is reduced to a downside only (i.e., dividend payment only) control in the form of (0,0;B,b) with a = A = 0. In this case, it is optimal for the mutual insurance firm to go bankrupt as soon as its reserve level reaches zero, rather than to jump restart by calling for additional contingent funds. This finding partially explains why many mutual insurance companies, that were once quite popular in the financial markets, are either disappeared or converted to non-mutual ones.

**Abstract**

We shall discuss on large deviation estimates of the probability of falling below a given target growth rate for controlled semimartingales, in relation to certain ergodic risk-sensitive stochastic control problems in the risk averse case. We present an expression of the limit value of the minimizing probability as the Legendre transform of the value of the stochastic control problem on infinite time horizon, which is characterized as the solution to the H-J-B equation of ergodic type. The problems discussed here are motivated by mathematical finance and they are called “downside risk minimization”. In this chapter, after reviewing the probabilistic meaning of the asymptotic analyses developed here and historical situation of the studies, we make an exposition about how we can obtain our duality theorem on the asymptotics.

**Abstract**

In continuous-time financial markets, several dynamic portfolio insurance techniques are introduced in generalized forms to construct self-financing portfolios, which satisfy the floor constraint, or a generalized drawdown constraint: Concretely, generalized CPPI (Constant Proportion Portfolio Insurance) methods, American OBPI (Option-Based Portfolio Insurance) method, and DFP (Dynamic Fund Protection) method are explained. Moreover, these portfolio insurance techniques are applied to solve the long-term risk-sensitized growth rate maximization problem subject to the floor constraint or the generalized drawdown constraint.

**Abstract**

We review methodological aspects of credit-risk models in the literature and their implications on credit-risk spreads. We have chosen five of representative structural models: Merton (1974), Longstaff and Schwartz (1995), Leland and Toft (1996), Collin-Dufresne and Goldstein (2001), and Chen, Collin-Dufresne and Goldstein (2009). Recent structural models suggest that credit risk spreads can be greatly influenced not only by the loss distribution of risky corporate securities but the representative investor's consumption habit formation, and by interaction between default losses and macroeconomic factors such as market prices of risk and stochastic interest rates.